Optimal. Leaf size=17 \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]
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Rubi [A] time = 0.0031938, antiderivative size = 17, 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} \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(1-x)^{3/2} \sqrt{1+x}} \, dx &=\frac{\sqrt{1+x}}{\sqrt{1-x}}\\ \end{align*}
Mathematica [A] time = 0.0031577, size = 17, normalized size = 1. \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, 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 = 1.48572, size = 22, normalized size = 1.29 \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 = 1.76067, size = 59, normalized size = 3.47 \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 = 1.04886, size = 29, normalized size = 1.71 \begin{align*} \begin{cases} \frac{1}{\sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{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 [A] time = 1.08491, size = 26, normalized size = 1.53 \begin{align*} -\frac{\sqrt{x + 1} \sqrt{-x + 1}}{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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