Optimal. Leaf size=90 \[ -\frac{\sqrt{\frac{1}{x}+1} x \sqrt{1-\frac{1}{x}}}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.109552, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac{\sqrt{\frac{1}{x}+1} x \sqrt{1-\frac{1}{x}}}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
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{\sqrt{1+x}}{(1-x)^2 \sqrt{x}} \, dx,x,\frac{1}{x}\right )}{(1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=-\frac{\sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{x} \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )}{2 (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=-\frac{\sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{(1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ &=-\frac{\sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0907516, size = 58, normalized size = 0.64 \[ \frac{\frac{2}{\sqrt{\frac{1}{x+1}}}+\sqrt{2} (x-1) \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{x+1}}\right )}{2 \sqrt{-\frac{(x-1)^2}{x^2}} x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.11, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{-2+2\,x}\sqrt{1-x} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1-x}} \right ) x-\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1-x}} \right ) +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 = 1.57579, size = 208, normalized size = 2.31 \begin{align*} -\frac{\sqrt{2}{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{x - 1}\right ) + 2 \,{\left (x + 1\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{2 \,{\left (x^{2} - 2 \, 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 [A] time = 1.13293, size = 59, normalized size = 0.66 \begin{align*} \frac{{\left (\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x - 1}\right ) + \frac{2 \, \sqrt{-x - 1}}{x - 1}\right )} \mathrm{sgn}\left (x\right )}{2 \, \mathrm{sgn}\left (-x - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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