Optimal. Leaf size=54 \[ \frac {2 e^{m x}}{m \left (m^2+4\right )}+\frac {m e^{m x} \cos ^2(x)}{m^2+4}+\frac {2 e^{m x} \sin (x) \cos (x)}{m^2+4} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4435, 2194} \[ \frac {2 e^{m x}}{m \left (m^2+4\right )}+\frac {m e^{m x} \cos ^2(x)}{m^2+4}+\frac {2 e^{m x} \sin (x) \cos (x)}{m^2+4} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4435
Rubi steps
\begin {align*} \int e^{m x} \cos ^2(x) \, dx &=\frac {e^{m x} m \cos ^2(x)}{4+m^2}+\frac {2 e^{m x} \cos (x) \sin (x)}{4+m^2}+\frac {2 \int e^{m x} \, dx}{4+m^2}\\ &=\frac {2 e^{m x}}{m \left (4+m^2\right )}+\frac {e^{m x} m \cos ^2(x)}{4+m^2}+\frac {2 e^{m x} \cos (x) \sin (x)}{4+m^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 0.72 \[ \frac {e^{m x} \left (m^2 \cos (2 x)+m^2+2 m \sin (2 x)+4\right )}{2 m \left (m^2+4\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \cos ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.04, size = 37, normalized size = 0.69 \[ \frac {2 \, m \cos \relax (x) e^{\left (m x\right )} \sin \relax (x) + {\left (m^{2} \cos \relax (x)^{2} + 2\right )} e^{\left (m x\right )}}{m^{3} + 4 \, m} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 43, normalized size = 0.80 \[ \frac {1}{2} \, {\left (\frac {m \cos \left (2 \, x\right )}{m^{2} + 4} + \frac {2 \, \sin \left (2 \, x\right )}{m^{2} + 4}\right )} e^{\left (m x\right )} + \frac {e^{\left (m x\right )}}{2 \, m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 41, normalized size = 0.76
method | result | size |
risch | \(\frac {{\mathrm e}^{m x}}{2 m}+\frac {{\mathrm e}^{\left (2 i+m \right ) x}}{8 i+4 m}+\frac {{\mathrm e}^{x \left (m -2 i\right )}}{4 m -8 i}\) | \(41\) |
norman | \(\frac {\frac {\left (m^{2}+2\right ) {\mathrm e}^{m x}}{m \left (m^{2}+4\right )}+\frac {\left (m^{2}+2\right ) {\mathrm e}^{m x} \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{m \left (m^{2}+4\right )}+\frac {4 \,{\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )}{m^{2}+4}-\frac {4 \,{\mathrm e}^{m x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{m^{2}+4}-\frac {2 \left (m^{2}-2\right ) {\mathrm e}^{m x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{m \left (m^{2}+4\right )}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 45, normalized size = 0.83 \[ \frac {m^{2} \cos \left (2 \, x\right ) e^{\left (m x\right )} + 2 \, m e^{\left (m x\right )} \sin \left (2 \, x\right ) + {\left (m^{2} + 4\right )} e^{\left (m x\right )}}{2 \, {\left (m^{3} + 4 \, m\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 0.69 \[ \frac {{\mathrm {e}}^{m\,x}}{2\,m}+\frac {{\mathrm {e}}^{m\,x}\,\left (2\,\sin \left (2\,x\right )+m\,\cos \left (2\,x\right )\right )}{2\,\left (m^2+4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.81, size = 265, normalized size = 4.91 \[ \begin {cases} \frac {x \sin ^{2}{\relax (x )}}{2} + \frac {x \cos ^{2}{\relax (x )}}{2} + \frac {\sin {\relax (x )} \cos {\relax (x )}}{2} & \text {for}\: m = 0 \\- \frac {x e^{- 2 i x} \sin ^{2}{\relax (x )}}{4} + \frac {i x e^{- 2 i x} \sin {\relax (x )} \cos {\relax (x )}}{2} + \frac {x e^{- 2 i x} \cos ^{2}{\relax (x )}}{4} - \frac {e^{- 2 i x} \sin {\relax (x )} \cos {\relax (x )}}{4} + \frac {i e^{- 2 i x} \cos ^{2}{\relax (x )}}{2} & \text {for}\: m = - 2 i \\- \frac {x e^{2 i x} \sin ^{2}{\relax (x )}}{4} - \frac {i x e^{2 i x} \sin {\relax (x )} \cos {\relax (x )}}{2} + \frac {x e^{2 i x} \cos ^{2}{\relax (x )}}{4} - \frac {e^{2 i x} \sin {\relax (x )} \cos {\relax (x )}}{4} - \frac {i e^{2 i x} \cos ^{2}{\relax (x )}}{2} & \text {for}\: m = 2 i \\\frac {m^{2} e^{m x} \cos ^{2}{\relax (x )}}{m^{3} + 4 m} + \frac {2 m e^{m x} \sin {\relax (x )} \cos {\relax (x )}}{m^{3} + 4 m} + \frac {2 e^{m x} \sin ^{2}{\relax (x )}}{m^{3} + 4 m} + \frac {2 e^{m x} \cos ^{2}{\relax (x )}}{m^{3} + 4 m} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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