Optimal. Leaf size=60 \[ x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a+\frac{1}{x}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a} \]
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Rubi [A] time = 0.778659, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {6168, 6742, 264, 266, 63, 208, 651} \[ x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a+\frac{1}{x}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6168
Rule 6742
Rule 264
Rule 266
Rule 63
Rule 208
Rule 651
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^2 \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{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3}{a x \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a (a+x) \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a+\frac{1}{x}}+\sqrt{1-\frac{1}{a^2 x^2}} x+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \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+\frac{1}{x}}+\sqrt{1-\frac{1}{a^2 x^2}} x-(3 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a+\frac{1}{x}}+\sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0466342, size = 54, normalized size = 0.9 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+5)}{a x+1}-\frac{3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.129, size = 248, normalized size = 4.1 \begin{align*} -{\frac{1}{ \left ( ax-1 \right ) a} \left ( 3\,\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}-3\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+6\,\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}+2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-6\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+3\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -3\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\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 = 1.03164, size = 150, normalized size = 2.5 \begin{align*} -a{\left (\frac{2 \, \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{4 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70101, size = 161, normalized size = 2.68 \begin{align*} \frac{{\left (a x + 5\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 )}{a} \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 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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