Optimal. Leaf size=22 \[ -\frac {2}{1-e^x}-2 \log \left (1-e^x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 45}
\begin {gather*} -\frac {2}{1-e^x}-2 \log \left (1-e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2320
Rubi steps
\begin {align*} \int \frac {e^x}{1-\cosh (x)} \, dx &=\text {Subst}\left (\int -\frac {2 x}{(1-x)^2} \, dx,x,e^x\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {x}{(1-x)^2} \, dx,x,e^x\right )\right )\\ &=-\left (2 \text {Subst}\left (\int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,e^x\right )\right )\\ &=-\frac {2}{1-e^x}-2 \log \left (1-e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 1.64 \begin {gather*} \frac {4 \left (\frac {1}{1-e^x}+\log \left (1-e^x\right )\right ) \sinh ^2\left (\frac {x}{2}\right )}{1-\cosh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 24, normalized size = 1.09
method | result | size |
risch | \(\frac {2}{-1+{\mathrm e}^{x}}-2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(17\) |
default | \(\frac {1}{\tanh \left (\frac {x}{2}\right )}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right )+2 \ln \left (-1+\tanh \left (\frac {x}{2}\right )\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.74, size = 16, normalized size = 0.73 \begin {gather*} \frac {2}{e^{x} - 1} - 2 \, \log \left (e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.64, size = 26, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left ({\left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 1\right )}}{\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {e^{x}}{\cosh {\left (x \right )} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 17, normalized size = 0.77 \begin {gather*} \frac {2}{e^{x} - 1} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 16, normalized size = 0.73 \begin {gather*} \frac {2}{{\mathrm {e}}^x-1}-2\,\ln \left ({\mathrm {e}}^x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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