Integrand size = 16, antiderivative size = 20 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx=-e^x+\frac {e^{2 x}}{2}+e^x x \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5767, 6874, 2207, 2225, 2320, 12, 14} \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx=e^x x-e^x+\frac {e^{2 x}}{2} \]
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Rule 12
Rule 14
Rule 2207
Rule 2225
Rule 2320
Rule 5767
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int e^x (x+\cosh (x)+\sinh (x)) \, dx \\ & = \int \left (e^x x+e^x \cosh (x)+e^x \sinh (x)\right ) \, dx \\ & = \int e^x x \, dx+\int e^x \cosh (x) \, dx+\int e^x \sinh (x) \, dx \\ & = e^x x-\int e^x \, dx+\text {Subst}\left (\int \frac {-1+x^2}{2 x} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {1+x^2}{2 x} \, dx,x,e^x\right ) \\ & = -e^x+e^x x+\frac {1}{2} \text {Subst}\left (\int \frac {-1+x^2}{x} \, dx,x,e^x\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,e^x\right ) \\ & = -e^x+e^x x+\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x}+x\right ) \, dx,x,e^x\right )+\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,e^x\right ) \\ & = -e^x+\frac {e^{2 x}}{2}+e^x x \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx=\frac {1}{2} \cosh (2 x)+(-1+x) \sinh (x)+\cosh (x) (-1+x+\sinh (x)) \]
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Time = 0.42 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\left (-1+x \right ) {\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{2}\) | \(14\) |
default | \(-\cosh \left (x \right )+x \sinh \left (x \right )+x \cosh \left (x \right )-\sinh \left (x \right )+\cosh \left (x \right ) \sinh \left (x \right )+\cosh ^{2}\left (x \right )\) | \(27\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx=\frac {2 \, x + \cosh \left (x\right ) + \sinh \left (x\right ) - 2}{2 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx=\frac {x}{- \sinh {\left (x \right )} + \cosh {\left (x \right )}} + \frac {\sinh {\left (x \right )}}{- \sinh {\left (x \right )} + \cosh {\left (x \right )}} - \frac {1}{- \sinh {\left (x \right )} + \cosh {\left (x \right )}} \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx={\left (x - 1\right )} e^{x} + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
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Time = 0.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx=\frac {1}{2} \, {\left (2 \, x + e^{x} - 2\right )} e^{x} \]
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Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx={\mathrm {e}}^x\,\left (x+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-1\right ) \]
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