Optimal. Leaf size=40 \[ \frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{1}{x}\right )+\frac{1}{2} a^2 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.036613, antiderivative size = 40, 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, 780, 216} \[ \frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{1}{x}\right )+\frac{1}{2} a^2 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (1-\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{1}{x}\right )+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{1}{x}\right )+\frac{1}{2} a^2 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0464496, size = 41, normalized size = 1.02 \[ \frac{a \left (\sqrt{1-\frac{1}{a^2 x^2}} (2 a x-1)+a x \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{2 x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.138, size = 260, normalized size = 6.5 \begin{align*} -{\frac{ax+1}{2\,{x}^{2}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -2\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+2\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +2\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-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}- \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\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 = 1.57412, size = 126, normalized size = 3.15 \begin{align*} -{\left (a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - \frac{3 \, a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + a \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{2 \,{\left (a x - 1\right )}}{a x + 1} + \frac{{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85432, size = 143, normalized size = 3.58 \begin{align*} -\frac{2 \, a^{2} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (2 \, a^{2} x^{2} + a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, x^{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*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1941, size = 212, normalized size = 5.3 \begin{align*} -a^{2} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right ) + \frac{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{3} a^{2} \mathrm{sgn}\left (a x + 1\right ) + 2 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} a{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) -{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )} a^{2} \mathrm{sgn}\left (a x + 1\right ) + 2 \, a{\left | a \right |} \mathrm{sgn}\left (a x + 1\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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