Optimal. Leaf size=109 \[ -\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^4}+\frac {\tanh ^{-1}(a x)^2}{2 a^4}-\frac {\tanh ^{-1}(a x)}{4 a^4}-\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4}+\frac {\tanh ^{-1}(a x)}{2 a^4 \left (1-a^2 x^2\right )}-\frac {x}{4 a^3 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 109, 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 = {6028, 5984, 5918, 2402, 2315, 5994, 199, 206} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {x}{4 a^3 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{2 a^4}-\frac {\tanh ^{-1}(a x)}{4 a^4}-\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 2315
Rule 2402
Rule 5918
Rule 5984
Rule 5994
Rule 6028
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {\int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}-\frac {\int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}\\ &=\frac {\tanh ^{-1}(a x)}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{2 a^4}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^3}-\frac {\int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{a^3}\\ &=-\frac {x}{4 a^3 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 a^4 \left (1-a^2 x^2\right )}+\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}{4 a^3}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}\\ &=-\frac {x}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{4 a^4}+\frac {\tanh ^{-1}(a x)}{2 a^4 \left (1-a^2 x^2\right )}+\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 {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^4}\\ &=-\frac {x}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{4 a^4}+\frac {\tanh ^{-1}(a x)}{2 a^4 \left (1-a^2 x^2\right )}+\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 = 0.17, size = 64, normalized size = 0.59 \[ -\frac {-4 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+4 \tanh ^{-1}(a x)^2+\sinh \left (2 \tanh ^{-1}(a x)\right )-2 \tanh ^{-1}(a x) \left (\cosh \left (2 \tanh ^{-1}(a x)\right )-4 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )}{8 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \operatorname {artanh}\left (a x\right )}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x^{3} \operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 203, normalized size = 1.86 \[ -\frac {\arctanh \left (a x \right )}{4 a^{4} \left (a x -1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2 a^{4}}+\frac {\arctanh \left (a x \right )}{4 a^{4} \left (a x +1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2 a^{4}}+\frac {\ln \left (a x -1\right )^{2}}{8 a^{4}}-\frac {\dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{2 a^{4}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4 a^{4}}-\frac {\ln \left (a x +1\right )^{2}}{8 a^{4}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4 a^{4}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4 a^{4}}+\frac {1}{8 a^{4} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{8 a^{4}}+\frac {1}{8 a^{4} \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{8 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 177, normalized size = 1.62 \[ -\frac {1}{8} \, a {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )}{a^{7} x^{2} - a^{5}} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} + \frac {\log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{2} \, {\left (\frac {1}{a^{6} x^{2} - a^{4}} - \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int \frac {x^3\,\mathrm {atanh}\left (a\,x\right )}{{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {atanh}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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