Optimal. Leaf size=37 \[ x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a} \]
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Rubi [A] time = 0.0385505, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6168, 807, 266, 63, 208} \[ x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6168
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=\sqrt{1-\frac{1}{a^2 x^2}} x-a \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0243888, size = 42, normalized size = 1.14 \[ x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{\log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.118, size = 98, normalized size = 2.7 \begin{align*}{\frac{ax+1}{a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( \sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }-a\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 ) \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 = 1.04831, size = 122, normalized size = 3.3 \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{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85872, size = 155, normalized size = 4.19 \begin{align*} \frac{{\left (a x + 1\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} \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 \sqrt{\frac{a x - 1}{a x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15539, size = 70, normalized size = 1.89 \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 )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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