Optimal. Leaf size=47 \[ \frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^2}+1}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{2 a^2}+\frac{x}{a} \]
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Rubi [A] time = 0.0264616, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6336, 8, 266, 47, 63, 208} \[ \frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^2}+1}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{2 a^2}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 8
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}(a x)} x \, dx &=\frac{\int 1 \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} x \, dx\\ &=\frac{x}{a}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{x}{a}+\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^2}} x^2-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^2}\\ &=\frac{x}{a}+\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^2}} x^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )\\ &=\frac{x}{a}+\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^2}} x^2+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0303789, size = 47, normalized size = 1. \[ \frac{a x \left (a x \sqrt{\frac{1}{a^2 x^2}+1}+2\right )+\log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.194, size = 85, normalized size = 1.8 \begin{align*}{\frac{x}{2\,{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( x\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+\ln \left ( x+\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}}+{\frac{x}{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967232, size = 105, normalized size = 2.23 \begin{align*} \frac{x}{a} + \frac{\sqrt{\frac{1}{a^{2} x^{2}} + 1}}{2 \,{\left (a^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - a^{2}\right )}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{4 \, a^{2}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.57032, size = 140, normalized size = 2.98 \begin{align*} \frac{a^{2} x^{2} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 2 \, a x - \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.4527, size = 29, normalized size = 0.62 \begin{align*} \frac{x \sqrt{a^{2} x^{2} + 1}}{2 a} + \frac{x}{a} + \frac{\operatorname{asinh}{\left (a x \right )}}{2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17104, size = 70, normalized size = 1.49 \begin{align*} \frac{\sqrt{a^{2} x^{2} + 1} x{\left | a \right |} \mathrm{sgn}\left (x\right )}{2 \, a^{2}} + \frac{x}{a} - \frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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