Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
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Rubi [A] time = 0.100635, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6175, 6178, 266, 63, 208} \[ -\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx &=-\frac{\int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right ) x} \, dx}{a c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c}\\ &=-\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}\\ &=-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c}\\ \end{align*}
Mathematica [A] time = 0.0383503, size = 34, normalized size = 1.48 \[ -\frac{\log \left (a x \left (\sqrt{\frac{a^2 x^2-1}{a^2 x^2}}+1\right )\right )}{a c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.181, size = 76, normalized size = 3.3 \begin{align*} -{\frac{ax+1}{c}\sqrt{{\frac{ax-1}{ax+1}}}\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \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 = 1.01962, size = 74, normalized size = 3.22 \begin{align*} -a{\left (\frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54787, size = 111, normalized size = 4.83 \begin{align*} -\frac{\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} \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{\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x - 1}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17058, size = 45, normalized size = 1.96 \begin{align*} \frac{\log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{c{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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