Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {x}{a^5 \text {arctanh}(a x)}-\frac {x}{a^5 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {2 x}{a^5 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {3 \text {Chi}(2 \text {arctanh}(a x))}{2 a^6}+\frac {\text {Chi}(4 \text {arctanh}(a x))}{2 a^6}+\frac {\text {Int}\left (\frac {1}{\text {arctanh}(a x)},x\right )}{a^5} \]
-x/a^5/arctanh(a*x)-x/a^5/(-a^2*x^2+1)^2/arctanh(a*x)+2*x/a^5/(-a^2*x^2+1) /arctanh(a*x)-3/2*Chi(2*arctanh(a*x))/a^6+1/2*Chi(4*arctanh(a*x))/a^6+Unin tegrable(1/arctanh(a*x),x)/a^5
Not integrable
Time = 8.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx \]
Not integrable
Time = 3.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6590, 6590, 6548, 6444, 6594, 6530, 3042, 3793, 2009, 6596, 3042, 25, 3793, 2009, 5971, 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.
| \(\displaystyle \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {\int \frac {x^3}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {x^3}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}}{a^2}-\frac {\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}dx}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 6548 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}}{a^2}-\frac {\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 6444 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}}{a^2}-\frac {\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx}{a}+3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 6530 |
| \(\displaystyle \frac {\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}+\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}+\frac {\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}+\frac {-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {1}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\) |
3.4.29.3.1 Defintions of rubi rules used
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]))
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]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Unintegrab le[(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, c, n, p}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x _Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && I LtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[x*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Si mp[1/(b*c*d*(p + 1)) Int[(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && !IGtQ[p, 0] && NeQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*A rcTanh[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcT anh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && In tegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^( p + 1)/(b*c*d*(p + 1))), x] + (Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^( m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x]) / ; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, - 1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sinh[x]^ m/Cosh[x]^(m + 2*(q + 1))), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (In tegerQ[q] || GtQ[d, 0])
Not integrable
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {x^{5}}{\left (-a^{2} x^{2}+1\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}d x\]
Not integrable
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 2.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=- \int \frac {x^{5}}{a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
-Integral(x**5/(a**6*x**6*atanh(a*x)**2 - 3*a**4*x**4*atanh(a*x)**2 + 3*a* *2*x**2*atanh(a*x)**2 - atanh(a*x)**2), x)
Not integrable
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.86 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \]
-2*x^5/((a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1) - (a^5*x^4 - 2*a^3*x^2 + a) *log(-a*x + 1)) - integrate(-2*(a^2*x^6 - 5*x^4)/((a^7*x^6 - 3*a^5*x^4 + 3 *a^3*x^2 - a)*log(a*x + 1) - (a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(-a* x + 1)), x)
Not integrable
Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \]
Not integrable
Time = 3.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\int \frac {x^5}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \]