\(\int \frac {(1-a^2 x^2)^{3/2} \text {arctanh}(a x)}{x^7} \, dx\) [458]

Optimal result

Integrand size = 22, antiderivative size = 243 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}+\frac {19 a^3 \sqrt {1-a^2 x^2}}{360 x^3}+\frac {31 a^5 \sqrt {1-a^2 x^2}}{720 x}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}+\frac {7 a^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{24 x^4}-\frac {a^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{16 x^2}-\frac {1}{8} a^6 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{16} a^6 \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{16} a^6 \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \] Output:

-1/30*a*(-a^2*x^2+1)^(1/2)/x^5+19/360*a^3*(-a^2*x^2+1)^(1/2)/x^3+31/720*a^ 
5*(-a^2*x^2+1)^(1/2)/x-1/6*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/x^6+7/24*a^2*(- 
a^2*x^2+1)^(1/2)*arctanh(a*x)/x^4-1/16*a^4*(-a^2*x^2+1)^(1/2)*arctanh(a*x) 
/x^2-1/8*a^6*arctanh(a*x)*arctanh((-a*x+1)^(1/2)/(a*x+1)^(1/2))+1/16*a^6*p 
olylog(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))
 

Mathematica [A] (warning: unable to verify)

Time = 4.15 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.95 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\frac {82 a^7 x^4 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+90 a^6 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+4 a^7 x^4 \text {csch}^4\left (\frac {1}{2} \text {arctanh}(a x)\right )-3 a^7 x^4 \text {csch}^6\left (\frac {1}{2} \text {arctanh}(a x)\right )-15 a^6 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \text {csch}^6\left (\frac {1}{2} \text {arctanh}(a x)\right )+360 a^6 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-360 a^6 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+360 a^6 x^3 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-360 a^6 x^3 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+328 a^5 x^2 \left (-1+a^2 x^2\right ) \sinh ^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+360 a^4 x \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \sinh ^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+64 a^3 \sinh ^4\left (\frac {1}{2} \text {arctanh}(a x)\right )-128 a^5 x^2 \sinh ^4\left (\frac {1}{2} \text {arctanh}(a x)\right )+64 a^7 x^4 \sinh ^4\left (\frac {1}{2} \text {arctanh}(a x)\right )-\frac {192 a \left (-1+a^2 x^2\right )^3 \sinh ^6\left (\frac {1}{2} \text {arctanh}(a x)\right )}{x^2}-\frac {960 \left (1-a^2 x^2\right )^{7/2} \text {arctanh}(a x) \sinh ^6\left (\frac {1}{2} \text {arctanh}(a x)\right )}{x^3}}{5760 x^3 \sqrt {1-a^2 x^2}} \] Input:

Integrate[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^7,x]
 

Output:

(82*a^7*x^4*Csch[ArcTanh[a*x]/2]^2 + 90*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[ 
a*x]*Csch[ArcTanh[a*x]/2]^2 + 4*a^7*x^4*Csch[ArcTanh[a*x]/2]^4 - 3*a^7*x^4 
*Csch[ArcTanh[a*x]/2]^6 - 15*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Csch[A 
rcTanh[a*x]/2]^6 + 360*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(- 
ArcTanh[a*x])] - 360*a^6*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-Ar 
cTanh[a*x])] + 360*a^6*x^3*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(-ArcTanh[a*x]) 
] - 360*a^6*x^3*Sqrt[1 - a^2*x^2]*PolyLog[2, E^(-ArcTanh[a*x])] + 328*a^5* 
x^2*(-1 + a^2*x^2)*Sinh[ArcTanh[a*x]/2]^2 + 360*a^4*x*(1 - a^2*x^2)^(3/2)* 
ArcTanh[a*x]*Sinh[ArcTanh[a*x]/2]^2 + 64*a^3*Sinh[ArcTanh[a*x]/2]^4 - 128* 
a^5*x^2*Sinh[ArcTanh[a*x]/2]^4 + 64*a^7*x^4*Sinh[ArcTanh[a*x]/2]^4 - (192* 
a*(-1 + a^2*x^2)^3*Sinh[ArcTanh[a*x]/2]^6)/x^2 - (960*(1 - a^2*x^2)^(7/2)* 
ArcTanh[a*x]*Sinh[ArcTanh[a*x]/2]^6)/x^3)/(5760*x^3*Sqrt[1 - a^2*x^2])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(737\) vs. \(2(243)=486\).

Time = 2.71 (sec) , antiderivative size = 737, normalized size of antiderivative = 3.03, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {6576, 6572, 245, 242, 245, 242, 6588, 245, 242, 245, 242, 6588, 242, 245, 242, 6580, 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.

\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx\)

\(\Big \downarrow \) 6576

\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^7}dx-a^2 \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^5}dx\)

\(\Big \downarrow \) 6572

\(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx-\left (a^2 \left (-\frac {1}{3} \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx+\frac {1}{3} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}\right )\right )+\frac {1}{5} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx-\left (a^2 \left (-\frac {1}{3} \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx+\frac {1}{3} a \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}\right )\right )+\frac {1}{5} a \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx-\left (a^2 \left (-\frac {1}{3} \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} a \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\)

\(\Big \downarrow \) 245

\(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx-\left (a^2 \left (-\frac {1}{3} \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}\)

\(\Big \downarrow \) 242

\(\displaystyle -\frac {1}{5} \int \frac {\text {arctanh}(a x)}{x^7 \sqrt {1-a^2 x^2}}dx-\left (a^2 \left (-\frac {1}{3} \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 6588

\(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {1}{4} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 245

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {1}{4} a \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 242

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \left (\frac {4}{5} a^2 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 245

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 242

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \int \frac {\text {arctanh}(a x)}{x^5 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 6588

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {1}{4} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 242

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {1}{4} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 245

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {1}{4} a \left (\frac {2}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 242

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 6580

\(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \int \frac {\text {arctanh}(a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 6588

\(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

\(\Big \downarrow \) 6580

\(\displaystyle -\left (a^2 \left (\frac {1}{3} \left (-\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}-\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 x^4}+\frac {1}{3} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )\right )+\frac {1}{5} \left (-\frac {5}{6} a^2 \left (\frac {3}{4} a^2 \left (\frac {1}{2} a^2 \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 x^2}-\frac {a \sqrt {1-a^2 x^2}}{2 x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 x^4}+\frac {1}{4} a \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 x^6}-\frac {1}{6} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{5 x^6}+\frac {1}{5} a \left (\frac {4}{5} a^2 \left (-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-a^2 x^2}}{5 x^5}\right )\)

Input:

Int[((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/x^7,x]
 

Output:

(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) - a^2*((a*(-1/3*Sqrt[1 - a^2*x^2]/x^3 - (2*a^2*Sqrt[1 - a^2*x^2])/(3* 
x)))/3 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*x^4) + (-1/4*(a*(-1/3*Sqrt[1 
- a^2*x^2]/x^3 - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x))) + (Sqrt[1 - a^2*x^2]*Ar 
cTanh[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, S 
qrt[1 - a*x]/Sqrt[1 + a*x]]))/2))/4)/3) + (-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]*ArcTanh[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
 

Defintions of rubi rules used

rule 242

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]
 

rule 245

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]
 

rule 6572

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]
 

rule 6576

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[d   Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a 
+ b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2)   Int[(f*x)^(m + 2)*(d + e*x 
^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, 
x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ 
[p, 1] && IntegerQ[q]))
 

rule 6580

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]
 

rule 6588

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]
 

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (-31 a^{5} x^{5}+45 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-38 a^{3} x^{3}-210 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+24 a x +120 \,\operatorname {arctanh}\left (a x \right )\right )}{720 x^{6}}+\frac {a^{6} \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}+\frac {a^{6} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}\)	\(184\)

Input:

int((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/720*(-(a*x-1)*(a*x+1))^(1/2)*(-31*a^5*x^5+45*a^4*x^4*arctanh(a*x)-38*a^ 
3*x^3-210*a^2*x^2*arctanh(a*x)+24*a*x+120*arctanh(a*x))/x^6+1/16*a^6*arcta 
nh(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*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}{x^{7}} \,d x } \] Input:

integrate((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^7,x, algorithm="fricas")
 

Output:

integral(-(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*arctanh(a*x)/x^7, x)
 

Sympy [F]

\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{7}}\, dx \] Input:

integrate((-a**2*x**2+1)**(3/2)*atanh(a*x)/x**7,x)
 

Output:

Integral((-(a*x - 1)*(a*x + 1))**(3/2)*atanh(a*x)/x**7, x)
 

Maxima [F]

\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}{x^{7}} \,d x } \] Input:

integrate((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^7,x, algorithm="maxima")
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)*arctanh(a*x)/x^7, x)
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*x^2+1)^(3/2)*arctanh(a*x)/x^7,x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^7} \,d x \] Input:

int((atanh(a*x)*(1 - a^2*x^2)^(3/2))/x^7,x)
 

Output:

int((atanh(a*x)*(1 - a^2*x^2)^(3/2))/x^7, x)
 

Reduce [F]

\[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^7} \, dx=\int \frac {\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )}{x^{7}}d x -\left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )}{x^{5}}d x \right ) a^{2} \] Input:

int((-a^2*x^2+1)^(3/2)*atanh(a*x)/x^7,x)
 

Output:

int((sqrt( - a**2*x**2 + 1)*atanh(a*x))/x**7,x) - int((sqrt( - a**2*x**2 + 
 1)*atanh(a*x))/x**5,x)*a**2