Optimal. Leaf size=103 \[ -\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3}+\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}-\frac {\tanh ^{-1}(a x)^3}{a^3}+\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3}-\frac {x \tanh ^{-1}(a x)^3}{a^2} \]
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Rubi [A] time = 0.27, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5980, 5910, 5984, 5918, 5948, 6058, 6610} \[ -\frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}-\frac {x \tanh ^{-1}(a x)^3}{a^2}-\frac {\tanh ^{-1}(a x)^3}{a^3}+\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3} \]
Antiderivative was successfully verified.
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Rule 5910
Rule 5918
Rule 5948
Rule 5980
Rule 5984
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx &=-\frac {\int \tanh ^{-1}(a x)^3 \, dx}{a^2}+\frac {\int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {x \tanh ^{-1}(a x)^3}{a^2}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}+\frac {3 \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^3}{a^3}-\frac {x \tanh ^{-1}(a x)^3}{a^2}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}+\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)^3}{a^3}-\frac {x \tanh ^{-1}(a x)^3}{a^2}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}-\frac {6 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)^3}{a^3}-\frac {x \tanh ^{-1}(a x)^3}{a^2}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3}-\frac {3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)^3}{a^3}-\frac {x \tanh ^{-1}(a x)^3}{a^2}+\frac {\tanh ^{-1}(a x)^4}{4 a^3}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 78, normalized size = 0.76 \[ \frac {-12 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-6 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x)^2 \left (\tanh ^{-1}(a x)^2+(4-4 a x) \tanh ^{-1}(a x)+12 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )}{4 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 788, normalized size = 7.65 \[ -\frac {x \arctanh \left (a x \right )^{3}}{a^{2}}-\frac {\arctanh \left (a x \right )^{3} \ln \left (a x -1\right )}{2 a^{3}}+\frac {\arctanh \left (a x \right )^{3} \ln \left (a x +1\right )}{2 a^{3}}-\frac {\arctanh \left (a x \right )^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3}}+\frac {\arctanh \left (a x \right )^{4}}{4 a^{3}}-\frac {\arctanh \left (a x \right )^{3}}{a^{3}}-\frac {3 \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3}}-\frac {i \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} \arctanh \left (a x \right )^{3}}{4 a^{3}}-\frac {i \arctanh \left (a x \right )^{3} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}}{4 a^{3}}+\frac {i \pi \arctanh \left (a x \right )^{3}}{2 a^{3}}-\frac {i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2} \arctanh \left (a x \right )^{3}}{2 a^{3}}+\frac {i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3} \arctanh \left (a x \right )^{3}}{2 a^{3}}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right ) \arctanh \left (a x \right )^{3}}{4 a^{3}}-\frac {i \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} \arctanh \left (a x \right )^{3}}{4 a^{3}}+\frac {i \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}}{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 ) \arctanh \left (a x \right )^{3}}{4 a^{3}}+\frac {i \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} \arctanh \left (a x \right )^{3}}{4 a^{3}}-\frac {i \arctanh \left (a x \right )^{3} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{2}}{2 a^{3}}+\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3}}+\frac {3 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, {\left (2 \, a x - \log \left (a x + 1\right ) - 2\right )} \log \left (-a x + 1\right )^{3} + \log \left (-a x + 1\right )^{4} - 6 \, {\left (4 \, {\left (a x + 1\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2}\right )} \log \left (-a x + 1\right )^{2}}{64 \, a^{3}} + \frac {1}{8} \, \int -\frac {2 \, a^{2} x^{2} \log \left (a x + 1\right )^{3} - 3 \, {\left ({\left (2 \, a^{2} x^{2} - a x - 1\right )} \log \left (a x + 1\right )^{2} + 4 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{2 \, {\left (a^{4} x^{2} - a^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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