Optimal. Leaf size=22 \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]
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Rubi [A] time = 0.0091146, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 190, 43} \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]
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 = 0.0077022, size = 22, normalized size = 1. \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 1.4 \begin{align*} 2\,\sqrt{1+x}+\ln \left ( -1+\sqrt{1+x} \right ) -\ln \left ( 1+\sqrt{1+x} \right ) -\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93172, size = 24, normalized size = 1.09 \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 = 1.66271, size = 54, normalized size = 2.45 \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 = 0.111209, size = 19, normalized size = 0.86 \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 = 1.05055, size = 24, normalized size = 1.09 \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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