Optimal. Leaf size=120 \[ -\frac {2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^2}+\frac {2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^2}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5994, 5950} \[ -\frac {2 i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^2}+\frac {2 i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^2}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 5950
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^2}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^2}-\frac {2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^2}+\frac {2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 104, normalized size = 0.87 \[ -\frac {\tanh ^{-1}(a x) \left (\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+2 i \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )+2 i \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-2 i \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )}{a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int \frac {x \operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 151, normalized size = 1.26 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right )^{2}}{a^{2}}-\frac {2 i \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{a^{2}}+\frac {2 i \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{a^{2}}-\frac {2 i \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}}+\frac {2 i \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2}} \]
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 \[ \int \frac {x \operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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