Optimal. Leaf size=18 \[ -\sqrt{1-x} \sqrt{x+1} \]
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Rubi [A] time = 0.0340615, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6129, 74} \[ -\sqrt{1-x} \sqrt{x+1} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 74
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(x)} x}{1+x} \, dx &=\int \frac{x}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}\\ \end{align*}
Mathematica [A] time = 0.0052176, size = 13, normalized size = 0.72 \[ -\sqrt{1-x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 17, normalized size = 0.9 \begin{align*}{ \left ( 1+x \right ) \left ( -1+x \right ){\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 = 0.937585, size = 15, normalized size = 0.83 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.75239, size = 23, normalized size = 1.28 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.139766, size = 8, normalized size = 0.44 \begin{align*} - \sqrt{1 - x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14251, size = 15, normalized size = 0.83 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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