Optimal. Leaf size=64 \[ \frac{4 e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2}-\frac{4 \cos \left (\frac{a}{2}+\frac{b x}{2}\right ) e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n} \]
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Rubi [A] time = 0.0374376, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {12, 2176, 2194} \[ \frac{4 e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2}-\frac{4 \cos \left (\frac{a}{2}+\frac{b x}{2}\right ) e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{n \cos \left (\frac{1}{2} (a+b x)\right )} \sin (a+b x) \, dx &=-\frac{2 \operatorname{Subst}\left (\int 2 e^{n x} x \, dx,x,\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\\ &=-\frac{4 \operatorname{Subst}\left (\int e^{n x} x \, dx,x,\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}\\ &=-\frac{4 e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )} \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}{b n}+\frac{4 \operatorname{Subst}\left (\int e^{n x} \, dx,x,\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b n}\\ &=\frac{4 e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}}{b n^2}-\frac{4 e^{n \cos \left (\frac{a}{2}+\frac{b x}{2}\right )} \cos \left (\frac{a}{2}+\frac{b x}{2}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0340926, size = 36, normalized size = 0.56 \[ -\frac{4 e^{n \cos \left (\frac{1}{2} (a+b x)\right )} \left (n \cos \left (\frac{1}{2} (a+b x)\right )-1\right )}{b n^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0., size = 124, normalized size = 1.9 \begin{align*} -2\,{\frac{{{\rm e}^{n\cos \left ( a/2 \right ) \cos \left ( 1/2\,bx \right ) -n\sin \left ( a/2 \right ) \sin \left ( 1/2\,bx \right ) }}{{\rm e}^{i/2bx}}{{\rm e}^{i/2a}}}{bn}}-2\,{\frac{{{\rm e}^{n\cos \left ( a/2 \right ) \cos \left ( 1/2\,bx \right ) -n\sin \left ( a/2 \right ) \sin \left ( 1/2\,bx \right ) }}{{\rm e}^{-i/2bx}}{{\rm e}^{-i/2a}}}{bn}}+4\,{\frac{{{\rm e}^{-n \left ( \sin \left ( a/2 \right ) \sin \left ( 1/2\,bx \right ) -\cos \left ( a/2 \right ) \cos \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 \cos \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 = 2.17702, size = 92, normalized size = 1.44 \begin{align*} -\frac{4 \,{\left (n \cos \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) - 1\right )} e^{\left (n \cos \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 \cos{\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 = 1.25338, size = 263, normalized size = 4.11 \begin{align*} \frac{4 \,{\left (n e^{\left (-\frac{n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} - n}{\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{n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} - n}{\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} - n e^{\left (-\frac{n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} - n}{\tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} + 1}\right )} + e^{\left (-\frac{n \tan \left (\frac{1}{4} \, b x + \frac{1}{4} \, a\right )^{2} - n}{\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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