Optimal. Leaf size=31 \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]
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Rubi [A] time = 0.0225466, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {632, 31} \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x-\sqrt{2+x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{-2-x+x^2} \, dx,x,\sqrt{2+x}\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{2+x}\right )+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-2+x} \, dx,x,\sqrt{2+x}\right )\\ &=\frac{4}{3} \log \left (2-\sqrt{2+x}\right )+\frac{2}{3} \log \left (1+\sqrt{2+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0101808, size = 31, normalized size = 1. \[ \frac{4}{3} \log \left (2-\sqrt{x+2}\right )+\frac{2}{3} \log \left (\sqrt{x+2}+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 54, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{3}}+{\frac{2\,\ln \left ( -2+x \right ) }{3}}-{\frac{2}{3}\ln \left ( \sqrt{2+x}+2 \right ) }+{\frac{1}{3}\ln \left ( 1+\sqrt{2+x} \right ) }-{\frac{1}{3}\ln \left ( -1+\sqrt{2+x} \right ) }+{\frac{2}{3}\ln \left ( \sqrt{2+x}-2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926835, size = 28, normalized size = 0.9 \begin{align*} \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left (\sqrt{x + 2} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89868, size = 72, normalized size = 2.32 \begin{align*} \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left (\sqrt{x + 2} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.31173, size = 36, normalized size = 1.16 \begin{align*} \log{\left (x - \sqrt{x + 2} \right )} + \frac{\log{\left (2 \sqrt{x + 2} - 4 \right )}}{3} - \frac{\log{\left (2 \sqrt{x + 2} + 2 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07866, size = 30, normalized size = 0.97 \begin{align*} \frac{2}{3} \, \log \left (\sqrt{x + 2} + 1\right ) + \frac{4}{3} \, \log \left ({\left | \sqrt{x + 2} - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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