3.4.41 \(\int \frac {1}{x (1-a^2 x^2)^3 \text {arctanh}(a x)^3} \, dx\) [341]

3.4.41.1 Optimal result

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*Unintegrable(1/x^2/arctanh(a*x)^2,x)/a
 

3.4.41.2 Mathematica [N/A]

Not integrable

Time = 4.24 (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]
 

3.4.41.3 Rubi [N/A]

Not integrable

Time = 2.92 (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 = {6592, 6592, 6552, 6468, 6558, 6594, 6528, 6590, 6528, 6596, 5971, 27, 2009, 3042, 26, 3779}

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
 

3.4.41.3.1 Defintions of rubi rules used

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 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 6468

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_S 
ymbol] :> Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^n])^p, x] /; FreeQ[{a, b, 
 c, d, m, n, p}, x]
 

rule 6528

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p 
 + 1))), x] + Simp[2*c*((q + 1)/(b*(p + 1)))   Int[x*(d + e*x^2)^q*(a + b*A 
rcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 
 0] && LtQ[q, -1] && LtQ[p, -1]
 

rule 6552

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d 
*(p + 1))), x] - Simp[f*(m/(b*c*d*(p + 1)))   Int[(f*x)^(m - 1)*(a + b*ArcT 
anh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + 
 e, 0] && LtQ[p, -1]
 

rule 6558

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2)^2 
, x_Symbol] :> Simp[x*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x 
^2))), x] + (Simp[(1 + c^2*x^2)*((a + b*ArcTanh[c*x])^(p + 2)/(b^2*e*(p + 1 
)*(p + 2)*(d + e*x^2))), x] + Simp[4/(b^2*(p + 1)*(p + 2))   Int[x*((a + b* 
ArcTanh[c*x])^(p + 2)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]
 

rule 6590

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]
 

rule 6592

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh 
[c*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers 
Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

rule 6594

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]
 

rule 6596

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])
 

3.4.41.4 Maple [N/A] (verified)

Not integrable

Time = 0.24 (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)
 

3.4.41.5 Fricas [N/A]

Not integrable

Time = 0.25 (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)
 

3.4.41.6 Sympy [N/A]

Not integrable

Time = 2.76 (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)
 

3.4.41.7 Maxima [N/A]

Not integrable

Time = 0.32 (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)
 

3.4.41.8 Giac [N/A]

Not integrable

Time = 0.38 (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)
 

3.4.41.9 Mupad [N/A]

Not integrable

Time = 3.74 (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)