Optimal. Leaf size=34 \[ -\frac{2}{3} a \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{2}{3 a^2 x^3}-\frac{1}{x} \]
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Rubi [A] time = 0.203554, antiderivative size = 54, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6338, 6715, 2117} \[ -\frac{1}{6} a \left (\sqrt{\frac{1}{a^2 x^2}+1}+\frac{1}{a x}\right )^3-\frac{1}{2} a \sqrt{\frac{1}{a^2 x^2}+1}-\frac{1}{2 x} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6715
Rule 2117
Rubi steps
\begin{align*} \int \frac{e^{2 \text{csch}^{-1}(a x)}}{x^2} \, dx &=\int \frac{\left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2}{x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \left (\frac{x}{a}+\sqrt{1+\frac{x^2}{a^2}}\right )^2 \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{2} a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )\right )\\ &=-\frac{1}{2} a \sqrt{1+\frac{1}{a^2 x^2}}-\frac{1}{6} a \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^3-\frac{1}{2 x}\\ \end{align*}
Mathematica [A] time = 0.04226, size = 46, normalized size = 1.35 \[ -\frac{3 a^2 x^2+2 a x \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+2}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.19, size = 63, normalized size = 1.9 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{{a}^{2}}{x}}-{\frac{1}{3\,{x}^{3}}} \right ) }-{\frac{2\,{a}^{2}{x}^{2}+2}{3\,a{x}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}-{\frac{1}{3\,{x}^{3}{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99973, size = 38, normalized size = 1.12 \begin{align*} -\frac{2}{3} \, a{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - \frac{1}{x} - \frac{2}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.4791, size = 127, normalized size = 3.74 \begin{align*} -\frac{2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} + a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 2}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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