Optimal. Leaf size=112 \[ \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]
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Rubi [A] time = 0.185961, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6030, 6018, 5994, 191} \[ \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6030
Rule 6018
Rule 5994
Rule 191
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-a \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 0.178578, size = 97, normalized size = 0.87 \[ \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )-\frac{a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.253, size = 157, normalized size = 1.4 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) -1}{2\,ax-2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{\it Artanh} \left ( ax \right ) +1}{2\,ax+2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\it Artanh} \left ( ax \right ) \ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{\it polylog} \left ( 2,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +{\it Artanh} \left ( ax \right ) \ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 2,{(ax+1){\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 )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}\,{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^{4} x^{5} - 2 \, a^{2} x^{3} + x}, 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 )}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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