Tue Feb 10 12:28:32 2004 run on Linux
*** 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);
si(y)*y - 1 + cos(y)
int(besselj(2,x^(1/4)),x,0,y);
1/4 1/4
- 2*((8*sqrt(y) - y)*besselj(0,y ) - besselj(2,y )*y
1/4 1/4 1/4 1/4
+ 4*y *(y + 2)*(y - 2)*besselj(1,y ))
int(x*besseli(2,x),x,0,y);
- (2*(besseli(0,y) - 1) - besseli(1,y)*y)
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);
y 2
(e - 1)
-----------
y
2*e
int(cosh(2*x),x,0,y);
2*y y y
(e + 1)*(e + 1)*(e - 1)
------------------------------
2*y
4*e
int(x*shi(x),x,0,y);
y y 2
- (e *(e *(y - 1) - 2*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);
sin(y) + 1 - cos(y)
int(sin x + sin(-2*x),x,0,y);
- (2*cos(y) - 1 - cos(2*y))
------------------------------
2
int(sin(n*x),x,0,y);
- (cos(n*y) - 1)
-------------------
n
int(heaviside(x-1),x,0,y);
(y - 1)*heaviside(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);
erf(i*k*s)*i
--------------
2 2
k *s
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(---------)*i*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 for test: 7010 ms, plus GC time: 300 ms