\(\int x^3 \operatorname {CosIntegral}(b x)^2 \, dx\) [78]

Optimal result

Integrand size = 10, antiderivative size = 163 \[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\frac {x^2}{4 b^2}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 \cos (b x) \operatorname {CosIntegral}(b x)}{b^4}-\frac {3 x^2 \cos (b x) \operatorname {CosIntegral}(b x)}{2 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2-\frac {3 \operatorname {CosIntegral}(2 b x)}{2 b^4}-\frac {3 \log (x)}{2 b^4}+\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {3 x \operatorname {CosIntegral}(b x) \sin (b x)}{b^3}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{2 b}-\frac {13 \sin ^2(b x)}{8 b^4}+\frac {x^2 \sin ^2(b x)}{4 b^2} \] Output:

1/4*x^2/b^2+3/8*cos(b*x)^2/b^4+3*cos(b*x)*Ci(b*x)/b^4-3/2*x^2*cos(b*x)*Ci( 
b*x)/b^2+1/4*x^4*Ci(b*x)^2-3/2*Ci(2*b*x)/b^4-3/2*ln(x)/b^4+x*cos(b*x)*sin( 
b*x)/b^3+3*x*Ci(b*x)*sin(b*x)/b^3-1/2*x^3*Ci(b*x)*sin(b*x)/b-13/8*sin(b*x) 
^2/b^4+1/4*x^2*sin(b*x)^2/b^2
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.66 \[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\frac {3 b^2 x^2+8 \cos (2 b x)-b^2 x^2 \cos (2 b x)+2 b^4 x^4 \operatorname {CosIntegral}(b x)^2-12 \operatorname {CosIntegral}(2 b x)-12 \log (x)-4 \operatorname {CosIntegral}(b x) \left (3 \left (-2+b^2 x^2\right ) \cos (b x)+b x \left (-6+b^2 x^2\right ) \sin (b x)\right )+4 b x \sin (2 b x)}{8 b^4} \] Input:

Integrate[x^3*CosIntegral[b*x]^2,x]
 

Output:

(3*b^2*x^2 + 8*Cos[2*b*x] - b^2*x^2*Cos[2*b*x] + 2*b^4*x^4*CosIntegral[b*x 
]^2 - 12*CosIntegral[2*b*x] - 12*Log[x] - 4*CosIntegral[b*x]*(3*(-2 + b^2* 
x^2)*Cos[b*x] + b*x*(-6 + b^2*x^2)*Sin[b*x]) + 4*b*x*Sin[2*b*x])/(8*b^4)
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.42, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.200, Rules used = {7062, 7068, 27, 3924, 3042, 3791, 15, 7074, 27, 3042, 3791, 15, 7068, 27, 3042, 3044, 15, 7072, 27, 3042, 3793, 2009}

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.

\(\displaystyle \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx\)

\(\Big \downarrow \) 7062

\(\displaystyle \frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2-\frac {1}{2} \int x^3 \cos (b x) \operatorname {CosIntegral}(b x)dx\)

\(\Big \downarrow \) 7068

\(\displaystyle \frac {1}{2} \left (\frac {3 \int x^2 \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\int \frac {x^2 \cos (b x) \sin (b x)}{b}dx-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {3 \int x^2 \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\frac {\int x^2 \cos (b x) \sin (b x)dx}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3924

\(\displaystyle \frac {1}{2} \left (\frac {3 \int x^2 \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\int x \sin ^2(b x)dx}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {3 \int x^2 \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\int x \sin (b x)^2dx}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\int xdx}{2}+\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}}{b}}{b}+\frac {3 \int x^2 \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} \left (\frac {3 \int x^2 \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 7074

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \int x \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\int \frac {x \cos ^2(b x)}{b}dx-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \int x \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\int x \cos ^2(b x)dx}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \int x \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\int x \sin \left (b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {\frac {\int xdx}{2}+\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}}{b}+\frac {2 \int x \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \int x \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 7068

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\int \frac {\cos (b x) \sin (b x)}{b}dx+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\int \cos (b x) \sin (b x)dx}{b}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\int \cos (b x) \sin (b x)dx}{b}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3044

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \sin (b x)d\sin (b x)}{b^2}-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 7072

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\int \frac {\cos ^2(b x)}{b x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \frac {\cos ^2(b x)}{x}dx}{b}-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin \left (b x+\frac {\pi }{2}\right )^2}{x}dx}{b}-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\frac {\int \left (\frac {\cos (2 b x)}{2 x}+\frac {1}{2 x}\right )dx}{b}-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (\frac {2 \left (-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {\frac {\operatorname {CosIntegral}(2 b x)}{2}+\frac {\log (x)}{2}}{b}-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{b}}{b}\right )}{b}+\frac {\frac {\cos ^2(b x)}{4 b^2}+\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}-\frac {x^2 \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\right )+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)^2\)

Input:

Int[x^3*CosIntegral[b*x]^2,x]
 

Output:

(x^4*CosIntegral[b*x]^2)/4 + (-((x^3*CosIntegral[b*x]*Sin[b*x])/b) + (3*(- 
((x^2*Cos[b*x]*CosIntegral[b*x])/b) + (x^2/4 + Cos[b*x]^2/(4*b^2) + (x*Cos 
[b*x]*Sin[b*x])/(2*b))/b + (2*(-((-((Cos[b*x]*CosIntegral[b*x])/b) + (CosI 
ntegral[2*b*x]/2 + Log[x]/2)/b)/b) + (x*CosIntegral[b*x]*Sin[b*x])/b - Sin 
[b*x]^2/(2*b^2)))/b))/b + ((x^2*Sin[b*x]^2)/(2*b) - (x^2/4 - (x*Cos[b*x]*S 
in[b*x])/(2*b) + Sin[b*x]^2/(4*b^2))/b)/b)/2
 

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 27

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[1/(a*f)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a 
*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

rule 3791

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[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 7062

Int[CosIntegral[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(CosI 
ntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*Cos[b*x]*CosIntegral 
[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

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 7072

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

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 = 7.58 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {b^{4} x^{4} \operatorname {Ci}\left (b x \right )^{2}}{4}-2 \,\operatorname {Ci}\left (b x \right ) \left (\frac {b^{3} x^{3} \sin \left (b x \right )}{4}+\frac {3 b^{2} x^{2} \cos \left (b x \right )}{4}-\frac {3 \cos \left (b x \right )}{2}-\frac {3 b x \sin \left (b x \right )}{2}\right )-\frac {\cos \left (b x \right )^{2} b^{2} x^{2}}{4}+2 b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )-\frac {b^{2} x^{2}}{2}-\frac {\sin \left (b x \right )^{2}}{2}-\frac {3 \ln \left (b x \right )}{2}-\frac {3 \,\operatorname {Ci}\left (2 b x \right )}{2}+\frac {3 \cos \left (b x \right )^{2}}{2}}{b^{4}}\)	\(135\)
default	\(\frac {\frac {b^{4} x^{4} \operatorname {Ci}\left (b x \right )^{2}}{4}-2 \,\operatorname {Ci}\left (b x \right ) \left (\frac {b^{3} x^{3} \sin \left (b x \right )}{4}+\frac {3 b^{2} x^{2} \cos \left (b x \right )}{4}-\frac {3 \cos \left (b x \right )}{2}-\frac {3 b x \sin \left (b x \right )}{2}\right )-\frac {\cos \left (b x \right )^{2} b^{2} x^{2}}{4}+2 b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )-\frac {b^{2} x^{2}}{2}-\frac {\sin \left (b x \right )^{2}}{2}-\frac {3 \ln \left (b x \right )}{2}-\frac {3 \,\operatorname {Ci}\left (2 b x \right )}{2}+\frac {3 \cos \left (b x \right )^{2}}{2}}{b^{4}}\)	\(135\)

Input:

int(x^3*Ci(b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b^4*(1/4*b^4*x^4*Ci(b*x)^2-2*Ci(b*x)*(1/4*b^3*x^3*sin(b*x)+3/4*b^2*x^2*c 
os(b*x)-3/2*cos(b*x)-3/2*b*x*sin(b*x))-1/4*cos(b*x)^2*b^2*x^2+2*b*x*(1/2*s 
in(b*x)*cos(b*x)+1/2*b*x)-1/2*b^2*x^2-1/2*sin(b*x)^2-3/2*ln(b*x)-3/2*Ci(2* 
b*x)+3/2*cos(b*x)^2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=-\frac {\pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, \pi b^{2} x^{2} + 6 \, \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - {\left (3 \, \pi + \pi ^{3} b^{4} x^{4}\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, \pi ^{3} b^{4}} \] Input:

integrate(x^3*fresnel_cos(b*x)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\int x^{3} \operatorname {Ci}^{2}{\left (b x \right )}\, dx \] Input:

integrate(x**3*Ci(b*x)**2,x)
 

Output:

Integral(x**3*Ci(b*x)**2, x)
 

Maxima [F]

\[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\int { x^{3} \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^3*fresnel_cos(b*x)^2,x, algorithm="maxima")
 

Output:

integrate(x^3*fresnel_cos(b*x)^2, x)
 

Giac [F]

\[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\int { x^{3} \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^3*fresnel_cos(b*x)^2,x, algorithm="giac")
 

Output:

integrate(x^3*fresnel_cos(b*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\int x^3\,{\mathrm {cosint}\left (b\,x\right )}^2 \,d x \] Input:

int(x^3*cosint(b*x)^2,x)
 

Output:

int(x^3*cosint(b*x)^2, x)
 

Reduce [F]

\[ \int x^3 \operatorname {CosIntegral}(b x)^2 \, dx=\int \mathit {ci} \left (b x \right )^{2} x^{3}d x \] Input:

int(x^3*Ci(b*x)^2,x)
                                                                                    
                                                                                    
 

Output:

int(ci(b*x)**2*x**3,x)