Optimal. Leaf size=15 \[ x-(1-x) \tanh \left (\frac {x}{2}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6742, 3318, 4184, 3475} \[ x-(1-x) \tanh \left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
Rule 6742
Rubi steps
\begin {align*} \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx &=\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*}
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Mathematica [A] time = 0.09, size = 20, normalized size = 1.33 \[ \frac {\sinh (x) \left (x+x \coth \left (\frac {x}{2}\right )-1\right )}{\cosh (x)+1} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.40, size = 20, normalized size = 1.33 \[ \frac {2 \, {\left (x \cosh \relax (x) + x \sinh \relax (x) + 1\right )}}{\cosh \relax (x) + \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 14, normalized size = 0.93 \[ \frac {2 \, {\left (x e^{x} + 1\right )}}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 16, normalized size = 1.07
method | result | size |
risch | \(2 x -\frac {2 \left (-1+x \right )}{1+{\mathrm e}^{x}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 35, normalized size = 2.33 \[ x + \frac {2 \, x e^{x}}{e^{x} + 1} - \frac {2}{e^{\left (-x\right )} + 1} + \log \left (\cosh \relax (x) + 1\right ) - 2 \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 17, normalized size = 1.13 \[ 2\,x-\frac {2\,x-2}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 12, normalized size = 0.80 \[ x \tanh {\left (\frac {x}{2} \right )} + x - \tanh {\left (\frac {x}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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