Optimal. Leaf size=9 \[ \frac {e^{2 x}}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2282, 30} \[ \frac {e^{2 x}}{2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2282
Rubi steps
\begin {align*} \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx &=\operatorname {Subst}\left (\int x \, dx,x,e^x\right )\\ &=\frac {e^{2 x}}{2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 9, normalized size = 1.00 \[ \frac {e^{2 x}}{2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x}{\cosh (x)-\sinh (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.93, size = 16, normalized size = 1.78 \[ \frac {\cosh \relax (x) + \sinh \relax (x)}{2 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 6, normalized size = 0.67 \[ \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 7, normalized size = 0.78
method | result | size |
risch | \(\frac {{\mathrm e}^{2 x}}{2}\) | \(7\) |
gosper | \(\frac {{\mathrm e}^{x}}{2 \cosh \relax (x )-2 \sinh \relax (x )}\) | \(14\) |
default | \(\frac {2}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {2}{\tanh \left (\frac {x}{2}\right )-1}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 6, normalized size = 0.67 \[ \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 6, normalized size = 0.67 \[ \frac {{\mathrm {e}}^{2\,x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.49, size = 12, normalized size = 1.33 \[ \frac {e^{x}}{- 2 \sinh {\relax (x )} + 2 \cosh {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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