Optimal. Leaf size=36 \[ \frac {e^{3 x}}{24}-\frac {1}{60} e^{3 x} \sin (6 x)-\frac {1}{120} e^{3 x} \cos (6 x) \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4469, 2194, 4433} \[ \frac {e^{3 x}}{24}-\frac {1}{60} e^{3 x} \sin (6 x)-\frac {1}{120} e^{3 x} \cos (6 x) \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4433
Rule 4469
Rubi steps
\begin {align*} \int e^{3 x} \cos ^2\left (\frac {3 x}{2}\right ) \sin ^2\left (\frac {3 x}{2}\right ) \, dx &=\int \left (\frac {e^{3 x}}{8}-\frac {1}{8} e^{3 x} \cos (6 x)\right ) \, dx\\ &=\frac {1}{8} \int e^{3 x} \, dx-\frac {1}{8} \int e^{3 x} \cos (6 x) \, dx\\ &=\frac {e^{3 x}}{24}-\frac {1}{120} e^{3 x} \cos (6 x)-\frac {1}{60} e^{3 x} \sin (6 x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 21, normalized size = 0.58 \[ -\frac {1}{120} e^{3 x} (2 \sin (6 x)+\cos (6 x)-5) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{3 x} \cos ^2\left (\frac {3 x}{2}\right ) \sin ^2\left (\frac {3 x}{2}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.37, size = 50, normalized size = 1.39 \[ -\frac {1}{15} \, {\left (2 \, \cos \left (\frac {3}{2} \, x\right )^{3} - \cos \left (\frac {3}{2} \, x\right )\right )} e^{\left (3 \, x\right )} \sin \left (\frac {3}{2} \, x\right ) - \frac {1}{30} \, {\left (2 \, \cos \left (\frac {3}{2} \, x\right )^{4} - 2 \, \cos \left (\frac {3}{2} \, x\right )^{2} - 1\right )} e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 24, normalized size = 0.67 \[ -\frac {1}{120} \, {\left (\cos \left (6 \, x\right ) + 2 \, \sin \left (6 \, x\right )\right )} e^{\left (3 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 42, normalized size = 1.17
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{24}-\frac {{\mathrm e}^{\left (3+6 i\right ) x}}{240}+\frac {i {\mathrm e}^{\left (3+6 i\right ) x}}{120}-\frac {{\mathrm e}^{\left (3-6 i\right ) x}}{240}-\frac {i {\mathrm e}^{\left (3-6 i\right ) x}}{120}\) | \(42\) |
norman | \(\frac {-\frac {2 \,{\mathrm e}^{3 x} \tan \left (\frac {3 x}{4}\right )}{15}+\frac {2 \,{\mathrm e}^{3 x} \left (\tan ^{2}\left (\frac {3 x}{4}\right )\right )}{5}+\frac {14 \,{\mathrm e}^{3 x} \left (\tan ^{3}\left (\frac {3 x}{4}\right )\right )}{15}-\frac {{\mathrm e}^{3 x} \left (\tan ^{4}\left (\frac {3 x}{4}\right )\right )}{3}-\frac {14 \,{\mathrm e}^{3 x} \left (\tan ^{5}\left (\frac {3 x}{4}\right )\right )}{15}+\frac {2 \,{\mathrm e}^{3 x} \left (\tan ^{6}\left (\frac {3 x}{4}\right )\right )}{5}+\frac {2 \,{\mathrm e}^{3 x} \left (\tan ^{7}\left (\frac {3 x}{4}\right )\right )}{15}+\frac {{\mathrm e}^{3 x} \left (\tan ^{8}\left (\frac {3 x}{4}\right )\right )}{30}+\frac {{\mathrm e}^{3 x}}{30}}{\left (1+\tan ^{2}\left (\frac {3 x}{4}\right )\right )^{4}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 27, normalized size = 0.75 \[ -\frac {1}{120} \, \cos \left (6 \, x\right ) e^{\left (3 \, x\right )} - \frac {1}{60} \, e^{\left (3 \, x\right )} \sin \left (6 \, x\right ) + \frac {1}{24} \, e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 18, normalized size = 0.50 \[ -\frac {{\mathrm {e}}^{3\,x}\,\left (\cos \left (6\,x\right )+2\,\sin \left (6\,x\right )-5\right )}{120} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.87, size = 99, normalized size = 2.75 \[ \frac {e^{3 x} \sin ^{4}{\left (\frac {3 x}{2} \right )}}{30} + \frac {e^{3 x} \sin ^{3}{\left (\frac {3 x}{2} \right )} \cos {\left (\frac {3 x}{2} \right )}}{15} + \frac {2 e^{3 x} \sin ^{2}{\left (\frac {3 x}{2} \right )} \cos ^{2}{\left (\frac {3 x}{2} \right )}}{15} - \frac {e^{3 x} \sin {\left (\frac {3 x}{2} \right )} \cos ^{3}{\left (\frac {3 x}{2} \right )}}{15} + \frac {e^{3 x} \cos ^{4}{\left (\frac {3 x}{2} \right )}}{30} \]
Verification of antiderivative is not currently implemented for this CAS.
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