Integrand size = 13, antiderivative size = 15 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=x-(1-x) \tanh \left (\frac {x}{2}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6874, 3399, 4269, 3556} \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=x-(1-x) \tanh \left (\frac {x}{2}\right ) \]
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Rule 3399
Rule 3556
Rule 4269
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x+\cosh (x)}{1+\cosh (x)}+\tanh \left (\frac {x}{2}\right )\right ) \, dx \\ & = \int \frac {x+\cosh (x)}{1+\cosh (x)} \, dx+\int \tanh \left (\frac {x}{2}\right ) \, dx \\ & = 2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\int \left (1+\frac {-1+x}{1+\cosh (x)}\right ) \, dx \\ & = x+2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\int \frac {-1+x}{1+\cosh (x)} \, dx \\ & = x+2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {1}{2} \int (-1+x) \text {sech}^2\left (\frac {x}{2}\right ) \, dx \\ & = x+2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-(1-x) \tanh \left (\frac {x}{2}\right )-\int \tanh \left (\frac {x}{2}\right ) \, dx \\ & = x-(1-x) \tanh \left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=\frac {\left (-1+x+x \coth \left (\frac {x}{2}\right )\right ) \sinh (x)}{1+\cosh (x)} \]
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Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
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none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=\frac {2 \, {\left (x \cosh \left (x\right ) + x \sinh \left (x\right ) + 1\right )}}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \]
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Time = 0.15 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=x \tanh {\left (\frac {x}{2} \right )} + x - \tanh {\left (\frac {x}{2} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (10) = 20\).
Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=x + \frac {2 \, x e^{x}}{e^{x} + 1} - \frac {2}{e^{\left (-x\right )} + 1} + \log \left (\cosh \left (x\right ) + 1\right ) - 2 \, \log \left (e^{x} + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=\frac {2 \, {\left (x e^{x} + 1\right )}}{e^{x} + 1} \]
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Time = 0.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx=2\,x-\frac {2\,x-2}{{\mathrm {e}}^x+1} \]
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