Optimal. Leaf size=15 \[ e^x-\frac {2}{1-e^x} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2282, 683} \[ e^x-\frac {2}{1-e^x} \]
Antiderivative was successfully verified.
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Rule 683
Rule 2282
Rubi steps
\begin {align*} \int \frac {e^x (1-\sinh (x))}{1-\cosh (x)} \, dx &=\operatorname {Subst}\left (\int \frac {-1-2 x+x^2}{(1-x)^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {2}{(-1+x)^2}\right ) \, dx,x,e^x\right )\\ &=e^x-\frac {2}{1-e^x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 20, normalized size = 1.33 \[ \frac {-e^x+e^{2 x}+2}{e^x-1} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1-\sinh (x))}{1-\cosh (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.71, size = 22, normalized size = 1.47 \[ -\frac {3 \, \cosh \relax (x) - \sinh \relax (x) - 1}{\cosh \relax (x) - \sinh \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 11, normalized size = 0.73 \[ \frac {2}{e^{x} - 1} + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 12, normalized size = 0.80
method | result | size |
risch | \({\mathrm e}^{x}+\frac {2}{-1+{\mathrm e}^{x}}\) | \(12\) |
default | \(\frac {1}{\tanh \left (\frac {x}{2}\right )}-\frac {2}{\tanh \left (\frac {x}{2}\right )-1}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 11, normalized size = 0.73 \[ \frac {2}{e^{x} - 1} + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 11, normalized size = 0.73 \[ {\mathrm {e}}^x+\frac {2}{{\mathrm {e}}^x-1} \]
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 \[ \int \frac {\left (\sinh {\relax (x )} - 1\right ) e^{x}}{\cosh {\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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