Optimal. Leaf size=36 \[ \frac {e^{2 x}}{16}-\frac {1}{80} e^{2 x} \cos (4 x)-\frac {1}{40} e^{2 x} \sin (4 x) \]
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Rubi [A]
time = 0.03, 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 = {4557, 2225,
4518} \begin {gather*} \frac {e^{2 x}}{16}-\frac {1}{40} e^{2 x} \sin (4 x)-\frac {1}{80} e^{2 x} \cos (4 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 4518
Rule 4557
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 = 0.03, size = 21, normalized size = 0.58 \begin {gather*} -\frac {1}{80} e^{2 x} (-5+\cos (4 x)+2 \sin (4 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, 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 = 2.39, size = 27, normalized size = 0.75 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 40, normalized size = 1.11 \begin {gather*} -\frac {1}{10} \, {\left (2 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} e^{\left (2 \, x\right )} \sin \left (x\right ) - \frac {1}{20} \, {\left (2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (29) = 58\).
time = 0.69, size = 70, normalized size = 1.94 \begin {gather*} \frac {e^{2 x} \sin ^{4}{\left (x \right )}}{20} + \frac {e^{2 x} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{10} + \frac {e^{2 x} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{5} - \frac {e^{2 x} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{10} + \frac {e^{2 x} \cos ^{4}{\left (x \right )}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 24, normalized size = 0.67 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} -\frac {{\mathrm {e}}^{2\,x}\,\left (\cos \left (4\,x\right )+2\,\sin \left (4\,x\right )-5\right )}{80} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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