Integrand size = 19, antiderivative size = 170 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Chi}(2 \text {arctanh}(a x))}{3 a}+\frac {4 \text {Chi}(4 \text {arctanh}(a x))}{3 a} \] Output:
-1/4/a/(-a^2*x^2+1)^2/arctanh(a*x)^4-1/3*x/(-a^2*x^2+1)^2/arctanh(a*x)^3-2 /3/a/(-a^2*x^2+1)^2/arctanh(a*x)^2+1/2/a/(-a^2*x^2+1)/arctanh(a*x)^2-8/3*x /(-a^2*x^2+1)^2/arctanh(a*x)+x/(-a^2*x^2+1)/arctanh(a*x)+1/3*Chi(2*arctanh (a*x))/a+4/3*Chi(4*arctanh(a*x))/a
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=-\frac {3+4 a x \text {arctanh}(a x)+2 \text {arctanh}(a x)^2+6 a^2 x^2 \text {arctanh}(a x)^2+20 a x \text {arctanh}(a x)^3+12 a^3 x^3 \text {arctanh}(a x)^3-4 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^4 \text {Chi}(2 \text {arctanh}(a x))-16 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^4 \text {Chi}(4 \text {arctanh}(a x))}{12 a \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^4} \] Input:
Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^5),x]
Output:
-1/12*(3 + 4*a*x*ArcTanh[a*x] + 2*ArcTanh[a*x]^2 + 6*a^2*x^2*ArcTanh[a*x]^ 2 + 20*a*x*ArcTanh[a*x]^3 + 12*a^3*x^3*ArcTanh[a*x]^3 - 4*(-1 + a^2*x^2)^2 *ArcTanh[a*x]^4*CoshIntegral[2*ArcTanh[a*x]] - 16*(-1 + a^2*x^2)^2*ArcTanh [a*x]^4*CoshIntegral[4*ArcTanh[a*x]])/(a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^4)
Leaf count is larger than twice the leaf count of optimal. \(409\) vs. \(2(170)=340\).
Time = 3.77 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.41, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.895, Rules used = {6528, 6594, 6528, 6590, 6528, 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.
| \(\displaystyle \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^4}dx-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx}{3 a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx+\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle a \left (a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx}{a^2}\right )+\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle a \left (\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle a \left (\frac {2 a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx}{a}+3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx}{a}+3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6530 |
| \(\displaystyle a \left (\frac {2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}+a \left (\frac {-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )}{3 a}+a \left (\frac {-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^4}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )}{a^2}-\frac {-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle a \left (\frac {2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}+\frac {3}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \left (\frac {2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (3 a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+a \left (\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+a \left (-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {3 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle a \left (\frac {2 a \left (\frac {3 \int \left (\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {1}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}+a \left (\frac {2 a \left (\frac {3 \int \left (\frac {\cosh (4 \text {arctanh}(a x))}{8 \text {arctanh}(a x)}-\frac {1}{8 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \left (a \left (\frac {2 a \left (\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{a^2}-\frac {a \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}}{a^2}\right )+\frac {2 a \left (\frac {3 \left (\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))-\frac {1}{8} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Chi}(4 \text {arctanh}(a x))+\frac {3}{8} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}}{3 a}-\frac {x}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}\right )-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\) |
Input:
Int[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^5),x]
Output:
-1/4*1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x]^4) + a*(-1/3*x/(a*(1 - a^2*x^2)^2*A rcTanh[a*x]^3) + a*((-1/2*1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + 2*a*(-(x/ (a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (3*(CoshIntegral[4*ArcTanh[a*x]]/8 - L og[ArcTanh[a*x]]/8))/a^2 + (CoshIntegral[2*ArcTanh[a*x]]/2 + CoshIntegral[ 4*ArcTanh[a*x]]/8 + (3*Log[ArcTanh[a*x]])/8)/a^2))/a^2 - (-1/2*1/(a*(1 - a ^2*x^2)*ArcTanh[a*x]^2) + a*(-(x/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + (CoshIn tegral[2*ArcTanh[a*x]]/2 - Log[ArcTanh[a*x]]/2)/a^2 + (CoshIntegral[2*ArcT anh[a*x]]/2 + Log[ArcTanh[a*x]]/2)/a^2))/a^2) + (-1/2*1/(a*(1 - a^2*x^2)^2 *ArcTanh[a*x]^2) + 2*a*(-(x/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (3*(CoshIn tegral[4*ArcTanh[a*x]]/8 - Log[ArcTanh[a*x]]/8))/a^2 + (CoshIntegral[2*Arc Tanh[a*x]]/2 + CoshIntegral[4*ArcTanh[a*x]]/8 + (3*Log[ArcTanh[a*x]])/8)/a ^2))/(3*a))
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_)]*(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]
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_)^(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])
Time = 2.03 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(\frac {-\frac {3}{32 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{12 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{12 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{6 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{3}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{24 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{12 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{3 \,\operatorname {arctanh}\left (a x \right )}+\frac {4 \,\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{3}}{a}\) | \(152\) |
| default | \(\frac {-\frac {3}{32 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{12 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{12 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{6 \,\operatorname {arctanh}\left (a x \right )}+\frac {\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{3}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{24 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{12 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{3 \,\operatorname {arctanh}\left (a x \right )}+\frac {4 \,\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{3}}{a}\) | \(152\) |
Input:
int(1/(-a^2*x^2+1)^3/arctanh(a*x)^5,x,method=_RETURNVERBOSE)
Output:
1/a*(-3/32/arctanh(a*x)^4-1/8/arctanh(a*x)^4*cosh(2*arctanh(a*x))-1/12*sin h(2*arctanh(a*x))/arctanh(a*x)^3-1/12/arctanh(a*x)^2*cosh(2*arctanh(a*x))- 1/6/arctanh(a*x)*sinh(2*arctanh(a*x))+1/3*Chi(2*arctanh(a*x))-1/32/arctanh (a*x)^4*cosh(4*arctanh(a*x))-1/24*sinh(4*arctanh(a*x))/arctanh(a*x)^3-1/12 /arctanh(a*x)^2*cosh(4*arctanh(a*x))-1/3*sinh(4*arctanh(a*x))/arctanh(a*x) +4/3*Chi(4*arctanh(a*x)))
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (151) = 302\).
Time = 0.08 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.78 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=\frac {{\left (4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} - 4 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 16 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 4 \, {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 24}{6 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4}} \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^5,x, algorithm="fricas")
Output:
1/6*((4*(a^4*x^4 - 2*a^2*x^2 + 1)*log_integral((a^2*x^2 + 2*a*x + 1)/(a^2* x^2 - 2*a*x + 1)) + 4*(a^4*x^4 - 2*a^2*x^2 + 1)*log_integral((a^2*x^2 - 2* a*x + 1)/(a^2*x^2 + 2*a*x + 1)) + (a^4*x^4 - 2*a^2*x^2 + 1)*log_integral(- (a*x + 1)/(a*x - 1)) + (a^4*x^4 - 2*a^2*x^2 + 1)*log_integral(-(a*x - 1)/( a*x + 1)))*log(-(a*x + 1)/(a*x - 1))^4 - 4*(3*a^3*x^3 + 5*a*x)*log(-(a*x + 1)/(a*x - 1))^3 - 16*a*x*log(-(a*x + 1)/(a*x - 1)) - 4*(3*a^2*x^2 + 1)*lo g(-(a*x + 1)/(a*x - 1))^2 - 24)/((a^5*x^4 - 2*a^3*x^2 + a)*log(-(a*x + 1)/ (a*x - 1))^4)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=- \int \frac {1}{a^{6} x^{6} \operatorname {atanh}^{5}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{5}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{5}{\left (a x \right )} - \operatorname {atanh}^{5}{\left (a x \right )}}\, dx \] Input:
integrate(1/(-a**2*x**2+1)**3/atanh(a*x)**5,x)
Output:
-Integral(1/(a**6*x**6*atanh(a*x)**5 - 3*a**4*x**4*atanh(a*x)**5 + 3*a**2* x**2*atanh(a*x)**5 - atanh(a*x)**5), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{5}} \,d x } \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^5,x, algorithm="maxima")
Output:
-2/3*((3*a^3*x^3 + 5*a*x)*log(a*x + 1)^3 - (3*a^3*x^3 + 5*a*x)*log(-a*x + 1)^3 + 4*a*x*log(a*x + 1) + (3*a^2*x^2 + 1)*log(a*x + 1)^2 + (3*a^2*x^2 + 3*(3*a^3*x^3 + 5*a*x)*log(a*x + 1) + 1)*log(-a*x + 1)^2 - (3*(3*a^3*x^3 + 5*a*x)*log(a*x + 1)^2 + 4*a*x + 2*(3*a^2*x^2 + 1)*log(a*x + 1))*log(-a*x + 1) + 6)/((a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^4 - 4*(a^5*x^4 - 2*a^3*x^ 2 + a)*log(a*x + 1)^3*log(-a*x + 1) + 6*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^2*log(-a*x + 1)^2 - 4*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)*log(-a*x + 1)^3 + (a^5*x^4 - 2*a^3*x^2 + a)*log(-a*x + 1)^4) + integrate(-2/3*(3*a ^4*x^4 + 24*a^2*x^2 + 5)/((a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1) - (a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-a*x + 1)), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{5}} \,d x } \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^5,x, algorithm="giac")
Output:
integrate(-1/((a^2*x^2 - 1)^3*arctanh(a*x)^5), x)
Timed out. \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=-\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^5\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-1/(atanh(a*x)^5*(a^2*x^2 - 1)^3),x)
Output:
-int(1/(atanh(a*x)^5*(a^2*x^2 - 1)^3), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5} \, dx=\frac {-12 \mathit {atanh} \left (a x \right )^{4} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{7} x^{4}+24 \mathit {atanh} \left (a x \right )^{4} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{5} x^{2}-12 \mathit {atanh} \left (a x \right )^{4} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{3}-8 \mathit {atanh} \left (a x \right )^{4} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{6} x^{4}+16 \mathit {atanh} \left (a x \right )^{4} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{4} x^{2}-8 \mathit {atanh} \left (a x \right )^{4} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{2} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{2} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{2}}d x \right ) a^{2}-2 \mathit {atanh} \left (a x \right )^{2}-4 \mathit {atanh} \left (a x \right ) a x -3}{12 \mathit {atanh} \left (a x \right )^{4} a \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:
int(1/(-a^2*x^2+1)^3/atanh(a*x)^5,x)
Output:
( - 12*atanh(a*x)**4*int(x**2/(atanh(a*x)**3*a**6*x**6 - 3*atanh(a*x)**3*a **4*x**4 + 3*atanh(a*x)**3*a**2*x**2 - atanh(a*x)**3),x)*a**7*x**4 + 24*at anh(a*x)**4*int(x**2/(atanh(a*x)**3*a**6*x**6 - 3*atanh(a*x)**3*a**4*x**4 + 3*atanh(a*x)**3*a**2*x**2 - atanh(a*x)**3),x)*a**5*x**2 - 12*atanh(a*x)* *4*int(x**2/(atanh(a*x)**3*a**6*x**6 - 3*atanh(a*x)**3*a**4*x**4 + 3*atanh (a*x)**3*a**2*x**2 - atanh(a*x)**3),x)*a**3 - 8*atanh(a*x)**4*int(x/(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)*a**6*x**4 + 16*atanh(a*x)**4*int(x/(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)*a**4*x**2 - 8*atanh(a*x)**4*int(x/(atanh(a*x)**2*a**6*x**6 - 3*at anh(a*x)**2*a**4*x**4 + 3*atanh(a*x)**2*a**2*x**2 - atanh(a*x)**2),x)*a**2 - 2*atanh(a*x)**2 - 4*atanh(a*x)*a*x - 3)/(12*atanh(a*x)**4*a*(a**4*x**4 - 2*a**2*x**2 + 1))