\(\int x^4 (1-a^2 x^2)^{3/2} \text {arctanh}(a x) \, dx\) [447]

Optimal result

Integrand size = 22, antiderivative size = 292 \[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\frac {3 \sqrt {1-a^2 x^2}}{128 a^5}+\frac {\left (1-a^2 x^2\right )^{3/2}}{192 a^5}-\frac {3 \left (1-a^2 x^2\right )^{5/2}}{80 a^5}+\frac {\left (1-a^2 x^2\right )^{7/2}}{56 a^5}-\frac {3 x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{128 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{64 a^2}+\frac {3}{16} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{8} a^2 x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {3 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{64 a^5}-\frac {3 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{128 a^5}+\frac {3 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{128 a^5} \] Output:

3/128*(-a^2*x^2+1)^(1/2)/a^5+1/192*(-a^2*x^2+1)^(3/2)/a^5-3/80*(-a^2*x^2+1 
)^(5/2)/a^5+1/56*(-a^2*x^2+1)^(7/2)/a^5-3/128*x*(-a^2*x^2+1)^(1/2)*arctanh 
(a*x)/a^4-1/64*x^3*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a^2+3/16*x^5*(-a^2*x^2+ 
1)^(1/2)*arctanh(a*x)-1/8*a^2*x^7*(-a^2*x^2+1)^(1/2)*arctanh(a*x)-3/64*arc 
tan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a^5-3/128*I*polylog(2,-I*(- 
a*x+1)^(1/2)/(a*x+1)^(1/2))/a^5+3/128*I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1) 
^(1/2))/a^5
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.93 \[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\frac {121 \sqrt {1-a^2 x^2}+218 a^2 x^2 \sqrt {1-a^2 x^2}+216 a^4 x^4 \sqrt {1-a^2 x^2}-240 a^6 x^6 \sqrt {1-a^2 x^2}-315 a x \sqrt {1-a^2 x^2} \text {arctanh}(a x)-210 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)+2520 a^5 x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-1680 a^7 x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-315 i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )+315 i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-315 i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )+315 i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )}{13440 a^5} \] Input:

Integrate[x^4*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]
 

Output:

(121*Sqrt[1 - a^2*x^2] + 218*a^2*x^2*Sqrt[1 - a^2*x^2] + 216*a^4*x^4*Sqrt[ 
1 - a^2*x^2] - 240*a^6*x^6*Sqrt[1 - a^2*x^2] - 315*a*x*Sqrt[1 - a^2*x^2]*A 
rcTanh[a*x] - 210*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 2520*a^5*x^5*Sq 
rt[1 - a^2*x^2]*ArcTanh[a*x] - 1680*a^7*x^7*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] 
 - (315*I)*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (315*I)*ArcTanh[a*x]*L 
og[1 + I/E^ArcTanh[a*x]] - (315*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (315* 
I)*PolyLog[2, I/E^ArcTanh[a*x]])/(13440*a^5)
 

Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(791\) vs. \(2(292)=584\).

Time = 2.80 (sec) , antiderivative size = 791, normalized size of antiderivative = 2.71, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {6576, 6572, 243, 53, 2009, 6578, 243, 53, 2009, 6578, 241, 243, 53, 2009, 6512, 6578, 241, 6512}

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.

\(\displaystyle \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx\)

\(\Big \downarrow \) 6576

\(\displaystyle \int x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)dx-a^2 \int x^6 \sqrt {1-a^2 x^2} \text {arctanh}(a x)dx\)

\(\Big \downarrow \) 6572

\(\displaystyle \frac {1}{6} \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{8} \int \frac {x^6 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{8} a \int \frac {x^7}{\sqrt {1-a^2 x^2}}dx+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )-\frac {1}{6} a \int \frac {x^5}{\sqrt {1-a^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{6} \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{8} \int \frac {x^6 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{16} a \int \frac {x^6}{\sqrt {1-a^2 x^2}}dx^2+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )-\frac {1}{12} a \int \frac {x^4}{\sqrt {1-a^2 x^2}}dx^2+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {1}{6} \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{8} \int \frac {x^6 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{16} a \int \left (-\frac {\left (1-a^2 x^2\right )^{5/2}}{a^6}+\frac {3 \left (1-a^2 x^2\right )^{3/2}}{a^6}-\frac {3 \sqrt {1-a^2 x^2}}{a^6}+\frac {1}{a^6 \sqrt {1-a^2 x^2}}\right )dx^2+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\right )-\frac {1}{12} a \int \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}+\frac {1}{a^4 \sqrt {1-a^2 x^2}}\right )dx^2+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx-a^2 \left (\frac {1}{8} \int \frac {x^6 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 6578

\(\displaystyle \frac {1}{6} \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3}{\sqrt {1-a^2 x^2}}dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )-a^2 \left (\frac {1}{8} \left (\frac {5 \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}+\frac {\int \frac {x^5}{\sqrt {1-a^2 x^2}}dx}{6 a}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{6} \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}}dx^2}{8 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )-a^2 \left (\frac {1}{8} \left (\frac {5 \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}+\frac {\int \frac {x^4}{\sqrt {1-a^2 x^2}}dx^2}{12 a}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {1}{6} \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \left (\frac {1}{a^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^2}\right )dx^2}{8 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )-a^2 \left (\frac {1}{8} \left (\frac {5 \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}+\frac {\int \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}+\frac {1}{a^4 \sqrt {1-a^2 x^2}}\right )dx^2}{12 a}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )-a^2 \left (\frac {1}{8} \left (\frac {5 \int \frac {x^4 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 6578

\(\displaystyle \frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )-a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3}{\sqrt {1-a^2 x^2}}dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 241

\(\displaystyle -a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3}{\sqrt {1-a^2 x^2}}dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle -a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}}dx^2}{8 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 53

\(\displaystyle -a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \left (\frac {1}{a^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^2}\right )dx^2}{8 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )-a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\)

\(\Big \downarrow \) 6512

\(\displaystyle -a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}+\frac {3 \left (\frac {-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}\right )\)

\(\Big \downarrow \) 6578

\(\displaystyle -a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}+\frac {3 \left (\frac {-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}\right )\)

\(\Big \downarrow \) 241

\(\displaystyle -a^2 \left (\frac {1}{8} \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}\right )+\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )\right )+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}+\frac {3 \left (\frac {-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}\right )\)

\(\Big \downarrow \) 6512

\(\displaystyle \frac {1}{6} x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{12} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}+\frac {3 \left (\frac {-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}\right )-a^2 \left (\frac {1}{8} x^7 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{16} a \left (\frac {2 \left (1-a^2 x^2\right )^{7/2}}{7 a^8}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^8}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a^8}-\frac {2 \sqrt {1-a^2 x^2}}{a^8}\right )+\frac {1}{8} \left (-\frac {x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a^2}+\frac {-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}}{12 a}+\frac {5 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{4 a^2}+\frac {\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}}{8 a}+\frac {3 \left (\frac {-\frac {2 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \text {arctanh}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{2 a^3}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\)

Input:

Int[x^4*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x],x]
 

Output:

-1/12*(a*((-2*Sqrt[1 - a^2*x^2])/a^6 + (4*(1 - a^2*x^2)^(3/2))/(3*a^6) - ( 
2*(1 - a^2*x^2)^(5/2))/(5*a^6))) + (x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/6 
+ (((-2*Sqrt[1 - a^2*x^2])/a^4 + (2*(1 - a^2*x^2)^(3/2))/(3*a^4))/(8*a) - 
(x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(4*a^2) + (3*(-1/2*Sqrt[1 - a^2*x^2]/ 
a^3 - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*a^2) + ((-2*ArcTan[Sqrt[1 - a* 
x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqr 
t[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a)/(2*a^2 
)))/(4*a^2))/6 - a^2*(-1/16*(a*((-2*Sqrt[1 - a^2*x^2])/a^8 + (2*(1 - a^2*x 
^2)^(3/2))/a^8 - (6*(1 - a^2*x^2)^(5/2))/(5*a^8) + (2*(1 - a^2*x^2)^(7/2)) 
/(7*a^8))) + (x^7*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/8 + (((-2*Sqrt[1 - a^2*x 
^2])/a^6 + (4*(1 - a^2*x^2)^(3/2))/(3*a^6) - (2*(1 - a^2*x^2)^(5/2))/(5*a^ 
6))/(12*a) - (x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(6*a^2) + (5*(((-2*Sqrt[ 
1 - a^2*x^2])/a^4 + (2*(1 - a^2*x^2)^(3/2))/(3*a^4))/(8*a) - (x^3*Sqrt[1 - 
 a^2*x^2]*ArcTanh[a*x])/(4*a^2) + (3*(-1/2*Sqrt[1 - a^2*x^2]/a^3 - (x*Sqrt 
[1 - a^2*x^2]*ArcTanh[a*x])/(2*a^2) + ((-2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a 
*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/ 
a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a)/(2*a^2)))/(4*a^2))) 
/(6*a^2))/8)
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 243

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6512

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol 
] :> Simp[-2*(a + b*ArcTanh[c*x])*(ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*S 
qrt[d])), x] + (-Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/( 
c*Sqrt[d])), x] + Simp[I*b*(PolyLog[2, I*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/(c* 
Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 
0]
 

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 6578

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) 
 + (e_.)*(x_)^2], x_Symbol] :> Simp[(-f)*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a 
+ b*ArcTanh[c*x])^p/(c^2*d*m)), x] + (Simp[b*f*(p/(c*m))   Int[(f*x)^(m - 1 
)*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[f^2*((m - 1 
)/(c^2*m))   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] && GtQ[p, 0] && 
GtQ[m, 1]
 

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.74

method	result	size
default	\(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (1680 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}+240 a^{6} x^{6}-2520 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-216 a^{4} x^{4}+210 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-218 a^{2} x^{2}+315 a x \,\operatorname {arctanh}\left (a x \right )-121\right )}{13440 a^{5}}-\frac {3 i \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 a^{5}}+\frac {3 i \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 a^{5}}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 a^{5}}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{128 a^{5}}\)	\(215\)

Input:

int(x^4*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x,method=_RETURNVERBOSE)
 

Output:

-1/13440/a^5*(-(a*x-1)*(a*x+1))^(1/2)*(1680*arctanh(a*x)*a^7*x^7+240*a^6*x 
^6-2520*arctanh(a*x)*a^5*x^5-216*a^4*x^4+210*a^3*x^3*arctanh(a*x)-218*a^2* 
x^2+315*a*x*arctanh(a*x)-121)-3/128*I*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))*a 
rctanh(a*x)/a^5+3/128*I*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^ 
5-3/128*I*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^5+3/128*I*dilog(1-I*(a*x 
+1)/(-a^2*x^2+1)^(1/2))/a^5
 

Fricas [F]

\[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:

integrate(x^4*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x, algorithm="fricas")
 

Output:

integral(-(a^2*x^6 - x^4)*sqrt(-a^2*x^2 + 1)*arctanh(a*x), x)
 

Sympy [F]

\[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}\, dx \] Input:

integrate(x**4*(-a**2*x**2+1)**(3/2)*atanh(a*x),x)
 

Output:

Integral(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)*atanh(a*x), x)
 

Maxima [F]

\[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:

integrate(x^4*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x, algorithm="maxima")
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)*x^4*arctanh(a*x), x)
 

Giac [F]

\[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4} \operatorname {artanh}\left (a x\right ) \,d x } \] Input:

integrate(x^4*(-a^2*x^2+1)^(3/2)*arctanh(a*x),x, algorithm="giac")
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)*x^4*arctanh(a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\int x^4\,\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2} \,d x \] Input:

int(x^4*atanh(a*x)*(1 - a^2*x^2)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int(x^4*atanh(a*x)*(1 - a^2*x^2)^(3/2), x)
 

Reduce [F]

\[ \int x^4 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=-\left (\int \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right ) x^{6}d x \right ) a^{2}+\int \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right ) x^{4}d x \] Input:

int(x^4*(-a^2*x^2+1)^(3/2)*atanh(a*x),x)
 

Output:

 - int(sqrt( - a**2*x**2 + 1)*atanh(a*x)*x**6,x)*a**2 + int(sqrt( - a**2*x 
**2 + 1)*atanh(a*x)*x**4,x)