Optimal. Leaf size=43 \[ \frac{2 x \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{2 \log (x)}{a^2}-\frac{2 \text{csch}^{-1}(a x)}{a^2}+\frac{x^2}{2} \]
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Rubi [A] time = 0.15791, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6338, 6742, 242, 277, 215} \[ \frac{2 x \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{2 \log (x)}{a^2}-\frac{2 \text{csch}^{-1}(a x)}{a^2}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6338
Rule 6742
Rule 242
Rule 277
Rule 215
Rubi steps
\begin{align*} \int e^{2 \text{csch}^{-1}(a x)} x \, dx &=\int \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2 x \, dx\\ &=\int \left (\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a}+\frac{2}{a^2 x}+x\right ) \, dx\\ &=\frac{x^2}{2}+\frac{2 \log (x)}{a^2}+\frac{2 \int \sqrt{1+\frac{1}{a^2 x^2}} \, dx}{a}\\ &=\frac{x^2}{2}+\frac{2 \log (x)}{a^2}-\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{2 \sqrt{1+\frac{1}{a^2 x^2}} x}{a}+\frac{x^2}{2}+\frac{2 \log (x)}{a^2}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3}\\ &=\frac{2 \sqrt{1+\frac{1}{a^2 x^2}} x}{a}+\frac{x^2}{2}-\frac{2 \text{csch}^{-1}(a x)}{a^2}+\frac{2 \log (x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0472147, size = 44, normalized size = 1.02 \[ \frac{a x \left (4 \sqrt{\frac{1}{a^2 x^2}+1}+a x\right )-4 \sinh ^{-1}\left (\frac{1}{a x}\right )+4 \log (x)}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.179, size = 120, normalized size = 2.8 \begin{align*}{\frac{{x}^{2}}{2}}+2\,{\frac{\ln \left ( x \right ) }{{a}^{2}}}+2\,{\frac{x}{{a}^{3}\sqrt{{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{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964158, size = 101, normalized size = 2.35 \begin{align*} \frac{1}{2} \, x^{2} + \frac{2 \, x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - \frac{\log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{a} + \frac{\log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{a}}{a} + \frac{2 \, \log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.49821, size = 235, normalized size = 5.47 \begin{align*} \frac{a^{2} x^{2} + 4 \, a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - 4 \, \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + 4 \, \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + 4 \, \log \left (x\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.36319, size = 63, normalized size = 1.47 \begin{align*} \frac{x^{2}}{2} + \frac{2 x}{a \sqrt{1 + \frac{1}{a^{2} x^{2}}}} + \frac{2 \log{\left (x \right )}}{a^{2}} - \frac{2 \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{a^{2}} + \frac{2}{a^{3} 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 [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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