Optimal. Leaf size=54 \[ \frac{2 e^{m x}}{m \left (m^2+4\right )}+\frac{m e^{m x} \cos ^2(x)}{m^2+4}+\frac{2 e^{m x} \sin (x) \cos (x)}{m^2+4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0230822, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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.
[In]
[Out]
Rule 4435
Rule 2194
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*}
Mathematica [A] time = 0.0351539, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 45, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{mx}}}{2\,m}}+{\frac{m{{\rm e}^{mx}}\cos \left ( 2\,x \right ) }{2\,{m}^{2}+8}}+{\frac{{{\rm e}^{mx}}\sin \left ( 2\,x \right ) }{{m}^{2}+4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.942968, size = 61, normalized size = 1.13 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.08634, size = 95, normalized size = 1.76 \begin{align*} \frac{2 \, m \cos \left (x\right ) e^{\left (m x\right )} \sin \left (x\right ) +{\left (m^{2} \cos \left (x\right )^{2} + 2\right )} e^{\left (m x\right )}}{m^{3} + 4 \, m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.42764, size = 265, normalized size = 4.91 \begin{align*} \begin{cases} \frac{x \sin ^{2}{\left (x \right )}}{2} + \frac{x \cos ^{2}{\left (x \right )}}{2} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} & \text{for}\: m = 0 \\- \frac{x e^{- 2 i x} \sin ^{2}{\left (x \right )}}{4} + \frac{i x e^{- 2 i x} \sin{\left (x \right )} \cos{\left (x \right )}}{2} + \frac{x e^{- 2 i x} \cos ^{2}{\left (x \right )}}{4} - \frac{e^{- 2 i x} \sin{\left (x \right )} \cos{\left (x \right )}}{4} + \frac{i e^{- 2 i x} \cos ^{2}{\left (x \right )}}{2} & \text{for}\: m = - 2 i \\- \frac{x e^{2 i x} \sin ^{2}{\left (x \right )}}{4} - \frac{i x e^{2 i x} \sin{\left (x \right )} \cos{\left (x \right )}}{2} + \frac{x e^{2 i x} \cos ^{2}{\left (x \right )}}{4} - \frac{e^{2 i x} \sin{\left (x \right )} \cos{\left (x \right )}}{4} - \frac{i e^{2 i x} \cos ^{2}{\left (x \right )}}{2} & \text{for}\: m = 2 i \\\frac{m^{2} e^{m x} \cos ^{2}{\left (x \right )}}{m^{3} + 4 m} + \frac{2 m e^{m x} \sin{\left (x \right )} \cos{\left (x \right )}}{m^{3} + 4 m} + \frac{2 e^{m x} \sin ^{2}{\left (x \right )}}{m^{3} + 4 m} + \frac{2 e^{m x} \cos ^{2}{\left (x \right )}}{m^{3} + 4 m} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09468, size = 58, normalized size = 1.07 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]