\(\int \frac {x^5}{(1-a^2 x^2)^3 \text {arctanh}(a x)^2} \, dx\) [329]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\frac {x}{a^5 \text {arctanh}(a x)}-\frac {x}{a^5 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {2 x}{a^5 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}-\frac {3 \text {Chi}(2 \text {arctanh}(a x))}{2 a^6}+\frac {\text {Chi}(4 \text {arctanh}(a x))}{2 a^6}+\frac {\text {Int}\left (\frac {1}{\text {arctanh}(a x)},x\right )}{a^5} \] Output:

-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+Defe 
r(Int)(1/arctanh(a*x),x)/a^5
 

Mathematica [N/A]

Not integrable

Time = 7.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx \] Input:

Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]
 

Output:

Integrate[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x]
 

Rubi [N/A]

Not integrable

Time = 3.23 (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.

\(\displaystyle \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx\)

\(\Big \downarrow \) 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

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}}{a^2}-\frac {\frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}-\frac {\frac {\int \frac {1}{\text {arctanh}(a x)}dx}{a}-\frac {x}{a \text {arctanh}(a x)}}{a^2}}{a^2}\)

Input:

Int[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2),x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {x^{5}}{\left (-a^{2} x^{2}+1\right )^{3} \operatorname {arctanh}\left (a x \right )^{2}}d x\]

Input:

int(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x)
 

Output:

int(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x)
 

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="fricas")
 

Output:

integral(-x^5/((a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*arctanh(a*x)^2), x)
 

Sympy [N/A]

Not integrable

Time = 2.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=- \int \frac {x^{5}}{a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \] Input:

integrate(x**5/(-a**2*x**2+1)**3/atanh(a*x)**2,x)
 

Output:

-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)
 

Maxima [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.86 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="maxima")
 

Output:

-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)
 

Giac [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=\int { -\frac {x^{5}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}} \,d x } \] Input:

integrate(x^5/(-a^2*x^2+1)^3/arctanh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(-x^5/((a^2*x^2 - 1)^3*arctanh(a*x)^2), x)
 

Mupad [N/A]

Not integrable

Time = 3.95 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\int \frac {x^5}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:

int(-x^5/(atanh(a*x)^2*(a^2*x^2 - 1)^3),x)
 

Output:

-int(x^5/(atanh(a*x)^2*(a^2*x^2 - 1)^3), x)
 

Reduce [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {x^5}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2} \, dx=-\left (\int \frac {x^{5}}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) \] Input:

int(x^5/(-a^2*x^2+1)^3/atanh(a*x)^2,x)
 

Output:

 - int(x**5/(atanh(a*x)**2*a**6*x**6 - 3*atanh(a*x)**2*a**4*x**4 + 3*atanh 
(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)