Integrand size = 8, antiderivative size = 54 \[ \int e^x \cos ^4(x) \, dx=\frac {24 e^x}{85}+\frac {12}{85} e^x \cos ^2(x)+\frac {1}{17} e^x \cos ^4(x)+\frac {24}{85} e^x \cos (x) \sin (x)+\frac {4}{17} e^x \cos ^3(x) \sin (x) \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4520, 2225} \[ \int e^x \cos ^4(x) \, dx=\frac {24 e^x}{85}+\frac {1}{17} e^x \cos ^4(x)+\frac {12}{85} e^x \cos ^2(x)+\frac {4}{17} e^x \sin (x) \cos ^3(x)+\frac {24}{85} e^x \sin (x) \cos (x) \]
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Rule 2225
Rule 4520
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int e^x \cos ^4(x) \, dx=\frac {1}{680} e^x (255+68 \cos (2 x)+5 \cos (4 x)+136 \sin (2 x)+20 \sin (4 x)) \]
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Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{x} \left (255+5 \cos \left (4 x \right )+68 \cos \left (2 x \right )+136 \sin \left (2 x \right )+20 \sin \left (4 x \right )\right )}{680}\) | \(31\) |
default | \(\frac {\left (\cos \left (x \right )+4 \sin \left (x \right )\right ) {\mathrm e}^{x} \cos \left (x \right )^{3}}{17}+\frac {12 \left (\cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{x} \cos \left (x \right )}{85}+\frac {24 \,{\mathrm e}^{x}}{85}\) | \(34\) |
risch | \(\frac {3 \,{\mathrm e}^{x}}{8}+\frac {{\mathrm e}^{\left (1+4 i\right ) x}}{272}-\frac {i {\mathrm e}^{\left (1+4 i\right ) x}}{68}+\frac {{\mathrm e}^{\left (1+2 i\right ) x}}{20}-\frac {i {\mathrm e}^{\left (1+2 i\right ) x}}{10}+\frac {{\mathrm e}^{\left (1-2 i\right ) x}}{20}+\frac {i {\mathrm e}^{\left (1-2 i\right ) x}}{10}+\frac {{\mathrm e}^{\left (1-4 i\right ) x}}{272}+\frac {i {\mathrm e}^{\left (1-4 i\right ) x}}{68}\) | \(74\) |
norman | \(\frac {\frac {88 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )}{85}+\frac {76 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{2}}{85}-\frac {72 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{3}}{85}+\frac {30 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{4}}{17}+\frac {72 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{5}}{85}+\frac {76 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{6}}{85}-\frac {88 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{7}}{85}+\frac {41 \,{\mathrm e}^{x} \tan \left (\frac {x}{2}\right )^{8}}{85}+\frac {41 \,{\mathrm e}^{x}}{85}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}\) | \(95\) |
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Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int e^x \cos ^4(x) \, dx=\frac {4}{85} \, {\left (5 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )\right )} e^{x} \sin \left (x\right ) + \frac {1}{85} \, {\left (5 \, \cos \left (x\right )^{4} + 12 \, \cos \left (x\right )^{2} + 24\right )} e^{x} \]
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Time = 0.48 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int e^x \cos ^4(x) \, dx=\frac {24 e^{x} \sin ^{4}{\left (x \right )}}{85} + \frac {24 e^{x} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{85} + \frac {12 e^{x} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{17} + \frac {44 e^{x} \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{85} + \frac {41 e^{x} \cos ^{4}{\left (x \right )}}{85} \]
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Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int e^x \cos ^4(x) \, dx=\frac {1}{136} \, \cos \left (4 \, x\right ) e^{x} + \frac {1}{10} \, \cos \left (2 \, x\right ) e^{x} + \frac {1}{34} \, e^{x} \sin \left (4 \, x\right ) + \frac {1}{5} \, e^{x} \sin \left (2 \, x\right ) + \frac {3}{8} \, e^{x} \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.65 \[ \int e^x \cos ^4(x) \, dx=\frac {1}{136} \, {\left (\cos \left (4 \, x\right ) + 4 \, \sin \left (4 \, x\right )\right )} e^{x} + \frac {1}{10} \, {\left (\cos \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} e^{x} + \frac {3}{8} \, e^{x} \]
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Time = 27.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.76 \[ \int e^x \cos ^4(x) \, dx=\frac {3\,{\mathrm {e}}^x}{8}+\frac {{\mathrm {e}}^x\,\left (\frac {4\,\cos \left (2\,x\right )}{5}+\frac {8\,\sin \left (2\,x\right )}{5}+\frac {2\,{\cos \left (2\,x\right )}^2}{17}+\frac {8\,\cos \left (2\,x\right )\,\sin \left (2\,x\right )}{17}-\frac {1}{17}\right )}{8} \]
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