Optimal. Leaf size=292 \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{128 a^5}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{128 a^5}+\frac{\left (1-a^2 x^2\right )^{7/2}}{56 a^5}-\frac{3 \left (1-a^2 x^2\right )^{5/2}}{80 a^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{192 a^5}+\frac{3 \sqrt{1-a^2 x^2}}{128 a^5}-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{128 a^4}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{64 a^5} \]
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Rubi [A] time = 0.817538, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6014, 6010, 6016, 266, 43, 261, 5950} \[ -\frac{3 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{128 a^5}+\frac{3 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{128 a^5}+\frac{\left (1-a^2 x^2\right )^{7/2}}{56 a^5}-\frac{3 \left (1-a^2 x^2\right )^{5/2}}{80 a^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{192 a^5}+\frac{3 \sqrt{1-a^2 x^2}}{128 a^5}-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{128 a^4}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{64 a^5} \]
Antiderivative was successfully verified.
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Rule 6014
Rule 6010
Rule 6016
Rule 266
Rule 43
Rule 261
Rule 5950
Rubi steps
\begin{align*} \int x^4 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^6 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\right )+\int x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\\ &=\frac{1}{6} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{1}{6} \int \frac{x^4 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{6} a \int \frac{x^5}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{8} a^2 \int \frac{x^6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\frac{1}{8} a^3 \int \frac{x^7}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{24 a^2}+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{5}{48} \int \frac{x^4 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\frac{\int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a^2}+\frac{\int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx}{24 a}-\frac{1}{48} a \int \frac{x^5}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{12} a \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )+\frac{1}{16} a^3 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{16 a^4}+\frac{\int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{16 a^3}-\frac{5 \int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{64 a^2}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )}{48 a}-\frac{5 \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx}{192 a}-\frac{1}{96} a \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )-\frac{1}{12} a \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{1-a^2 x}}-\frac{2 \sqrt{1-a^2 x}}{a^4}+\frac{\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )+\frac{1}{16} a^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^6 \sqrt{1-a^2 x}}-\frac{3 \sqrt{1-a^2 x}}{a^6}+\frac{3 \left (1-a^2 x\right )^{3/2}}{a^6}-\frac{\left (1-a^2 x\right )^{5/2}}{a^6}\right ) \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{48 a^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{72 a^5}-\frac{\left (1-a^2 x^2\right )^{5/2}}{24 a^5}+\frac{\left (1-a^2 x^2\right )^{7/2}}{56 a^5}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{128 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{8 a^5}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{16 a^5}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{16 a^5}-\frac{5 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{128 a^4}-\frac{5 \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{128 a^3}-\frac{5 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )}{384 a}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{48 a}-\frac{1}{96} a \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{1-a^2 x}}-\frac{2 \sqrt{1-a^2 x}}{a^4}+\frac{\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{384 a^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{72 a^5}-\frac{3 \left (1-a^2 x^2\right )^{5/2}}{80 a^5}+\frac{\left (1-a^2 x^2\right )^{7/2}}{56 a^5}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{128 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{64 a^5}-\frac{3 i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{128 a^5}+\frac{3 i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{128 a^5}-\frac{5 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{384 a}\\ &=\frac{3 \sqrt{1-a^2 x^2}}{128 a^5}+\frac{\left (1-a^2 x^2\right )^{3/2}}{192 a^5}-\frac{3 \left (1-a^2 x^2\right )^{5/2}}{80 a^5}+\frac{\left (1-a^2 x^2\right )^{7/2}}{56 a^5}-\frac{3 x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{128 a^4}-\frac{x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{64 a^2}+\frac{3}{16} x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} a^2 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{3 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{64 a^5}-\frac{3 i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{128 a^5}+\frac{3 i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{128 a^5}\\ \end{align*}
Mathematica [A] time = 1.33109, size = 272, normalized size = 0.93 \[ \frac{-315 i \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )+315 i \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )-240 a^6 x^6 \sqrt{1-a^2 x^2}+216 a^4 x^4 \sqrt{1-a^2 x^2}+218 a^2 x^2 \sqrt{1-a^2 x^2}+121 \sqrt{1-a^2 x^2}-1680 a^7 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+2520 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-210 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-315 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-315 i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+315 i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )}{13440 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.213, size = 215, normalized size = 0.7 \begin{align*} -{\frac{1680\,{\it Artanh} \left ( ax \right ){x}^{7}{a}^{7}+240\,{x}^{6}{a}^{6}-2520\,{\it Artanh} \left ( ax \right ){x}^{5}{a}^{5}-216\,{x}^{4}{a}^{4}+210\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -218\,{a}^{2}{x}^{2}+315\,ax{\it Artanh} \left ( ax \right ) -121}{13440\,{a}^{5}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\frac{3\,i}{128}}{\it Artanh} \left ( ax \right ) }{{a}^{5}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{3\,i}{128}}{\it Artanh} \left ( ax \right ) }{{a}^{5}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{3\,i}{128}}}{{a}^{5}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{3\,i}{128}}}{{a}^{5}}{\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{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{4} \operatorname{artanh}\left (a x\right )\,{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 (-{\left (a^{2} x^{6} - x^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right ), 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{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{4} \operatorname{artanh}\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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