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 = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 41
Rule 50
Rule 80
Rule 216
Rule 6129
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.47, size = 45, normalized size = 0.52 \[ -\frac {1}{8} \, {\left (2 \, x^{3} + 8 \, x^{2} + 15 \, x + 24\right )} \sqrt {-x^{2} + 1} - \frac {15}{4} \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 28, normalized size = 0.32 \[ -\frac {1}{8} \, {\left ({\left (2 \, {\left (x + 4\right )} x + 15\right )} x + 24\right )} \sqrt {-x^{2} + 1} + \frac {15}{8} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 0.66 \[ -\frac {x^{3} \sqrt {-x^{2}+1}}{4}-\frac {15 x \sqrt {-x^{2}+1}}{8}+\frac {15 \arcsin \relax (x )}{8}-x^{2} \sqrt {-x^{2}+1}-3 \sqrt {-x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 56, normalized size = 0.64 \[ -\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} - \sqrt {-x^{2} + 1} x^{2} - \frac {15}{8} \, \sqrt {-x^{2} + 1} x - 3 \, \sqrt {-x^{2} + 1} + \frac {15}{8} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 29, normalized size = 0.33 \[ \frac {15\,\mathrm {asin}\relax (x)}{8}-\sqrt {1-x^2}\,\left (\frac {x^3}{4}+x^2+\frac {15\,x}{8}+3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 54, normalized size = 0.62 \[ - \frac {x^{3} \sqrt {1 - x^{2}}}{4} - x^{2} \sqrt {1 - x^{2}} - \frac {15 x \sqrt {1 - x^{2}}}{8} - 3 \sqrt {1 - x^{2}} + \frac {15 \operatorname {asin}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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