Optimal. Leaf size=25 \[ a \left (-\csc ^{-1}(a x)\right )-a \sqrt{1-\frac{1}{a^2 x^2}} \]
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Rubi [A] time = 0.0267834, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6169, 641, 216} \[ a \left (-\csc ^{-1}(a x)\right )-a \sqrt{1-\frac{1}{a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6169
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-a \sqrt{1-\frac{1}{a^2 x^2}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-a \sqrt{1-\frac{1}{a^2 x^2}}-a \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0201107, size = 26, normalized size = 1.04 \[ -a \left (\sqrt{1-\frac{1}{a^2 x^2}}+\sin ^{-1}\left (\frac{1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.128, size = 220, normalized size = 8.8 \begin{align*}{\frac{ax+1}{x}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2}-ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-\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 ) x{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 = 1.5047, size = 74, normalized size = 2.96 \begin{align*} -2 \, a{\left (\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\frac{a x - 1}{a x + 1} + 1} - \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86385, size = 112, normalized size = 4.48 \begin{align*} \frac{2 \, a x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1525, size = 84, normalized size = 3.36 \begin{align*} 2 \, a \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right ) - \frac{2 \,{\left | a \right |} \mathrm{sgn}\left (a x + 1\right )}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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