Optimal. Leaf size=22 \[ \frac{e^{n \cosh (c (a+b x))}}{b c n} \]
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Rubi [A] time = 0.0167992, antiderivative size = 22, 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 = {4337, 2194} \[ \frac{e^{n \cosh (c (a+b x))}}{b c n} \]
Antiderivative was successfully verified.
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Rule 4337
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cosh (a c+b c x)} \sinh (c (a+b x)) \, dx &=\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\cosh (c (a+b x))\right )}{b c}\\ &=\frac{e^{n \cosh (c (a+b x))}}{b c n}\\ \end{align*}
Mathematica [A] time = 0.208852, size = 22, normalized size = 1. \[ \frac{e^{n \cosh (c (a+b x))}}{b c n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 23, normalized size = 1.1 \begin{align*}{\frac{{{\rm e}^{n\cosh \left ( bcx+ac \right ) }}}{cbn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984368, size = 30, normalized size = 1.36 \begin{align*} \frac{e^{\left (n \cosh \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 = 2.3356, size = 88, normalized size = 4. \begin{align*} \frac{\cosh \left (n \cosh \left (b c x + a c\right )\right ) + \sinh \left (n \cosh \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{n \cosh{\left (a c + b c x \right )}} \sinh{\left (a c + b c x \right )}\, dx \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 e^{\left (n \cosh \left (b c x + a c\right )\right )} \sinh \left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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