Integrand size = 11, antiderivative size = 32 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx=-x+\sqrt {1+x}+\frac {1}{2} \log (1+x)+x \log \left (1+x+\sqrt {1+x}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2628} \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx=-x+\sqrt {x+1}+x \log \left (x+\sqrt {x+1}+1\right )+\frac {1}{2} \log (x+1) \]
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Rule 2628
Rubi steps \begin{align*} \text {integral}& = 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 \text {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*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx=-x+\sqrt {1+x}-\log \left (1+\sqrt {1+x}\right )+(1+x) \log \left (1+x+\sqrt {1+x}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
method | result | size |
parts | \(x \ln \left (1+x +\sqrt {x +1}\right )+\sqrt {x +1}+\frac {\ln \left (x +1\right )}{2}-x -1\) | \(28\) |
derivativedivides | \(\left (x +1\right ) \ln \left (1+x +\sqrt {x +1}\right )-x -1+\sqrt {x +1}-\ln \left (\sqrt {x +1}+1\right )\) | \(34\) |
default | \(\left (x +1\right ) \ln \left (1+x +\sqrt {x +1}\right )-x -1+\sqrt {x +1}-\ln \left (\sqrt {x +1}+1\right )\) | \(34\) |
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Time = 0.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx={\left (x - 1\right )} \log \left (x + \sqrt {x + 1} + 1\right ) - x + \sqrt {x + 1} + \log \left (\sqrt {x + 1} + 1\right ) + 2 \, \log \left (\sqrt {x + 1}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (27) = 54\).
Time = 0.41 (sec) , antiderivative size = 184, normalized size of antiderivative = 5.75 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx=\frac {x \sqrt {x + 1} \log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} - \frac {x \sqrt {x + 1}}{\sqrt {x + 1} + 1} + \frac {x \log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} - \frac {\sqrt {x + 1} \log {\left (\sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} + \frac {\sqrt {x + 1} \log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} + \frac {\sqrt {x + 1}}{\sqrt {x + 1} + 1} - \frac {\log {\left (\sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} + \frac {\log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} + \frac {1}{\sqrt {x + 1} + 1} \]
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Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx={\left (x + 1\right )} \log \left (x + \sqrt {x + 1} + 1\right ) - x + \sqrt {x + 1} - \log \left (\sqrt {x + 1} + 1\right ) - 1 \]
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Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx={\left (x + 1\right )} \log \left (x + \sqrt {x + 1} + 1\right ) - x + \sqrt {x + 1} - \log \left (\sqrt {x + 1} + 1\right ) - 1 \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \log \left (1+x+\sqrt {1+x}\right ) \, dx=\ln \left (\sqrt {x+1}\right )-x+\sqrt {x+1}+x\,\ln \left (x+\sqrt {x+1}+1\right ) \]
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