Integrand size = 22, antiderivative size = 243 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}-\frac {11 a^3 \sqrt {1-a^2 x^2}}{360 x^3}+\frac {a^5 \sqrt {1-a^2 x^2}}{720 x}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}+\frac {a^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{24 x^4}+\frac {a^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{16 x^2}+\frac {1}{8} a^6 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{16} a^6 \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{16} a^6 \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
1/8*a^6*arctanh(a*x)*arctanh((-a*x+1)^(1/2)/(a*x+1)^(1/2))-1/16*a^6*polylo g(2,-(-a*x+1)^(1/2)/(a*x+1)^(1/2))+1/16*a^6*polylog(2,(-a*x+1)^(1/2)/(a*x+ 1)^(1/2))-1/30*a*(-a^2*x^2+1)^(1/2)/x^5-11/360*a^3*(-a^2*x^2+1)^(1/2)/x^3+ 1/720*a^5*(-a^2*x^2+1)^(1/2)/x-1/6*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^6+1/2 4*a^2*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^4+1/16*a^4*arctanh(a*x)*(-a^2*x^2+ 1)^(1/2)/x^2
Time = 2.49 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=\frac {a^6 \left (-76 \coth \left (\frac {1}{2} \text {arctanh}(a x)\right )-90 \text {arctanh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )-\frac {26 a x \text {csch}^4\left (\frac {1}{2} \text {arctanh}(a x)\right )}{\sqrt {1-a^2 x^2}}-90 \text {arctanh}(a x) \text {csch}^4\left (\frac {1}{2} \text {arctanh}(a x)\right )-\frac {3 a x \text {csch}^6\left (\frac {1}{2} \text {arctanh}(a x)\right )}{\sqrt {1-a^2 x^2}}-15 \text {arctanh}(a x) \text {csch}^6\left (\frac {1}{2} \text {arctanh}(a x)\right )-360 \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )+360 \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )-360 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+360 \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )-90 \text {arctanh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+90 \text {arctanh}(a x) \text {sech}^4\left (\frac {1}{2} \text {arctanh}(a x)\right )-15 \text {arctanh}(a x) \text {sech}^6\left (\frac {1}{2} \text {arctanh}(a x)\right )-\frac {416 \left (1-a^2 x^2\right )^{3/2} \sinh ^4\left (\frac {1}{2} \text {arctanh}(a x)\right )}{a^3 x^3}+76 \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )+6 \text {sech}^4\left (\frac {1}{2} \text {arctanh}(a x)\right ) \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )}{5760} \]
(a^6*(-76*Coth[ArcTanh[a*x]/2] - 90*ArcTanh[a*x]*Csch[ArcTanh[a*x]/2]^2 - (26*a*x*Csch[ArcTanh[a*x]/2]^4)/Sqrt[1 - a^2*x^2] - 90*ArcTanh[a*x]*Csch[A rcTanh[a*x]/2]^4 - (3*a*x*Csch[ArcTanh[a*x]/2]^6)/Sqrt[1 - a^2*x^2] - 15*A rcTanh[a*x]*Csch[ArcTanh[a*x]/2]^6 - 360*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[ a*x])] + 360*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 360*PolyLog[2, -E^( -ArcTanh[a*x])] + 360*PolyLog[2, E^(-ArcTanh[a*x])] - 90*ArcTanh[a*x]*Sech [ArcTanh[a*x]/2]^2 + 90*ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^4 - 15*ArcTanh[a *x]*Sech[ArcTanh[a*x]/2]^6 - (416*(1 - a^2*x^2)^(3/2)*Sinh[ArcTanh[a*x]/2] ^4)/(a^3*x^3) + 76*Tanh[ArcTanh[a*x]/2] + 6*Sech[ArcTanh[a*x]/2]^4*Tanh[Ar cTanh[a*x]/2]))/5760
Time = 1.28 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.80, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6572, 245, 245, 242, 6588, 245, 245, 242, 6588, 245, 242, 6588, 242, 6580}
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.
| \(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx\) |
| \(\Big \downarrow \) 6572 |
| \(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx+\frac {1}{5} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx+\frac {1}{5} a \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 6588 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 6588 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {1}{4} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {1}{4} a \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 6588 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
| \(\Big \downarrow \) 6580 |
| \(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\) |
(a*(-1/5*Sqrt[1 - a^2*x^2]/x^5 + (4*a^2*(-1/3*Sqrt[1 - a^2*x^2]/x^3 - (2*a ^2*Sqrt[1 - a^2*x^2])/(3*x)))/5))/5 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(5* x^6) + (-1/6*(a*(-1/5*Sqrt[1 - a^2*x^2]/x^5 + (4*a^2*(-1/3*Sqrt[1 - a^2*x^ 2]/x^3 - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x)))/5)) + (Sqrt[1 - a^2*x^2]*ArcTan h[a*x])/(6*x^6) - (5*a^2*((a*(-1/3*Sqrt[1 - a^2*x^2]/x^3 - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x)))/4 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(4*x^4) + (3*a^2*( -1/2*(a*Sqrt[1 - a^2*x^2])/x - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) + (a^2*(-2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -( Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]))/ 2))/4))/6)/5
3.5.37.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 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_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x _Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTanh[c*x])*ArcTanh[Sqrt[1 - c*x]/Sq rt[1 + c*x]], x] + (Simp[(b/Sqrt[d])*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x ]], x] - Simp[(b/Sqrt[d])*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]], x]) /; F reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
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 = 0.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (a^{5} x^{5}+45 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-22 a^{3} x^{3}+30 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-24 a x -120 \,\operatorname {arctanh}\left (a x \right )\right )}{720 x^{6}}-\frac {a^{6} \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}+\frac {a^{6} \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}+\frac {a^{6} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}\) | \(183\) |
1/720*(-(a*x-1)*(a*x+1))^(1/2)*(a^5*x^5+45*a^4*x^4*arctanh(a*x)-22*a^3*x^3 +30*a^2*x^2*arctanh(a*x)-24*a*x-120*arctanh(a*x))/x^6-1/16*a^6*arctanh(a*x )*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-1/16*a^6*polylog(2,(a*x+1)/(-a^2*x^2+1) ^(1/2))+1/16*a^6*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/16*a^6*po lylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{7}} \,d x } \]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x^{7}}\, dx \]
\[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{7}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x^7} \,d x \]