Optimal. Leaf size=38 \[ -\sqrt{\frac{1}{a^2 x^2}+1}+\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )-\frac{1}{a x} \]
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Rubi [A] time = 0.034632, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6336, 30, 266, 50, 63, 208} \[ -\sqrt{\frac{1}{a^2 x^2}+1}+\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(a x)}}{x} \, dx &=\frac{\int \frac{1}{x^2} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x} \, dx\\ &=-\frac{1}{a x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\sqrt{1+\frac{1}{a^2 x^2}}-\frac{1}{a x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\sqrt{1+\frac{1}{a^2 x^2}}-\frac{1}{a x}-a^2 \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )\\ &=-\sqrt{1+\frac{1}{a^2 x^2}}-\frac{1}{a x}+\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0275416, size = 42, normalized size = 1.11 \[ -\sqrt{\frac{1}{a^2 x^2}+1}+\log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )-\frac{1}{a x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.18, size = 107, normalized size = 2.8 \begin{align*}{\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{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{x}^{2}{a}^{2}+\ln \left ( x+\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}} \right ) x \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}}-{\frac{1}{ax}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00055, size = 73, normalized size = 1.92 \begin{align*} -\sqrt{\frac{1}{a^{2} x^{2}} + 1} - \frac{1}{a x} + \frac{1}{2} \, \log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.01638, size = 142, normalized size = 3.74 \begin{align*} -\frac{a x \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right ) + a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + a x + 1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.26471, size = 41, normalized size = 1.08 \begin{align*} - \frac{a x}{\sqrt{a^{2} x^{2} + 1}} + \operatorname{asinh}{\left (a x \right )} - \frac{1}{a x} - \frac{1}{a x \sqrt{a^{2} x^{2} + 1}} \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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