Optimal. Leaf size=18 \[ \frac{x}{\sqrt{1-x} \sqrt{x+1}} \]
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Rubi [A] time = 0.0016086, 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 = {39} \[ \frac{x}{\sqrt{1-x} \sqrt{x+1}} \]
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx &=\frac{x}{\sqrt{1-x} \sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.0027818, size = 13, normalized size = 0.72 \[ \frac{x}{\sqrt{1-x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{x{\frac{1}{\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 = 1.01873, size = 15, normalized size = 0.83 \begin{align*} \frac{x}{\sqrt{-x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74749, size = 53, normalized size = 2.94 \begin{align*} -\frac{\sqrt{x + 1} x \sqrt{-x + 1}}{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.71814, size = 65, normalized size = 3.61 \begin{align*} \begin{cases} \frac{1}{\sqrt{-1 + \frac{2}{x + 1}}} - \frac{1}{\sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{x - 1} + \frac{i \sqrt{1 - \frac{2}{x + 1}}}{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 = 1.06418, size = 84, normalized size = 4.67 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{4 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1} \sqrt{-x + 1}}{2 \,{\left (x - 1\right )}} - \frac{\sqrt{x + 1}}{4 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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