Optimal. Leaf size=87 \[ -\frac {x}{2 a^3}+\frac {\tanh ^{-1}(a x)}{2 a^4}-\frac {x^2 \tanh ^{-1}(a x)}{2 a^2}-\frac {\tanh ^{-1}(a x)^2}{2 a^4}+\frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6127, 6037,
327, 212, 6131, 6055, 2449, 2352} \begin {gather*} \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {\tanh ^{-1}(a x)^2}{2 a^4}+\frac {\tanh ^{-1}(a x)}{2 a^4}+\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4}-\frac {x}{2 a^3}-\frac {x^2 \tanh ^{-1}(a x)}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 6037
Rule 6055
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx &=-\frac {\int x \tanh ^{-1}(a x) \, dx}{a^2}+\frac {\int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {x^2 \tanh ^{-1}(a x)}{2 a^2}-\frac {\tanh ^{-1}(a x)^2}{2 a^4}+\frac {\int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{a^3}+\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{2 a}\\ &=-\frac {x}{2 a^3}-\frac {x^2 \tanh ^{-1}(a x)}{2 a^2}-\frac {\tanh ^{-1}(a x)^2}{2 a^4}+\frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{2 a^3}-\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}\\ &=-\frac {x}{2 a^3}+\frac {\tanh ^{-1}(a x)}{2 a^4}-\frac {x^2 \tanh ^{-1}(a x)}{2 a^2}-\frac {\tanh ^{-1}(a x)^2}{2 a^4}+\frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^4}\\ &=-\frac {x}{2 a^3}+\frac {\tanh ^{-1}(a x)}{2 a^4}-\frac {x^2 \tanh ^{-1}(a x)}{2 a^2}-\frac {\tanh ^{-1}(a x)^2}{2 a^4}+\frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 60, normalized size = 0.69 \begin {gather*} \frac {-a x+\tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (1-a^2 x^2+2 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )\right )-\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 131, normalized size = 1.51
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} x^{2} \arctanh \left (a x \right )}{2}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}}{a^{4}}\) | \(131\) |
default | \(\frac {-\frac {a^{2} x^{2} \arctanh \left (a x \right )}{2}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}}{a^{4}}\) | \(131\) |
risch | \(\frac {\ln \left (-a x +1\right ) x^{2}}{4 a^{2}}-\frac {\ln \left (-a x +1\right )}{4 a^{4}}-\frac {x}{2 a^{3}}+\frac {\ln \left (-a x +1\right )^{2}}{8 a^{4}}+\frac {\ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{4 a^{4}}-\frac {\dilog \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{4}}-\frac {\ln \left (a x +1\right ) x^{2}}{4 a^{2}}+\frac {\ln \left (a x +1\right )}{4 a^{4}}-\frac {\ln \left (a x +1\right )^{2}}{8 a^{4}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4 a^{4}}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{4 a^{4}}\) | \(148\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 120, normalized size = 1.38 \begin {gather*} -\frac {1}{8} \, a {\left (\frac {4 \, a x - \log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2} + 2 \, \log \left (a x - 1\right )}{a^{5}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} - \frac {2 \, \log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{2} \, {\left (\frac {x^{2}}{a^{2}} + \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \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^{3} \operatorname {atanh}{\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^3\,\mathrm {atanh}\left (a\,x\right )}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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