@@ -1,1127 +1,1127 @@ -REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... - - - -*** ci already defined as operator - -*** si already defined as operator -% Test cases for definite integration. - -int(x/(x+2),x,2,6); - - -2*( - log(2) + 2) - - -int(sin x,x,0,pi/2); - - -1 - - -int(log(x),x,1,5); - - -5*log(5) - 4 - - -int((1+x**2/p**2)**(1/2),x,0,p); - - - p*(sqrt(2) + log(sqrt(2) + 1)) --------------------------------- - 2 - - -int(x**9+y+y**x+x,x,0,2); - - - 2 - 10*log(y)*y + 522*log(y) + 5*y - 5 -------------------------------------- - 5*log(y) - - - -% Collected by Kerry Gaskell, ZIB, 1993/94. - -int(x^2*log(1+x),x,0,infinity); - - - 2 -int(x *log(1 + x),x,0,infinity) - - -int(x*e^(-1/2x),x,0,infinity); - - -4 - - -int(x/4*e^(-1/2x),x,0,infinity); - - -1 - - -int(sqrt(2)*x^(1/2)*e^(-1/2x),x,0,infinity); - - -2*sqrt(pi) - - -int(x^(3/2)*e^(-x),x,0,infinity); - - - 3*sqrt(pi) ------------- - 4 - - -int(sqrt(pi)*x^(3/2)*e^(-x),x,0,infinity); - - - 3*pi ------- - 4 - - -int(x*log(1+1/x),x,0,infinity); - - - 1 -int(x*log(1 + ---),x,0,infinity) - x - - -int(si(1/x),x,0,infinity); - - - 1 -int(si(---),x,0,infinity) - x - - -int(cos(1/x),x,0,infinity); - - - 1 -int(cos(---),x,0,infinity) - x - - -int(sin(x^2),x,0,infinity); - - - sqrt(pi)*sqrt(2) ------------------- - 4 - - -int(sin(x^(3/2)),x,0,infinity); - - - 2/3 5 - sqrt(pi)*2 *gamma(---) - 6 --------------------------- - 2 - 3*gamma(---) - 3 - - -int(besselj(2,x),x,0,infinity); - - -1 - - -int(besselj(2,y^(5/4)),y,0,infinity); - - - 4/5 7 - 2*2 *gamma(---) - 5 -------------------- - 8 - 5*gamma(---) - 5 - - -int(x^(-1)*besselj(2,sqrt(x)),x,0,infinity); - - -1 - - -int(bessely(2,x),x,0,infinity); - - -int(bessely(2,x),x,0,infinity) - - -int(x*besseli(2,x),x,0,infinity); - - -int(x*besseli(2,x),x,0,infinity) - - -int(besselk(0,x),x,0,infinity); - - - pi ----- - 2 - - -int(x^2*besselk(2,x),x,0,infinity); - - - 3*pi ------- - 2 - - -int(sinh(x),x,0,infinity); - - -int(sinh(x),x,0,infinity) - - -int(cosh(2*x),x,0,infinity); - - -int(cosh(2*x),x,0,infinity) - - -int(-3*ei(-x),x,0,infinity); - - -3 - - -int(x*shi(x),x,0,infinity); - - -int(x*shi(x),x,0,infinity) - - -int(x*fresnel_c(x),x,0,infinity); - - -int(x*fresnel_c(x),x,0,infinity) - - -int(x^3*e^(-2*x),x,0,infinity); - - - 3 ---- - 8 - - -int(x^(-1)*sin(x/3),x,0,infinity); - - - pi ----- - 2 - - -int(x^(-1/2)*sin(x),x,0,infinity); - - - sqrt(pi)*sqrt(2) ------------------- - 2 - - -int(2*x^(-1/2)*cos(x),x,0,infinity); - - -sqrt(pi)*sqrt(2) - - -int(sin x + cos x,x,0,infinity); - - -int(sin(x) + cos(x),x,0,infinity) - - -int(ei(-x) + sin(x^2),x,0,infinity); - - - sqrt(pi)*sqrt(2) - 4 ----------------------- - 4 - - -int(x^(-1)*(sin (-2*x) + sin(x^2)),x,0,infinity); - - - - pi -------- - 4 - - -int(x^(-1)*(cos(x/2) - cos(x/3)),x,0,infinity); - - - 3 - - log(---) - 2 - - -int(x^(-1)*(cos x - cos(2*x)),x,0,infinity); - - -log(2) - - -int(x^(-1)*(cos(x) - cos(x)),x,0,infinity); - - -0 - - -int(2,x,0,infinity); - - -int(2,x,0,infinity) - - -int(cos(x)*si(x),x,0,infinity); - - -int(cos(x)*si(x),x,0,infinity) - - -int(2*cos(x)*e^(-x),x,0,infinity); - - -1 - - -int(x/2*cos(x)*e^(-x),x,0,infinity); - - -0 - - -int(x^2/4*cos(x)*e^(-2*x),x,0,infinity); - - - 1 ------ - 125 - - -int(1/(2*x)*sin(x)*e^(-3*x),x,0,infinity); - - - 1 - atan(---) - 3 ------------ - 2 - - -int(3/x^2*sin(x)*e^(-x),x,0,infinity); - - - 3 - x -int(----*sin(x)*e ,x,0,infinity) - 2 - x - - -int(cos(sqrt(x))*e^(-x),x,0,infinity); - - - i 1/4 - sqrt( - pi)*erf(---) + 2*e - 2 -------------------------------- - 1/4 - 2*e - - -int(e^(-x)*besselj(2,x),x,0,infinity); - - - - 2*sqrt(2) + 3 ------------------- - sqrt(2) - - -int(cos(x^2)*e^(-x),x,0,infinity); - - - 1 1 1 1 1 -(pi*( - 2*cos(---)*fresnel_s(---) + cos(---) + 2*fresnel_c(---)*sin(---) - 4 4 4 4 4 - - 1 - - sin(---)))/(2*sqrt(pi)*sqrt(2)) - 4 - - -int(erf(x)*e^(-x),x,0,infinity); - - - 1/4 1 -e *( - erf(---) + 1) - 2 - - -int(besseli(2,x)*e^(-x),x,0,infinity); - - - - 1 1 -2*hypergeometric({------},{},1) + hypergeometric({---},{},1) - 2 - 2 2 - - -int(e^(-x^2)*cos(x),x,0,infinity); - - - sqrt(pi) ----------- - 1/4 - 2*e - - -int(x^(-1)*sin(x)*cos(x),x,0,infinity); - - - pi ----- - 4 - - -int(x^(-1)*sin(x)*cos(2*x),x,0,infinity); - - -0 - - -int(x^(-1)*sin(x)*cos(x/2),x,0,infinity); - - - pi ----- - 2 - - -int(e^x,x,0,infinity); - - - x -int(e ,x,0,infinity) - - -int(e^(-x^2 - x),x,0,infinity); - - - 1/4 1 - e *pi*( - erf(---) + 1) - 2 ---------------------------- - 2*sqrt(pi) - - -int(e^(-(x+x^2+1)),x,0,infinity); - - - 1/4 1 - e *pi*( - erf(---) + 1) - 2 ---------------------------- - 2*sqrt(pi)*e - - -int(e^(-(x+1/x)^2),x,0,infinity); - - - sqrt(pi) ----------- - 4 - 2*e - - -int(e^(-(x+2))*sin(x),x,0,infinity); - - - 1 ------- - 2 - 2*e - - -int(-3*x*e^(-1/2x),x,0,infinity); - - --12 - - -int(x*e^(-1/2*x^2),x,0,infinity); - - -1 - - -int(x^2*besselj(2,x),x,0,infinity); - - - 2 -int(x *besselj(2,x),x,0,infinity) - - -int(x*besselk(1,x),x,0,infinity); - - - pi ----- - 2 - - -int(-3*ei(-x),x,0,infinity); - - -3 - - -int(x^3*e^(-2*x^2),x,0,infinity); - - - 1 ---- - 8 - - -int(sqrt(2)/2*x^(-3/2)*sin x,x,0,infinity); - - -sqrt(pi) - - -int(x^(-1)*(sin(-2*x) + sin(x^2)),x,0,infinity); - - - - pi -------- - 4 - - -int(x^(-1)*(cos(3*x) - cos(x/2)),x,0,infinity); - - - - log(6) - - -int(x^(-1)*(sin x - sin(2*x)),x,0,infinity); - - -0 - - -int(1/x*sin(x)*e^(-3*x),x,0,infinity); - - - 1 -atan(---) - 3 - - -int(sin(x)*e^(-x),x,0,infinity); - - - 1 ---- - 2 - - -int(x^(-1)*sin(x)*cos(x),x,0,infinity); - - - pi ----- - 4 - - -int(e^(1-x)*e^(2-x^2),x,0,infinity); - - - 1/4 3 1 - e *e *pi*( - erf(---) + 1) - 2 ------------------------------- - 2*sqrt(pi) - - -int(e^(-1/2x),x,0,y); - - - y/2 - 2*(e - 1) --------------- - y/2 - e - - -int(si(x),x,0,y); - - -cos(y) + si(y)*y - 1 - - -int(besselj(2,x^(1/4)),x,0,y); - - - 1/4 - 4*besselj(3,y )*y ---------------------- - 1/4 - y - - -int(x*besseli(2,x),x,0,y); - - -besseli(1,y)*y - 2*besseli(0,y) + 2 - - -int(x^(3/2)*e^(-x),x,0,y); - - - y - 3*sqrt(pi)*e *erf(sqrt(y)) - 4*sqrt(y)*y - 6*sqrt(y) ------------------------------------------------------- - y - 4*e - - -int(sinh(x),x,0,y); - - - 2*y y - e - 2*e + 1 ------------------ - y - 2*e - - -int(cosh(2*x),x,0,y); - - - 4*y - e - 1 ----------- - 2*y - 4*e - - -int(x*shi(x),x,0,y); - - - 2*y 2*y y 2 - - e *y + e + 2*e *shi(y)*y - y - 1 -------------------------------------------- - y - 4*e - - -int(x^2*e^(-x^2),x,0,y); - - - 2 - y - sqrt(pi)*e *erf(y) - 2*y ---------------------------- - 2 - y - 4*e - - -int(x^(-1)/2*sin(x),x,0,y); - - - si(y) -------- - 2 - - -int(sin x + cos x,x,0,y); - - - - cos(y) + sin(y) + 1 - - -int(sin x + sin(-2*x),x,0,y); - - - cos(2*y) - 2*cos(y) + 1 -------------------------- - 2 - - -int(sin(n*x),x,0,y); - - - - cos(n*y) + 1 ------------------ - n - - -int(heaviside(x-1),x,0,y); - - -heaviside(y - 1)*(y - 1) - - - -% Tests of transformations defined in defint package. - -laplace_transform(1,x); - - - 1 ---- - s - - -laplace_transform(x,x); - - - 1 ----- - 2 - s - - -laplace_transform(x^a/factorial(a),x); - - - 1 ------- - a - s *s - - -laplace_transform(x,e^(-a*x),x); - - - 1 ------------------ - 2 2 - a + 2*a*s + s - - -laplace_transform(x^k,e^(-a*x),x); - - - gamma(k + 1) -------------------------- - k k - (a + s) *a + (a + s) *s - - -laplace_transform(cosh(a*x),x); - - - - s ---------- - 2 2 - a - s - - -laplace_transform(1/(2*a^3),sinh(a*x)-sin(a*x),x); - - - - 1 ---------- - 4 4 - a - s - - -laplace_transform(1/(a^2),1-cos(a*x),x); - - - 1 ------------ - 2 3 - a *s + s - - -laplace_transform(1/(b^2-a^2),cos(a*x)-cos(b*x),x); - - - s ----------------------------- - 2 2 2 2 2 2 4 - a *b + a *s + b *s + s - - -laplace_transform(besselj(0,2*sqrt(k*x)),x); - - - 1 --------- - k/s - e *s - - -laplace_transform(Heaviside(x-1),x); - - - 1 ------- - s - e *s - - -laplace_transform(1/x,sin(k*x),x); - - - k -atan(---) - s - - -laplace_transform(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); - - - 2 2 2 2 - k *s k *s - - e *erf(k*s) + e - - -laplace_transform(1/k,e^(-k^2/(4*x)),x); - - - besselk(1,sqrt(s)*k) ----------------------- - sqrt(s) - - -laplace_transform(2/(sqrt(pi*x)),besselk(0,2*sqrt(2*k*x)),x); - - - k/s k - e *besselk(0,---) - s ---------------------- - sqrt(s) - - -hankel_transform(x,x); - - - n + 4 - gamma(-------) - 2 -------------------- - n - 2 2 - gamma(-------)*s - 2 - - -Y_transform(x,x); - - - - n + 4 n + 4 - gamma(----------)*gamma(-------) - 2 2 -------------------------------------- - - n + 3 n - 1 2 - gamma(----------)*gamma(-------)*s - 2 2 - - -K_transform(x,x); - - - - n + 4 n + 4 - gamma(----------)*gamma(-------) - 2 2 ----------------------------------- - 2 - 2*s - - -struveh_transform(x,x); - - - - n - 3 n + 5 - gamma(----------)*gamma(-------) - 2 2 -------------------------------------- - - n - 2 n - 2 2 - gamma(----------)*gamma(-------)*s - 2 2 - - -fourier_sin(e^(-x),x); - - - s --------- - 2 - s + 1 - - -fourier_sin(sqrt(x),e^(-1/2*x),x); - - - 3*atan(2*s) - 2*sin(-------------)*pi - 2 --------------------------------- - 2 3/4 - sqrt(pi)*(4*s + 1) *sqrt(2) - - -fourier_sin(1/x,e^(-a*x),x); - - - s -atan(---) - a - - -fourier_sin(x^k,x); - - - k/2 - k k - 4 *gamma(------)*gamma(---)*k - 2 2 ---------------------------------- - k k - 4*s *2 *gamma( - k)*s - - -fourier_sin(1/(b-a),(e^(-a*x)-e^(-b*x)),x); - - - a*s + b*s ----------------------------- - 2 2 2 2 2 2 4 - a *b + a *s + b *s + s - - -fourier_sin(besselj(0,a*x),x); - - - 2 2 - - a + s - heaviside(------------) - 2 - a -------------------------- - 2 2 - sqrt( - a + s ) - - -fourier_sin(1/sqrt(pi*x),cos(2*sqrt(k*x)),x); - - - k k - sqrt(s)*sqrt(2)*cos(---) - sqrt(s)*sqrt(2)*sin(---) - s s ------------------------------------------------------ - 2*s - - -fourier_sin(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); - - - sqrt( - pi)*erf(i*k*s) ------------------------- - 2 2 - k *s - sqrt(pi)*e - - -fourier_cos(e^(-1/2x),x); - - - 2 ----------- - 2 - 4*s + 1 - - -fourier_cos(x,e^(-x),x); - - - 2 - - s + 1 ---------------- - 4 2 - s + 2*s + 1 - - -fourier_cos(x,e^(-1/2*x^2),x); - - - 2 - i*s s /2 - sqrt( - pi)*erf(---------)*s + e *sqrt(2) - sqrt(2) ----------------------------------------------- - 2 - s /2 - e *sqrt(2) - - -fourier_cos(2*x^2,e^(-1/2x),x); - - - 2 - - 384*s + 32 ---------------------------- - 6 4 2 - 64*s + 48*s + 12*s + 1 - - -fourier_cos(x,e^(-a*x),x); - - - 2 2 - a - s -------------------- - 4 2 2 4 - a + 2*a *s + s - - -fourier_cos(x^n,e^(-a*x),x); - - - s s - cos(atan(---)*n + atan(---))*gamma(n + 1) - a a -------------------------------------------- - 2 2 (n + 1)/2 - (a + s ) - - -fourier_cos(1/x,sin(k*x),x); - - - 2 2 - sign(k - s )*pi + pi ------------------------ - 4 - - -fourier_cos(1/sqrt(pi*x),cos(2*sqrt(k*x)),x); - - - k k - sqrt(s)*sqrt(2)*cos(---) + sqrt(s)*sqrt(2)*sin(---) - s s ------------------------------------------------------ - 2*s - - -fourier_cos(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); - - - 1 --------- - 2 2 - k *s - e - - -fourier_cos(1/(pi*x),sin(2*k*sqrt(x)),x); - - - 2 2 - k k -intfc(----) + intfs(----) - s s - - -fourier_cos(1/(sqrt(pi*x)),e^(-2*k*sqrt(x)),x); - - - 2 2 2 - k k k -( - 2*sqrt(s)*cos(----)*fresnel_s(----) + sqrt(s)*cos(----) - s s s - - 2 2 2 - k k k - + 2*sqrt(s)*fresnel_c(----)*sin(----) - sqrt(s)*sin(----))/(sqrt(2)*s) - s s s - - -laplace_transform(x^n/factorial(n)*e^(-a*x),x); - - - 1 -------------------------- - n n - (a + s) *a + (a + s) *s - - -laplace_transform(1/(2*a^2)*(cosh(a*x)-cos(a*x)),x); - - - - s ---------- - 4 4 - a - s - - -laplace_transform(k*a^k/x*besselj(k,a*x),x); - - - 2*k - a ----------------------- - 2 2 k - (sqrt(a + s ) + s) - - -fourier_sin(1/x*e^(-3*x),x); - - - s -atan(---) - 3 - - -fourier_sin(1/(pi*x)*sin(2*k*sqrt(x)),x); - - - 2 2 - k k -intfc(----) - intfs(----) - s s - - -fourier_cos(x^n*e^(-a*x),x); - - - s s - cos(atan(---)*n + atan(---))*gamma(n + 1) - a a -------------------------------------------- - 2 2 (n + 1)/2 - (a + s ) - - -fourier_cos(1/(k*sqrt(pi))*e^(-x^2/(4*k^2)),x); - - - 1 --------- - 2 2 - k *s - e - - -end; -(TIME: defint 163620 182910) +REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... + + + +*** ci already defined as operator + +*** si already defined as operator +% Test cases for definite integration. + +int(x/(x+2),x,2,6); + + +2*( - log(2) + 2) + + +int(sin x,x,0,pi/2); + + +1 + + +int(log(x),x,1,5); + + +5*log(5) - 4 + + +int((1+x**2/p**2)**(1/2),x,0,p); + + + p*(sqrt(2) + log(sqrt(2) + 1)) +-------------------------------- + 2 + + +int(x**9+y+y**x+x,x,0,2); + + + 2 + 10*log(y)*y + 522*log(y) + 5*y - 5 +------------------------------------- + 5*log(y) + + + +% Collected by Kerry Gaskell, ZIB, 1993/94. + +int(x^2*log(1+x),x,0,infinity); + + + 2 +int(x *log(1 + x),x,0,infinity) + + +int(x*e^(-1/2x),x,0,infinity); + + +4 + + +int(x/4*e^(-1/2x),x,0,infinity); + + +1 + + +int(sqrt(2)*x^(1/2)*e^(-1/2x),x,0,infinity); + + +2*sqrt(pi) + + +int(x^(3/2)*e^(-x),x,0,infinity); + + + 3*sqrt(pi) +------------ + 4 + + +int(sqrt(pi)*x^(3/2)*e^(-x),x,0,infinity); + + + 3*pi +------ + 4 + + +int(x*log(1+1/x),x,0,infinity); + + + 1 +int(x*log(1 + ---),x,0,infinity) + x + + +int(si(1/x),x,0,infinity); + + + 1 +int(si(---),x,0,infinity) + x + + +int(cos(1/x),x,0,infinity); + + + 1 +int(cos(---),x,0,infinity) + x + + +int(sin(x^2),x,0,infinity); + + + sqrt(pi)*sqrt(2) +------------------ + 4 + + +int(sin(x^(3/2)),x,0,infinity); + + + 2/3 5 + sqrt(pi)*2 *gamma(---) + 6 +-------------------------- + 2 + 3*gamma(---) + 3 + + +int(besselj(2,x),x,0,infinity); + + +1 + + +int(besselj(2,y^(5/4)),y,0,infinity); + + + 4/5 7 + 2*2 *gamma(---) + 5 +------------------- + 8 + 5*gamma(---) + 5 + + +int(x^(-1)*besselj(2,sqrt(x)),x,0,infinity); + + +1 + + +int(bessely(2,x),x,0,infinity); + + +int(bessely(2,x),x,0,infinity) + + +int(x*besseli(2,x),x,0,infinity); + + +int(x*besseli(2,x),x,0,infinity) + + +int(besselk(0,x),x,0,infinity); + + + pi +---- + 2 + + +int(x^2*besselk(2,x),x,0,infinity); + + + 3*pi +------ + 2 + + +int(sinh(x),x,0,infinity); + + +int(sinh(x),x,0,infinity) + + +int(cosh(2*x),x,0,infinity); + + +int(cosh(2*x),x,0,infinity) + + +int(-3*ei(-x),x,0,infinity); + + +3 + + +int(x*shi(x),x,0,infinity); + + +int(x*shi(x),x,0,infinity) + + +int(x*fresnel_c(x),x,0,infinity); + + +int(x*fresnel_c(x),x,0,infinity) + + +int(x^3*e^(-2*x),x,0,infinity); + + + 3 +--- + 8 + + +int(x^(-1)*sin(x/3),x,0,infinity); + + + pi +---- + 2 + + +int(x^(-1/2)*sin(x),x,0,infinity); + + + sqrt(pi)*sqrt(2) +------------------ + 2 + + +int(2*x^(-1/2)*cos(x),x,0,infinity); + + +sqrt(pi)*sqrt(2) + + +int(sin x + cos x,x,0,infinity); + + +int(sin(x) + cos(x),x,0,infinity) + + +int(ei(-x) + sin(x^2),x,0,infinity); + + + sqrt(pi)*sqrt(2) - 4 +---------------------- + 4 + + +int(x^(-1)*(sin (-2*x) + sin(x^2)),x,0,infinity); + + + - pi +------- + 4 + + +int(x^(-1)*(cos(x/2) - cos(x/3)),x,0,infinity); + + + 3 + - log(---) + 2 + + +int(x^(-1)*(cos x - cos(2*x)),x,0,infinity); + + +log(2) + + +int(x^(-1)*(cos(x) - cos(x)),x,0,infinity); + + +0 + + +int(2,x,0,infinity); + + +int(2,x,0,infinity) + + +int(cos(x)*si(x),x,0,infinity); + + +int(cos(x)*si(x),x,0,infinity) + + +int(2*cos(x)*e^(-x),x,0,infinity); + + +1 + + +int(x/2*cos(x)*e^(-x),x,0,infinity); + + +0 + + +int(x^2/4*cos(x)*e^(-2*x),x,0,infinity); + + + 1 +----- + 125 + + +int(1/(2*x)*sin(x)*e^(-3*x),x,0,infinity); + + + 1 + atan(---) + 3 +----------- + 2 + + +int(3/x^2*sin(x)*e^(-x),x,0,infinity); + + + 3 - x +int(----*sin(x)*e ,x,0,infinity) + 2 + x + + +int(cos(sqrt(x))*e^(-x),x,0,infinity); + + + i 1/4 + sqrt( - pi)*erf(---) + 2*e + 2 +------------------------------- + 1/4 + 2*e + + +int(e^(-x)*besselj(2,x),x,0,infinity); + + + - 2*sqrt(2) + 3 +------------------ + sqrt(2) + + +int(cos(x^2)*e^(-x),x,0,infinity); + + + 1 1 1 1 1 +(pi*( - 2*cos(---)*fresnel_s(---) + cos(---) + 2*fresnel_c(---)*sin(---) + 4 4 4 4 4 + + 1 + - sin(---)))/(2*sqrt(pi)*sqrt(2)) + 4 + + +int(erf(x)*e^(-x),x,0,infinity); + + + 1/4 1 +e *( - erf(---) + 1) + 2 + + +int(besseli(2,x)*e^(-x),x,0,infinity); + + + - 1 1 +2*hypergeometric({------},{},1) + hypergeometric({---},{},1) - 2 + 2 2 + + +int(e^(-x^2)*cos(x),x,0,infinity); + + + sqrt(pi) +---------- + 1/4 + 2*e + + +int(x^(-1)*sin(x)*cos(x),x,0,infinity); + + + pi +---- + 4 + + +int(x^(-1)*sin(x)*cos(2*x),x,0,infinity); + + +0 + + +int(x^(-1)*sin(x)*cos(x/2),x,0,infinity); + + + pi +---- + 2 + + +int(e^x,x,0,infinity); + + + x +int(e ,x,0,infinity) + + +int(e^(-x^2 - x),x,0,infinity); + + + 1/4 1 + e *pi*( - erf(---) + 1) + 2 +--------------------------- + 2*sqrt(pi) + + +int(e^(-(x+x^2+1)),x,0,infinity); + + + 1/4 1 + e *pi*( - erf(---) + 1) + 2 +--------------------------- + 2*sqrt(pi)*e + + +int(e^(-(x+1/x)^2),x,0,infinity); + + + sqrt(pi) +---------- + 4 + 2*e + + +int(e^(-(x+2))*sin(x),x,0,infinity); + + + 1 +------ + 2 + 2*e + + +int(-3*x*e^(-1/2x),x,0,infinity); + + +-12 + + +int(x*e^(-1/2*x^2),x,0,infinity); + + +1 + + +int(x^2*besselj(2,x),x,0,infinity); + + + 2 +int(x *besselj(2,x),x,0,infinity) + + +int(x*besselk(1,x),x,0,infinity); + + + pi +---- + 2 + + +int(-3*ei(-x),x,0,infinity); + + +3 + + +int(x^3*e^(-2*x^2),x,0,infinity); + + + 1 +--- + 8 + + +int(sqrt(2)/2*x^(-3/2)*sin x,x,0,infinity); + + +sqrt(pi) + + +int(x^(-1)*(sin(-2*x) + sin(x^2)),x,0,infinity); + + + - pi +------- + 4 + + +int(x^(-1)*(cos(3*x) - cos(x/2)),x,0,infinity); + + + - log(6) + + +int(x^(-1)*(sin x - sin(2*x)),x,0,infinity); + + +0 + + +int(1/x*sin(x)*e^(-3*x),x,0,infinity); + + + 1 +atan(---) + 3 + + +int(sin(x)*e^(-x),x,0,infinity); + + + 1 +--- + 2 + + +int(x^(-1)*sin(x)*cos(x),x,0,infinity); + + + pi +---- + 4 + + +int(e^(1-x)*e^(2-x^2),x,0,infinity); + + + 1/4 3 1 + e *e *pi*( - erf(---) + 1) + 2 +------------------------------ + 2*sqrt(pi) + + +int(e^(-1/2x),x,0,y); + + + y/2 + 2*(e - 1) +-------------- + y/2 + e + + +int(si(x),x,0,y); + + +cos(y) + si(y)*y - 1 + + +int(besselj(2,x^(1/4)),x,0,y); + + + 1/4 + 4*besselj(3,y )*y +--------------------- + 1/4 + y + + +int(x*besseli(2,x),x,0,y); + + +besseli(1,y)*y - 2*besseli(0,y) + 2 + + +int(x^(3/2)*e^(-x),x,0,y); + + + y + 3*sqrt(pi)*e *erf(sqrt(y)) - 4*sqrt(y)*y - 6*sqrt(y) +------------------------------------------------------ + y + 4*e + + +int(sinh(x),x,0,y); + + + 2*y y + e - 2*e + 1 +----------------- + y + 2*e + + +int(cosh(2*x),x,0,y); + + + 4*y + e - 1 +---------- + 2*y + 4*e + + +int(x*shi(x),x,0,y); + + + 2*y 2*y y 2 + - e *y + e + 2*e *shi(y)*y - y - 1 +------------------------------------------- + y + 4*e + + +int(x^2*e^(-x^2),x,0,y); + + + 2 + y + sqrt(pi)*e *erf(y) - 2*y +--------------------------- + 2 + y + 4*e + + +int(x^(-1)/2*sin(x),x,0,y); + + + si(y) +------- + 2 + + +int(sin x + cos x,x,0,y); + + + - cos(y) + sin(y) + 1 + + +int(sin x + sin(-2*x),x,0,y); + + + cos(2*y) - 2*cos(y) + 1 +------------------------- + 2 + + +int(sin(n*x),x,0,y); + + + - cos(n*y) + 1 +----------------- + n + + +int(heaviside(x-1),x,0,y); + + +heaviside(y - 1)*(y - 1) + + + +% Tests of transformations defined in defint package. + +laplace_transform(1,x); + + + 1 +--- + s + + +laplace_transform(x,x); + + + 1 +---- + 2 + s + + +laplace_transform(x^a/factorial(a),x); + + + 1 +------ + a + s *s + + +laplace_transform(x,e^(-a*x),x); + + + 1 +----------------- + 2 2 + a + 2*a*s + s + + +laplace_transform(x^k,e^(-a*x),x); + + + gamma(k + 1) +------------------------- + k k + (a + s) *a + (a + s) *s + + +laplace_transform(cosh(a*x),x); + + + - s +--------- + 2 2 + a - s + + +laplace_transform(1/(2*a^3),sinh(a*x)-sin(a*x),x); + + + - 1 +--------- + 4 4 + a - s + + +laplace_transform(1/(a^2),1-cos(a*x),x); + + + 1 +----------- + 2 3 + a *s + s + + +laplace_transform(1/(b^2-a^2),cos(a*x)-cos(b*x),x); + + + s +---------------------------- + 2 2 2 2 2 2 4 + a *b + a *s + b *s + s + + +laplace_transform(besselj(0,2*sqrt(k*x)),x); + + + 1 +-------- + k/s + e *s + + +laplace_transform(Heaviside(x-1),x); + + + 1 +------ + s + e *s + + +laplace_transform(1/x,sin(k*x),x); + + + k +atan(---) + s + + +laplace_transform(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); + + + 2 2 2 2 + k *s k *s + - e *erf(k*s) + e + + +laplace_transform(1/k,e^(-k^2/(4*x)),x); + + + besselk(1,sqrt(s)*k) +---------------------- + sqrt(s) + + +laplace_transform(2/(sqrt(pi*x)),besselk(0,2*sqrt(2*k*x)),x); + + + k/s k + e *besselk(0,---) + s +--------------------- + sqrt(s) + + +hankel_transform(x,x); + + + n + 4 + gamma(-------) + 2 +------------------- + n - 2 2 + gamma(-------)*s + 2 + + +Y_transform(x,x); + + + - n + 4 n + 4 + gamma(----------)*gamma(-------) + 2 2 +------------------------------------- + - n + 3 n - 1 2 + gamma(----------)*gamma(-------)*s + 2 2 + + +K_transform(x,x); + + + - n + 4 n + 4 + gamma(----------)*gamma(-------) + 2 2 +---------------------------------- + 2 + 2*s + + +struveh_transform(x,x); + + + - n - 3 n + 5 + gamma(----------)*gamma(-------) + 2 2 +------------------------------------- + - n - 2 n - 2 2 + gamma(----------)*gamma(-------)*s + 2 2 + + +fourier_sin(e^(-x),x); + + + s +-------- + 2 + s + 1 + + +fourier_sin(sqrt(x),e^(-1/2*x),x); + + + 3*atan(2*s) + 2*sin(-------------)*pi + 2 +-------------------------------- + 2 3/4 + sqrt(pi)*(4*s + 1) *sqrt(2) + + +fourier_sin(1/x,e^(-a*x),x); + + + s +atan(---) + a + + +fourier_sin(x^k,x); + + + k/2 - k k + 4 *gamma(------)*gamma(---)*k + 2 2 +--------------------------------- + k k + 4*s *2 *gamma( - k)*s + + +fourier_sin(1/(b-a),(e^(-a*x)-e^(-b*x)),x); + + + a*s + b*s +---------------------------- + 2 2 2 2 2 2 4 + a *b + a *s + b *s + s + + +fourier_sin(besselj(0,a*x),x); + + + 2 2 + - a + s + heaviside(------------) + 2 + a +------------------------- + 2 2 + sqrt( - a + s ) + + +fourier_sin(1/sqrt(pi*x),cos(2*sqrt(k*x)),x); + + + k k + sqrt(s)*sqrt(2)*cos(---) - sqrt(s)*sqrt(2)*sin(---) + s s +----------------------------------------------------- + 2*s + + +fourier_sin(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); + + + sqrt( - pi)*erf(i*k*s) +------------------------ + 2 2 + k *s + sqrt(pi)*e + + +fourier_cos(e^(-1/2x),x); + + + 2 +---------- + 2 + 4*s + 1 + + +fourier_cos(x,e^(-x),x); + + + 2 + - s + 1 +--------------- + 4 2 + s + 2*s + 1 + + +fourier_cos(x,e^(-1/2*x^2),x); + + + 2 + i*s s /2 + sqrt( - pi)*erf(---------)*s + e *sqrt(2) + sqrt(2) +---------------------------------------------- + 2 + s /2 + e *sqrt(2) + + +fourier_cos(2*x^2,e^(-1/2x),x); + + + 2 + - 384*s + 32 +--------------------------- + 6 4 2 + 64*s + 48*s + 12*s + 1 + + +fourier_cos(x,e^(-a*x),x); + + + 2 2 + a - s +------------------- + 4 2 2 4 + a + 2*a *s + s + + +fourier_cos(x^n,e^(-a*x),x); + + + s s + cos(atan(---)*n + atan(---))*gamma(n + 1) + a a +------------------------------------------- + 2 2 (n + 1)/2 + (a + s ) + + +fourier_cos(1/x,sin(k*x),x); + + + 2 2 + sign(k - s )*pi + pi +----------------------- + 4 + + +fourier_cos(1/sqrt(pi*x),cos(2*sqrt(k*x)),x); + + + k k + sqrt(s)*sqrt(2)*cos(---) + sqrt(s)*sqrt(2)*sin(---) + s s +----------------------------------------------------- + 2*s + + +fourier_cos(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x); + + + 1 +-------- + 2 2 + k *s + e + + +fourier_cos(1/(pi*x),sin(2*k*sqrt(x)),x); + + + 2 2 + k k +intfc(----) + intfs(----) + s s + + +fourier_cos(1/(sqrt(pi*x)),e^(-2*k*sqrt(x)),x); + + + 2 2 2 + k k k +( - 2*sqrt(s)*cos(----)*fresnel_s(----) + sqrt(s)*cos(----) + s s s + + 2 2 2 + k k k + + 2*sqrt(s)*fresnel_c(----)*sin(----) - sqrt(s)*sin(----))/(sqrt(2)*s) + s s s + + +laplace_transform(x^n/factorial(n)*e^(-a*x),x); + + + 1 +------------------------- + n n + (a + s) *a + (a + s) *s + + +laplace_transform(1/(2*a^2)*(cosh(a*x)-cos(a*x)),x); + + + - s +--------- + 4 4 + a - s + + +laplace_transform(k*a^k/x*besselj(k,a*x),x); + + + 2*k + a +---------------------- + 2 2 k + (sqrt(a + s ) + s) + + +fourier_sin(1/x*e^(-3*x),x); + + + s +atan(---) + 3 + + +fourier_sin(1/(pi*x)*sin(2*k*sqrt(x)),x); + + + 2 2 + k k +intfc(----) - intfs(----) + s s + + +fourier_cos(x^n*e^(-a*x),x); + + + s s + cos(atan(---)*n + atan(---))*gamma(n + 1) + a a +------------------------------------------- + 2 2 (n + 1)/2 + (a + s ) + + +fourier_cos(1/(k*sqrt(pi))*e^(-x^2/(4*k^2)),x); + + + 1 +-------- + 2 2 + k *s + e + + +end; +(TIME: defint 163620 182910)