Optimal. Leaf size=18 \[ -\frac{e^{n \cos (a+b x)}}{b n} \]
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Rubi [A] time = 0.0144725, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4335, 2194} \[ -\frac{e^{n \cos (a+b x)}}{b n} \]
Antiderivative was successfully verified.
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Rule 4335
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cos (a+b x)} \sin (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int e^{n x} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{e^{n \cos (a+b x)}}{b n}\\ \end{align*}
Mathematica [A] time = 0.0510995, size = 18, normalized size = 1. \[ -\frac{e^{n \cos (a+b x)}}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 18, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{n\cos \left ( bx+a \right ) }}}{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984035, size = 23, normalized size = 1.28 \begin{align*} -\frac{e^{\left (n \cos \left (b x + a\right )\right )}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00534, size = 36, normalized size = 2. \begin{align*} -\frac{e^{\left (n \cos \left (b x + a\right )\right )}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.512365, size = 39, normalized size = 2.17 \begin{align*} \begin{cases} x \sin{\left (a \right )} & \text{for}\: b = 0 \wedge n = 0 \\x e^{n \cos{\left (a \right )}} \sin{\left (a \right )} & \text{for}\: b = 0 \\- \frac{\cos{\left (a + b x \right )}}{b} & \text{for}\: n = 0 \\- \frac{e^{n \cos{\left (a + b x \right )}}}{b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09581, size = 23, normalized size = 1.28 \begin{align*} -\frac{e^{\left (n \cos \left (b x + a\right )\right )}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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