Optimal. Leaf size=36 \[ \frac {e^{2 x}}{16}-\frac {1}{40} e^{2 x} \sin (4 x)-\frac {1}{80} e^{2 x} \cos (4 x) \]
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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 = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4469, 2194, 4433} \[ \frac {e^{2 x}}{16}-\frac {1}{40} e^{2 x} \sin (4 x)-\frac {1}{80} e^{2 x} \cos (4 x) \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4433
Rule 4469
Rubi steps
\begin {align*} \int e^{2 x} \cos ^2(x) \sin ^2(x) \, dx &=\int \left (\frac {e^{2 x}}{8}-\frac {1}{8} e^{2 x} \cos (4 x)\right ) \, dx\\ &=\frac {1}{8} \int e^{2 x} \, dx-\frac {1}{8} \int e^{2 x} \cos (4 x) \, dx\\ &=\frac {e^{2 x}}{16}-\frac {1}{80} e^{2 x} \cos (4 x)-\frac {1}{40} e^{2 x} \sin (4 x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 21, normalized size = 0.58 \[ -\frac {1}{80} e^{2 x} (2 \sin (4 x)+\cos (4 x)-5) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{2 x} \cos ^2(x) \sin ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.41, size = 40, normalized size = 1.11 \[ -\frac {1}{10} \, {\left (2 \, \cos \relax (x)^{3} - \cos \relax (x)\right )} e^{\left (2 \, x\right )} \sin \relax (x) - \frac {1}{20} \, {\left (2 \, \cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 24, normalized size = 0.67 \[ -\frac {1}{80} \, {\left (\cos \left (4 \, x\right ) + 2 \, \sin \left (4 \, x\right )\right )} e^{\left (2 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 28, normalized size = 0.78
method | result | size |
default | \(\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{2 x} \cos \left (4 x \right )}{80}-\frac {{\mathrm e}^{2 x} \sin \left (4 x \right )}{40}\) | \(28\) |
risch | \(\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{\left (2+4 i\right ) x}}{160}+\frac {i {\mathrm e}^{\left (2+4 i\right ) x}}{80}-\frac {{\mathrm e}^{\left (2-4 i\right ) x}}{160}-\frac {i {\mathrm e}^{\left (2-4 i\right ) x}}{80}\) | \(42\) |
norman | \(\frac {-\frac {{\mathrm e}^{2 x} \tan \left (\frac {x}{2}\right )}{5}+\frac {3 \,{\mathrm e}^{2 x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{5}+\frac {7 \,{\mathrm e}^{2 x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{5}-\frac {{\mathrm e}^{2 x} \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}-\frac {7 \,{\mathrm e}^{2 x} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}+\frac {3 \,{\mathrm e}^{2 x} \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{5}+\frac {{\mathrm e}^{2 x} \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{5}+\frac {{\mathrm e}^{2 x} \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{20}+\frac {{\mathrm e}^{2 x}}{20}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 27, normalized size = 0.75 \[ -\frac {1}{80} \, \cos \left (4 \, x\right ) e^{\left (2 \, x\right )} - \frac {1}{40} \, e^{\left (2 \, x\right )} \sin \left (4 \, x\right ) + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 18, normalized size = 0.50 \[ -\frac {{\mathrm {e}}^{2\,x}\,\left (\cos \left (4\,x\right )+2\,\sin \left (4\,x\right )-5\right )}{80} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.06, size = 70, normalized size = 1.94 \[ \frac {e^{2 x} \sin ^{4}{\relax (x )}}{20} + \frac {e^{2 x} \sin ^{3}{\relax (x )} \cos {\relax (x )}}{10} + \frac {e^{2 x} \sin ^{2}{\relax (x )} \cos ^{2}{\relax (x )}}{5} - \frac {e^{2 x} \sin {\relax (x )} \cos ^{3}{\relax (x )}}{10} + \frac {e^{2 x} \cos ^{4}{\relax (x )}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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