Optimal. Leaf size=24 \[ -2 \sqrt{x+1}-2 \log \left (1-\sqrt{x+1}\right ) \]
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Rubi [A] time = 0.010346, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 190, 43} \[ -2 \sqrt{x+1}-2 \log \left (1-\sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{1-\sqrt{1+x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{x}} \, dx,x,1+x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{1-x} \, dx,x,\sqrt{1+x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-1+\frac{1}{1-x}\right ) \, dx,x,\sqrt{1+x}\right )\\ &=-2 \sqrt{1+x}-2 \log \left (1-\sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0100692, size = 24, normalized size = 1. \[ -2 \sqrt{x+1}-2 \log \left (1-\sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 31, normalized size = 1.3 \begin{align*} -\ln \left ( x \right ) -2\,\sqrt{1+x}-\ln \left ( -1+\sqrt{1+x} \right ) +\ln \left ( 1+\sqrt{1+x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93336, size = 24, normalized size = 1. \begin{align*} -2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65158, size = 55, normalized size = 2.29 \begin{align*} -2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.118422, size = 20, normalized size = 0.83 \begin{align*} - 2 \sqrt{x + 1} - 2 \log{\left (\sqrt{x + 1} - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12016, size = 26, normalized size = 1.08 \begin{align*} -2 \, \sqrt{x + 1} - 2 \, \log \left ({\left | \sqrt{x + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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