Optimal. Leaf size=47 \[ -\frac {1}{2} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{2} \sqrt {1-x} \sqrt {x+1}+\frac {3}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6129, 50, 41, 216} \[ -\frac {1}{2} \sqrt {1-x} (x+1)^{3/2}-\frac {3}{2} \sqrt {1-x} \sqrt {x+1}+\frac {3}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 216
Rule 6129
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} (1+x) \, dx &=\int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3}{2} \sqrt {1-x} \sqrt {1+x}-\frac {1}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {3}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.79 \[ -\frac {1}{2} \sqrt {1-x^2} (x+4)-3 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 33, normalized size = 0.70 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} - 3 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 19, normalized size = 0.40 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} + \frac {3}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 0.62 \[ -\frac {x \sqrt {-x^{2}+1}}{2}+\frac {3 \arcsin \relax (x )}{2}-2 \sqrt {-x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 28, normalized size = 0.60 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} x - 2 \, \sqrt {-x^{2} + 1} + \frac {3}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 21, normalized size = 0.45 \[ \frac {3\,\mathrm {asin}\relax (x)}{2}-\left (\frac {x}{2}+2\right )\,\sqrt {1-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 27, normalized size = 0.57 \[ - \frac {x \sqrt {1 - x^{2}}}{2} - 2 \sqrt {1 - x^{2}} + \frac {3 \operatorname {asin}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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