Optimal. Leaf size=61 \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0319597, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {6129, 80, 50, 41, 216} \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6129
Rule 80
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(x)} x (1+x) \, dx &=\int \frac{x (1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{2}{3} \int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x} \sqrt{1+x}-\frac{1}{3} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0296669, size = 42, normalized size = 0.69 \[ -\frac{1}{3} \sqrt{1-x^2} \left (x^2+3 x+5\right )-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.036, size = 41, normalized size = 0.7 \begin{align*} -{\frac{{x}^{2}}{3}\sqrt{-{x}^{2}+1}}-{\frac{5}{3}\sqrt{-{x}^{2}+1}}-x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42634, size = 54, normalized size = 0.89 \begin{align*} -\frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \sqrt{-x^{2} + 1} x - \frac{5}{3} \, \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7552, size = 97, normalized size = 1.59 \begin{align*} -\frac{1}{3} \,{\left (x^{2} + 3 \, x + 5\right )} \sqrt{-x^{2} + 1} - 2 \, \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 = 0.420643, size = 37, normalized size = 0.61 \begin{align*} - \frac{x^{2} \sqrt{1 - x^{2}}}{3} - x \sqrt{1 - x^{2}} - \frac{5 \sqrt{1 - x^{2}}}{3} + \operatorname{asin}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15787, size = 28, normalized size = 0.46 \begin{align*} -\frac{1}{3} \,{\left ({\left (x + 3\right )} x + 5\right )} \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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