Optimal. Leaf size=54 \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \cos ^4(x)+\frac{12}{85} e^x \cos ^2(x)+\frac{4}{17} e^x \sin (x) \cos ^3(x)+\frac{24}{85} e^x \sin (x) \cos (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0289402, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4435, 2194} \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \cos ^4(x)+\frac{12}{85} e^x \cos ^2(x)+\frac{4}{17} e^x \sin (x) \cos ^3(x)+\frac{24}{85} e^x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4435
Rule 2194
Rubi steps
\begin{align*} \int e^x \cos ^4(x) \, dx &=\frac{1}{17} e^x \cos ^4(x)+\frac{4}{17} e^x \cos ^3(x) \sin (x)+\frac{12}{17} \int e^x \cos ^2(x) \, dx\\ &=\frac{12}{85} e^x \cos ^2(x)+\frac{1}{17} e^x \cos ^4(x)+\frac{24}{85} e^x \cos (x) \sin (x)+\frac{4}{17} e^x \cos ^3(x) \sin (x)+\frac{24 \int e^x \, dx}{85}\\ &=\frac{24 e^x}{85}+\frac{12}{85} e^x \cos ^2(x)+\frac{1}{17} e^x \cos ^4(x)+\frac{24}{85} e^x \cos (x) \sin (x)+\frac{4}{17} e^x \cos ^3(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.026743, size = 33, normalized size = 0.61 \[ \frac{1}{680} e^x (136 \sin (2 x)+20 \sin (4 x)+68 \cos (2 x)+5 \cos (4 x)+255) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 34, normalized size = 0.6 \begin{align*}{\frac{ \left ( \cos \left ( x \right ) +4\,\sin \left ( x \right ) \right ){{\rm e}^{x}} \left ( \cos \left ( x \right ) \right ) ^{3}}{17}}+{\frac{ \left ( 12\,\cos \left ( x \right ) +24\,\sin \left ( x \right ) \right ){{\rm e}^{x}}\cos \left ( x \right ) }{85}}+{\frac{24\,{{\rm e}^{x}}}{85}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02456, size = 50, normalized size = 0.93 \begin{align*} \frac{1}{136} \, \cos \left (4 \, x\right ) e^{x} + \frac{1}{10} \, \cos \left (2 \, x\right ) e^{x} + \frac{1}{34} \, e^{x} \sin \left (4 \, x\right ) + \frac{1}{5} \, e^{x} \sin \left (2 \, x\right ) + \frac{3}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.468187, size = 113, normalized size = 2.09 \begin{align*} \frac{4}{85} \,{\left (5 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )\right )} e^{x} \sin \left (x\right ) + \frac{1}{85} \,{\left (5 \, \cos \left (x\right )^{4} + 12 \, \cos \left (x\right )^{2} + 24\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 8.32891, size = 70, normalized size = 1.3 \begin{align*} \frac{24 e^{x} \sin ^{4}{\left (x \right )}}{85} + \frac{24 e^{x} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{85} + \frac{12 e^{x} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{17} + \frac{44 e^{x} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{85} + \frac{41 e^{x} \cos ^{4}{\left (x \right )}}{85} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15466, size = 47, normalized size = 0.87 \begin{align*} \frac{1}{136} \,{\left (\cos \left (4 \, x\right ) + 4 \, \sin \left (4 \, x\right )\right )} e^{x} + \frac{1}{10} \,{\left (\cos \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} e^{x} + \frac{3}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]