Optimal. Leaf size=27 \[ \frac {1}{10} e^x \cos (4+3 x)+\frac {3}{10} e^x \sin (4+3 x) \]
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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 {3}{10} e^x \sin (3 x+4)+\frac {1}{10} e^x \cos (3 x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 4518
Rubi steps
\begin {align*} \int e^x \cos (4+3 x) \, dx &=\frac {1}{10} e^x \cos (4+3 x)+\frac {3}{10} e^x \sin (4+3 x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{10} e^x (\cos (4+3 x)+3 \sin (4+3 x)) \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 | \(\frac {{\mathrm e}^{x} \cos \left (3 x +4\right )}{10}+\frac {3 \,{\mathrm e}^{x} \sin \left (3 x +4\right )}{10}\) | \(22\) |
risch | \(\left (\frac {1}{20}-\frac {3 i}{20}\right ) {\mathrm e}^{x} {\mathrm e}^{3 i x} {\mathrm e}^{4 i}+\left (\frac {1}{20}+\frac {3 i}{20}\right ) {\mathrm e}^{x} {\mathrm e}^{-3 i x} {\mathrm e}^{-4 i}\) | \(30\) |
norman | \(\frac {\frac {3 \,{\mathrm e}^{x} \tan \left (\frac {3 x}{2}+2\right )}{5}-\frac {{\mathrm e}^{x} \left (\tan ^{2}\left (\frac {3 x}{2}+2\right )\right )}{10}+\frac {{\mathrm e}^{x}}{10}}{1+\tan ^{2}\left (\frac {3 x}{2}+2\right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.11, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{10} \, \cos \left (3 \, x + 4\right ) e^{x} + \frac {3}{10} \, e^{x} \sin \left (3 \, x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 24, normalized size = 0.89 \begin {gather*} \frac {3 e^{x} \sin {\left (3 x + 4 \right )}}{10} + \frac {e^{x} \cos {\left (3 x + 4 \right )}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{10} \, {\left (\cos \left (3 \, x + 4\right ) + 3 \, \sin \left (3 \, x + 4\right )\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^x\,\left (\cos \left (3\,x+4\right )+3\,\sin \left (3\,x+4\right )\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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