Optimal. Leaf size=24 \[ -\frac{e^{n \cos (a c+b c x)}}{b c n} \]
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Rubi [A] time = 0.0144949, antiderivative size = 24, 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 = {4335, 2194} \[ -\frac{e^{n \cos (a c+b c x)}}{b c n} \]
Antiderivative was successfully verified.
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Rule 4335
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cos (c (a+b x))} \sin (a c+b c x) \, dx &=-\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\cos (a c+b c x)\right )}{b c}\\ &=-\frac{e^{n \cos (a c+b c x)}}{b c n}\\ \end{align*}
Mathematica [A] time = 0.0425428, size = 23, normalized size = 0.96 \[ -\frac{e^{n \cos (c (a+b x))}}{b c n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 24, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{n\cos \left ( bcx+ac \right ) }}}{cbn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960272, size = 31, normalized size = 1.29 \begin{align*} -\frac{e^{\left (n \cos \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 = 2.0295, size = 45, normalized size = 1.88 \begin{align*} -\frac{e^{\left (n \cos \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 = 2.76905, size = 51, normalized size = 2.12 \begin{align*} \begin{cases} 0 & \text{for}\: b = 0 \wedge c = 0 \wedge n = 0 \\x e^{n \cos{\left (a c \right )}} \sin{\left (a c \right )} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\- \frac{\cos{\left (a c + b c x \right )}}{b c} & \text{for}\: n = 0 \\- \frac{e^{n \cos{\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 [A] time = 1.14893, size = 31, normalized size = 1.29 \begin{align*} -\frac{e^{\left (n \cos \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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