Integrand size = 7, antiderivative size = 30 \[ \int e^x x \cos (x) \, dx=\frac {1}{2} e^x x \cos (x)-\frac {1}{2} e^x \sin (x)+\frac {1}{2} e^x x \sin (x) \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4518, 4554, 4517} \[ \int e^x x \cos (x) \, dx=-\frac {1}{2} e^x \sin (x)+\frac {1}{2} e^x x \sin (x)+\frac {1}{2} e^x x \cos (x) \]
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Rule 4517
Rule 4518
Rule 4554
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} e^x x \cos (x)+\frac {1}{2} e^x x \sin (x)-\int \left (\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x)\right ) \, dx \\ & = \frac {1}{2} e^x x \cos (x)+\frac {1}{2} e^x x \sin (x)-\frac {1}{2} \int e^x \cos (x) \, dx-\frac {1}{2} \int e^x \sin (x) \, dx \\ & = \frac {1}{2} e^x x \cos (x)-\frac {1}{2} e^x \sin (x)+\frac {1}{2} e^x x \sin (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60 \[ \int e^x x \cos (x) \, dx=\frac {1}{2} e^x (x \cos (x)+(-1+x) \sin (x)) \]
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Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{x} \left (\left (-1+x \right ) \sin \left (x \right )+x \cos \left (x \right )\right )}{2}\) | \(16\) |
default | \(\frac {{\mathrm e}^{x} x \cos \left (x \right )}{2}-\left (-\frac {x}{2}+\frac {1}{2}\right ) {\mathrm e}^{x} \sin \left (x \right )\) | \(20\) |
risch | \(\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-1+i+2 x \right ) {\mathrm e}^{\left (1+i\right ) x}+\left (\frac {1}{8}+\frac {i}{8}\right ) \left (-1-i+2 x \right ) {\mathrm e}^{\left (1-i\right ) x}\) | \(36\) |
norman | \(\frac {{\mathrm e}^{x} x \tan \left (\frac {x}{2}\right )+\frac {{\mathrm e}^{x} x}{2}-{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )-\frac {{\mathrm e}^{x} x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(45\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int e^x x \cos (x) \, dx=\frac {1}{2} \, x \cos \left (x\right ) e^{x} + \frac {1}{2} \, {\left (x - 1\right )} e^{x} \sin \left (x\right ) \]
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Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int e^x x \cos (x) \, dx=\frac {x e^{x} \sin {\left (x \right )}}{2} + \frac {x e^{x} \cos {\left (x \right )}}{2} - \frac {e^{x} \sin {\left (x \right )}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int e^x x \cos (x) \, dx=\frac {1}{2} \, x \cos \left (x\right ) e^{x} + \frac {1}{2} \, {\left (x - 1\right )} e^{x} \sin \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.50 \[ \int e^x x \cos (x) \, dx=\frac {1}{2} \, {\left (x \cos \left (x\right ) + {\left (x - 1\right )} \sin \left (x\right )\right )} e^{x} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int e^x x \cos (x) \, dx=\frac {{\mathrm {e}}^x\,\left (x\,\cos \left (x\right )-\sin \left (x\right )+x\,\sin \left (x\right )\right )}{2} \]
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