Optimal. Leaf size=27 \[ \frac {3}{34} e^{3 x} \cos (5 x)+\frac {5}{34} e^{3 x} \sin (5 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4518}
\begin {gather*} \frac {5}{34} e^{3 x} \sin (5 x)+\frac {3}{34} e^{3 x} \cos (5 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4518
Rubi steps
\begin {align*} \int e^{3 x} \cos (5 x) \, dx &=\frac {3}{34} e^{3 x} \cos (5 x)+\frac {5}{34} e^{3 x} \sin (5 x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{34} e^{3 x} (3 \cos (5 x)+5 \sin (5 x)) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.89, size = 20, normalized size = 0.74 \begin {gather*} \frac {\left (3 \text {Cos}\left [5 x\right ]+5 \text {Sin}\left [5 x\right ]\right ) E^{3 x}}{34} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 22, normalized size = 0.81
method | result | size |
default | \(\frac {3 \,{\mathrm e}^{3 x} \cos \left (5 x \right )}{34}+\frac {5 \,{\mathrm e}^{3 x} \sin \left (5 x \right )}{34}\) | \(22\) |
risch | \(\frac {3 \,{\mathrm e}^{\left (3+5 i\right ) x}}{68}-\frac {5 i {\mathrm e}^{\left (3+5 i\right ) x}}{68}+\frac {3 \,{\mathrm e}^{\left (3-5 i\right ) x}}{68}+\frac {5 i {\mathrm e}^{\left (3-5 i\right ) x}}{68}\) | \(36\) |
norman | \(\frac {\frac {5 \,{\mathrm e}^{3 x} \tan \left (\frac {5 x}{2}\right )}{17}-\frac {3 \,{\mathrm e}^{3 x} \left (\tan ^{2}\left (\frac {5 x}{2}\right )\right )}{34}+\frac {3 \,{\mathrm e}^{3 x}}{34}}{1+\tan ^{2}\left (\frac {5 x}{2}\right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{34} \, {\left (3 \, \cos \left (5 \, x\right ) + 5 \, \sin \left (5 \, x\right )\right )} e^{\left (3 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 21, normalized size = 0.78 \begin {gather*} \frac {3}{34} \, \cos \left (5 \, x\right ) e^{\left (3 \, x\right )} + \frac {5}{34} \, e^{\left (3 \, x\right )} \sin \left (5 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 26, normalized size = 0.96 \begin {gather*} \frac {5 e^{3 x} \sin {\left (5 x \right )}}{34} + \frac {3 e^{3 x} \cos {\left (5 x \right )}}{34} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 23, normalized size = 0.85 \begin {gather*} \mathrm {e}^{3 x} \left (\frac {3}{34} \cos \left (5 x\right )+\frac {5}{34} \sin \left (5 x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^{3\,x}\,\left (3\,\cos \left (5\,x\right )+5\,\sin \left (5\,x\right )\right )}{34} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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