Optimal. Leaf size=20 \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+\csc ^{-1}(a x) \]
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Rubi [A] time = 0.0475338, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6169, 844, 216, 266, 63, 208} \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+\csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\csc ^{-1}(a x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )\\ &=\csc ^{-1}(a x)+a^2 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\csc ^{-1}(a x)+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0145184, size = 34, normalized size = 1.7 \[ \log \left (x \left (\sqrt{\frac{a^2 x^2-1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac{1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.125, size = 130, normalized size = 6.5 \begin{align*}{(ax+1)\sqrt{{\frac{ax-1}{ax+1}}} \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \sqrt{{a}^{2}}+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 ) -\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 [B] time = 1.59906, size = 95, normalized size = 4.75 \begin{align*} -a{\left (\frac{2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a} + \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84833, size = 151, normalized size = 7.55 \begin{align*} -2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) \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 \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16105, size = 80, normalized size = 4. \begin{align*} -2 \, \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right ) - \frac{a \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 |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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