Integrand size = 21, antiderivative size = 64 \[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=\frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2}-\frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )} \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}{b n} \]
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Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2207, 2225} \[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=\frac {4 e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n^2}-\frac {4 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) e^{n \cos \left (\frac {a}{2}+\frac {b x}{2}\right )}}{b n} \]
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Rule 12
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int 2 e^{n x} x \, dx,x,\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \\ & = -\frac {4 \text {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 \text {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*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.56 \[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=-\frac {4 e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \left (-1+n \cos \left (\frac {1}{2} (a+b x)\right )\right )}{b n^2} \]
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Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{n \cos \left (\frac {a}{2}\right ) \cos \left (\frac {x b}{2}\right )-n \sin \left (\frac {a}{2}\right ) \sin \left (\frac {x b}{2}\right )} {\mathrm e}^{-\frac {i b x}{2}} {\mathrm e}^{-\frac {i a}{2}}}{b n}-\frac {2 \,{\mathrm e}^{n \cos \left (\frac {a}{2}\right ) \cos \left (\frac {x b}{2}\right )-n \sin \left (\frac {a}{2}\right ) \sin \left (\frac {x b}{2}\right )} {\mathrm e}^{\frac {i b x}{2}} {\mathrm e}^{\frac {i a}{2}}}{b n}+\frac {4 \,{\mathrm e}^{n \left (\cos \left (\frac {a}{2}\right ) \cos \left (\frac {x b}{2}\right )-\sin \left (\frac {a}{2}\right ) \sin \left (\frac {x b}{2}\right )\right )}}{b \,n^{2}}\) | \(123\) |
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none
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52 \[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=-\frac {4 \, {\left (n \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 1\right )} e^{\left (n \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}}{b n^{2}} \]
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\[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=\int e^{n \cos {\left (\frac {a}{2} + \frac {b x}{2} \right )}} \sin {\left (a + b x \right )}\, dx \]
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\[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=\int { e^{\left (n \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )} \sin \left (b x + a\right ) \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (50) = 100\).
Time = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.05 \[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=\frac {4 \, {\left (n e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )} + e^{\left (-\frac {n \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} - n}{\tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + 1}\right )}\right )}}{b n^{2} \tan \left (\frac {1}{4} \, b x + \frac {1}{4} \, a\right )^{2} + b n^{2}} \]
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Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52 \[ \int e^{n \cos \left (\frac {1}{2} (a+b x)\right )} \sin (a+b x) \, dx=-\frac {4\,{\mathrm {e}}^{n\,\cos \left (\frac {a}{2}+\frac {b\,x}{2}\right )}\,\left (n\,\cos \left (\frac {a}{2}+\frac {b\,x}{2}\right )-1\right )}{b\,n^2} \]
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