For comparison with other CAS choose from: Axiom Derive Macsyma Maple Mathematica
(a1*cos(wt) + a3*cos(3*wt) + b1*sin(wt) + b3*sin(3*wt))**3
where cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2,
cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2,
sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2,
cos(~x)**2 => (1+cos(2*x))/2,
sin(~x)**2 => (1-cos(2*x))/2;
factor cos,sin;
2 2
(cos(9*wt)*a3*(a3 - 3*b3 )
2 2
+ 3*cos(7*wt)*(a1*a3 - a1*b3 - 2*a3*b1*b3) + 3*cos(5*wt)*
2 2 2 2
(a1 *a3 + a1*a3 - 2*a1*b1*b3 - a1*b3 - a3*b1 + 2*a3*b1*b3) +
3 2 2 3 2 2
cos(3*wt)*(a1 + 6*a1 *a3 - 3*a1*b1 + 3*a3 + 6*a3*b1 + 3*a3*b3 )
+ 3*cos(wt)*
3 2 2 2 2 2
(a1 + a1 *a3 + 2*a1*a3 + a1*b1 + 2*a1*b1*b3 + 2*a1*b3 - a3*b1 )
2 2
+ sin(9*wt)*b3*(3*a3 - b3 )
2 2
+ 3*sin(7*wt)*(2*a1*a3*b3 + a3 *b1 - b1*b3 ) + 3*sin(5*wt)*
2 2 2 2
(a1 *b3 + 2*a1*a3*b1 + 2*a1*a3*b3 - a3 *b1 - b1 *b3 + b1*b3 ) +
2 2 2 3 2 3
sin(3*wt)*(3*a1 *b1 + 6*a1 *b3 + 3*a3 *b3 - b1 + 6*b1 *b3 + 3*b3 )
+ 3*sin(wt)*
2 2 2 3 2 2
(a1 *b1 + a1 *b3 - 2*a1*a3*b1 + 2*a3 *b1 + b1 - b1 *b3 + 2*b1*b3 )
)/4
operator integrate;
linear integrate;
let integrate(~x**~p,x) => x**(p+1)/(p+1) when df(p,x)=0,
integrate(~x,x) => x**2/2,
integrate(1,~x) => x$
integrate(a^2*b+a^b+3*a-5,a);
b 2 2 2
a*(6*a + 2*a *b + 2*a *b + 9*a*b + 9*a - 30*b - 30)
-------------------------------------------------------
6*(b + 1)
integrate(a^(a+1),a);
a
integrate(a *a,a)