Optimal. Leaf size=18 \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]
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Rubi [A] time = 0.0181478, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6129, 37} \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{\sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.0039239, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 14, normalized size = 0.8 \begin{align*}{(-1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41851, size = 22, normalized size = 1.22 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73147, size = 47, normalized size = 2.61 \begin{align*} -\frac{x + \sqrt{-x^{2} + 1} + 1}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27452, size = 28, normalized size = 1.56 \begin{align*} \frac{2}{\frac{\sqrt{-x^{2} + 1} - 1}{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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