Optimal. Leaf size=73 \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}-\frac{a x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left (a-\frac{1}{x}\right )}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
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Rubi [A] time = 0.11113, antiderivative size = 73, 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, 857, 807, 266, 63, 208} \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}-\frac{a x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left (a-\frac{1}{x}\right )}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 857
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (c-\frac{c x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{a \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \left (a-\frac{1}{x}\right )}+\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{a \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \left (a-\frac{1}{x}\right )}-\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{a \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \left (a-\frac{1}{x}\right )}-\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{a \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \left (a-\frac{1}{x}\right )}+\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{a \sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \left (a-\frac{1}{x}\right )}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.0348603, size = 69, normalized size = 0.95 \[ \frac{a^2 x^2+a x \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-a x-2}{a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.137, size = 256, normalized size = 3.5 \begin{align*}{\frac{ax+1}{2\,a{c}^{2} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{ax-1}{ax+1}}} \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 ){\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 = 1.04304, size = 162, normalized size = 2.22 \begin{align*} -a{\left (\frac{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} - \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}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90645, size = 220, normalized size = 3.01 \begin{align*} \frac{{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) -{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a^{2} x^{2} - a x - 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \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*} \frac{a^{2} \int \frac{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \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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