Optimal. Leaf size=23 \[ -\frac {1}{2} e^{-x} \sin (x)-\frac {1}{2} e^{-x} \cos (x) \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4432} \[ -\frac {1}{2} e^{-x} \sin (x)-\frac {1}{2} e^{-x} \cos (x) \]
Antiderivative was successfully verified.
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Rule 4432
Rubi steps
\begin {align*} \int e^{-x} \sin (x) \, dx &=-\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^{-x} \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 14, normalized size = 0.61 \[ -\frac {1}{2} e^{-x} (\sin (x)+\cos (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-x} \sin (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.88, size = 17, normalized size = 0.74 \[ -\frac {1}{2} \, \cos \relax (x) e^{\left (-x\right )} - \frac {1}{2} \, e^{\left (-x\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 11, normalized size = 0.48 \[ -\frac {1}{2} \, {\left (\cos \relax (x) + \sin \relax (x)\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.78
method | result | size |
default | \(-\frac {{\mathrm e}^{-x} \cos \relax (x )}{2}-\frac {{\mathrm e}^{-x} \sin \relax (x )}{2}\) | \(18\) |
norman | \(\frac {\left (-\frac {1}{2}+\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-x}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(32\) |
risch | \(-\frac {{\mathrm e}^{\left (-1+i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (-1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (-1-i\right ) x}}{4}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 11, normalized size = 0.48 \[ -\frac {1}{2} \, {\left (\cos \relax (x) + \sin \relax (x)\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 11, normalized size = 0.48 \[ -\frac {{\mathrm {e}}^{-x}\,\left (\cos \relax (x)+\sin \relax (x)\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 17, normalized size = 0.74 \[ - \frac {e^{- x} \sin {\relax (x )}}{2} - \frac {e^{- x} \cos {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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