Optimal. Leaf size=138 \[ -\frac {1}{2} a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )-a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}-\frac {a \tanh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.36, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5982, 5916, 266, 36, 29, 31, 5948, 5988, 5932, 6056, 6610} \[ -\frac {1}{2} a^2 \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}-\frac {a \tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5916
Rule 5932
Rule 5948
Rule 5982
Rule 5988
Rule 6056
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )+a \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+a^2 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+a^3 \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^4 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a \tanh ^{-1}(a x)}{x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [C] time = 0.37, size = 133, normalized size = 0.96 \[ -a^2 \left (-\log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )+\frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{2 a^2 x^2}-\tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{3} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)}{a x}-\tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac {i \pi ^3}{24}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{5} - x^{3}}, x\right ) \]
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 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.89, size = 1360, normalized size = 9.86 \[ \text {result too large to display} \]
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 \[ -\frac {a^{2} x^{2} \log \left (-a x + 1\right )^{3} + 3 \, {\left (a^{2} x^{2} \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{2}}{24 \, x^{2}} + \frac {1}{4} \, \int -\frac {\log \left (a x + 1\right )^{2} - {\left (a^{2} x^{2} + a x + {\left (a^{4} x^{4} + a^{3} x^{3} + 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{a^{2} x^{5} - x^{3}}\,{d x} \]
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 \[ -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^3\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{5} - x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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