Optimal. Leaf size=24 \[ -\frac {e^{n \cos (a c+b c x)}}{b c n} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 2194
Rule 4335
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.90, size = 23, normalized size = 0.96 \[ -\frac {e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 0.96 \[ -\frac {e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 1.00 \[ -\frac {{\mathrm e}^{n \cos \left (b c x +a c \right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 23, normalized size = 0.96 \[ -\frac {e^{\left (n \cos \left (b c x + a c\right )\right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.02, size = 23, normalized size = 0.96 \[ -\frac {{\mathrm {e}}^{n\,\cos \left (a\,c+b\,c\,x\right )}}{b\,c\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.23, size = 54, normalized size = 2.25 \[ \begin {cases} 0 & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \wedge \left (c = 0 \vee n = 0\right ) \\x e^{n \cos {\left (a c \right )}} \sin {\left (a c \right )} & \text {for}\: b = 0 \\- \frac {\cos {\left (a c + b c x \right )}}{b c} & \text {for}\: n = 0 \\- \frac {e^{n \cos {\left (a c + b c x \right )}}}{b c n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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