Optimal. Leaf size=21 \[ \frac{\sqrt{\frac{x-1}{x}}}{\sqrt{\frac{1}{x}+1}} \]
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Rubi [A] time = 0.0651539, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6175, 6180, 37} \[ \frac{\sqrt{\frac{x-1}{x}}}{\sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac{e^{\coth ^{-1}(x)}}{\left (1+\frac{1}{x}\right )^2 x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{\frac{-1+x}{x}}}{\sqrt{1+\frac{1}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0109699, size = 18, normalized size = 0.86 \[ \frac{\sqrt{1-\frac{1}{x^2}} x}{x+1} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.058, size = 21, normalized size = 1. \begin{align*}{\frac{-1+x}{1+x}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03692, size = 15, normalized size = 0.71 \begin{align*} \sqrt{\frac{x - 1}{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59365, size = 31, normalized size = 1.48 \begin{align*} \sqrt{\frac{x - 1}{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.0712, size = 8, normalized size = 0.38 \begin{align*} \sqrt{\frac{x - 1}{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10328, size = 15, normalized size = 0.71 \begin{align*} \sqrt{\frac{x - 1}{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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