Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x}}{\sqrt {x+1}} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx &=-\frac {\sqrt {1-x}}{\sqrt {1+x}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 23, normalized size = 1.28 \begin {gather*} -\frac {x + \sqrt {x + 1} \sqrt {-x + 1} + 1}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 43, normalized size = 2.39 \begin {gather*} \frac {\sqrt {2} - \sqrt {-x + 1}}{2 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {-x +1}}{\sqrt {x +1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 29, normalized size = 1.61 \begin {gather*} \begin {cases} - \sqrt {-1 + \frac {2}{x + 1}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- i \sqrt {1 - \frac {2}{x + 1}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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