Integrand size = 22, antiderivative size = 150 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}-\frac {a^3 \sqrt {1-a^2 x^2}}{24 x^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{15 x^3}+\frac {2 a^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{15 x}+\frac {11}{120} a^5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
11/120*a^5*arctanh((-a^2*x^2+1)^(1/2))-1/20*a*(-a^2*x^2+1)^(1/2)/x^4-1/24* a^3*(-a^2*x^2+1)^(1/2)/x^2-1/5*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^5+1/15*a^ 2*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^3+2/15*a^4*arctanh(a*x)*(-a^2*x^2+1)^( 1/2)/x
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=\frac {1}{120} \left (-\frac {a \sqrt {1-a^2 x^2} \left (6+5 a^2 x^2\right )}{x^4}+\frac {8 \sqrt {1-a^2 x^2} \left (-3+a^2 x^2+2 a^4 x^4\right ) \text {arctanh}(a x)}{x^5}-11 a^5 \log (x)+11 a^5 \log \left (1+\sqrt {1-a^2 x^2}\right )\right ) \]
(-((a*Sqrt[1 - a^2*x^2]*(6 + 5*a^2*x^2))/x^4) + (8*Sqrt[1 - a^2*x^2]*(-3 + a^2*x^2 + 2*a^4*x^4)*ArcTanh[a*x])/x^5 - 11*a^5*Log[x] + 11*a^5*Log[1 + S qrt[1 - a^2*x^2]])/120
Leaf count is larger than twice the leaf count of optimal. \(331\) vs. \(2(150)=300\).
Time = 1.09 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.21, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {6572, 243, 52, 52, 73, 221, 6588, 243, 52, 52, 73, 221, 6588, 243, 52, 73, 221, 6570, 243, 73, 221}
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.
| \(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx\) |
| \(\Big \downarrow \) 6572 |
| \(\displaystyle -\frac {1}{4} \int \frac {\text {arctanh}(a x)}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {1}{4} a \int \frac {1}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{4} \int \frac {\text {arctanh}(a x)}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {1}{8} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {1}{4} \int \frac {\text {arctanh}(a x)}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {1}{8} a \left (\frac {3}{4} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {1}{4} \int \frac {\text {arctanh}(a x)}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {1}{4} \int \frac {\text {arctanh}(a x)}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (-\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {1}{4} \int \frac {\text {arctanh}(a x)}{x^6 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 6588 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \int \frac {\text {arctanh}(a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {1}{5} a \int \frac {1}{x^5 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \int \frac {\text {arctanh}(a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {1}{10} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}}dx^2+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \int \frac {\text {arctanh}(a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {1}{10} a \left (\frac {3}{4} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \int \frac {\text {arctanh}(a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \int \frac {\text {arctanh}(a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (-\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \int \frac {\text {arctanh}(a x)}{x^4 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 6588 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {1}{3} a \int \frac {1}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {1}{6} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {1}{6} a \left (\frac {1}{2} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {1}{6} a \left (-\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {\text {arctanh}(a x)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {1}{6} a \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 6570 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \left (a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}\right )+\frac {1}{6} a \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^5}+\frac {1}{8} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )+\frac {1}{4} \left (\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^5}-\frac {1}{10} a \left (\frac {3}{4} a^2 \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^4}\right )-\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+\frac {1}{6} a \left (a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )-\frac {\sqrt {1-a^2 x^2}}{x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^3}\right )\right )\) |
-1/4*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x^5 + (a*(-1/2*Sqrt[1 - a^2*x^2]/x^4 + (3*a^2*(-(Sqrt[1 - a^2*x^2]/x^2) - a^2*ArcTanh[Sqrt[1 - a^2*x^2]]))/4)) /8 + ((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(5*x^5) - (4*a^2*(-1/3*(Sqrt[1 - a^ 2*x^2]*ArcTanh[a*x])/x^3 + (2*a^2*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]))/3 + (a*(-(Sqrt[1 - a^2*x^2]/x^2) - a^2*Arc Tanh[Sqrt[1 - a^2*x^2]]))/6))/5 - (a*(-1/2*Sqrt[1 - a^2*x^2]/x^4 + (3*a^2* (-(Sqrt[1 - a^2*x^2]/x^2) - a^2*ArcTanh[Sqrt[1 - a^2*x^2]]))/4))/10)/4
3.5.36.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1)) Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTanh[c *x])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTanh[c*x]) /Sqrt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sq rt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && NeQ[m, -2]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*A rcTanh[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1))) Int[(f*x)^ (m + 1)*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[c^2*( (m + 2)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && G tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (16 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-5 a^{3} x^{3}+8 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-6 a x -24 \,\operatorname {arctanh}\left (a x \right )\right )}{120 x^{5}}-\frac {11 a^{5} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{120}+\frac {11 a^{5} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{120}\) | \(116\) |
1/120*(-(a*x-1)*(a*x+1))^(1/2)*(16*a^4*x^4*arctanh(a*x)-5*a^3*x^3+8*a^2*x^ 2*arctanh(a*x)-6*a*x-24*arctanh(a*x))/x^5-11/120*a^5*ln((a*x+1)/(-a^2*x^2+ 1)^(1/2)-1)+11/120*a^5*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=-\frac {11 \, a^{5} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (5 \, a^{3} x^{3} + 6 \, a x - 4 \, {\left (2 \, a^{4} x^{4} + a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, x^{5}} \]
-1/120*(11*a^5*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (5*a^3*x^3 + 6*a*x - 4*(2*a^4*x^4 + a^2*x^2 - 3)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1)) /x^5
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=\frac {1}{120} \, {\left (3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{4} + 8 \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \sqrt {-a^{2} x^{2} + 1} a^{2} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2}}\right )} a^{2} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{x^{2}} - \frac {6 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{4}}\right )} a - \frac {1}{15} \, {\left (\frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{x^{3}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{5}}\right )} \operatorname {artanh}\left (a x\right ) \]
1/120*(3*a^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 3*sqrt(-a^2*x^2 + 1)*a^4 + 8*(a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - sqrt(-a^2 *x^2 + 1)*a^2 - (-a^2*x^2 + 1)^(3/2)/x^2)*a^2 - 3*(-a^2*x^2 + 1)^(3/2)*a^2 /x^2 - 6*(-a^2*x^2 + 1)^(3/2)/x^4)*a - 1/15*(2*(-a^2*x^2 + 1)^(3/2)*a^2/x^ 3 + 3*(-a^2*x^2 + 1)^(3/2)/x^5)*arctanh(a*x)
Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^6} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x^6} \,d x \]