Optimal. Leaf size=90 \[ \frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3} \]
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Rubi [A] time = 0.100033, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6169, 835, 807, 266, 63, 208} \[ \frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 6169
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{3}{a}-\frac{2 x}{a^2}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{\frac{4}{a^2}-\frac{3 x}{a^3}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^3}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^3}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 a}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0486425, size = 60, normalized size = 0.67 \[ \frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-3 a x+4\right )-3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{6 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.128, size = 173, normalized size = 1.9 \begin{align*} -{\frac{ax+1}{6\,{a}^{3}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 3\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-3\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a-6\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+6\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01539, size = 224, normalized size = 2.49 \begin{align*} -\frac{1}{6} \, a{\left (\frac{2 \,{\left (9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 4 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{4}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92685, size = 198, normalized size = 2.2 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} - a^{2} x^{2} + a x + 4\right )} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{6 \, a^{3}} \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 x^{2} \sqrt{\frac{a x - 1}{a x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19654, size = 116, normalized size = 1.29 \begin{align*} \frac{1}{6} \, \sqrt{a^{2} x^{2} - 1}{\left (x{\left (\frac{2 \, x \mathrm{sgn}\left (a x + 1\right )}{a} - \frac{3 \, \mathrm{sgn}\left (a x + 1\right )}{a^{2}}\right )} + \frac{4 \, \mathrm{sgn}\left (a x + 1\right )}{a^{3}}\right )} + \frac{\log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{2 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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