Optimal. Leaf size=47 \[ -\frac{1}{2} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{2} \sqrt{1-x} \sqrt{x+1}+\frac{3}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0174384, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6129, 50, 41, 216} \[ -\frac{1}{2} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{2} \sqrt{1-x} \sqrt{x+1}+\frac{3}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} (1+x) \, dx &=\int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{2} \sqrt{1-x} (1+x)^{3/2}+\frac{3}{2} \int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-\frac{3}{2} \sqrt{1-x} \sqrt{1+x}-\frac{1}{2} \sqrt{1-x} (1+x)^{3/2}+\frac{3}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{3}{2} \sqrt{1-x} \sqrt{1+x}-\frac{1}{2} \sqrt{1-x} (1+x)^{3/2}+\frac{3}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{3}{2} \sqrt{1-x} \sqrt{1+x}-\frac{1}{2} \sqrt{1-x} (1+x)^{3/2}+\frac{3}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0119762, size = 37, normalized size = 0.79 \[ -\frac{1}{2} \sqrt{1-x^2} (x+4)-3 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.034, size = 29, normalized size = 0.6 \begin{align*} -{\frac{x}{2}\sqrt{-{x}^{2}+1}}+{\frac{3\,\arcsin \left ( x \right ) }{2}}-2\,\sqrt{-{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43312, size = 38, normalized size = 0.81 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1} x - 2 \, \sqrt{-x^{2} + 1} + \frac{3}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79934, size = 86, normalized size = 1.83 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x + 4\right )} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.236391, size = 27, normalized size = 0.57 \begin{align*} - \frac{x \sqrt{1 - x^{2}}}{2} - 2 \sqrt{1 - x^{2}} + \frac{3 \operatorname{asin}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12413, size = 26, normalized size = 0.55 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x + 4\right )} + \frac{3}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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