\(\int x^2 \text {Si}(a+b x)^2 \, dx\) [26]

Optimal result

Integrand size = 12, antiderivative size = 334 \[ \int x^2 \text {Si}(a+b x)^2 \, dx=\frac {2 x}{3 b^2}-\frac {a \cos (2 a+2 b x)}{6 b^3}-\frac {(a-b x) \cos (2 a+2 b x)}{6 b^3}+\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}-\frac {2 \cos (a+b x) \sin (a+b x)}{3 b^3}-\frac {\sin (2 a+2 b x)}{12 b^3}-\frac {4 \cos (a+b x) \text {Si}(a+b x)}{3 b^3}+\frac {2 a^2 \cos (a+b x) \text {Si}(a+b x)}{3 b^3}-\frac {2 a x \cos (a+b x) \text {Si}(a+b x)}{3 b^2}+\frac {2 x^2 \cos (a+b x) \text {Si}(a+b x)}{3 b}+\frac {2 a \sin (a+b x) \text {Si}(a+b x)}{3 b^3}-\frac {4 x \sin (a+b x) \text {Si}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Si}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Si}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Si}(a+b x)^2}{3 b}+\frac {2 \text {Si}(2 a+2 b x)}{3 b^3}-\frac {a^2 \text {Si}(2 a+2 b x)}{b^3} \] Output:

2/3*x/b^2-1/6*a*cos(2*b*x+2*a)/b^3-1/6*(-b*x+a)*cos(2*b*x+2*a)/b^3+a*Ci(2* 
b*x+2*a)/b^3-a*ln(b*x+a)/b^3-2/3*cos(b*x+a)*sin(b*x+a)/b^3-1/12*sin(2*b*x+ 
2*a)/b^3-4/3*cos(b*x+a)*Si(b*x+a)/b^3+2/3*a^2*cos(b*x+a)*Si(b*x+a)/b^3-2/3 
*a*x*cos(b*x+a)*Si(b*x+a)/b^2+2/3*x^2*cos(b*x+a)*Si(b*x+a)/b+2/3*a*sin(b*x 
+a)*Si(b*x+a)/b^3-4/3*x*sin(b*x+a)*Si(b*x+a)/b^2+1/3*a^2*(b*x+a)*Si(b*x+a) 
^2/b^3-1/3*a*x*(b*x+a)*Si(b*x+a)^2/b^2+1/3*x^2*(b*x+a)*Si(b*x+a)^2/b+2/3*S 
i(2*b*x+2*a)/b^3-a^2*Si(2*b*x+2*a)/b^3
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.47 \[ \int x^2 \text {Si}(a+b x)^2 \, dx=\frac {8 a+8 b x-4 a \cos (2 (a+b x))+2 b x \cos (2 (a+b x))+12 a \operatorname {CosIntegral}(2 (a+b x))-12 a \log (a+b x)-5 \sin (2 (a+b x))+8 \left (\left (-2+a^2-a b x+b^2 x^2\right ) \cos (a+b x)+(a-2 b x) \sin (a+b x)\right ) \text {Si}(a+b x)+4 \left (a^3+b^3 x^3\right ) \text {Si}(a+b x)^2+8 \text {Si}(2 (a+b x))-12 a^2 \text {Si}(2 (a+b x))}{12 b^3} \] Input:

Integrate[x^2*SinIntegral[a + b*x]^2,x]
 

Output:

(8*a + 8*b*x - 4*a*Cos[2*(a + b*x)] + 2*b*x*Cos[2*(a + b*x)] + 12*a*CosInt 
egral[2*(a + b*x)] - 12*a*Log[a + b*x] - 5*Sin[2*(a + b*x)] + 8*((-2 + a^2 
 - a*b*x + b^2*x^2)*Cos[a + b*x] + (a - 2*b*x)*Sin[a + b*x])*SinIntegral[a 
 + b*x] + 4*(a^3 + b^3*x^3)*SinIntegral[a + b*x]^2 + 8*SinIntegral[2*(a + 
b*x)] - 12*a^2*SinIntegral[2*(a + b*x)])/(12*b^3)
 

Rubi [A] (verified)

Time = 4.47 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.30, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.917, Rules used = {7063, 7063, 7059, 7065, 4906, 27, 3042, 3780, 7067, 5084, 7071, 3042, 3793, 2009, 7073, 7065, 4906, 27, 3042, 3780, 7292, 7293, 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.

\(\displaystyle \int x^2 \text {Si}(a+b x)^2 \, dx\)

\(\Big \downarrow \) 7063

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

\(\Big \downarrow \) 7063

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

\(\Big \downarrow \) 7059

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

\(\Big \downarrow \) 7065

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7067

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

\(\Big \downarrow \) 5084

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

\(\Big \downarrow \) 7071

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7073

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

\(\Big \downarrow \) 7065

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^2*SinIntegral[a + b*x]^2,x]
 

Output:

(x^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b) - (2*a*((x*Cos[a + b*x]*SinIn 
tegral[a + b*x])/b + (x*(a + b*x)*SinIntegral[a + b*x]^2)/(2*b) - (CosInte 
gral[2*a + 2*b*x]/(2*b) - Log[a + b*x]/(2*b) + (Sin[a + b*x]*SinIntegral[a 
 + b*x])/b)/b + (Cos[2*a + 2*b*x]/(2*b^2) + (a*SinIntegral[2*a + 2*b*x])/b 
^2)/2 - (a*(((a + b*x)*SinIntegral[a + b*x]^2)/b - 2*(-((Cos[a + b*x]*SinI 
ntegral[a + b*x])/b) + SinIntegral[2*a + 2*b*x]/(2*b))))/(2*b)))/(3*b) - ( 
2*(-((x^2*Cos[a + b*x]*SinIntegral[a + b*x])/b) + ((a*Cos[2*a + 2*b*x])/(2 
*b^3) - (x*Cos[2*a + 2*b*x])/(2*b^2) + Sin[2*a + 2*b*x]/(4*b^3) + (a^2*Sin 
Integral[2*a + 2*b*x])/b^3)/2 + (2*(-1/2*x/b - (a*CosIntegral[2*a + 2*b*x] 
)/(2*b^2) + (a*Log[a + b*x])/(2*b^2) + (Cos[a + b*x]*Sin[a + b*x])/(2*b^2) 
 + (x*Sin[a + b*x]*SinIntegral[a + b*x])/b - (-((Cos[a + b*x]*SinIntegral[ 
a + b*x])/b) + SinIntegral[2*a + 2*b*x]/(2*b))/b))/b))/3
 

Defintions of rubi rules used

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 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 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 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 5084

Int[Cos[w_]^(p_.)*(u_.)*Sin[v_]^(p_.), x_Symbol] :> Simp[1/2^p   Int[u*Sin[ 
2*v]^p, x], x] /; EqQ[w, v] && IntegerQ[p]
 

rule 7059

Int[SinIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(SinInte 
gral[a + b*x]^2/b), x] - Simp[2   Int[Sin[a + b*x]*SinIntegral[a + b*x], x] 
, x] /; FreeQ[{a, b}, x]
 

rule 7063

Int[((c_.) + (d_.)*(x_))^(m_.)*SinIntegral[(a_) + (b_.)*(x_)]^2, x_Symbol] 
:> Simp[(a + b*x)*(c + d*x)^m*(SinIntegral[a + b*x]^2/(b*(m + 1))), x] + (- 
Simp[2/(m + 1)   Int[(c + d*x)^m*Sin[a + b*x]*SinIntegral[a + b*x], x], x] 
+ Simp[(b*c - a*d)*(m/(b*(m + 1)))   Int[(c + d*x)^(m - 1)*SinIntegral[a + 
b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 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 7071

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

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]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int x^{2} \operatorname {Si}\left (b x +a \right )^{2}d x\]

Input:

int(x^2*Si(b*x+a)^2,x)
 

Output:

int(x^2*Si(b*x+a)^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.45 \[ \int x^2 \text {Si}(a+b x)^2 \, dx=\frac {2 \, {\left (b x - 2 \, a\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (b^{2} x^{2} - a b x + a^{2} - 2\right )} \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \operatorname {Si}\left (b x + a\right )^{2} + 3 \, b x + 6 \, a \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) - 6 \, a \log \left (b x + a\right ) - {\left (4 \, {\left (2 \, b x - a\right )} \operatorname {Si}\left (b x + a\right ) + 5 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right ) - 2 \, {\left (3 \, a^{2} - 2\right )} \operatorname {Si}\left (2 \, b x + 2 \, a\right )}{6 \, b^{3}} \] Input:

integrate(x^2*sin_integral(b*x+a)^2,x, algorithm="fricas")
 

Output:

1/6*(2*(b*x - 2*a)*cos(b*x + a)^2 + 4*(b^2*x^2 - a*b*x + a^2 - 2)*cos(b*x 
+ a)*sin_integral(b*x + a) + 2*(b^3*x^3 + a^3)*sin_integral(b*x + a)^2 + 3 
*b*x + 6*a*cos_integral(2*b*x + 2*a) - 6*a*log(b*x + a) - (4*(2*b*x - a)*s 
in_integral(b*x + a) + 5*cos(b*x + a))*sin(b*x + a) - 2*(3*a^2 - 2)*sin_in 
tegral(2*b*x + 2*a))/b^3
 

Sympy [F]

\[ \int x^2 \text {Si}(a+b x)^2 \, dx=\int x^{2} \operatorname {Si}^{2}{\left (a + b x \right )}\, dx \] Input:

integrate(x**2*Si(b*x+a)**2,x)
 

Output:

Integral(x**2*Si(a + b*x)**2, x)
 

Maxima [F]

\[ \int x^2 \text {Si}(a+b x)^2 \, dx=\int { x^{2} \operatorname {Si}\left (b x + a\right )^{2} \,d x } \] Input:

integrate(x^2*sin_integral(b*x+a)^2,x, algorithm="maxima")
 

Output:

integrate(x^2*sin_integral(b*x + a)^2, x)
 

Giac [F]

\[ \int x^2 \text {Si}(a+b x)^2 \, dx=\int { x^{2} \operatorname {Si}\left (b x + a\right )^{2} \,d x } \] Input:

integrate(x^2*sin_integral(b*x+a)^2,x, algorithm="giac")
 

Output:

integrate(x^2*sin_integral(b*x + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \text {Si}(a+b x)^2 \, dx=\int x^2\,{\mathrm {sinint}\left (a+b\,x\right )}^2 \,d x \] Input:

int(x^2*sinint(a + b*x)^2,x)
 

Output:

int(x^2*sinint(a + b*x)^2, x)
 

Reduce [F]

\[ \int x^2 \text {Si}(a+b x)^2 \, dx=\int \mathit {si} \left (b x +a \right )^{2} x^{2}d x \] Input:

int(x^2*Si(b*x+a)^2,x)
                                                                                    
                                                                                    
 

Output:

int(si(a + b*x)**2*x**2,x)