Optimal. Leaf size=19 \[ -\frac{2}{\sqrt{x+1}}-\frac{2}{\sqrt{x-1}} \]
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Rubi [A] time = 0.270052, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {6688} \[ -\frac{2}{\sqrt{x+1}}-\frac{2}{\sqrt{x-1}} \]
Antiderivative was successfully verified.
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Rule 6688
Rubi steps
\begin{align*} \int \frac{(-1+x)^{3/2}+(1+x)^{3/2}}{(-1+x)^{3/2} (1+x)^{3/2}} \, dx &=\int \left (\frac{1}{(-1+x)^{3/2}}+\frac{1}{(1+x)^{3/2}}\right ) \, dx\\ &=-\frac{2}{\sqrt{-1+x}}-\frac{2}{\sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.0174567, size = 19, normalized size = 1. \[ -\frac{2}{\sqrt{x+1}}-\frac{2}{\sqrt{x-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 16, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{\sqrt{-1+x}}}-2\,{\frac{1}{\sqrt{1+x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.933505, size = 20, normalized size = 1.05 \begin{align*} -\frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97151, size = 78, normalized size = 4.11 \begin{align*} -\frac{2 \,{\left ({\left (x + 1\right )} \sqrt{x - 1} + \sqrt{x + 1}{\left (x - 1\right )}\right )}}{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.48919, size = 56, normalized size = 2.95 \begin{align*} - \frac{2 x \sqrt{x - 1}}{x^{2} - 1} - \frac{2 x \sqrt{x + 1}}{x^{2} - 1} - \frac{2 \sqrt{x - 1}}{x^{2} - 1} + \frac{2 \sqrt{x + 1}}{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09706, size = 20, normalized size = 1.05 \begin{align*} -\frac{2}{\sqrt{x + 1}} - \frac{2}{\sqrt{x - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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