Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\frac {1}{2 a x \text {arctanh}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {3}{2} \text {Shi}(2 \text {arctanh}(a x))+\text {Shi}(4 \text {arctanh}(a x))-\frac {\text {Int}\left (\frac {1}{x^2 \text {arctanh}(a x)^2},x\right )}{2 a} \] Output:
-1/2/a/x/arctanh(a*x)^2-1/2*a*x/(-a^2*x^2+1)^2/arctanh(a*x)^2-1/2*a*x/(-a^ 2*x^2+1)/arctanh(a*x)^2-2/(-a^2*x^2+1)^2/arctanh(a*x)+3/2/(-a^2*x^2+1)/arc tanh(a*x)-1/2*(a^2*x^2+1)/(-a^2*x^2+1)/arctanh(a*x)+3/2*Shi(2*arctanh(a*x) )+Shi(4*arctanh(a*x))-1/2*Defer(Int)(1/x^2/arctanh(a*x)^2,x)/a
Not integrable
Time = 3.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx \] Input:
Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]
Output:
Integrate[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3), x]
Not integrable
Time = 3.00 (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 = {}
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.
| \(\displaystyle \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx+\int \frac {1}{x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx+\int \frac {1}{x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}dx\) |
| \(\Big \downarrow \) 6552 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6468 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6558 |
| \(\displaystyle a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx+a^2 \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle a^2 \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{2 a}+\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )+a^2 \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle a^2 \left (\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )+a^2 \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle a^2 \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\right )+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )+a^2 \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle a^2 \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle a^2 \left (\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle a^2 \left (\frac {2 \int \frac {\sinh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {\sinh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^2 \left (\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \left (\frac {\sinh (2 \text {arctanh}(a x))}{4 \text {arctanh}(a x)}+\frac {\sinh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 \left (\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )+a^2 \left (\frac {\int \frac {\sinh (2 \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \left (\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\int -\frac {i \sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )+a^2 \left (\frac {\int -\frac {i \sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle a^2 \left (\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {i \int \frac {\sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )+a^2 \left (-\frac {i \int \frac {\sin (2 i \text {arctanh}(a x))}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle -\frac {\int \frac {1}{x^2 \text {arctanh}(a x)^2}dx}{2 a}+a^2 \left (\frac {\text {Shi}(2 \text {arctanh}(a x))}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )+a^2 \left (\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\text {Shi}(2 \text {arctanh}(a x))}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{2 a x \text {arctanh}(a x)^2}\) |
Input:
Int[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3),x]
Output:
$Aborted
Not integrable
Time = 1.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \operatorname {arctanh}\left (a x \right )^{3}}d x\]
Input:
int(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x)
Output:
int(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x)
Not integrable
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="fricas")
Output:
integral(-1/((a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x)*arctanh(a*x)^3), x)
Not integrable
Time = 2.96 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=- \int \frac {1}{a^{6} x^{7} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{3}{\left (a x \right )} - x \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \] Input:
integrate(1/x/(-a**2*x**2+1)**3/atanh(a*x)**3,x)
Output:
-Integral(1/(a**6*x**7*atanh(a*x)**3 - 3*a**4*x**5*atanh(a*x)**3 + 3*a**2* x**3*atanh(a*x)**3 - x*atanh(a*x)**3), x)
Not integrable
Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 11.59 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="maxima")
Output:
-(2*a*x + (5*a^2*x^2 - 1)*log(a*x + 1) - (5*a^2*x^2 - 1)*log(-a*x + 1))/(( a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*log(a*x + 1)^2 - 2*(a^6*x^6 - 2*a^4*x^4 + a ^2*x^2)*log(a*x + 1)*log(-a*x + 1) + (a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*log(- a*x + 1)^2) + integrate(-2*(10*a^4*x^4 - 3*a^2*x^2 + 1)/((a^8*x^9 - 3*a^6* x^7 + 3*a^4*x^5 - a^2*x^3)*log(a*x + 1) - (a^8*x^9 - 3*a^6*x^7 + 3*a^4*x^5 - a^2*x^3)*log(-a*x + 1)), x)
Not integrable
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{3}} \,d x } \] Input:
integrate(1/x/(-a^2*x^2+1)^3/arctanh(a*x)^3,x, algorithm="giac")
Output:
integrate(-1/((a^2*x^2 - 1)^3*x*arctanh(a*x)^3), x)
Not integrable
Time = 3.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-1/(x*atanh(a*x)^3*(a^2*x^2 - 1)^3),x)
Output:
-int(1/(x*atanh(a*x)^3*(a^2*x^2 - 1)^3), x)
Not integrable
Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3} \, dx=-\left (\int \frac {1}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{7}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{5}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{3}-\mathit {atanh} \left (a x \right )^{3} x}d x \right ) \] Input:
int(1/x/(-a^2*x^2+1)^3/atanh(a*x)^3,x)
Output:
- int(1/(atanh(a*x)**3*a**6*x**7 - 3*atanh(a*x)**3*a**4*x**5 + 3*atanh(a* x)**3*a**2*x**3 - atanh(a*x)**3*x),x)