Optimal. Leaf size=53 \[ \frac{2 \left (a-\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 a \sqrt{1-\frac{1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \]
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Rubi [A] time = 0.0775462, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6169, 853, 669, 641, 216} \[ \frac{2 \left (a-\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 a \sqrt{1-\frac{1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 853
Rule 669
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 \left (a-\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=3 a \sqrt{1-\frac{1}{a^2 x^2}}+\frac{2 \left (a-\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=3 a \sqrt{1-\frac{1}{a^2 x^2}}+\frac{2 \left (a-\frac{1}{x}\right )^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 a \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0673987, size = 41, normalized size = 0.77 \[ \frac{a \sqrt{1-\frac{1}{a^2 x^2}} (5 a x+1)}{a x+1}+3 a \sin ^{-1}\left (\frac{1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.131, size = 592, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5446, size = 97, normalized size = 1.83 \begin{align*} 2 \, a{\left (2 \, \sqrt{\frac{a x - 1}{a x + 1}} + \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\frac{a x - 1}{a x + 1} + 1} - 3 \, \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 [A] time = 1.51105, size = 116, normalized size = 2.19 \begin{align*} -\frac{6 \, a x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (5 \, 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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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