Optimal. Leaf size=40 \[ -\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{2 x}-\frac{1}{2 a x^2}-\frac{1}{2} a \text{csch}^{-1}(a x) \]
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Rubi [A] time = 0.028842, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6336, 30, 335, 195, 215} \[ -\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{2 x}-\frac{1}{2 a x^2}-\frac{1}{2} a \text{csch}^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 335
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(a x)}}{x^2} \, dx &=\frac{\int \frac{1}{x^3} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^2} \, dx\\ &=-\frac{1}{2 a x^2}-\operatorname{Subst}\left (\int \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2 a x^2}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2 a x^2}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a \text{csch}^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0272924, size = 43, normalized size = 1.08 \[ -\frac{a x \sqrt{\frac{1}{a^2 x^2}+1}+a^2 x^2 \sinh ^{-1}\left (\frac{1}{a x}\right )+1}{2 a x^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.188, size = 145, normalized size = 3.6 \begin{align*} -{\frac{1}{2\,x}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ({a}^{2} \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{-2}}-\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}\sqrt{{a}^{-2}}{x}^{2}{a}^{2}+\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ){x}^{2} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}{\frac{1}{\sqrt{{a}^{-2}}}}}-{\frac{1}{2\,a{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0079, size = 116, normalized size = 2.9 \begin{align*} -\frac{a^{2} x \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{2 \,{\left (a^{2} x^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - 1\right )}} - \frac{1}{4} \, a \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) + \frac{1}{4} \, a \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right ) - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.94851, size = 232, normalized size = 5.8 \begin{align*} -\frac{a^{2} x^{2} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.07544, size = 36, normalized size = 0.9 \begin{align*} - \frac{a \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{2} - \frac{\sqrt{1 + \frac{1}{a^{2} x^{2}}}}{2 x} - \frac{1}{2 a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13513, size = 111, normalized size = 2.78 \begin{align*} -\frac{a^{4}{\left | a \right |} \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) \mathrm{sgn}\left (x\right ) - a^{4}{\left | a \right |} \log \left (\sqrt{a^{2} x^{2} + 1} - 1\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \,{\left (\sqrt{a^{2} x^{2} + 1} a^{4}{\left | a \right |} \mathrm{sgn}\left (x\right ) + a^{5}\right )}}{a^{2} x^{2}}}{4 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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