Optimal. Leaf size=74 \[ -\frac{x \left (a-\frac{1}{x}\right )}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
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Rubi [A] time = 0.101428, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6177, 823, 807, 266, 63, 208} \[ -\frac{x \left (a-\frac{1}{x}\right )}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 823
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{c-\frac{c x}{a}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{\left (a-\frac{1}{x}\right ) x}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{2 c}{a^2}-\frac{c x}{a^3}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{\left (a-\frac{1}{x}\right ) x}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{\left (a-\frac{1}{x}\right ) x}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c^2}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{\left (a-\frac{1}{x}\right ) x}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^2}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{\left (a-\frac{1}{x}\right ) x}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.0372498, size = 69, normalized size = 0.93 \[ \frac{a^2 x^2-a x \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+a x-2}{a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.133, size = 250, normalized size = 3.4 \begin{align*} -{\frac{1}{2\,a{c}^{2} \left ( ax-1 \right ) } \left ( 2\,\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}+4\,\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}+ \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-6\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+2\,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 [A] time = 1.03599, size = 169, normalized size = 2.28 \begin{align*} -a{\left (\frac{2 \, \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2} c^{2}}{a x + 1} - a^{2} c^{2}} + \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} - \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93056, size = 163, normalized size = 2.2 \begin{align*} \frac{{\left (a x + 2\right )} \sqrt{\frac{a x - 1}{a x + 1}} - \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a c^{2}} \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*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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