Optimal. Leaf size=23 \[ -\frac {2 \sqrt {1-x}}{\sqrt {x+1}}-\sin ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ -\frac {2 \sqrt {1-x}}{\sqrt {x+1}}-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 47
Rule 216
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-x}}{\sqrt {1+x}}-\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 \sqrt {1-x}}{\sqrt {1+x}}-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 \sqrt {1-x}}{\sqrt {1+x}}-\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 1.48 \[ 2 \left (\frac {x-1}{\sqrt {1-x^2}}+\sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 50, normalized size = 2.17 \[ \frac {2 \, {\left ({\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - x - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.69, size = 55, normalized size = 2.39 \[ \frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 67, normalized size = 2.91 \[ -\frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {2 \left (x -1\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{\sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 21, normalized size = 0.91 \[ -\frac {2 \, \sqrt {-x^{2} + 1}}{x + 1} - \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {1-x}}{{\left (x+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.54, size = 104, normalized size = 4.52 \[ \begin {cases} 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {4 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {4}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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