Optimal. Leaf size=20 \[ \frac {\sqrt {1-x}}{\sqrt {x+1}}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6129, 78, 41, 216} \[ \frac {\sqrt {1-x}}{\sqrt {x+1}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 78
Rule 216
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)} x}{(1+x)^2} \, dx &=\int \frac {x}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-x}}{\sqrt {1+x}}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1-x}}{\sqrt {1+x}}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {\sqrt {1-x}}{\sqrt {1+x}}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 1.00 \[ \frac {\sqrt {1-x}}{\sqrt {x+1}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 44, normalized size = 2.20 \[ -\frac {2 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - x - \sqrt {-x^{2} + 1} - 1}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 24, normalized size = 1.20 \[ -\frac {2}{\frac {\sqrt {-x^{2} + 1} - 1}{x} - 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 1.20 \[ \arcsin \relax (x )+\frac {\sqrt {-\left (1+x \right )^{2}+2 x +2}}{1+x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 18, normalized size = 0.90 \[ \frac {\sqrt {-x^{2} + 1}}{x + 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 18, normalized size = 0.90 \[ \mathrm {asin}\relax (x)+\frac {\sqrt {1-x^2}}{x+1} \]
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 \[ \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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