Optimal. Leaf size=25 \[ \frac{2}{3} (1-x)^{3/2}-2 \sqrt{1-x} \]
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Rubi [A] time = 0.0390745, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {6129, 43} \[ \frac{2}{3} (1-x)^{3/2}-2 \sqrt{1-x} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(x)} x}{\sqrt{1+x}} \, dx &=\int \frac{x}{\sqrt{1-x}} \, dx\\ &=\int \left (\frac{1}{\sqrt{1-x}}-\sqrt{1-x}\right ) \, dx\\ &=-2 \sqrt{1-x}+\frac{2}{3} (1-x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0066284, size = 16, normalized size = 0.64 \[ -\frac{2}{3} \sqrt{1-x} (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 23, normalized size = 0.9 \begin{align*}{\frac{ \left ( -2+2\,x \right ) \left ( x+2 \right ) }{3}\sqrt{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 = 0.961726, size = 20, normalized size = 0.8 \begin{align*} \frac{2 \,{\left (x^{2} + x - 2\right )}}{3 \, \sqrt{-x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05257, size = 55, normalized size = 2.2 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}{\left (x + 2\right )}}{3 \, \sqrt{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{x \sqrt{x + 1}}{\sqrt{- \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 = 1.15428, size = 32, normalized size = 1.28 \begin{align*} \frac{2}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{2} - 2 \, \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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