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 = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.98, size = 33, normalized size = 0.52 \[ \frac {4 \, {\left (n \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 1\right )} e^{\left (n \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}}{b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 138, normalized size = 2.16 \[ \frac {4 \, {\left (2 \, n e^{\left (\frac {2 \, n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right ) - e^{\left (\frac {2 \, n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - e^{\left (\frac {2 \, n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )}\right )}}{b n^{2} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + b n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 122, normalized size = 1.91 \[ \frac {2 i {\mathrm e}^{n \sin \left (\frac {b x}{2}\right ) \cos \left (\frac {a}{2}\right )+n \cos \left (\frac {b x}{2}\right ) \sin \left (\frac {a}{2}\right )} {\mathrm e}^{-\frac {i b x}{2}} {\mathrm e}^{-\frac {i a}{2}}}{n b}-\frac {2 i {\mathrm e}^{n \sin \left (\frac {b x}{2}\right ) \cos \left (\frac {a}{2}\right )+n \cos \left (\frac {b x}{2}\right ) \sin \left (\frac {a}{2}\right )} {\mathrm e}^{\frac {i b x}{2}} {\mathrm e}^{\frac {i a}{2}}}{n b}-\frac {4 \,{\mathrm e}^{n \left (\sin \left (\frac {b x}{2}\right ) \cos \left (\frac {a}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin \left (\frac {a}{2}\right )\right )}}{n^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (n \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.18, size = 33, normalized size = 0.52 \[ \frac {4\,{\mathrm {e}}^{n\,\sin \left (\frac {a}{2}+\frac {b\,x}{2}\right )}\,\left (n\,\sin \left (\frac {a}{2}+\frac {b\,x}{2}\right )-1\right )}{b\,n^2} \]
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 \[ \int e^{n \sin {\left (\frac {a}{2} + \frac {b x}{2} \right )}} \sin {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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