Optimal. Leaf size=18 \[ 2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right ) \]
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Rubi [A] time = 0.0059158, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {190, 43} \[ 2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{1+\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{x}-2 \log \left (1+\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0066791, size = 18, normalized size = 1. \[ 2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 1.5 \begin{align*} 2\,\sqrt{x}+\ln \left ( \sqrt{x}-1 \right ) -\ln \left ( \sqrt{x}+1 \right ) -\ln \left ( -1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928509, size = 20, normalized size = 1.11 \begin{align*} 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) + 2 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79365, size = 43, normalized size = 2.39 \begin{align*} 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.109201, size = 15, normalized size = 0.83 \begin{align*} 2 \sqrt{x} - 2 \log{\left (\sqrt{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04502, size = 19, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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