Optimal. Leaf size=13 \[ \frac{e^{(m-1) x}}{m-1} \]
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Rubi [A] time = 0.0291286, antiderivative size = 19, normalized size of antiderivative = 1.46, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5648, 2227, 2194} \[ -\frac{e^{-(1-m) x}}{1-m} \]
Antiderivative was successfully verified.
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Rule 5648
Rule 2227
Rule 2194
Rubi steps
\begin{align*} \int \frac{e^{m x}}{\cosh (x)+\sinh (x)} \, dx &=\int e^{-x+m x} \, dx\\ &=\int e^{-(1-m) x} \, dx\\ &=-\frac{e^{-(1-m) x}}{1-m}\\ \end{align*}
Mathematica [A] time = 0.0302059, size = 18, normalized size = 1.38 \[ \frac{e^{m x} (\cosh (x)-\sinh (x))}{m-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 1.4 \begin{align*}{\frac{{{\rm e}^{mx}}}{ \left ( -1+m \right ) \left ( \cosh \left ( x \right ) +\sinh \left ( x \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11526, size = 82, normalized size = 6.31 \begin{align*} \frac{\cosh \left (m x\right ) + \sinh \left (m x\right )}{{\left (m - 1\right )} \cosh \left (x\right ) +{\left (m - 1\right )} \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.54703, size = 32, normalized size = 2.46 \begin{align*} \begin{cases} \frac{e^{m x}}{m \sinh{\left (x \right )} + m \cosh{\left (x \right )} - \sinh{\left (x \right )} - \cosh{\left (x \right )}} & \text{for}\: m \neq 1 \\\frac{x e^{x}}{\sinh{\left (x \right )} + \cosh{\left (x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11001, size = 22, normalized size = 1.69 \begin{align*} \frac{e^{\left (m x\right )}}{m e^{x} - e^{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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