Optimal. Leaf size=23 \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]
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Rubi [A] time = 0.0225083, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {6689} \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 6689
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+x}+\sqrt{1+x}} \, dx &=-\left (\frac{1}{2} \int \left (\sqrt{-1+x}-\sqrt{1+x}\right ) \, dx\right )\\ &=-\frac{1}{3} (-1+x)^{3/2}+\frac{1}{3} (1+x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0202374, size = 23, normalized size = 1. \[ \frac{1}{3} (x+1)^{3/2}-\frac{1}{3} (x-1)^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 16, normalized size = 0.7 \begin{align*} -{\frac{1}{3} \left ( x-1 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + 1} + \sqrt{x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22195, size = 53, normalized size = 2.3 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.3948, size = 51, normalized size = 2.22 \begin{align*} \frac{4 x}{3 \sqrt{x - 1} + 3 \sqrt{x + 1}} + \frac{2 \sqrt{x - 1} \sqrt{x + 1}}{3 \sqrt{x - 1} + 3 \sqrt{x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1578, size = 20, normalized size = 0.87 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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