Integrand size = 21, antiderivative size = 277 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac {413312 x}{385875 \sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {12 \text {arctanh}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \text {arctanh}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {32 \text {arctanh}(a x)}{35 a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \text {arctanh}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \text {arctanh}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \text {arctanh}(a x)^2}{35 \sqrt {1-a^2 x^2}} \] Output:
2/343*x/(-a^2*x^2+1)^(7/2)+888/42875*x/(-a^2*x^2+1)^(5/2)+30256/385875*x/( -a^2*x^2+1)^(3/2)+413312/385875*x/(-a^2*x^2+1)^(1/2)-2/49*arctanh(a*x)/a/( -a^2*x^2+1)^(7/2)-12/175*arctanh(a*x)/a/(-a^2*x^2+1)^(5/2)-16/105*arctanh( a*x)/a/(-a^2*x^2+1)^(3/2)-32/35*arctanh(a*x)/a/(-a^2*x^2+1)^(1/2)+1/7*x*ar ctanh(a*x)^2/(-a^2*x^2+1)^(7/2)+6/35*x*arctanh(a*x)^2/(-a^2*x^2+1)^(5/2)+8 /35*x*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2)+16/35*x*arctanh(a*x)^2/(-a^2*x^2+1 )^(1/2)
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.43 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {2 a x \left (226905-654220 a^2 x^2+635096 a^4 x^4-206656 a^6 x^6\right )+210 \left (-2161+5726 a^2 x^2-5320 a^4 x^4+1680 a^6 x^6\right ) \text {arctanh}(a x)-11025 a x \left (-35+70 a^2 x^2-56 a^4 x^4+16 a^6 x^6\right ) \text {arctanh}(a x)^2}{385875 a \left (1-a^2 x^2\right )^{7/2}} \] Input:
Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(9/2),x]
Output:
(2*a*x*(226905 - 654220*a^2*x^2 + 635096*a^4*x^4 - 206656*a^6*x^6) + 210*( -2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6)*ArcTanh[a*x] - 11025*a *x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[a*x]^2)/(385875*a* (1 - a^2*x^2)^(7/2))
Time = 1.00 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.55, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6526, 209, 209, 209, 208, 6526, 209, 209, 208, 6526, 209, 208, 6524, 208}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {2}{49} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2}}dx+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {2}{49} \left (\frac {6}{7} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {2}{49} \left (\frac {6}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {2}{49} \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2}{25} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2}{25} \left (\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2}{25} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {2}{9} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \text {arctanh}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {2}{9} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \text {arctanh}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \text {arctanh}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{9} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 6524 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a \sqrt {1-a^2 x^2}}\right )+\frac {x \text {arctanh}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \text {arctanh}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{9} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )+\frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \text {arctanh}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \text {arctanh}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \text {arctanh}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {2 x}{\sqrt {1-a^2 x^2}}\right )+\frac {2}{9} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {2}{25} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )+\frac {2}{49} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )\) |
Input:
Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^(9/2),x]
Output:
(2*(x/(7*(1 - a^2*x^2)^(7/2)) + (6*(x/(5*(1 - a^2*x^2)^(5/2)) + (4*(x/(3*( 1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2])))/5))/7))/49 - (2*ArcTan h[a*x])/(49*a*(1 - a^2*x^2)^(7/2)) + (x*ArcTanh[a*x]^2)/(7*(1 - a^2*x^2)^( 7/2)) + (6*((2*(x/(5*(1 - a^2*x^2)^(5/2)) + (4*(x/(3*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2])))/5))/25 - (2*ArcTanh[a*x])/(25*a*(1 - a^2*x ^2)^(5/2)) + (x*ArcTanh[a*x]^2)/(5*(1 - a^2*x^2)^(5/2)) + (4*((2*(x/(3*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2])))/9 - (2*ArcTanh[a*x])/(9* a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x]^2)/(3*(1 - a^2*x^2)^(3/2)) + (2*( (2*x)/Sqrt[1 - a^2*x^2] - (2*ArcTanh[a*x])/(a*Sqrt[1 - a^2*x^2]) + (x*ArcT anh[a*x]^2)/Sqrt[1 - a^2*x^2]))/3))/5))/7
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) ), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 *p*(p - 1) Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(4 *c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p /(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1 )*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int [(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Time = 0.48 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (176400 \operatorname {arctanh}\left (a x \right )^{2} a^{7} x^{7}+413312 a^{7} x^{7}-352800 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}-617400 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{5}-1270192 a^{5} x^{5}+1117200 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+771750 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x^{3}+1308440 a^{3} x^{3}-1202460 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-385875 \operatorname {arctanh}\left (a x \right )^{2} a x -453810 a x +453810 \,\operatorname {arctanh}\left (a x \right )\right )}{385875 a \left (a^{2} x^{2}-1\right )^{4}}\) | \(152\) |
| orering | \(\frac {\left (\frac {413312}{8575} a^{10} x^{11}-\frac {8428176}{42875} a^{8} x^{9}+\frac {12911712}{42875} a^{6} x^{7}-\frac {179314}{875} a^{4} x^{5}+\frac {62669}{1225} a^{2} x^{3}+x \right ) \operatorname {arctanh}\left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {\left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (826624 a^{8} x^{8}-2505104 a^{6} x^{6}+2505160 a^{4} x^{4}-787374 a^{2} x^{2}-45381\right ) \left (\frac {2 \,\operatorname {arctanh}\left (a x \right ) a}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {9 \operatorname {arctanh}\left (a x \right )^{2} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}\right )}{77175 a^{2}}+\frac {x \left (206656 a^{6} x^{6}-635096 a^{4} x^{4}+654220 a^{2} x^{2}-226905\right ) \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (\frac {2 a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {40 \,\operatorname {arctanh}\left (a x \right ) a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {99 \operatorname {arctanh}\left (a x \right )^{2} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {9 \operatorname {arctanh}\left (a x \right )^{2} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}\right )}{385875 a^{2}}\) | \(295\) |
Input:
int(arctanh(a*x)^2/(-a^2*x^2+1)^(9/2),x,method=_RETURNVERBOSE)
Output:
-1/385875/a*(-a^2*x^2+1)^(1/2)*(176400*arctanh(a*x)^2*a^7*x^7+413312*a^7*x ^7-352800*arctanh(a*x)*a^6*x^6-617400*arctanh(a*x)^2*a^5*x^5-1270192*a^5*x ^5+1117200*a^4*x^4*arctanh(a*x)+771750*arctanh(a*x)^2*a^3*x^3+1308440*a^3* x^3-1202460*a^2*x^2*arctanh(a*x)-385875*arctanh(a*x)^2*a*x-453810*a*x+4538 10*arctanh(a*x))/(a^2*x^2-1)^4
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.61 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {{\left (1653248 \, a^{7} x^{7} - 5080768 \, a^{5} x^{5} + 5233760 \, a^{3} x^{3} + 11025 \, {\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 1815240 \, a x - 420 \, {\left (1680 \, a^{6} x^{6} - 5320 \, a^{4} x^{4} + 5726 \, a^{2} x^{2} - 2161\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{1543500 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(9/2),x, algorithm="fricas")
Output:
-1/1543500*(1653248*a^7*x^7 - 5080768*a^5*x^5 + 5233760*a^3*x^3 + 11025*(1 6*a^7*x^7 - 56*a^5*x^5 + 70*a^3*x^3 - 35*a*x)*log(-(a*x + 1)/(a*x - 1))^2 - 1815240*a*x - 420*(1680*a^6*x^6 - 5320*a^4*x^4 + 5726*a^2*x^2 - 2161)*lo g(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1)/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x ^4 - 4*a^3*x^2 + a)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}\, dx \] Input:
integrate(atanh(a*x)**2/(-a**2*x**2+1)**(9/2),x)
Output:
Integral(atanh(a*x)**2/(-(a*x - 1)*(a*x + 1))**(9/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 751 vs. \(2 (229) = 458\).
Time = 0.19 (sec) , antiderivative size = 751, normalized size of antiderivative = 2.71 \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(9/2),x, algorithm="maxima")
Output:
1/35*(16*x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) + 6*x/(-a^2*x^2 + 1)^(5/2) + 5*x/(-a^2*x^2 + 1)^(7/2))*arctanh(a*x)^2 + 1/385875*a*(225*(16 *x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) - 5/((-a^2*x^2 + 1)^(5/2) *a^2*x + (-a^2*x^2 + 1)^(5/2)*a) + 6*x/(-a^2*x^2 + 1)^(5/2))/a + 225*(16*x /sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) - 5/((-a^2*x^2 + 1)^(5/2)*a ^2*x - (-a^2*x^2 + 1)^(5/2)*a) + 6*x/(-a^2*x^2 + 1)^(5/2))/a + 882*(8*x/sq rt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) - 3/((-a^2*x^2 + 1)^(3/2)*a^2* x + (-a^2*x^2 + 1)^(3/2)*a))/a + 882*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x ^2 + 1)^(3/2) - 3/((-a^2*x^2 + 1)^(3/2)*a^2*x - (-a^2*x^2 + 1)^(3/2)*a))/a + 9800*(2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x + sqrt(-a^2* x^2 + 1)*a))/a + 9800*(2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2* x - sqrt(-a^2*x^2 + 1)*a))/a - 176400*sqrt(-a^2*x^2 + 1)/((a^2*x + a)*a) - 176400*sqrt(-a^2*x^2 + 1)/((a^2*x - a)*a) - 176400*log(a*x + 1)/(sqrt(-a^ 2*x^2 + 1)*a^2) + 176400*log(-a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) - 29400*lo g(a*x + 1)/((-a^2*x^2 + 1)^(3/2)*a^2) + 29400*log(-a*x + 1)/((-a^2*x^2 + 1 )^(3/2)*a^2) - 13230*log(a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2) + 13230*log(- a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2) - 7875*log(a*x + 1)/((-a^2*x^2 + 1)^(7 /2)*a^2) + 7875*log(-a*x + 1)/((-a^2*x^2 + 1)^(7/2)*a^2))
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(9/2),x, algorithm="giac")
Output:
integrate(arctanh(a*x)^2/(-a^2*x^2 + 1)^(9/2), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \] Input:
int(atanh(a*x)^2/(1 - a^2*x^2)^(9/2),x)
Output:
int(atanh(a*x)^2/(1 - a^2*x^2)^(9/2), x)
\[ \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int \frac {\mathit {atanh} \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}\, a^{8} x^{8}-4 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+6 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-4 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(atanh(a*x)^2/(-a^2*x^2+1)^(9/2),x)
Output:
int(atanh(a*x)**2/(sqrt( - a**2*x**2 + 1)*a**8*x**8 - 4*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 6*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 4*sqrt( - a**2*x**2 + 1)*a**2*x**2 + sqrt( - a**2*x**2 + 1)),x)