Integrand size = 19, antiderivative size = 257 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}-\frac {x}{5 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}-\frac {4}{15 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}+\frac {1}{5 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}-\frac {8 x}{15 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}+\frac {x}{5 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {32}{15 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}+\frac {8}{5 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {2 \text {Shi}(2 \text {arctanh}(a x))}{15 a}+\frac {16 \text {Shi}(4 \text {arctanh}(a x))}{15 a} \] Output:
-1/5/a/(-a^2*x^2+1)^2/arctanh(a*x)^5-1/5*x/(-a^2*x^2+1)^2/arctanh(a*x)^4-4 /15/a/(-a^2*x^2+1)^2/arctanh(a*x)^3+1/5/a/(-a^2*x^2+1)/arctanh(a*x)^3-8/15 *x/(-a^2*x^2+1)^2/arctanh(a*x)^2+1/5*x/(-a^2*x^2+1)/arctanh(a*x)^2-32/15/a /(-a^2*x^2+1)^2/arctanh(a*x)+8/5/a/(-a^2*x^2+1)/arctanh(a*x)+1/5*(a^2*x^2+ 1)/a/(-a^2*x^2+1)/arctanh(a*x)+2/15*Shi(2*arctanh(a*x))/a+16/15*Shi(4*arct anh(a*x))/a
Time = 0.38 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=-\frac {3+3 a x \text {arctanh}(a x)+\text {arctanh}(a x)^2+3 a^2 x^2 \text {arctanh}(a x)^2+5 a x \text {arctanh}(a x)^3+3 a^3 x^3 \text {arctanh}(a x)^3+5 \text {arctanh}(a x)^4+24 a^2 x^2 \text {arctanh}(a x)^4+3 a^4 x^4 \text {arctanh}(a x)^4-2 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^5 \text {Shi}(2 \text {arctanh}(a x))-16 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^5 \text {Shi}(4 \text {arctanh}(a x))}{15 a \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^5} \] Input:
Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^6),x]
Output:
-1/15*(3 + 3*a*x*ArcTanh[a*x] + ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 5*a*x*ArcTanh[a*x]^3 + 3*a^3*x^3*ArcTanh[a*x]^3 + 5*ArcTanh[a*x]^4 + 24* a^2*x^2*ArcTanh[a*x]^4 + 3*a^4*x^4*ArcTanh[a*x]^4 - 2*(-1 + a^2*x^2)^2*Arc Tanh[a*x]^5*SinhIntegral[2*ArcTanh[a*x]] - 16*(-1 + a^2*x^2)^2*ArcTanh[a*x ]^5*SinhIntegral[4*ArcTanh[a*x]])/(a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^5)
Leaf count is larger than twice the leaf count of optimal. \(621\) vs. \(2(257)=514\).
Time = 4.80 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.42, 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, 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.
| \(\displaystyle \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {4}{5} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^5}dx-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^4}dx}{4 a}+\frac {3}{4} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^4}dx-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {4}{5} a \left (\frac {3}{4} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^4}dx+\frac {\frac {4}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {4}{5} a \left (\frac {3}{4} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^4}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}dx}{a^2}\right )+\frac {\frac {4}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6558 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3}dx-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{2 a}+\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{2 a}+\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \left (\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \left (\frac {3}{2} a \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6590 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\right )+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \left (\frac {3}{2} a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a^2}\right )+\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {4 a \int \frac {x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {2 a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}+\frac {3}{2} a \left (\frac {\frac {4 \int \frac {a x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (\frac {2 \int \frac {a x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} 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)}}{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 {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} 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)}}{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 {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \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 )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} 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)}}{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 {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} 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)}}{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 {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \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 {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{5} a \left (\frac {\frac {4}{3} a \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 )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}+\frac {3}{4} a \left (\frac {\frac {4}{3} a \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 )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \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 {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}+\frac {4}{5} a \left (\frac {-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}+\frac {4}{3} a \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 )}{4 a}+\frac {3}{4} a \left (\frac {-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}+\frac {4}{3} a \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}-\frac {-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}+\frac {2}{3} a \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 )}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}+\frac {4}{5} a \left (\frac {-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}+\frac {4}{3} a \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 )}{4 a}+\frac {3}{4} a \left (\frac {-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}+\frac {4}{3} a \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}-\frac {-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}+\frac {2}{3} a \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 )}{a^2}\right )-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {4}{5} a \left (\frac {3}{4} a \left (\frac {\frac {4}{3} a \left (\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\text {Shi}(2 \text {arctanh}(a x))}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{a^2}-\frac {\frac {2}{3} a \left (\frac {\text {Shi}(2 \text {arctanh}(a x))}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {a^2 x^2+1}{2 a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{3 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}}{a^2}\right )+\frac {\frac {4}{3} a \left (\frac {3}{2} a \left (\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{a^2}-\frac {\frac {\text {Shi}(2 \text {arctanh}(a x))}{a}-\frac {1}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a^2}\right )+\frac {\frac {4 \left (\frac {1}{4} \text {Shi}(2 \text {arctanh}(a x))+\frac {1}{8} \text {Shi}(4 \text {arctanh}(a x))\right )}{a}-\frac {1}{a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}}{2 a}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}\right )-\frac {1}{3 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3}}{4 a}-\frac {x}{4 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^4}\right )-\frac {1}{5 a \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^5}\) |
Input:
Int[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^6),x]
Output:
-1/5*1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x]^5) + (4*a*(-1/4*x/(a*(1 - a^2*x^2)^ 2*ArcTanh[a*x]^4) + (3*a*(-((-1/3*1/(a*(1 - a^2*x^2)*ArcTanh[a*x]^3) + (2* a*(-1/2*x/(a*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^2*(1 - a^2 *x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^2))/3)/a^2) + (-1/3*1 /(a*(1 - a^2*x^2)^2*ArcTanh[a*x]^3) + (4*a*(-1/2*x/(a*(1 - a^2*x^2)^2*ArcT anh[a*x]^2) + (3*a*(-((-(1/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + SinhIntegral[ 2*ArcTanh[a*x]]/a)/a^2) + (-(1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(Sin hIntegral[2*ArcTanh[a*x]]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/a^2))/2 + (-(1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(SinhIntegral[2*ArcTanh[a*x] ]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/(2*a)))/3)/a^2))/4 + (-1/3*1/(a* (1 - a^2*x^2)^2*ArcTanh[a*x]^3) + (4*a*(-1/2*x/(a*(1 - a^2*x^2)^2*ArcTanh[ a*x]^2) + (3*a*(-((-(1/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + SinhIntegral[2*Ar cTanh[a*x]]/a)/a^2) + (-(1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(SinhInt egral[2*ArcTanh[a*x]]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/a^2))/2 + (- (1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + (4*(SinhIntegral[2*ArcTanh[a*x]]/4 + SinhIntegral[4*ArcTanh[a*x]]/8))/a)/(2*a)))/3)/(4*a)))/5
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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_)*(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]
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 = 1.83 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {-\frac {3}{40 \operatorname {arctanh}\left (a x \right )^{5}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{10 \operatorname {arctanh}\left (a x \right )^{5}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{20 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{30 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{30 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{15 \,\operatorname {arctanh}\left (a x \right )}+\frac {2 \,\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{15}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{40 \operatorname {arctanh}\left (a x \right )^{5}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{40 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{30 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{15 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {4 \cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{15 \,\operatorname {arctanh}\left (a x \right )}+\frac {16 \,\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{15}}{a}\) | \(182\) |
| default | \(\frac {-\frac {3}{40 \operatorname {arctanh}\left (a x \right )^{5}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{10 \operatorname {arctanh}\left (a x \right )^{5}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{20 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{30 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{30 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{15 \,\operatorname {arctanh}\left (a x \right )}+\frac {2 \,\operatorname {Shi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{15}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{40 \operatorname {arctanh}\left (a x \right )^{5}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{40 \operatorname {arctanh}\left (a x \right )^{4}}-\frac {\cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{30 \operatorname {arctanh}\left (a x \right )^{3}}-\frac {\sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{15 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {4 \cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{15 \,\operatorname {arctanh}\left (a x \right )}+\frac {16 \,\operatorname {Shi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{15}}{a}\) | \(182\) |
Input:
int(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x,method=_RETURNVERBOSE)
Output:
1/a*(-3/40/arctanh(a*x)^5-1/10/arctanh(a*x)^5*cosh(2*arctanh(a*x))-1/20/ar ctanh(a*x)^4*sinh(2*arctanh(a*x))-1/30/arctanh(a*x)^3*cosh(2*arctanh(a*x)) -1/30/arctanh(a*x)^2*sinh(2*arctanh(a*x))-1/15/arctanh(a*x)*cosh(2*arctanh (a*x))+2/15*Shi(2*arctanh(a*x))-1/40/arctanh(a*x)^5*cosh(4*arctanh(a*x))-1 /40/arctanh(a*x)^4*sinh(4*arctanh(a*x))-1/30/arctanh(a*x)^3*cosh(4*arctanh (a*x))-1/15*sinh(4*arctanh(a*x))/arctanh(a*x)^2-4/15/arctanh(a*x)*cosh(4*a rctanh(a*x))+16/15*Shi(4*arctanh(a*x)))
Time = 0.09 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=\frac {{\left (8 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 8 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{5} - 2 \, {\left (3 \, a^{4} x^{4} + 24 \, a^{2} x^{2} + 5\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} - 4 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 48 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 8 \, {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 96}{15 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{5}} \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x, algorithm="fricas")
Output:
1/15*((8*(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)) - 8*(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))^5 - 2*(3*a^4*x^4 + 24*a^2*x^2 + 5)*l og(-(a*x + 1)/(a*x - 1))^4 - 4*(3*a^3*x^3 + 5*a*x)*log(-(a*x + 1)/(a*x - 1 ))^3 - 48*a*x*log(-(a*x + 1)/(a*x - 1)) - 8*(3*a^2*x^2 + 1)*log(-(a*x + 1) /(a*x - 1))^2 - 96)/((a^5*x^4 - 2*a^3*x^2 + a)*log(-(a*x + 1)/(a*x - 1))^5 )
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=- \int \frac {1}{a^{6} x^{6} \operatorname {atanh}^{6}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{6}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{6}{\left (a x \right )} - \operatorname {atanh}^{6}{\left (a x \right )}}\, dx \] Input:
integrate(1/(-a**2*x**2+1)**3/atanh(a*x)**6,x)
Output:
-Integral(1/(a**6*x**6*atanh(a*x)**6 - 3*a**4*x**4*atanh(a*x)**6 + 3*a**2* x**2*atanh(a*x)**6 - atanh(a*x)**6), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{6}} \,d x } \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x, algorithm="maxima")
Output:
-2/15*((3*a^4*x^4 + 24*a^2*x^2 + 5)*log(a*x + 1)^4 + (3*a^4*x^4 + 24*a^2*x ^2 + 5)*log(-a*x + 1)^4 + 2*(3*a^3*x^3 + 5*a*x)*log(a*x + 1)^3 - 2*(3*a^3* x^3 + 5*a*x + 2*(3*a^4*x^4 + 24*a^2*x^2 + 5)*log(a*x + 1))*log(-a*x + 1)^3 + 24*a*x*log(a*x + 1) + 4*(3*a^2*x^2 + 1)*log(a*x + 1)^2 + 2*(6*a^2*x^2 + 3*(3*a^4*x^4 + 24*a^2*x^2 + 5)*log(a*x + 1)^2 + 3*(3*a^3*x^3 + 5*a*x)*log (a*x + 1) + 2)*log(-a*x + 1)^2 - 2*(2*(3*a^4*x^4 + 24*a^2*x^2 + 5)*log(a*x + 1)^3 + 3*(3*a^3*x^3 + 5*a*x)*log(a*x + 1)^2 + 12*a*x + 4*(3*a^2*x^2 + 1 )*log(a*x + 1))*log(-a*x + 1) + 48)/((a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1 )^5 - 5*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^4*log(-a*x + 1) + 10*(a^5*x ^4 - 2*a^3*x^2 + a)*log(a*x + 1)^3*log(-a*x + 1)^2 - 10*(a^5*x^4 - 2*a^3*x ^2 + a)*log(a*x + 1)^2*log(-a*x + 1)^3 + 5*(a^5*x^4 - 2*a^3*x^2 + a)*log(a *x + 1)*log(-a*x + 1)^4 - (a^5*x^4 - 2*a^3*x^2 + a)*log(-a*x + 1)^5) + int egrate(-8/15*(15*a^3*x^3 + 17*a*x)/((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)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{6}} \,d x } \] Input:
integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x, algorithm="giac")
Output:
integrate(-1/((a^2*x^2 - 1)^3*arctanh(a*x)^6), x)
Timed out. \[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=-\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^6\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-1/(atanh(a*x)^6*(a^2*x^2 - 1)^3),x)
Output:
-int(1/(atanh(a*x)^6*(a^2*x^2 - 1)^3), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^6} \, dx=\frac {-9 \mathit {atanh} \left (a x \right )^{5} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{4} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{4} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{4} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{4}}d x \right ) a^{7} x^{4}+18 \mathit {atanh} \left (a x \right )^{5} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{4} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{4} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{4} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{4}}d x \right ) a^{5} x^{2}-9 \mathit {atanh} \left (a x \right )^{5} \left (\int \frac {x^{2}}{\mathit {atanh} \left (a x \right )^{4} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{4} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{4} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{4}}d x \right ) a^{3}-4 \mathit {atanh} \left (a x \right )^{5} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{6} x^{4}+8 \mathit {atanh} \left (a x \right )^{5} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{4} x^{2}-4 \mathit {atanh} \left (a x \right )^{5} \left (\int \frac {x}{\mathit {atanh} \left (a x \right )^{3} a^{6} x^{6}-3 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+3 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {atanh} \left (a x \right )^{3}}d x \right ) a^{2}-\mathit {atanh} \left (a x \right )^{2}-3 \mathit {atanh} \left (a x \right ) a x -3}{15 \mathit {atanh} \left (a x \right )^{5} a \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:
int(1/(-a^2*x^2+1)^3/atanh(a*x)^6,x)
Output:
( - 9*atanh(a*x)**5*int(x**2/(atanh(a*x)**4*a**6*x**6 - 3*atanh(a*x)**4*a* *4*x**4 + 3*atanh(a*x)**4*a**2*x**2 - atanh(a*x)**4),x)*a**7*x**4 + 18*ata nh(a*x)**5*int(x**2/(atanh(a*x)**4*a**6*x**6 - 3*atanh(a*x)**4*a**4*x**4 + 3*atanh(a*x)**4*a**2*x**2 - atanh(a*x)**4),x)*a**5*x**2 - 9*atanh(a*x)**5 *int(x**2/(atanh(a*x)**4*a**6*x**6 - 3*atanh(a*x)**4*a**4*x**4 + 3*atanh(a *x)**4*a**2*x**2 - atanh(a*x)**4),x)*a**3 - 4*atanh(a*x)**5*int(x/(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**6*x**4 + 8*atanh(a*x)**5*int(x/(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**4*x**2 - 4*atanh(a*x)**5*int(x/(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**2 - atanh(a*x)**2 - 3*atanh(a*x)*a*x - 3)/(15*atanh(a*x)**5*a*(a**4*x**4 - 2*a **2*x**2 + 1))