Optimal. Leaf size=179 \[ \frac{3}{2} i a \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{3}{2} i a \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+3 a \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.284488, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6014, 6008, 266, 63, 208, 5950, 5942} \[ \frac{3}{2} i a \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{3}{2} i a \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+3 a \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6014
Rule 6008
Rule 266
Rule 63
Rule 208
Rule 5950
Rule 5942
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^2} \, dx &=-\left (a^2 \int \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\right )+\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^2} \, dx\\ &=-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{2} a^2 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx-a^2 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\int \frac{\tanh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)+\frac{3}{2} i a \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} i a \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+a \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)+\frac{3}{2} i a \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} i a \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)+\frac{3}{2} i a \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} i a \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a}\\ &=-\frac{1}{2} a \sqrt{1-a^2 x^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{1}{2} a^2 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\frac{3}{2} i a \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} i a \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 0.578006, size = 168, normalized size = 0.94 \[ \frac{1}{2} \left (3 i a \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-3 i a \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )-a \sqrt{1-a^2 x^2}+a^2 (-x) \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+3 i a \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-3 i a \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+2 a \log \left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.221, size = 205, normalized size = 1.2 \begin{align*} -{\frac{{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +ax+2\,{\it Artanh} \left ( ax \right ) }{2\,x}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+a\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-1 \right ) -a\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +{\frac{3\,i}{2}}a\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\it Artanh} \left ( ax \right ) -{\frac{3\,i}{2}}a\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\it Artanh} \left ( ax \right ) +{\frac{3\,i}{2}}a{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{\frac{3\,i}{2}}a{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}{x^{2}}\,{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{{\left (a^{2} x^{2} - 1\right )} \sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{x^{2}}, 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{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \operatorname{atanh}{\left (a x \right )}}{x^{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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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