Optimal. Leaf size=243 \[ \frac{1}{16} a^6 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{1}{16} a^6 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{31 a^5 \sqrt{1-a^2 x^2}}{720 x}+\frac{19 a^3 \sqrt{1-a^2 x^2}}{360 x^3}-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}-\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac{7 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]
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Rubi [A] time = 0.772984, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6014, 6010, 6026, 271, 264, 6018} \[ \frac{1}{16} a^6 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{1}{16} a^6 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )+\frac{31 a^5 \sqrt{1-a^2 x^2}}{720 x}+\frac{19 a^3 \sqrt{1-a^2 x^2}}{360 x^3}-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}-\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac{7 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6014
Rule 6010
Rule 6026
Rule 271
Rule 264
Rule 6018
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^7} \, dx &=-\left (a^2 \int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^5} \, dx\right )+\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^7} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^6}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^4}-\frac{1}{5} \int \frac{\tanh ^{-1}(a x)}{x^7 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{5} a \int \frac{1}{x^6 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{3} a^2 \int \frac{\tanh ^{-1}(a x)}{x^5 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{3} a^3 \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{25 x^5}+\frac{a^3 \sqrt{1-a^2 x^2}}{9 x^3}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}-\frac{1}{30} a \int \frac{1}{x^6 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{6} a^2 \int \frac{\tanh ^{-1}(a x)}{x^5 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{12} a^3 \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{25} \left (4 a^3\right ) \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{4} a^4 \int \frac{\tanh ^{-1}(a x)}{x^3 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{9} \left (2 a^5\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{100 x^3}+\frac{2 a^5 \sqrt{1-a^2 x^2}}{9 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{7 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-\frac{1}{75} \left (2 a^3\right ) \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{24} a^3 \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{8} a^4 \int \frac{\tanh ^{-1}(a x)}{x^3 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{18} a^5 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{75} \left (8 a^5\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{8} a^5 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{8} a^6 \int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{19 a^3 \sqrt{1-a^2 x^2}}{360 x^3}-\frac{13 a^5 \sqrt{1-a^2 x^2}}{200 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{7 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac{1}{4} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\frac{1}{8} a^6 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{8} a^6 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{225} \left (4 a^5\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{36} a^5 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{16} a^5 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx-\frac{1}{16} a^6 \int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{30 x^5}+\frac{19 a^3 \sqrt{1-a^2 x^2}}{360 x^3}+\frac{31 a^5 \sqrt{1-a^2 x^2}}{720 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac{7 a^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac{a^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac{1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\frac{1}{16} a^6 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{16} a^6 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 6.80202, size = 474, normalized size = 1.95 \[ \frac{360 a^6 x^3 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-360 a^6 x^3 \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+360 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-360 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+64 a^7 x^4 \sinh ^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-128 a^5 x^2 \sinh ^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+328 a^5 x^2 \left (a^2 x^2-1\right ) \sinh ^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+360 a^4 x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \sinh ^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-\frac{192 a \left (a^2 x^2-1\right )^3 \sinh ^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{x^2}-\frac{960 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x) \sinh ^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{x^3}-3 a^7 x^4 \text{csch}^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+4 a^7 x^4 \text{csch}^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+82 a^7 x^4 \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-15 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{csch}^6\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+90 a^6 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )+64 a^3 \sinh ^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{5760 x^3 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.203, size = 184, normalized size = 0.8 \begin{align*} -{\frac{-31\,{x}^{5}{a}^{5}+45\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) -38\,{x}^{3}{a}^{3}-210\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +24\,ax+120\,{\it Artanh} \left ( ax \right ) }{720\,{x}^{6}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{a}^{6}{\it Artanh} \left ( ax \right ) }{16}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{a}^{6}}{16}{\it polylog} \left ( 2,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{a}^{6}{\it Artanh} \left ( ax \right ) }{16}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{a}^{6}}{16}{\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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}{x^{7}}\,{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^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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