Optimal. Leaf size=30 \[ -\frac {1}{2} e^x \sin (x)+\frac {1}{2} e^x x \sin (x)+\frac {1}{2} e^x x \cos (x) \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4433, 4466, 4432} \[ -\frac {1}{2} e^x \sin (x)+\frac {1}{2} e^x x \sin (x)+\frac {1}{2} e^x x \cos (x) \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4433
Rule 4466
Rubi steps
\begin {align*} \int e^x x \cos (x) \, dx &=\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*}
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Mathematica [A] time = 0.03, size = 18, normalized size = 0.60 \[ \frac {1}{2} e^x ((x-1) \sin (x)+x \cos (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^x x \cos (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.38, size = 17, normalized size = 0.57 \[ \frac {1}{2} \, x \cos \relax (x) e^{x} + \frac {1}{2} \, {\left (x - 1\right )} e^{x} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 15, normalized size = 0.50 \[ \frac {1}{2} \, {\left (x \cos \relax (x) + {\left (x - 1\right )} \sin \relax (x)\right )} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.67
method | result | size |
default | \(\frac {{\mathrm e}^{x} x \cos \relax (x )}{2}-\left (-\frac {x}{2}+\frac {1}{2}\right ) {\mathrm e}^{x} \sin \relax (x )\) | \(20\) |
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 | \(\frac {{\mathrm e}^{x} x \tan \left (\frac {x}{2}\right )+\frac {{\mathrm e}^{x} x}{2}-{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )-\frac {{\mathrm e}^{x} x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 17, normalized size = 0.57 \[ \frac {1}{2} \, x \cos \relax (x) e^{x} + \frac {1}{2} \, {\left (x - 1\right )} e^{x} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 17, normalized size = 0.57 \[ \frac {{\mathrm {e}}^x\,\left (x\,\cos \relax (x)-\sin \relax (x)+x\,\sin \relax (x)\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 27, normalized size = 0.90 \[ \frac {x e^{x} \sin {\relax (x )}}{2} + \frac {x e^{x} \cos {\relax (x )}}{2} - \frac {e^{x} \sin {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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