Optimal. Leaf size=78 \[ -\frac {\tanh ^{-1}(a x)^3}{3 a^2}+\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^2}+\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^2}-\frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^2} \]
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Rubi [A]
time = 0.11, 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 = {6131, 6055,
6095, 6205, 6745} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 6055
Rule 6095
Rule 6131
Rule 6205
Rule 6745
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*}
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Mathematica [A]
time = 0.05, size = 68, normalized size = 0.87 \begin {gather*} -\frac {-\frac {1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 31.49, size = 638, normalized size = 8.18
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\arctanh \left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\arctanh \left (a x \right )^{3}}{3}+\frac {\left (i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 i \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}-i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )-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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+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 )+2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi +4 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{2}}{4}+\arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}}{a^{2}}\) | \(638\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )^{2} \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\arctanh \left (a x \right )^{2} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\arctanh \left (a x \right )^{3}}{3}+\frac {\left (i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{a^{2} x^{2}-1}\right )^{3}+2 i \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}-i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )-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 (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+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 )+2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )^{2}}{\left (a^{2} x^{2}-1\right ) \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2}+2 i \pi +4 \ln \left (2\right )\right ) \arctanh \left (a x \right )^{2}}{4}+\arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}}{a^{2}}\) | \(638\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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