Optimal. Leaf size=64 \[ \frac{4 \sin \left (\frac{a}{2}+\frac{b x}{2}\right ) e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n}-\frac{4 e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2} \]
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Rubi [A] time = 0.0362031, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {12, 2176, 2194} \[ \frac{4 \sin \left (\frac{a}{2}+\frac{b x}{2}\right ) e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n}-\frac{4 e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )} \sin (a+b x) \, dx &=\frac{2 \operatorname{Subst}\left (\int 2 e^{n x} x \, dx,x,\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\\ &=\frac{4 \operatorname{Subst}\left (\int e^{n x} x \, dx,x,\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\\ &=\frac{4 e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )} \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}{b n}-\frac{4 \operatorname{Subst}\left (\int e^{n x} \, dx,x,\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b n}\\ &=-\frac{4 e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2}+\frac{4 e^{n \sin \left (\frac{a}{2}+\frac{b x}{2}\right )} \sin \left (\frac{a}{2}+\frac{b x}{2}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.067914, size = 36, normalized size = 0.56 \[ \frac{4 e^{n \sin \left (\frac{1}{2} (a+b x)\right )} \left (n \sin \left (\frac{1}{2} (a+b x)\right )-1\right )}{b n^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.081, size = 122, normalized size = 1.9 \begin{align*}{\frac{2\,i{{\rm e}^{-{\frac{i}{2}}bx}}{{\rm e}^{-{\frac{i}{2}}a}}}{nb}{{\rm e}^{n\sin \left ({\frac{a}{2}} \right ) \cos \left ({\frac{bx}{2}} \right ) +n\cos \left ({\frac{a}{2}} \right ) \sin \left ({\frac{bx}{2}} \right ) }}}-{\frac{2\,i{{\rm e}^{{\frac{i}{2}}bx}}{{\rm e}^{{\frac{i}{2}}a}}}{nb}{{\rm e}^{n\sin \left ({\frac{a}{2}} \right ) \cos \left ({\frac{bx}{2}} \right ) +n\cos \left ({\frac{a}{2}} \right ) \sin \left ({\frac{bx}{2}} \right ) }}}-4\,{\frac{{{\rm e}^{n \left ( \sin \left ( a/2 \right ) \cos \left ( 1/2\,bx \right ) +\cos \left ( a/2 \right ) \sin \left ( 1/2\,bx \right ) \right ) }}}{b{n}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (n \sin \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right )} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13023, size = 90, normalized size = 1.41 \begin{align*} \frac{4 \,{\left (n \sin \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) - 1\right )} e^{\left (n \sin \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right )}}{b n^{2}} \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 \sin{\left (\frac{a}{2} + \frac{b x}{2} \right )}} \sin{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24357, size = 186, normalized size = 2.91 \begin{align*} \frac{4 \,{\left (2 \, n e^{\left (\frac{2 \, n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )}{\tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right ) - e^{\left (\frac{2 \, n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )}{\tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} - e^{\left (\frac{2 \, n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )}{\tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} + 1}\right )}\right )}}{b n^{2} \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} + b n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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