Optimal. Leaf size=47 \[ -\frac{2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{a}-\frac{2}{a^2 x}+x \]
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Rubi [A] time = 0.0718435, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6333, 6742, 266, 50, 63, 208} \[ -\frac{2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{a}-\frac{2}{a^2 x}+x \]
Antiderivative was successfully verified.
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Rule 6333
Rule 6742
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \text{csch}^{-1}(a x)} \, dx &=\int \left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2 \, dx\\ &=\int \left (1+\frac{2}{a^2 x^2}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a x}\right ) \, dx\\ &=-\frac{2}{a^2 x}+x+\frac{2 \int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x} \, dx}{a}\\ &=-\frac{2}{a^2 x}+x-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=-\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a}-\frac{2}{a^2 x}+x-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=-\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a}-\frac{2}{a^2 x}+x-(2 a) \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )\\ &=-\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a}-\frac{2}{a^2 x}+x+\frac{2 \tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0349868, size = 52, normalized size = 1.11 \[ -\frac{2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{2 \log \left (a x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{a}-\frac{2}{a^2 x}+x \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.128, size = 112, normalized size = 2.4 \begin{align*} x-2\,{\frac{1}{{a}^{2}x}}+2\,{\frac{1}{a}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( -{a}^{2} \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{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}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965249, size = 80, normalized size = 1.7 \begin{align*} x - \frac{2 \, \sqrt{\frac{1}{a^{2} x^{2}} + 1} - \log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) + \log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{a} - \frac{2}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53242, size = 165, normalized size = 3.51 \begin{align*} \frac{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 x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2 \, a x - 2}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.21986, size = 49, normalized size = 1.04 \begin{align*} x - \frac{2 x}{\sqrt{a^{2} x^{2} + 1}} + \frac{2 \operatorname{asinh}{\left (a x \right )}}{a} - \frac{2}{a^{2} x} - \frac{2}{a^{2} 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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