Optimal. Leaf size=24 \[ \frac{\log (x)}{a}+x e^{\text{csch}^{-1}(a x)}-\frac{\text{csch}^{-1}(a x)}{a} \]
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Rubi [A] time = 0.015288, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6331, 29, 242, 277, 215} \[ x \sqrt{\frac{1}{a^2 x^2}+1}+\frac{\log (x)}{a}-\frac{\text{csch}^{-1}(a x)}{a} \]
Warning: Unable to verify antiderivative.
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Rule 6331
Rule 29
Rule 242
Rule 277
Rule 215
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}(a x)} \, dx &=\frac{\int \frac{1}{x} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} \, dx\\ &=\frac{\log (x)}{a}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{1+\frac{1}{a^2 x^2}} x+\frac{\log (x)}{a}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=\sqrt{1+\frac{1}{a^2 x^2}} x-\frac{\text{csch}^{-1}(a x)}{a}+\frac{\log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0147928, size = 35, normalized size = 1.46 \[ \frac{a x \sqrt{\frac{1}{a^2 x^2}+1}+\log (a x)-\sinh ^{-1}\left (\frac{1}{a x}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.169, size = 113, normalized size = 4.7 \begin{align*} -{\frac{x}{{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( -\sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{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 ) \right ){\frac{1}{\sqrt{{a}^{-2}}}}{\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}}+{\frac{\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00209, size = 86, normalized size = 3.58 \begin{align*} x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - \frac{\log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{2 \, a} + \frac{\log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.77596, size = 203, normalized size = 8.46 \begin{align*} \frac{a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + \log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.01817, size = 48, normalized size = 2. \begin{align*} \frac{x}{\sqrt{1 + \frac{1}{a^{2} x^{2}}}} + \frac{\log{\left (x \right )}}{a} - \frac{\operatorname{asinh}{\left (\frac{1}{a x} \right )}}{a} + \frac{1}{a^{2} x \sqrt{1 + \frac{1}{a^{2} x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16093, size = 82, normalized size = 3.42 \begin{align*} \frac{{\left (2 \, \sqrt{a^{2} x^{2} + 1} - \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) + \log \left (\sqrt{a^{2} x^{2} + 1} - 1\right )\right )}{\left | a \right |} \mathrm{sgn}\left (x\right )}{2 \, a^{2}} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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