Optimal. Leaf size=23 \[ \frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6128, 26, 43} \[ \frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 26
Rule 43
Rule 6128
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} \sqrt {1-x} x \, dx &=\int \frac {x \sqrt {1-x^2}}{\sqrt {1-x}} \, dx\\ &=\int x \sqrt {1+x} \, dx\\ &=\int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx\\ &=-\frac {2}{3} (1+x)^{3/2}+\frac {2}{5} (1+x)^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.70 \[ \frac {2}{15} (x+1)^{3/2} (3 x-2) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 31, normalized size = 1.35 \[ -\frac {2 \, {\left (3 \, x^{2} + x - 2\right )} \sqrt {-x^{2} + 1} \sqrt {-x + 1}}{15 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 20, normalized size = 0.87 \[ \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{15} \, \sqrt {2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 1.26 \[ \frac {2 \left (1+x \right )^{2} \left (3 x -2\right ) \sqrt {1-x}}{15 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 38, normalized size = 1.65 \[ \frac {2 \, {\left (3 \, x^{3} - x^{2} + 4 \, x + 8\right )}}{15 \, \sqrt {x + 1}} + \frac {2 \, {\left (x^{2} - x - 2\right )}}{3 \, \sqrt {x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 42, normalized size = 1.83 \[ \frac {4\,\sqrt {1-x^2}}{15\,\sqrt {1-x}}-\frac {2\,\left (3\,x+4\right )\,\sqrt {1-x^2}\,\sqrt {1-x}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {1 - x} \left (x + 1\right )}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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