Optimal. Leaf size=75 \[ \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}-\frac {\tanh ^{-1}(a x)^2}{a^3}+\frac {2 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {x \tanh ^{-1}(a x)^2}{a^2} \]
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Rubi [A] time = 0.17, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5980, 5910, 5984, 5918, 2402, 2315, 5948} \[ \frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}-\frac {x \tanh ^{-1}(a x)^2}{a^2}-\frac {\tanh ^{-1}(a x)^2}{a^3}+\frac {2 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5910
Rule 5918
Rule 5948
Rule 5980
Rule 5984
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx &=-\frac {\int \tanh ^{-1}(a x)^2 \, dx}{a^2}+\frac {\int \frac {\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {x \tanh ^{-1}(a x)^2}{a^2}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}+\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(a x)^2}{a^3}-\frac {x \tanh ^{-1}(a x)^2}{a^2}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)^2}{a^3}-\frac {x \tanh ^{-1}(a x)^2}{a^2}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}+\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3}-\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)^2}{a^3}-\frac {x \tanh ^{-1}(a x)^2}{a^2}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}+\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^3}\\ &=-\frac {\tanh ^{-1}(a x)^2}{a^3}-\frac {x \tanh ^{-1}(a x)^2}{a^2}+\frac {\tanh ^{-1}(a x)^3}{3 a^3}+\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 59, normalized size = 0.79 \[ -\frac {\text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-\frac {1}{3} \tanh ^{-1}(a x) \left (-3 a x \tanh ^{-1}(a x)+\left (\tanh ^{-1}(a x)+3\right ) \tanh ^{-1}(a x)+6 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )}{a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{2} \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.55, size = 5573, normalized size = 74.31 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 200, normalized size = 2.67 \[ -\frac {1}{2} \, {\left (\frac {2 \, x}{a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} - \frac {\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a}}{24 \, a^{2}} + \frac {{\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \operatorname {artanh}\left (a x\right )}{4 \, a^{3}} \]
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 \[ -\int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^2}{a^2\,x^2-1} \,d x \]
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 \[ - \int \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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