Optimal. Leaf size=67 \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{5}{6} \sqrt{1-x} (x+1)^{3/2}-\frac{5}{2} \sqrt{1-x} \sqrt{x+1}+\frac{5}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0290799, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6129, 50, 41, 216} \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{5}{6} \sqrt{1-x} (x+1)^{3/2}-\frac{5}{2} \sqrt{1-x} \sqrt{x+1}+\frac{5}{2} \sin ^{-1}(x) \]
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)^2 \, dx &=\int \frac{(1+x)^{5/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{5}{3} \int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{5}{6} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{5}{2} \int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-\frac{5}{2} \sqrt{1-x} \sqrt{1+x}-\frac{5}{6} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{5}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{5}{2} \sqrt{1-x} \sqrt{1+x}-\frac{5}{6} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{5}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{5}{2} \sqrt{1-x} \sqrt{1+x}-\frac{5}{6} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{3} \sqrt{1-x} (1+x)^{5/2}+\frac{5}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0216032, size = 44, normalized size = 0.66 \[ -\frac{1}{6} \sqrt{1-x^2} \left (2 x^2+9 x+22\right )-5 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.045, size = 43, normalized size = 0.6 \begin{align*} -{\frac{{x}^{2}}{3}\sqrt{-{x}^{2}+1}}-{\frac{11}{3}\sqrt{-{x}^{2}+1}}-{\frac{3\,x}{2}\sqrt{-{x}^{2}+1}}+{\frac{5\,\arcsin \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44777, size = 57, normalized size = 0.85 \begin{align*} -\frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{3}{2} \, \sqrt{-x^{2} + 1} x - \frac{11}{3} \, \sqrt{-x^{2} + 1} + \frac{5}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73717, size = 101, normalized size = 1.51 \begin{align*} -\frac{1}{6} \,{\left (2 \, x^{2} + 9 \, x + 22\right )} \sqrt{-x^{2} + 1} - 5 \, \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 = 0.418913, size = 44, normalized size = 0.66 \begin{align*} - \frac{x^{2} \sqrt{1 - x^{2}}}{3} - \frac{3 x \sqrt{1 - x^{2}}}{2} - \frac{11 \sqrt{1 - x^{2}}}{3} + \frac{5 \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 = 1.17571, size = 34, normalized size = 0.51 \begin{align*} -\frac{1}{6} \,{\left ({\left (2 \, x + 9\right )} x + 22\right )} \sqrt{-x^{2} + 1} + \frac{5}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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