Optimal. Leaf size=87 \[ -\frac{1}{4} \sqrt{1-x} (x+1)^{7/2}-\frac{1}{4} \sqrt{1-x} (x+1)^{5/2}-\frac{5}{8} \sqrt{1-x} (x+1)^{3/2}-\frac{15}{8} \sqrt{1-x} \sqrt{x+1}+\frac{15}{8} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0482521, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6129, 80, 50, 41, 216} \[ -\frac{1}{4} \sqrt{1-x} (x+1)^{7/2}-\frac{1}{4} \sqrt{1-x} (x+1)^{5/2}-\frac{5}{8} \sqrt{1-x} (x+1)^{3/2}-\frac{15}{8} \sqrt{1-x} \sqrt{x+1}+\frac{15}{8} \sin ^{-1}(x) \]
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)^2 \, dx &=\int \frac{x (1+x)^{5/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{4} \sqrt{1-x} (1+x)^{7/2}+\frac{3}{4} \int \frac{(1+x)^{5/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{4} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{7/2}+\frac{5}{4} \int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{5}{8} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{7/2}+\frac{15}{8} \int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-\frac{15}{8} \sqrt{1-x} \sqrt{1+x}-\frac{5}{8} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{7/2}+\frac{15}{8} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{15}{8} \sqrt{1-x} \sqrt{1+x}-\frac{5}{8} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{7/2}+\frac{15}{8} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{15}{8} \sqrt{1-x} \sqrt{1+x}-\frac{5}{8} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} (1+x)^{7/2}+\frac{15}{8} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0310331, size = 51, normalized size = 0.59 \[ \frac{1}{8} \left (-\sqrt{1-x^2} \left (2 x^3+8 x^2+15 x+24\right )-30 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.035, size = 57, normalized size = 0.7 \begin{align*} -{\frac{{x}^{3}}{4}\sqrt{-{x}^{2}+1}}-{\frac{15\,x}{8}\sqrt{-{x}^{2}+1}}+{\frac{15\,\arcsin \left ( x \right ) }{8}}-{x}^{2}\sqrt{-{x}^{2}+1}-3\,\sqrt{-{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43023, size = 76, normalized size = 0.87 \begin{align*} -\frac{1}{4} \, \sqrt{-x^{2} + 1} x^{3} - \sqrt{-x^{2} + 1} x^{2} - \frac{15}{8} \, \sqrt{-x^{2} + 1} x - 3 \, \sqrt{-x^{2} + 1} + \frac{15}{8} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86501, size = 117, normalized size = 1.34 \begin{align*} -\frac{1}{8} \,{\left (2 \, x^{3} + 8 \, x^{2} + 15 \, x + 24\right )} \sqrt{-x^{2} + 1} - \frac{15}{4} \, \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 = 0.865044, size = 54, normalized size = 0.62 \begin{align*} - \frac{x^{3} \sqrt{1 - x^{2}}}{4} - x^{2} \sqrt{1 - x^{2}} - \frac{15 x \sqrt{1 - x^{2}}}{8} - 3 \sqrt{1 - x^{2}} + \frac{15 \operatorname{asin}{\left (x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28611, size = 38, normalized size = 0.44 \begin{align*} -\frac{1}{8} \,{\left ({\left (2 \,{\left (x + 4\right )} x + 15\right )} x + 24\right )} \sqrt{-x^{2} + 1} + \frac{15}{8} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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