Optimal. Leaf size=161 \[ \frac {1}{4 a^4 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4}+\frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^4} \]
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Rubi [A]
time = 0.22, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6175, 6131,
6055, 6095, 6205, 6745, 6141, 6103, 267} \begin {gather*} \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}-\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^4}+\frac {1}{4 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6055
Rule 6095
Rule 6103
Rule 6131
Rule 6141
Rule 6175
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {\int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}-\frac {\int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a^2}\\ &=\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a^3}-\frac {\int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a^3}\\ &=-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {2 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}+\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}\\ &=\frac {1}{4 a^4 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^4}+\frac {\int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}\\ &=\frac {1}{4 a^4 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^4}+\frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 103, normalized size = 0.64 \begin {gather*} \frac {-\frac {1}{3} \tanh ^{-1}(a x)^3+\frac {1}{8} \left (1+2 \tanh ^{-1}(a x)^2\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )-\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 )-\frac {1}{4} \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )}{a^4} \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 = 64.10, size = 735, normalized size = 4.57 Too large to display
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^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, 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 )}^2}{{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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