Optimal. Leaf size=32 \[ -x+\sqrt{x+1}+x \log \left (x+\sqrt{x+1}+1\right )+\frac{1}{2} \log (x+1) \]
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Rubi [A] time = 0.205253, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2548} \[ -x+\sqrt{x+1}+x \log \left (x+\sqrt{x+1}+1\right )+\frac{1}{2} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 2548
Rubi steps
\begin{align*} \int \log \left (1+x+\sqrt{1+x}\right ) \, dx &=x \log \left (1+x+\sqrt{1+x}\right )-\int \frac{x \left (1+\frac{1}{2 \sqrt{1+x}}\right )}{1+x+\sqrt{1+x}} \, dx\\ &=x \log \left (1+x+\sqrt{1+x}\right )-2 \operatorname{Subst}\left (\int \left (-\frac{1}{2}-\frac{1}{2 x}+x\right ) \, dx,x,\sqrt{1+x}\right )\\ &=-x+\sqrt{1+x}+\frac{1}{2} \log (1+x)+x \log \left (1+x+\sqrt{1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0138862, size = 38, normalized size = 1.19 \[ -x+\sqrt{x+1}-\log \left (\sqrt{x+1}+1\right )+(x+1) \log \left (x+\sqrt{x+1}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 34, normalized size = 1.1 \begin{align*} \left ( 1+x \right ) \ln \left ( 1+x+\sqrt{1+x} \right ) -x-1+\sqrt{1+x}-\ln \left ( \sqrt{1+x}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06802, size = 45, normalized size = 1.41 \begin{align*}{\left (x + 1\right )} \log \left (x + \sqrt{x + 1} + 1\right ) - x + \sqrt{x + 1} - \log \left (\sqrt{x + 1} + 1\right ) - 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81003, size = 130, normalized size = 4.06 \begin{align*}{\left (x - 1\right )} \log \left (x + \sqrt{x + 1} + 1\right ) - x + \sqrt{x + 1} + \log \left (\sqrt{x + 1} + 1\right ) + 2 \, \log \left (\sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.05152, size = 158, normalized size = 4.94 \begin{align*} \frac{x \sqrt{x + 1} \log{\left (x + \sqrt{x + 1} + 1 \right )}}{\sqrt{x + 1} + 1} - \frac{x \sqrt{x + 1}}{\sqrt{x + 1} + 1} + \frac{x \log{\left (x + \sqrt{x + 1} + 1 \right )}}{\sqrt{x + 1} + 1} - \frac{\sqrt{x + 1} \log{\left (\sqrt{x + 1} + 1 \right )}}{\sqrt{x + 1} + 1} + \frac{\sqrt{x + 1} \log{\left (x + \sqrt{x + 1} + 1 \right )}}{\sqrt{x + 1} + 1} - \frac{\log{\left (\sqrt{x + 1} + 1 \right )}}{\sqrt{x + 1} + 1} + \frac{\log{\left (x + \sqrt{x + 1} + 1 \right )}}{\sqrt{x + 1} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31745, size = 45, normalized size = 1.41 \begin{align*}{\left (x + 1\right )} \log \left (x + \sqrt{x + 1} + 1\right ) - x + \sqrt{x + 1} - \log \left (\sqrt{x + 1} + 1\right ) - 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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