Optimal. Leaf size=25 \[ \frac {2}{3} (1-x)^{3/2}-2 \sqrt {1-x} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6129, 43} \[ \frac {2}{3} (1-x)^{3/2}-2 \sqrt {1-x} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)} x}{\sqrt {1+x}} \, dx &=\int \frac {x}{\sqrt {1-x}} \, dx\\ &=\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx\\ &=-2 \sqrt {1-x}+\frac {2}{3} (1-x)^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.64 \[ -\frac {2}{3} \sqrt {1-x} (x+2) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 19, normalized size = 0.76 \[ -\frac {2 \, \sqrt {-x^{2} + 1} {\left (x + 2\right )}}{3 \, \sqrt {x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 23, normalized size = 0.92 \[ \frac {2 \left (-1+x \right ) \left (x +2\right ) \sqrt {1+x}}{3 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 15, normalized size = 0.60 \[ \frac {2 \, {\left (x^{2} + x - 2\right )}}{3 \, \sqrt {-x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 19, normalized size = 0.76 \[ -\frac {2\,\sqrt {1-x^2}\,\left (x+2\right )}{3\,\sqrt {x+1}} \]
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 \[ \int \frac {x \sqrt {x + 1}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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