Optimal. Leaf size=41 \[ -\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^x \cos (x)-\frac {1}{2} e^{-x} \sin (x)+\frac {1}{2} e^x \sin (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4517}
\begin {gather*} -\frac {1}{2} e^{-x} \sin (x)+\frac {1}{2} e^x \sin (x)-\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^x \cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4517
Rubi steps
\begin {align*} \int \left (e^{-x} \sin (x)+e^x \sin (x)\right ) \, dx &=\int e^{-x} \sin (x) \, dx+\int e^x \sin (x) \, dx\\ &=-\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^x \cos (x)-\frac {1}{2} e^{-x} \sin (x)+\frac {1}{2} e^x \sin (x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 33, normalized size = 0.80 \begin {gather*} -\frac {1}{2} e^x \left (1+e^{-2 x}\right ) \cos (x)-\frac {1}{2} e^x \left (-1+e^{-2 x}\right ) \sin (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 30, normalized size = 0.73
method | result | size |
default | \(-\frac {{\mathrm e}^{x} \cos \left (x \right )}{2}+\frac {{\mathrm e}^{x} \sin \left (x \right )}{2}-\frac {{\mathrm e}^{-x} \cos \left (x \right )}{2}-\frac {{\mathrm e}^{-x} \sin \left (x \right )}{2}\) | \(30\) |
norman | \(\frac {\left (-\frac {1}{2}+{\mathrm e}^{2 x} \tan \left (\frac {x}{2}\right )-\frac {{\mathrm e}^{2 x}}{2}+\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}+\frac {{\mathrm e}^{2 x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-x}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(59\) |
risch | \(-\frac {{\mathrm e}^{\left (-1+i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (-1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {{\mathrm e}^{\left (1+i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (1-i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (1-i\right ) x}}{4}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 23, normalized size = 0.56 \begin {gather*} -\frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} - \frac {1}{2} \, {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.47, size = 26, normalized size = 0.63 \begin {gather*} -\frac {1}{2} \, {\left (\cos \left (x\right ) e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} - 1\right )} \sin \left (x\right ) + \cos \left (x\right )\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 32, normalized size = 0.78 \begin {gather*} \frac {e^{x} \sin {\left (x \right )}}{2} - \frac {e^{x} \cos {\left (x \right )}}{2} - \frac {e^{- x} \sin {\left (x \right )}}{2} - \frac {e^{- x} \cos {\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 23, normalized size = 0.56 \begin {gather*} -\frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} - \frac {1}{2} \, {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.41, size = 31, normalized size = 0.76 \begin {gather*} -{\mathrm {e}}^{-x}\,\left (\frac {\cos \left (x\right )}{2}+\frac {\sin \left (x\right )}{2}+\frac {{\mathrm {e}}^{2\,x}\,\cos \left (x\right )}{2}-\frac {{\mathrm {e}}^{2\,x}\,\sin \left (x\right )}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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