Optimal. Leaf size=23 \[ \frac{e^{n \sinh (a c+b c x)}}{b c n} \]
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Rubi [A] time = 0.0135506, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4336, 2194} \[ \frac{e^{n \sinh (a c+b c x)}}{b c n} \]
Antiderivative was successfully verified.
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Rule 4336
Rule 2194
Rubi steps
\begin{align*} \int e^{n \sinh (c (a+b x))} \cosh (a c+b c x) \, dx &=\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\sinh (a c+b c x)\right )}{b c}\\ &=\frac{e^{n \sinh (a c+b c x)}}{b c n}\\ \end{align*}
Mathematica [A] time = 0.0416347, size = 23, normalized size = 1. \[ \frac{e^{n \sinh (a c+b c x)}}{b c n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 23, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{n\sinh \left ( bcx+ac \right ) }}}{cbn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05436, size = 30, normalized size = 1.3 \begin{align*} \frac{e^{\left (n \sinh \left (b c x + a c\right )\right )}}{b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01133, size = 88, normalized size = 3.83 \begin{align*} \frac{\cosh \left (n \sinh \left (b c x + a c\right )\right ) + \sinh \left (n \sinh \left (b c x + a c\right )\right )}{b c n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.69556, size = 48, normalized size = 2.09 \begin{align*} \begin{cases} x & \text{for}\: b = 0 \wedge c = 0 \wedge n = 0 \\x e^{n \sinh{\left (a c \right )}} \cosh{\left (a c \right )} & \text{for}\: b = 0 \\\frac{\sinh{\left (a c + b c x \right )}}{b c} & \text{for}\: n = 0 \\x & \text{for}\: c = 0 \\\frac{e^{n \sinh{\left (a c + b c x \right )}}}{b c n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b c x + a c\right ) e^{\left (n \sinh \left ({\left (b x + a\right )} c\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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