Integrand size = 16, antiderivative size = 79 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {48 \cos \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}-\frac {2 \cos ^3\left (\frac {x}{3}\right )}{5 \sqrt {e^x}}+\frac {32 \sin \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}+\frac {4 \cos ^2\left (\frac {x}{3}\right ) \sin \left (\frac {x}{3}\right )}{5 \sqrt {e^x}} \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2319, 4520, 4518} \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {32 \sin \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}-\frac {2 \cos ^3\left (\frac {x}{3}\right )}{5 \sqrt {e^x}}-\frac {48 \cos \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}+\frac {4 \sin \left (\frac {x}{3}\right ) \cos ^2\left (\frac {x}{3}\right )}{5 \sqrt {e^x}} \]
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Rule 2319
Rule 4518
Rule 4520
Rubi steps \begin{align*} \text {integral}& = \frac {e^{x/2} \int e^{-x/2} \cos ^3\left (\frac {x}{3}\right ) \, dx}{\sqrt {e^x}} \\ & = -\frac {2 \cos ^3\left (\frac {x}{3}\right )}{5 \sqrt {e^x}}+\frac {4 \cos ^2\left (\frac {x}{3}\right ) \sin \left (\frac {x}{3}\right )}{5 \sqrt {e^x}}+\frac {\left (8 e^{x/2}\right ) \int e^{-x/2} \cos \left (\frac {x}{3}\right ) \, dx}{15 \sqrt {e^x}} \\ & = -\frac {48 \cos \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}-\frac {2 \cos ^3\left (\frac {x}{3}\right )}{5 \sqrt {e^x}}+\frac {32 \sin \left (\frac {x}{3}\right )}{65 \sqrt {e^x}}+\frac {4 \cos ^2\left (\frac {x}{3}\right ) \sin \left (\frac {x}{3}\right )}{5 \sqrt {e^x}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.46 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {-135 \cos \left (\frac {x}{3}\right )-13 \cos (x)+90 \sin \left (\frac {x}{3}\right )+26 \sin (x)}{130 \sqrt {e^x}} \]
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Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.35
method | result | size |
parallelrisch | \(\frac {-13 \cos \left (x \right )-135 \cos \left (\frac {x}{3}\right )+26 \sin \left (x \right )+90 \sin \left (\frac {x}{3}\right )}{130 \sqrt {{\mathrm e}^{x}}}\) | \(28\) |
default | \(-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (x \right )}{10}+\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (x \right )}{5}-\frac {27 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {x}{3}\right )}{26}+\frac {9 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {x}{3}\right )}{13}\) | \(38\) |
risch | \(\frac {\left (-\frac {1}{1300}-\frac {i}{650}\right ) \left (-52 i {\mathrm e}^{-i x}+65 \,{\mathrm e}^{i x}-39 \,{\mathrm e}^{-i x}+\left (270-540 i\right ) \cos \left (\frac {x}{3}\right )+\left (-180+360 i\right ) \sin \left (\frac {x}{3}\right )\right )}{\sqrt {{\mathrm e}^{x}}}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.53 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {4}{65} \, {\left (13 \, \cos \left (\frac {1}{3} \, x\right )^{2} + 8\right )} e^{\left (-\frac {1}{2} \, x\right )} \sin \left (\frac {1}{3} \, x\right ) - \frac {2}{65} \, {\left (13 \, \cos \left (\frac {1}{3} \, x\right )^{3} + 24 \, \cos \left (\frac {1}{3} \, x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.44 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=\frac {32 \sin ^{3}{\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} - \frac {48 \sin ^{2}{\left (\frac {x}{3} \right )} \cos {\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} + \frac {84 \sin {\left (\frac {x}{3} \right )} \cos ^{2}{\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} - \frac {74 \cos ^{3}{\left (\frac {x}{3} \right )}}{65 \sqrt {e^{x}}} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.34 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {1}{130} \, {\left (135 \, \cos \left (\frac {1}{3} \, x\right ) + 13 \, \cos \left (x\right ) - 90 \, \sin \left (\frac {1}{3} \, x\right ) - 26 \, \sin \left (x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.42 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {9}{26} \, {\left (3 \, \cos \left (\frac {1}{3} \, x\right ) - 2 \, \sin \left (\frac {1}{3} \, x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} - \frac {1}{10} \, {\left (\cos \left (x\right ) - 2 \, \sin \left (x\right )\right )} e^{\left (-\frac {1}{2} \, x\right )} \]
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Time = 0.38 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \frac {\cos ^3\left (\frac {x}{3}\right )}{\sqrt {e^x}} \, dx=-\frac {{\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {8\,{\cos \left (\frac {x}{3}\right )}^3}{5}-\frac {16\,\sin \left (\frac {x}{3}\right )\,{\cos \left (\frac {x}{3}\right )}^2}{5}+\frac {192\,\cos \left (\frac {x}{3}\right )}{65}-\frac {128\,\sin \left (\frac {x}{3}\right )}{65}\right )}{4} \]
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