Optimal. Leaf size=43 \[ \frac {2 \sin (a+b x) e^{n \sin (a+b x)}}{b n}-\frac {2 e^{n \sin (a+b x)}}{b n^2} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2176, 2194} \[ \frac {2 \sin (a+b x) e^{n \sin (a+b x)}}{b n}-\frac {2 e^{n \sin (a+b x)}}{b n^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{n \sin (a+b x)} \sin (2 a+2 b x) \, dx &=\frac {\operatorname {Subst}\left (\int 2 e^{n x} x \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int e^{n x} x \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {2 e^{n \sin (a+b x)} \sin (a+b x)}{b n}-\frac {2 \operatorname {Subst}\left (\int e^{n x} \, dx,x,\sin (a+b x)\right )}{b n}\\ &=-\frac {2 e^{n \sin (a+b x)}}{b n^2}+\frac {2 e^{n \sin (a+b x)} \sin (a+b x)}{b n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 28, normalized size = 0.65 \[ \frac {2 e^{n \sin (a+b x)} (n \sin (a+b x)-1)}{b n^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 27, normalized size = 0.63 \[ \frac {2 \, {\left (n \sin \left (b x + a\right ) - 1\right )} e^{\left (n \sin \left (b x + a\right )\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \sin \left (b x + a\right )\right )} \sin \left (2 \, b x + 2 \, a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 104, normalized size = 2.42 \[ -\frac {i {\mathrm e}^{n \sin \left (b x \right ) \cos \relax (a )+n \cos \left (b x \right ) \sin \relax (a )} {\mathrm e}^{i b x} {\mathrm e}^{i a}}{n b}+\frac {i {\mathrm e}^{n \sin \left (b x \right ) \cos \relax (a )+n \cos \left (b x \right ) \sin \relax (a )} {\mathrm e}^{-i b x} {\mathrm e}^{-i a}}{n b}-\frac {2 \,{\mathrm e}^{n \left (\sin \left (b x \right ) \cos \relax (a )+\cos \left (b x \right ) \sin \relax (a )\right )}}{n^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 37, normalized size = 0.86 \[ \frac {2 \, {\left (n e^{\left (n \sin \left (b x + a\right )\right )} \sin \left (b x + a\right ) - e^{\left (n \sin \left (b x + a\right )\right )}\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.18, size = 27, normalized size = 0.63 \[ \frac {2\,{\mathrm {e}}^{n\,\sin \left (a+b\,x\right )}\,\left (n\,\sin \left (a+b\,x\right )-1\right )}{b\,n^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n \sin {\left (a + b x \right )}} \sin {\left (2 a + 2 b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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