Optimal. Leaf size=189 \[ -i a^3 \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )+i a^3 \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{7}{6} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.31227, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6014, 6008, 266, 47, 63, 208, 5950} \[ -i a^3 \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )+i a^3 \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{7}{6} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6014
Rule 6008
Rule 266
Rule 47
Rule 63
Rule 208
Rule 5950
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^4} \, dx &=-\left (a^2 \int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^2} \, dx\right )+\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^4} \, dx\\ &=-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{3} a \int \frac{\sqrt{1-a^2 x^2}}{x^3} \, dx-a^2 \int \frac{\tanh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx+a^4 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)-i a^3 \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+i a^3 \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+\frac{1}{6} a \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x}}{x^2} \, dx,x,x^2\right )-a^3 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)-i a^3 \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+i a^3 \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{12} a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )-\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)-i a^3 \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+i a^3 \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+\frac{1}{6} a \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 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)+\frac{7}{6} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-i a^3 \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )+i a^3 \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 1.1868, size = 199, normalized size = 1.05 \[ -\frac{\left (1-a^2 x^2\right )^{3/2} \left (\log \left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right ) \left (\sinh \left (3 \tanh ^{-1}(a x)\right )-\frac{3 a x}{\sqrt{1-a^2 x^2}}\right )+8 \tanh ^{-1}(a x)+2 \sinh \left (2 \tanh ^{-1}(a x)\right )\right )}{24 x^3}+a^3 \left (-\left (i \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-i \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a x}+i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+\log \left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.223, size = 220, normalized size = 1.2 \begin{align*}{\frac{8\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) -ax-2\,{\it Artanh} \left ( ax \right ) }{6\,{x}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{7\,{a}^{3}}{6}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-1 \right ) }+{\frac{7\,{a}^{3}}{6}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-i{a}^{3}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\it Artanh} \left ( ax \right ) +i{a}^{3}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\it Artanh} \left ( ax \right ) -i{a}^{3}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +i{a}^{3}{\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^{4}}\,{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^{4}}, 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^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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