Optimal. Leaf size=12 \[ x-\frac {2 \sinh (x)}{1+\cosh (x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2759, 8}
\begin {gather*} x-\frac {2 \sinh (x)}{\cosh (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2759
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(1+\cosh (x))^2} \, dx &=-\frac {2 \sinh (x)}{1+\cosh (x)}+\int 1 \, dx\\ &=x-\frac {2 \sinh (x)}{1+\cosh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.50 \begin {gather*} 2 \tanh ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 24, normalized size = 2.00
method | result | size |
risch | \(x +\frac {4}{{\mathrm e}^{x}+1}\) | \(11\) |
default | \(-2 \tanh \left (\frac {x}{2}\right )-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 12, normalized size = 1.00 \begin {gather*} x - \frac {4}{e^{\left (-x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 20, normalized size = 1.67 \begin {gather*} \frac {x \cosh \left (x\right ) + x \sinh \left (x\right ) + x + 4}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 7, normalized size = 0.58 \begin {gather*} x - 2 \tanh {\left (\frac {x}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 10, normalized size = 0.83 \begin {gather*} x + \frac {4}{e^{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 10, normalized size = 0.83 \begin {gather*} x+\frac {4}{{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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