Optimal. Leaf size=78 \[ -\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^2}+\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^2}-\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5984, 5918, 5948, 6058, 6610} \[ -\frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^2}+\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2}-\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5918
Rule 5948
Rule 5984
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx &=-\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2}-\frac {2 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^2}-\frac {\int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^2}-\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 68, normalized size = 0.87 \[ -\frac {\tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\frac {1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{a^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.52, size = 741, normalized size = 9.50 \[ -\frac {\arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{2 a^{2}}-\frac {\arctanh \left (a x \right )^{2} \ln \left (a x +1\right )}{2 a^{2}}+\frac {\arctanh \left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}-\frac {\arctanh \left (a x \right )^{3}}{3 a^{2}}+\frac {\arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{2}}-\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi }{2 a^{2}}-\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )}{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2}}{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2}}{4 a^{2}}-\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{2}}{4 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}}{4 a^{2}}-\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2}}{2 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3}}{2 a^{2}}+\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}\right )^{3}}{4 a^{2}}+\frac {\arctanh \left (a x \right )^{2} \ln \relax (2)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {3 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{2} + \log \left (-a x + 1\right )^{3}}{24 \, a^{2}} + \frac {1}{4} \, \int -\frac {a x \log \left (a x + 1\right )^{2} - {\left (3 \, a x + 1\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{3} x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________