Optimal. Leaf size=161 \[ \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}-\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^4}+\frac {1}{4 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )} \]
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Rubi [A] time = 0.29, 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 = {6028, 5984, 5918, 5948, 6058, 6610, 5994, 5956, 261} \[ \frac {\text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^4}+\frac {1}{4 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{2 a^4 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{3 a^4}-\frac {\tanh ^{-1}(a x)^2}{4 a^4}-\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^4} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5918
Rule 5948
Rule 5956
Rule 5984
Rule 5994
Rule 6028
Rule 6058
Rule 6610
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 = 0.18, size = 103, normalized size = 0.64 \[ \frac {\tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\frac {1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-\frac {1}{4} \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )+\frac {1}{8} \left (2 \tanh ^{-1}(a x)^2+1\right ) \cosh \left (2 \tanh ^{-1}(a x)\right )}{a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \operatorname {artanh}\left (a x\right )^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x^{3} \operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.70, size = 907, normalized size = 5.63 \[ \text {result too large to display} \]
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 \[ -\frac {3}{4} \, a^{3} \int \frac {x^{3} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\,{d x} - \frac {1}{4} \, a^{2} \int \frac {x^{2} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\,{d x} - \frac {1}{32} \, {\left (a {\left (\frac {2}{a^{7} x - a^{6}} - \frac {\log \left (a x + 1\right )}{a^{6}} + \frac {\log \left (a x - 1\right )}{a^{6}}\right )} + \frac {4 \, \log \left (-a x + 1\right )}{a^{7} x^{2} - a^{5}}\right )} a + \frac {1}{4} \, a \int \frac {x \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\,{d x} + \frac {{\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )^{3} + 3 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) - 1\right )} \log \left (-a x + 1\right )^{2}}{24 \, {\left (a^{6} x^{2} - a^{4}\right )}} + \frac {1}{4} \, \int \frac {a^{3} x^{3} \log \left (a x + 1\right )^{2}}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\,{d x} + \frac {1}{4} \, \int \frac {\log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\,{d x} + \frac {1}{4} \, \int \frac {\log \left (-a x + 1\right )}{a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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