Optimal. Leaf size=19 \[ -\frac {1}{2} e^x \cos (x)+\frac {1}{2} e^x \sin (x) \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4517}
\begin {gather*} \frac {1}{2} e^x \sin (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 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 = 0.01, size = 14, normalized size = 0.74 \begin {gather*} \frac {1}{2} e^x (-\cos (x)+\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 14, normalized size = 0.74
method | result | size |
default | \(-\frac {{\mathrm e}^{x} \cos \left (x \right )}{2}+\frac {{\mathrm e}^{x} \sin \left (x \right )}{2}\) | \(14\) |
norman | \(\frac {{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )+\frac {{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {{\mathrm e}^{x}}{2}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(34\) |
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 = 2.87, size = 11, normalized size = 0.58 \begin {gather*} -\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 = 1.40, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{2} \, \cos \left (x\right ) e^{x} + \frac {1}{2} \, e^{x} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 15, normalized size = 0.79 \begin {gather*} \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 = 0.42, size = 11, normalized size = 0.58 \begin {gather*} -\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 = 0.02, size = 11, normalized size = 0.58 \begin {gather*} -\frac {{\mathrm {e}}^x\,\left (\cos \left (x\right )-\sin \left (x\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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