3.1.78 \(\int x^3 \text {Chi}(b x)^2 \, dx\) [78]

3.1.78.1 Optimal result

Integrand size = 10, antiderivative size = 164 \[ \int x^3 \text {Chi}(b x)^2 \, dx=-\frac {x^2}{4 b^2}+\frac {3 \cosh ^2(b x)}{8 b^4}+\frac {3 \cosh (b x) \text {Chi}(b x)}{b^4}+\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Chi}(b x)^2-\frac {3 \text {Chi}(2 b x)}{2 b^4}-\frac {3 \log (x)}{2 b^4}-\frac {x \cosh (b x) \sinh (b x)}{b^3}-\frac {3 x \text {Chi}(b x) \sinh (b x)}{b^3}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{2 b}+\frac {13 \sinh ^2(b x)}{8 b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2} \]

output

-1/4*x^2/b^2+1/4*x^4*Chi(b*x)^2-3/2*Chi(2*b*x)/b^4+3*Chi(b*x)*cosh(b*x)/b^ 
4+3/2*x^2*Chi(b*x)*cosh(b*x)/b^2+3/8*cosh(b*x)^2/b^4-3/2*ln(x)/b^4-3*x*Chi 
(b*x)*sinh(b*x)/b^3-1/2*x^3*Chi(b*x)*sinh(b*x)/b-x*cosh(b*x)*sinh(b*x)/b^3 
+13/8*sinh(b*x)^2/b^4+1/4*x^2*sinh(b*x)^2/b^2
 

3.1.78.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.65 \[ \int x^3 \text {Chi}(b x)^2 \, dx=\frac {-3 b^2 x^2+8 \cosh (2 b x)+b^2 x^2 \cosh (2 b x)+2 b^4 x^4 \text {Chi}(b x)^2-12 \text {Chi}(2 b x)-12 \log (x)-4 \text {Chi}(b x) \left (-3 \left (2+b^2 x^2\right ) \cosh (b x)+b x \left (6+b^2 x^2\right ) \sinh (b x)\right )-4 b x \sinh (2 b x)}{8 b^4} \]

input

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

output

(-3*b^2*x^2 + 8*Cosh[2*b*x] + b^2*x^2*Cosh[2*b*x] + 2*b^4*x^4*CoshIntegral 
[b*x]^2 - 12*CoshIntegral[2*b*x] - 12*Log[x] - 4*CoshIntegral[b*x]*(-3*(2 
+ b^2*x^2)*Cosh[b*x] + b*x*(6 + b^2*x^2)*Sinh[b*x]) - 4*b*x*Sinh[2*b*x])/( 
8*b^4)
 

3.1.78.3 Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.41, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.400, Rules used = {7091, 7097, 27, 5895, 3042, 25, 3791, 15, 7103, 27, 3042, 3791, 15, 7097, 27, 3042, 26, 3044, 15, 7101, 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.

\(\displaystyle \int x^3 \text {Chi}(b x)^2 \, dx\)

\(\Big \downarrow \) 7091

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2-\frac {1}{2} \int x^3 \cosh (b x) \text {Chi}(b x)dx\)

\(\Big \downarrow \) 7097

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5895

\(\displaystyle \frac {1}{2} \left (\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {\frac {x^2 \sinh ^2(b x)}{2 b}-\frac {\int x \sinh ^2(b x)dx}{b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )+\frac {1}{4} x^4 \text {Chi}(b x)^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {\frac {x^2 \sinh ^2(b x)}{2 b}-\frac {\int -x \sin (i b x)^2dx}{b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {\frac {x^2 \sinh ^2(b x)}{2 b}+\frac {\int x \sin (i b x)^2dx}{b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\int xdx}{2}+\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )+\frac {1}{4} x^4 \text {Chi}(b x)^2\)

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7103

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \left (-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}-\frac {\int x \sin \left (i b x+\frac {\pi }{2}\right )^2dx}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}+\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {1}{2} \left (\frac {3 \left (-\frac {\frac {\int xdx}{2}-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}}{b}-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}+\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )+\frac {1}{4} x^4 \text {Chi}(b x)^2\)

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7097

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\int -i \cos (i b x) \sin (i b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}+\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {i \int \cos (i b x) \sin (i b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}+\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 3044

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \left (-\frac {2 \left (\frac {\int i \sinh (b x)d(i \sinh (b x))}{b^2}-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}+\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7101

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} x^4 \text {Chi}(b x)^2+\frac {1}{2} \left (\frac {3 \left (-\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sin \left (i b x+\frac {\pi }{2}\right )^2}{x}dx}{b}}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}+\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\right )\)

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

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

output

(x^4*CoshIntegral[b*x]^2)/4 + (-((x^3*CoshIntegral[b*x]*Sinh[b*x])/b) + (3 
*((x^2*Cosh[b*x]*CoshIntegral[b*x])/b - (x^2/4 - Cosh[b*x]^2/(4*b^2) + (x* 
Cosh[b*x]*Sinh[b*x])/(2*b))/b - (2*(-(((Cosh[b*x]*CoshIntegral[b*x])/b - ( 
CoshIntegral[2*b*x]/2 + Log[x]/2)/b)/b) + (x*CoshIntegral[b*x]*Sinh[b*x])/ 
b - Sinh[b*x]^2/(2*b^2)))/b))/b + ((x^2*Sinh[b*x]^2)/(2*b) + (x^2/4 - (x*C 
osh[b*x]*Sinh[b*x])/(2*b) + Sinh[b*x]^2/(4*b^2))/b)/b)/2
 

3.1.78.3.1 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 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 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 5895

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

rule 7091

Int[CoshIntegral[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Cos 
hIntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*Cosh[b*x]*CoshInte 
gral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

rule 7097

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

rule 7101

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

rule 7103

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

3.1.78.4 Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\frac {b^{4} x^{4} \operatorname {Chi}\left (b x \right )^{2}}{4}-2 \,\operatorname {Chi}\left (b x \right ) \left (\frac {b^{3} x^{3} \sinh \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \cosh \left (b x \right )}{4}+\frac {3 b x \sinh \left (b x \right )}{2}-\frac {3 \cosh \left (b x \right )}{2}\right )+\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{4}-b x \cosh \left (b x \right ) \sinh \left (b x \right )-\frac {b^{2} x^{2}}{2}+2 \cosh \left (b x \right )^{2}-\frac {3 \ln \left (b x \right )}{2}-\frac {3 \,\operatorname {Chi}\left (2 b x \right )}{2}}{b^{4}}\)	\(120\)
default	\(\frac {\frac {b^{4} x^{4} \operatorname {Chi}\left (b x \right )^{2}}{4}-2 \,\operatorname {Chi}\left (b x \right ) \left (\frac {b^{3} x^{3} \sinh \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \cosh \left (b x \right )}{4}+\frac {3 b x \sinh \left (b x \right )}{2}-\frac {3 \cosh \left (b x \right )}{2}\right )+\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{4}-b x \cosh \left (b x \right ) \sinh \left (b x \right )-\frac {b^{2} x^{2}}{2}+2 \cosh \left (b x \right )^{2}-\frac {3 \ln \left (b x \right )}{2}-\frac {3 \,\operatorname {Chi}\left (2 b x \right )}{2}}{b^{4}}\)	\(120\)

input

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

output

1/b^4*(1/4*b^4*x^4*Chi(b*x)^2-2*Chi(b*x)*(1/4*b^3*x^3*sinh(b*x)-3/4*b^2*x^ 
2*cosh(b*x)+3/2*b*x*sinh(b*x)-3/2*cosh(b*x))+1/4*b^2*x^2*cosh(b*x)^2-b*x*c 
osh(b*x)*sinh(b*x)-1/2*b^2*x^2+2*cosh(b*x)^2-3/2*ln(b*x)-3/2*Chi(2*b*x))
 

3.1.78.5 Fricas [F]

\[ \int x^3 \text {Chi}(b x)^2 \, dx=\int { x^{3} {\rm Chi}\left (b x\right )^{2} \,d x } \]

input

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

output

integral(x^3*cosh_integral(b*x)^2, x)
 

3.1.78.6 Sympy [F]

\[ \int x^3 \text {Chi}(b x)^2 \, dx=\int x^{3} \operatorname {Chi}^{2}\left (b x\right )\, dx \]

input

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

output

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

3.1.78.7 Maxima [F]

\[ \int x^3 \text {Chi}(b x)^2 \, dx=\int { x^{3} {\rm Chi}\left (b x\right )^{2} \,d x } \]

input

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

output

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

3.1.78.8 Giac [F]

\[ \int x^3 \text {Chi}(b x)^2 \, dx=\int { x^{3} {\rm Chi}\left (b x\right )^{2} \,d x } \]

input

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

output

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

3.1.78.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \text {Chi}(b x)^2 \, dx=\int x^3\,{\mathrm {coshint}\left (b\,x\right )}^2 \,d x \]

input

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

output

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