Optimal. Leaf size=118 \[ \frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3} \]
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Rubi [A] time = 1.07141, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6169, 6742, 651, 271, 264, 266, 51, 63, 208} \[ \frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 6169
Rule 6742
Rule 651
Rule 271
Rule 264
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^2}{x^4 \left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{4}{a^3 (a-x) \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{x^4 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{a x^3 \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^2 x^2 \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^3 x \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}} x}{a^2}+\frac{1}{3} \sqrt{1-\frac{1}{a^2 x^2}} x^3-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{14 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}+\frac{3 \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{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^3}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{14 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}+\frac{3 \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{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^3}+\frac{3 \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{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left (a-\frac{1}{x}\right )}+\frac{14 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^2}+\frac{3 \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{11 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0800755, size = 75, normalized size = 0.64 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3+7 a^2 x^2+19 a x-52\right )}{a x-1}+33 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{6 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.166, size = 471, normalized size = 4. \begin{align*}{\frac{1}{6\,{a}^{3} \left ( ax+1 \right ) } \left ( 9\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+2\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-18\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-4\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa+42\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+42\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+9\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+18\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-10\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-84\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-84\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a+42\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+42\,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 ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00047, size = 246, normalized size = 2.08 \begin{align*} -\frac{1}{6} \, a{\left (\frac{2 \,{\left (\frac{75 \,{\left (a x - 1\right )}}{a x + 1} - \frac{88 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{33 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 12\right )}}{a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 3 \, a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 3 \, a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{4}} + \frac{33 \, \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 = 1.98187, size = 267, normalized size = 2.26 \begin{align*} \frac{33 \,{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 33 \,{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{4} x^{4} + 9 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 33 \, a x - 52\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \,{\left (a^{4} x - a^{3}\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{x^{2}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17711, size = 231, normalized size = 1.96 \begin{align*} \frac{1}{6} \, a{\left (\frac{33 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{4}} - \frac{33 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{4}} - \frac{24}{a^{4} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{2 \,{\left (\frac{52 \,{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{21 \,{\left (a x - 1\right )}^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 39 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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