\(\int x^3 \cos (b x) \text {Si}(b x) \, dx\) [52]

Optimal result

Integrand size = 12, antiderivative size = 128 \[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=\frac {4 x}{b^3}-\frac {x^3}{6 b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {2 x \sin ^2(b x)}{b^3}-\frac {6 \cos (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \cos (b x) \text {Si}(b x)}{b^2}-\frac {6 x \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \sin (b x) \text {Si}(b x)}{b}+\frac {3 \text {Si}(2 b x)}{b^4} \] Output:

4*x/b^3-1/6*x^3/b-4*cos(b*x)*sin(b*x)/b^4+1/2*x^2*cos(b*x)*sin(b*x)/b^2-2* 
x*sin(b*x)^2/b^3-6*cos(b*x)*Si(b*x)/b^4+3*x^2*cos(b*x)*Si(b*x)/b^2-6*x*sin 
(b*x)*Si(b*x)/b^3+x^3*sin(b*x)*Si(b*x)/b+3*Si(2*b*x)/b^4
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=\frac {36 b x-2 b^3 x^3+12 b x \cos (2 b x)-24 \sin (2 b x)+3 b^2 x^2 \sin (2 b x)+12 \left (3 \left (-2+b^2 x^2\right ) \cos (b x)+b x \left (-6+b^2 x^2\right ) \sin (b x)\right ) \text {Si}(b x)+36 \text {Si}(2 b x)}{12 b^4} \] Input:

Integrate[x^3*Cos[b*x]*SinIntegral[b*x],x]
 

Output:

(36*b*x - 2*b^3*x^3 + 12*b*x*Cos[2*b*x] - 24*Sin[2*b*x] + 3*b^2*x^2*Sin[2* 
b*x] + 12*(3*(-2 + b^2*x^2)*Cos[b*x] + b*x*(-6 + b^2*x^2)*Sin[b*x])*SinInt 
egral[b*x] + 36*SinIntegral[2*b*x])/(12*b^4)
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.83, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.083, Rules used = {7073, 27, 3042, 3792, 15, 3042, 3115, 24, 7067, 27, 3924, 3042, 3115, 24, 7073, 27, 3042, 3115, 24, 7065, 27, 4906, 27, 3042, 3780}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \text {Si}(b x) \cos (b x) \, dx\)

\(\Big \downarrow \) 7073

\(\displaystyle -\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}-\int \frac {x^2 \sin ^2(b x)}{b}dx+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int x^2 \sin ^2(b x)dx}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int x^2 \sin (b x)^2dx}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {-\frac {\int \sin ^2(b x)dx}{2 b^2}+\frac {\int x^2dx}{2}+\frac {x \sin ^2(b x)}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}}{b}-\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {-\frac {\int \sin ^2(b x)dx}{2 b^2}+\frac {x \sin ^2(b x)}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}-\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int \sin (b x)^2dx}{2 b^2}+\frac {x \sin ^2(b x)}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}-\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}+\frac {x \sin ^2(b x)}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}-\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {3 \int x^2 \sin (b x) \text {Si}(b x)dx}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 7067

\(\displaystyle -\frac {3 \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\int \frac {x \cos (b x) \sin (b x)}{b}dx-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\int x \cos (b x) \sin (b x)dx}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3924

\(\displaystyle -\frac {3 \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\int \sin ^2(b x)dx}{2 b}}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\int \sin (b x)^2dx}{2 b}}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {3 \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {3 \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 7073

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\int \frac {\sin ^2(b x)}{b}dx+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int \sin ^2(b x)dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int \sin (b x)^2dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 7065

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\int \frac {\cos (b x) \sin (b x)}{b x}dx-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \frac {\cos (b x) \sin (b x)}{x}dx}{b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin (2 b x)}{2 x}dx}{b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}-\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {\frac {x \sin ^2(b x)}{2 b^2}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b^2}-\frac {x^2 \sin (b x) \cos (b x)}{2 b}+\frac {x^3}{6}}{b}+\frac {x^3 \text {Si}(b x) \sin (b x)}{b}-\frac {3 \left (-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {2 \left (\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )}{b}\)

Input:

Int[x^3*Cos[b*x]*SinIntegral[b*x],x]
 

Output:

-((x^3/6 - (x^2*Cos[b*x]*Sin[b*x])/(2*b) + (x*Sin[b*x]^2)/(2*b^2) - (x/2 - 
 (Cos[b*x]*Sin[b*x])/(2*b))/(2*b^2))/b) + (x^3*Sin[b*x]*SinIntegral[b*x])/ 
b - (3*(((x*Sin[b*x]^2)/(2*b) - (x/2 - (Cos[b*x]*Sin[b*x])/(2*b))/(2*b))/b 
 - (x^2*Cos[b*x]*SinIntegral[b*x])/b + (2*(-((x/2 - (Cos[b*x]*Sin[b*x])/(2 
*b))/b) + (x*Sin[b*x]*SinIntegral[b*x])/b - (-((Cos[b*x]*SinIntegral[b*x]) 
/b) + SinIntegral[2*b*x]/(2*b))/b))/b))/b
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3924

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^ 
(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1) 
)), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sin[a + b*x^n]^(p + 
 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 7065

Int[Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[(-Cos[a + b*x])*(SinIntegral[c + d*x]/b), x] + Simp[d/b   Int[Cos[a + b 
*x]*(Sin[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7067

Int[((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + 
(d_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*Cos[a + b*x]*(SinIntegral[c + 
 d*x]/b), x] + (Simp[d/b   Int[(e + f*x)^m*Cos[a + b*x]*(Sin[c + d*x]/(c + 
d*x)), x], x] + Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegr 
al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

rule 7073

Int[Cos[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinIntegral[(c_.) + 
(d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^m*Sin[a + b*x]*(SinIntegral[c + d* 
x]/b), x] + (-Simp[d/b   Int[(e + f*x)^m*Sin[a + b*x]*(Sin[c + d*x]/(c + d* 
x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sin[a + b*x]*SinIntegral 
[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 5.91 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (b x \right ) \left (b^{3} x^{3} \sin \left (b x \right )+3 b^{2} x^{2} \cos \left (b x \right )-6 \cos \left (b x \right )-6 b x \sin \left (b x \right )\right )-b^{2} x^{2} \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+2 \cos \left (b x \right )^{2} b x -4 \sin \left (b x \right ) \cos \left (b x \right )+2 b x +\frac {b^{3} x^{3}}{3}+3 \,\operatorname {Si}\left (2 b x \right )}{b^{4}}\)	\(111\)
default	\(\frac {\operatorname {Si}\left (b x \right ) \left (b^{3} x^{3} \sin \left (b x \right )+3 b^{2} x^{2} \cos \left (b x \right )-6 \cos \left (b x \right )-6 b x \sin \left (b x \right )\right )-b^{2} x^{2} \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+2 \cos \left (b x \right )^{2} b x -4 \sin \left (b x \right ) \cos \left (b x \right )+2 b x +\frac {b^{3} x^{3}}{3}+3 \,\operatorname {Si}\left (2 b x \right )}{b^{4}}\)	\(111\)

Input:

int(x^3*cos(b*x)*Si(b*x),x,method=_RETURNVERBOSE)
 

Output:

1/b^4*(Si(b*x)*(b^3*x^3*sin(b*x)+3*b^2*x^2*cos(b*x)-6*cos(b*x)-6*b*x*sin(b 
*x))-b^2*x^2*(-1/2*sin(b*x)*cos(b*x)+1/2*b*x)+2*cos(b*x)^2*b*x-4*sin(b*x)* 
cos(b*x)+2*b*x+1/3*b^3*x^3+3*Si(2*b*x))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.72 \[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=-\frac {b^{3} x^{3} - 12 \, b x \cos \left (b x\right )^{2} - 18 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - 12 \, b x - 3 \, {\left ({\left (b^{2} x^{2} - 8\right )} \cos \left (b x\right ) + 2 \, {\left (b^{3} x^{3} - 6 \, b x\right )} \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) - 18 \, \operatorname {Si}\left (2 \, b x\right )}{6 \, b^{4}} \] Input:

integrate(x^3*cos(b*x)*sin_integral(b*x),x, algorithm="fricas")
 

Output:

-1/6*(b^3*x^3 - 12*b*x*cos(b*x)^2 - 18*(b^2*x^2 - 2)*cos(b*x)*sin_integral 
(b*x) - 12*b*x - 3*((b^2*x^2 - 8)*cos(b*x) + 2*(b^3*x^3 - 6*b*x)*sin_integ 
ral(b*x))*sin(b*x) - 18*sin_integral(2*b*x))/b^4
 

Sympy [F]

\[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=\int x^{3} \cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \] Input:

integrate(x**3*cos(b*x)*Si(b*x),x)
 

Output:

Integral(x**3*cos(b*x)*Si(b*x), x)
 

Maxima [F]

\[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=\int { x^{3} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) \,d x } \] Input:

integrate(x^3*cos(b*x)*sin_integral(b*x),x, algorithm="maxima")
 

Output:

integrate(x^3*cos(b*x)*sin_integral(b*x), x)
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.41 \[ \int x^3 \cos (b x) \text {Si}(b x) \, dx={\left (\frac {3 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{4}} + \frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Si}\left (b x\right ) - \frac {b^{3} x^{3} \tan \left (b x\right )^{2} + b^{3} x^{3} - 3 \, b^{2} x^{2} \tan \left (b x\right ) - 12 \, b x \tan \left (b x\right )^{2} - 9 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 9 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 18 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} - 24 \, b x - 9 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) + 9 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) - 18 \, \operatorname {Si}\left (2 \, b x\right ) + 24 \, \tan \left (b x\right )}{6 \, {\left (b^{4} \tan \left (b x\right )^{2} + b^{4}\right )}} \] Input:

integrate(x^3*cos(b*x)*sin_integral(b*x),x, algorithm="giac")
 

Output:

(3*(b^2*x^2 - 2)*cos(b*x)/b^4 + (b^3*x^3 - 6*b*x)*sin(b*x)/b^4)*sin_integr 
al(b*x) - 1/6*(b^3*x^3*tan(b*x)^2 + b^3*x^3 - 3*b^2*x^2*tan(b*x) - 12*b*x* 
tan(b*x)^2 - 9*imag_part(cos_integral(2*b*x))*tan(b*x)^2 + 9*imag_part(cos 
_integral(-2*b*x))*tan(b*x)^2 - 18*sin_integral(2*b*x)*tan(b*x)^2 - 24*b*x 
 - 9*imag_part(cos_integral(2*b*x)) + 9*imag_part(cos_integral(-2*b*x)) - 
18*sin_integral(2*b*x) + 24*tan(b*x))/(b^4*tan(b*x)^2 + b^4)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=\int x^3\,\mathrm {sinint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \] Input:

int(x^3*sinint(b*x)*cos(b*x),x)
 

Output:

int(x^3*sinint(b*x)*cos(b*x), x)
 

Reduce [F]

\[ \int x^3 \cos (b x) \text {Si}(b x) \, dx=\int \cos \left (b x \right ) \mathit {si} \left (b x \right ) x^{3}d x \] Input:

int(x^3*cos(b*x)*Si(b*x),x)
 

Output:

int(cos(b*x)*si(b*x)*x**3,x)