3.5.39 \(\int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx\) [439]

3.5.39.1 Optimal result

Integrand size = 24, antiderivative size = 281 \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {11 \sqrt {1-a^2 x^2}}{60 a^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{30 a^4}+\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{12 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{10 a}-\frac {11 \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)}{30 a^4}-\frac {2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2-\frac {11 i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{60 a^4}+\frac {11 i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{60 a^4} \]

output

-1/30*(-a^2*x^2+1)^(3/2)/a^4-11/30*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*ar 
ctanh(a*x)/a^4-11/60*I*polylog(2,-I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^4+11/6 
0*I*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^4+11/60*(-a^2*x^2+1)^(1/2) 
/a^4+1/12*x*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/a^3+1/10*x^3*arctanh(a*x)*(-a^ 
2*x^2+1)^(1/2)/a-2/15*arctanh(a*x)^2*(-a^2*x^2+1)^(1/2)/a^4-1/15*x^2*arcta 
nh(a*x)^2*(-a^2*x^2+1)^(1/2)/a^2+1/5*x^4*arctanh(a*x)^2*(-a^2*x^2+1)^(1/2)
 

3.5.39.2 Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.62 \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {\sqrt {1-a^2 x^2} \left (11+11 a x \text {arctanh}(a x)+6 a x \left (-1+a^2 x^2\right ) \text {arctanh}(a x)+12 \left (-1+a^2 x^2\right )^2 \text {arctanh}(a x)^2+2 \left (-1+a^2 x^2\right ) \left (1+10 \text {arctanh}(a x)^2\right )-\frac {11 i \left (\text {arctanh}(a x) \left (\log \left (1-i e^{-\text {arctanh}(a x)}\right )-\log \left (1+i e^{-\text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{60 a^4} \]

input

Integrate[x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]
 

output

(Sqrt[1 - a^2*x^2]*(11 + 11*a*x*ArcTanh[a*x] + 6*a*x*(-1 + a^2*x^2)*ArcTan 
h[a*x] + 12*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2 + 2*(-1 + a^2*x^2)*(1 + 10*Arc 
Tanh[a*x]^2) - ((11*I)*(ArcTanh[a*x]*(Log[1 - I/E^ArcTanh[a*x]] - Log[1 + 
I/E^ArcTanh[a*x]]) + PolyLog[2, (-I)/E^ArcTanh[a*x]] - PolyLog[2, I/E^ArcT 
anh[a*x]]))/Sqrt[1 - a^2*x^2]))/(60*a^4)
 

3.5.39.3 Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(926\) vs. \(2(281)=562\).

Time = 4.18 (sec) , antiderivative size = 926, normalized size of antiderivative = 3.30, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6576, 6578, 6556, 6512, 6578, 241, 243, 53, 2009, 6512, 6556, 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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6578

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6512

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

input

Int[x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]
 

output

-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2 + (2*(-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)))/(3*a) + (2*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2) + (2*((-2*Arc 
Tan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, ((-I)*Sqr 
t[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a 
*x]])/a))/a))/(3*a^2) - a^2*(-1/5*(x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a 
^2 + (2*(((-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)))/(5*a) + (4*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 
)/a^2 + (2*(-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)))/(3*a) + (2*(-((Sqrt[1 - a^2*x^2]*Ar 
cTanh[a*x]^2)/a^2) + (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*Poly...
 

3.5.39.3.1 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 6556

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q 
_.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q 
+ 1))), x] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[p, 0] && NeQ[q, -1]
 

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]
 

3.5.39.4 Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (12 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+6 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-4 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+2 a^{2} x^{2}+5 a x \,\operatorname {arctanh}\left (a x \right )-8 \operatorname {arctanh}\left (a x \right )^{2}+9\right )}{60 a^{4}}-\frac {11 i \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {arctanh}\left (a x \right )}{60 a^{4}}+\frac {11 i \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \operatorname {arctanh}\left (a x \right )}{60 a^{4}}-\frac {11 i \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{60 a^{4}}+\frac {11 i \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{60 a^{4}}\)	\(211\)

input

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

output

1/60/a^4*(-(a*x-1)*(a*x+1))^(1/2)*(12*a^4*x^4*arctanh(a*x)^2+6*a^3*x^3*arc 
tanh(a*x)-4*a^2*x^2*arctanh(a*x)^2+2*a^2*x^2+5*a*x*arctanh(a*x)-8*arctanh( 
a*x)^2+9)-11/60*I*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^4+11/6 
0*I*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^4-11/60*I*dilog(1+I* 
(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4+11/60*I*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/ 
2))/a^4
 

3.5.39.5 Fricas [F]

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

input

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

output

integral(sqrt(-a^2*x^2 + 1)*x^3*arctanh(a*x)^2, x)
 

3.5.39.6 Sympy [F]

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

input

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

output

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

3.5.39.7 Maxima [F]

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

input

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

output

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

3.5.39.8 Giac [F(-2)]

Exception generated. \[ \int x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]

input

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

3.5.39.9 Mupad [F(-1)]

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

input

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

output

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