3.5.49 \(\int x^2 (1-a^2 x^2)^{3/2} \text {arctanh}(a x) \, dx\) [449]

3.5.49.1 Optimal result

Integrand size = 22, antiderivative size = 243 \[ \int x^2 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\frac {\sqrt {1-a^2 x^2}}{16 a^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{72 a^3}-\frac {\left (1-a^2 x^2\right )^{5/2}}{30 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{16 a^2}+\frac {7}{24} x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {1}{6} a^2 x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-\frac {\arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{8 a^3}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{16 a^3}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{16 a^3} \]

output

1/72*(-a^2*x^2+1)^(3/2)/a^3-1/30*(-a^2*x^2+1)^(5/2)/a^3-1/8*arctan((-a*x+1 
)^(1/2)/(a*x+1)^(1/2))*arctanh(a*x)/a^3-1/16*I*polylog(2,-I*(-a*x+1)^(1/2) 
/(a*x+1)^(1/2))/a^3+1/16*I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^3+1 
/16*(-a^2*x^2+1)^(1/2)/a^3-1/16*x*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/a^2+7/24 
*x^3*arctanh(a*x)*(-a^2*x^2+1)^(1/2)-1/6*a^2*x^5*arctanh(a*x)*(-a^2*x^2+1) 
^(1/2)
 

3.5.49.2 Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.92 \[ \int x^2 \left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x) \, dx=\frac {31 \sqrt {1-a^2 x^2}+38 a^2 x^2 \sqrt {1-a^2 x^2}-24 a^4 x^4 \sqrt {1-a^2 x^2}-45 a x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+210 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-120 a^5 x^5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)-45 i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )+45 i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-45 i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )+45 i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )}{720 a^3} \]

input

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

output

(31*Sqrt[1 - a^2*x^2] + 38*a^2*x^2*Sqrt[1 - a^2*x^2] - 24*a^4*x^4*Sqrt[1 - 
 a^2*x^2] - 45*a*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 210*a^3*x^3*Sqrt[1 - a 
^2*x^2]*ArcTanh[a*x] - 120*a^5*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (45*I) 
*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x]] + (45*I)*ArcTanh[a*x]*Log[1 + I/E^ 
ArcTanh[a*x]] - (45*I)*PolyLog[2, (-I)/E^ArcTanh[a*x]] + (45*I)*PolyLog[2, 
 I/E^ArcTanh[a*x]])/(720*a^3)
 

3.5.49.3 Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(562\) vs. \(2(243)=486\).

Time = 1.82 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.31, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6576, 6572, 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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6572

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6512

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

input

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

output

-1/8*(a*((-2*Sqrt[1 - a^2*x^2])/a^4 + (2*(1 - a^2*x^2)^(3/2))/(3*a^4))) + 
(x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/4 + (-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*(-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])/ 
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)
 

3.5.49.3.1 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]
 

3.5.49.4 Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (120 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}+24 a^{4} x^{4}-210 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-38 a^{2} x^{2}+45 a x \,\operatorname {arctanh}\left (a x \right )-31\right )}{720 a^{3}}-\frac {i \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {arctanh}\left (a x \right )}{16 a^{3}}+\frac {i \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {arctanh}\left (a x \right )}{16 a^{3}}-\frac {i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{3}}+\frac {i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a^{3}}\)	\(195\)

input

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

output

-1/720/a^3*(-(a*x-1)*(a*x+1))^(1/2)*(120*arctanh(a*x)*a^5*x^5+24*a^4*x^4-2 
10*a^3*x^3*arctanh(a*x)-38*a^2*x^2+45*a*x*arctanh(a*x)-31)-1/16*I*ln(1+I*( 
a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^3+1/16*I*ln(1-I*(a*x+1)/(-a^2*x^ 
2+1)^(1/2))*arctanh(a*x)/a^3-1/16*I*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))/ 
a^3+1/16*I*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3
 

3.5.49.5 Fricas [F]

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

input

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

output

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

3.5.49.6 Sympy [F]

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

input

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

output

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

3.5.49.7 Maxima [F]

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

input

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

output

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

3.5.49.8 Giac [F]

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

input

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

output

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

3.5.49.9 Mupad [F(-1)]

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

input

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

output

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