\(\int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx\) [113]

Optimal result

Integrand size = 12, antiderivative size = 147 \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=-\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {3 \text {Si}(2 b x)}{b^4} \] Output:

-5/2*x/b^3+1/6*x^3/b+1/2*x*cos(b*x)^2/b^3+6*x*cos(b*x)*Ci(b*x)/b^3-x^3*cos 
(b*x)*Ci(b*x)/b-4*cos(b*x)*sin(b*x)/b^4+1/2*x^2*cos(b*x)*sin(b*x)/b^2-6*Ci 
(b*x)*sin(b*x)/b^4+3*x^2*Ci(b*x)*sin(b*x)/b^2-3/2*x*sin(b*x)^2/b^3+3*Si(2* 
b*x)/b^4
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\frac {-36 b x+2 b^3 x^3+12 b x \cos (2 b x)-12 \operatorname {CosIntegral}(b x) \left (b x \left (-6+b^2 x^2\right ) \cos (b x)-3 \left (-2+b^2 x^2\right ) \sin (b x)\right )-24 \sin (2 b x)+3 b^2 x^2 \sin (2 b x)+36 \text {Si}(2 b x)}{12 b^4} \] Input:

Integrate[x^3*CosIntegral[b*x]*Sin[b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.58, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.083, Rules used = {7074, 27, 3042, 3792, 15, 3042, 3115, 24, 7068, 27, 3924, 3042, 3115, 24, 7074, 27, 3042, 3115, 24, 7066, 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.

\(\displaystyle \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx\)

\(\Big \downarrow \) 7074

\(\displaystyle \frac {3 \int x^2 \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\int \frac {x^2 \cos ^2(b x)}{b}dx-\frac {x^3 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \int x^2 \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\int x^2 \cos ^2(b x)dx}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \int x^2 \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\int x^2 \sin \left (b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3924

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7074

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

Input:

Int[x^3*CosIntegral[b*x]*Sin[b*x],x]
 

Output:

-((x^3*Cos[b*x]*CosIntegral[b*x])/b) + (x^3/6 + (x*Cos[b*x]^2)/(2*b^2) + ( 
x^2*Cos[b*x]*Sin[b*x])/(2*b) - (x/2 + (Cos[b*x]*Sin[b*x])/(2*b))/(2*b^2))/ 
b + (3*((x^2*CosIntegral[b*x]*Sin[b*x])/b - ((x*Sin[b*x]^2)/(2*b) - (x/2 - 
 (Cos[b*x]*Sin[b*x])/(2*b))/(2*b))/b - (2*(-((x*Cos[b*x]*CosIntegral[b*x]) 
/b) + (x/2 + (Cos[b*x]*Sin[b*x])/(2*b))/b + ((CosIntegral[b*x]*Sin[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 7066

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

rule 7068

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

rule 7074

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

Maple [A] (verified)

Time = 5.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76

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

Input:

int(x^3*Ci(b*x)*sin(b*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (137) = 274\).

Time = 0.11 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.46 \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=-\frac {2 \, \pi b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \cos \left (b x\right ) + 2 \, {\left (\pi ^{3} b^{4} x^{3} - 6 \, \pi ^{3} b^{2} x\right )} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) + {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + 2 \, {\left (3 \, \pi ^{2} b^{2} x \sin \left (b x\right ) - {\left (\pi ^{2} b^{3} x^{2} - 6 \, \pi ^{2} b + b\right )} \cos \left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 3 \, {\left (\pi ^{3} b^{3} x^{2} - 2 \, \pi ^{3} b\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (b x\right )}{2 \, \pi ^{3} b^{5}} \] Input:

integrate(x^3*fresnel_cos(b*x)*sin(b*x),x, algorithm="fricas")
 

Output:

-1/2*(2*pi*b*cos(1/2*pi*b^2*x^2)*cos(b*x) + 2*(pi^3*b^4*x^3 - 6*pi^3*b^2*x 
)*cos(b*x)*fresnel_cos(b*x) + (6*pi^3*sin(1/2/pi) - (3*pi^2 - 1)*cos(1/2/p 
i))*sqrt(b^2)*fresnel_cos((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - (6*pi^3*sin(1/2 
/pi) - (3*pi^2 - 1)*cos(1/2/pi))*sqrt(b^2)*fresnel_cos((pi*b*x - 1)*sqrt(b 
^2)/(pi*b)) - (6*pi^3*cos(1/2/pi) + (3*pi^2 - 1)*sin(1/2/pi))*sqrt(b^2)*fr 
esnel_sin((pi*b*x + 1)*sqrt(b^2)/(pi*b)) + (6*pi^3*cos(1/2/pi) + (3*pi^2 - 
 1)*sin(1/2/pi))*sqrt(b^2)*fresnel_sin((pi*b*x - 1)*sqrt(b^2)/(pi*b)) + 2* 
(3*pi^2*b^2*x*sin(b*x) - (pi^2*b^3*x^2 - 6*pi^2*b + b)*cos(b*x))*sin(1/2*p 
i*b^2*x^2) + 2*(pi*b^2*x*cos(1/2*pi*b^2*x^2) - 3*(pi^3*b^3*x^2 - 2*pi^3*b) 
*fresnel_cos(b*x))*sin(b*x))/(pi^3*b^5)
 

Sympy [F]

\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int x^{3} \sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \] Input:

integrate(x**3*Ci(b*x)*sin(b*x),x)
 

Output:

Integral(x**3*sin(b*x)*Ci(b*x), x)
 

Maxima [F]

\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \sin \left (b x\right ) \,d x } \] Input:

integrate(x^3*fresnel_cos(b*x)*sin(b*x),x, algorithm="maxima")
 

Output:

integrate(x^3*fresnel_cos(b*x)*sin(b*x), x)
 

Giac [F]

\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \sin \left (b x\right ) \,d x } \] Input:

integrate(x^3*fresnel_cos(b*x)*sin(b*x),x, algorithm="giac")
 

Output:

integrate(x^3*fresnel_cos(b*x)*sin(b*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int x^3\,\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \] Input:

int(x^3*cosint(b*x)*sin(b*x),x)
 

Output:

int(x^3*cosint(b*x)*sin(b*x), x)
 

Reduce [F]

\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int \mathit {ci} \left (b x \right ) \sin \left (b x \right ) x^{3}d x \] Input:

int(x^3*Ci(b*x)*sin(b*x),x)
                                                                                    
                                                                                    
 

Output:

int(ci(b*x)*sin(b*x)*x**3,x)