Optimal. Leaf size=58 \[ -\frac{\sqrt{2} \left (\frac{1}{x}+1\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (x+1)^{3/2}} \]
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Rubi [A] time = 0.0965376, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6176, 6181, 93, 203} \[ -\frac{\sqrt{2} \left (\frac{1}{x}+1\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 93
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)}}{(1+x)^{3/2}} \, dx &=\frac{\left (\left (1+\frac{1}{x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{\coth ^{-1}(x)}}{\left (1+\frac{1}{x}\right )^{3/2} x^{3/2}} \, dx}{(1+x)^{3/2}}\\ &=-\frac{\left (1+\frac{1}{x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{x} (1+x)} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}}\\ &=-\frac{\left (2 \left (1+\frac{1}{x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{\frac{-1+x}{x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}}\\ &=-\frac{\sqrt{2} \left (1+\frac{1}{x}\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (1+x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0191591, size = 41, normalized size = 0.71 \[ \sqrt{2} \sqrt{\frac{1}{x+1}} \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{\frac{x-1}{x^2}} x}{\sqrt{2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.111, size = 37, normalized size = 0.6 \begin{align*}{\sqrt{2}\sqrt{-1+x}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1+x}} \right ){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{\frac{x - 1}{x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61836, size = 85, normalized size = 1.47 \begin{align*} \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1} \sqrt{\frac{x - 1}{x + 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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