Optimal. Leaf size=95 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.0264132, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5950} \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 5950
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{a}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0733041, size = 76, normalized size = 0.8 \[ -\frac{i \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.247, size = 366, normalized size = 3.9 \begin{align*}{\frac{{\frac{i}{2}}{\it Artanh} \left ( ax \right ) }{a}\ln \left ({-i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i{\it Artanh} \left ( ax \right ) }{a}\ln \left ( \left ( 1-i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1+i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) }-{\frac{{\frac{i}{2}}{\it Artanh} \left ( ax \right ) }{a}\ln \left ({i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{i{\it Artanh} \left ( ax \right ) }{a}\ln \left ( \left ( 1+i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1-i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) }+{\frac{i}{a}\ln \left ( \left ( 1-i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1+i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) \ln \left ({-i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i}{a}\ln \left ( \left ( 1+i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1-i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) \ln \left ({i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{i}{a}{\it dilog} \left ({-i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i}{a}{\it dilog} \left ({i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{a^{2} x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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