Optimal. Leaf size=18 \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]
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Rubi [A] time = 0.0015644, antiderivative size = 18, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]
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*}
Mathematica [A] time = 0.0034147, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\sqrt{1-x}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55, size = 22, normalized size = 1.22 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7723, size = 61, normalized size = 3.39 \begin{align*} -\frac{x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07564, size = 29, normalized size = 1.61 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07944, size = 58, normalized size = 3.22 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{2 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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