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

Optimal result

Integrand size = 12, antiderivative size = 333 \[ \int x^2 \text {Shi}(a+b x)^2 \, dx=\frac {2 x}{3 b^2}-\frac {a \cosh (2 a+2 b x)}{6 b^3}-\frac {(a-b x) \cosh (2 a+2 b x)}{6 b^3}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {\sinh (2 a+2 b x)}{12 b^3}-\frac {4 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}-\frac {2 a^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {2 a \sinh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3} \] Output:

2/3*x/b^2-1/6*a*cosh(2*b*x+2*a)/b^3-1/6*(-b*x+a)*cosh(2*b*x+2*a)/b^3+a*Chi 
(2*b*x+2*a)/b^3-a*ln(b*x+a)/b^3-2/3*cosh(b*x+a)*sinh(b*x+a)/b^3-1/12*sinh( 
2*b*x+2*a)/b^3-4/3*cosh(b*x+a)*Shi(b*x+a)/b^3-2/3*a^2*cosh(b*x+a)*Shi(b*x+ 
a)/b^3+2/3*a*x*cosh(b*x+a)*Shi(b*x+a)/b^2-2/3*x^2*cosh(b*x+a)*Shi(b*x+a)/b 
-2/3*a*sinh(b*x+a)*Shi(b*x+a)/b^3+4/3*x*sinh(b*x+a)*Shi(b*x+a)/b^2+1/3*a^2 
*(b*x+a)*Shi(b*x+a)^2/b^3-1/3*a*x*(b*x+a)*Shi(b*x+a)^2/b^2+1/3*x^2*(b*x+a) 
*Shi(b*x+a)^2/b+2/3*Shi(2*b*x+2*a)/b^3+a^2*Shi(2*b*x+2*a)/b^3
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.47 \[ \int x^2 \text {Shi}(a+b x)^2 \, dx=\frac {8 a+8 b x-4 a \cosh (2 (a+b x))+2 b x \cosh (2 (a+b x))+12 a \text {Chi}(2 (a+b x))-12 a \log (a+b x)-5 \sinh (2 (a+b x))-8 \left (\left (2+a^2-a b x+b^2 x^2\right ) \cosh (a+b x)+(a-2 b x) \sinh (a+b x)\right ) \text {Shi}(a+b x)+4 \left (a^3+b^3 x^3\right ) \text {Shi}(a+b x)^2+8 \text {Shi}(2 (a+b x))+12 a^2 \text {Shi}(2 (a+b x))}{12 b^3} \] Input:

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

Output:

(8*a + 8*b*x - 4*a*Cosh[2*(a + b*x)] + 2*b*x*Cosh[2*(a + b*x)] + 12*a*Cosh 
Integral[2*(a + b*x)] - 12*a*Log[a + b*x] - 5*Sinh[2*(a + b*x)] - 8*((2 + 
a^2 - a*b*x + b^2*x^2)*Cosh[a + b*x] + (a - 2*b*x)*Sinh[a + b*x])*SinhInte 
gral[a + b*x] + 4*(a^3 + b^3*x^3)*SinhIntegral[a + b*x]^2 + 8*SinhIntegral 
[2*(a + b*x)] + 12*a^2*SinhIntegral[2*(a + b*x)])/(12*b^3)
 

Rubi [A] (verified)

Time = 4.75 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.30, number of steps used = 26, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.167, Rules used = {7092, 7092, 7088, 7094, 5971, 27, 3042, 26, 3779, 7096, 6151, 7100, 3042, 25, 3793, 2009, 7102, 7094, 5971, 27, 3042, 26, 3779, 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.

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

\(\Big \downarrow \) 7092

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

\(\Big \downarrow \) 7092

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

\(\Big \downarrow \) 7088

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

\(\Big \downarrow \) 7094

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{3} \int x^2 \sinh (a+b x) \text {Shi}(a+b x)dx-\frac {2 a \left (-\int x \sinh (a+b x) \text {Shi}(a+b x)dx-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{a+b x}dx\right )\right )}{2 b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2}{3} \int x^2 \sinh (a+b x) \text {Shi}(a+b x)dx-\frac {2 a \left (-\int x \sinh (a+b x) \text {Shi}(a+b x)dx-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {1}{2} \int -\frac {i \sin (2 i a+2 i b x)}{a+b x}dx\right )\right )}{2 b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2}{3} \int x^2 \sinh (a+b x) \text {Shi}(a+b x)dx-\frac {2 a \left (-\int x \sinh (a+b x) \text {Shi}(a+b x)dx-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}+\frac {1}{2} i \int \frac {\sin (2 i a+2 i b x)}{a+b x}dx\right )\right )}{2 b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 7096

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

\(\Big \downarrow \) 6151

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x)dx}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {\int \cosh (a+b x) \text {Shi}(a+b x)dx}{b}+\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7100

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x)dx}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}-\int \frac {\sinh ^2(a+b x)}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x)dx}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}-\int -\frac {\sin (i a+i b x)^2}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x)dx}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\int \frac {\sin (i a+i b x)^2}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {2}{3} \left (-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x)dx}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {\int \left (\frac {1}{2 (a+b x)}-\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right )dx+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}}{b}+\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x)dx}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7102

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\frac {\int \sinh (a+b x) \text {Shi}(a+b x)dx}{b}-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7094

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x}dx}{b}-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\int \frac {\sinh (2 a+2 b x)}{2 (a+b x)}dx}{b}-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{a+b x}dx}{b}-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {1}{2} \int -\frac {i \sin (2 i a+2 i b x)}{a+b x}dx}{b}-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}+\frac {1}{2} i \int \frac {\sin (2 i a+2 i b x)}{a+b x}dx}{b}-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {2 a \left (\frac {1}{2} \int \frac {x \sinh (2 a+2 b x)}{a+b x}dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}-\frac {2}{3} \left (-\frac {2 \left (-\int \frac {x \sinh ^2(a+b x)}{a+b x}dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}}{b}\right )}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 a+2 b x)}{a+b x}dx+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {2}{3} \left (-\frac {1}{2} \int \left (\frac {\sinh (2 a+2 b x) a^2}{b^2 (a+b x)}-\frac {\sinh (2 a+2 b x) a}{b^2}+\frac {x \sinh (2 a+2 b x)}{b}\right )dx-\frac {2 \left (-\int \left (\frac {\sinh ^2(a+b x)}{b}-\frac {a \sinh ^2(a+b x)}{b (a+b x)}\right )dx+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {1}{2} \int \left (\frac {\sinh (2 a+2 b x)}{b}+\frac {a \sinh (2 a+2 b x)}{b (-a-b x)}\right )dx+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{3} \left (\frac {1}{2} \left (-\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{4 b^3}+\frac {a \cosh (2 a+2 b x)}{2 b^3}-\frac {x \cosh (2 a+2 b x)}{2 b^2}\right )-\frac {2 \left (\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}}{b}+\frac {x}{2 b}\right )}{b}+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\right )-\frac {2 a \left (\frac {1}{2} \left (\frac {\cosh (2 a+2 b x)}{2 b^2}-\frac {a \text {Shi}(2 a+2 b x)}{b^2}\right )+\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {\log (a+b x)}{2 b}}{b}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {a \left (\frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \left (\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}\right )\right )}{2 b}\right )}{3 b}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}\)

Input:

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

Output:

(x^2*(a + b*x)*SinhIntegral[a + b*x]^2)/(3*b) - (2*a*(-((x*Cosh[a + b*x]*S 
inhIntegral[a + b*x])/b) + (x*(a + b*x)*SinhIntegral[a + b*x]^2)/(2*b) + ( 
-1/2*CoshIntegral[2*a + 2*b*x]/b + Log[a + b*x]/(2*b) + (Sinh[a + b*x]*Sin 
hIntegral[a + b*x])/b)/b + (Cosh[2*a + 2*b*x]/(2*b^2) - (a*SinhIntegral[2* 
a + 2*b*x])/b^2)/2 - (a*(((a + b*x)*SinhIntegral[a + b*x]^2)/b - 2*((Cosh[ 
a + b*x]*SinhIntegral[a + b*x])/b - SinhIntegral[2*a + 2*b*x]/(2*b))))/(2* 
b)))/(3*b) - (2*((x^2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b + ((a*Cosh[2* 
a + 2*b*x])/(2*b^3) - (x*Cosh[2*a + 2*b*x])/(2*b^2) + Sinh[2*a + 2*b*x]/(4 
*b^3) - (a^2*SinhIntegral[2*a + 2*b*x])/b^3)/2 - (2*(x/(2*b) + (a*CoshInte 
gral[2*a + 2*b*x])/(2*b^2) - (a*Log[a + b*x])/(2*b^2) - (Cosh[a + b*x]*Sin 
h[a + b*x])/(2*b^2) + (x*Sinh[a + b*x]*SinhIntegral[a + b*x])/b - ((Cosh[a 
 + b*x]*SinhIntegral[a + b*x])/b - SinhIntegral[2*a + 2*b*x]/(2*b))/b))/b) 
)/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 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 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 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 5971

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

rule 6151

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

rule 7088

Int[SinhIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(SinhIn 
tegral[a + b*x]^2/b), x] - Simp[2   Int[Sinh[a + b*x]*SinhIntegral[a + b*x] 
, x], x] /; FreeQ[{a, b}, x]
 

rule 7092

Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_) + (b_.)*(x_)]^2, x_Symbol] 
 :> Simp[(a + b*x)*(c + d*x)^m*(SinhIntegral[a + b*x]^2/(b*(m + 1))), x] + 
(-Simp[2/(m + 1)   Int[(c + d*x)^m*Sinh[a + b*x]*SinhIntegral[a + b*x], x], 
 x] + Simp[(b*c - a*d)*(m/(b*(m + 1)))   Int[(c + d*x)^(m - 1)*SinhIntegral 
[a + b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
 

rule 7094

Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> 
 Simp[Cosh[a + b*x]*(SinhIntegral[c + d*x]/b), x] - Simp[d/b   Int[Cosh[a + 
 b*x]*(Sinh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7096

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

rule 7100

Int[Cosh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> 
 Simp[Sinh[a + b*x]*(SinhIntegral[c + d*x]/b), x] - Simp[d/b   Int[Sinh[a + 
 b*x]*(Sinh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7102

Int[Cosh[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinhIntegral[(c_.) 
+ (d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^m*Sinh[a + b*x]*(SinhIntegral[c 
+ d*x]/b), x] + (-Simp[d/b   Int[(e + f*x)^m*Sinh[a + b*x]*(Sinh[c + d*x]/( 
c + d*x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*Sinh 
Integral[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x^2*sinh_integral(b*x + a)^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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