Integrand size = 18, antiderivative size = 69 \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\frac {625}{32} \arcsin \left (\frac {2 \sin (x)}{\sqrt {5}}\right )+\frac {125}{16} \sin (x) \sqrt {5-4 \sin ^2(x)}+\frac {25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac {1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4441, 201, 222} \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\frac {625}{32} \arcsin \left (\frac {2 \sin (x)}{\sqrt {5}}\right )+\frac {1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac {25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac {125}{16} \sin (x) \sqrt {5-4 \sin ^2(x)} \]
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Rule 201
Rule 222
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (5-4 x^2\right )^{5/2} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac {25}{6} \text {Subst}\left (\int \left (5-4 x^2\right )^{3/2} \, dx,x,\sin (x)\right ) \\ & = \frac {25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac {1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac {125}{8} \text {Subst}\left (\int \sqrt {5-4 x^2} \, dx,x,\sin (x)\right ) \\ & = \frac {125}{16} \sin (x) \sqrt {5-4 \sin ^2(x)}+\frac {25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac {1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac {625}{16} \text {Subst}\left (\int \frac {1}{\sqrt {5-4 x^2}} \, dx,x,\sin (x)\right ) \\ & = \frac {625}{32} \arcsin \left (\frac {2 \sin (x)}{\sqrt {5}}\right )+\frac {125}{16} \sin (x) \sqrt {5-4 \sin ^2(x)}+\frac {25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac {1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\frac {1}{96} \left (1875 \arcsin \left (\frac {2 \sin (x)}{\sqrt {5}}\right )+2 \sqrt {3+2 \cos (2 x)} (515 \sin (x)+90 \sin (3 x)+8 \sin (5 x))\right ) \]
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Time = 2.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {625 \arcsin \left (\frac {2 \sin \left (x \right ) \sqrt {5}}{5}\right )}{32}+\frac {25 \sin \left (x \right ) {\left (5-4 \left (\sin ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}{24}+\frac {\sin \left (x \right ) {\left (5-4 \left (\sin ^{2}\left (x \right )\right )\right )}^{\frac {5}{2}}}{6}+\frac {125 \sin \left (x \right ) \sqrt {5-4 \left (\sin ^{2}\left (x \right )\right )}}{16}\) | \(54\) |
default | \(\frac {625 \arcsin \left (\frac {2 \sin \left (x \right ) \sqrt {5}}{5}\right )}{32}+\frac {25 \sin \left (x \right ) {\left (5-4 \left (\sin ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}{24}+\frac {\sin \left (x \right ) {\left (5-4 \left (\sin ^{2}\left (x \right )\right )\right )}^{\frac {5}{2}}}{6}+\frac {125 \sin \left (x \right ) \sqrt {5-4 \left (\sin ^{2}\left (x \right )\right )}}{16}\) | \(54\) |
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28 \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\frac {1}{48} \, {\left (128 \, \cos \left (x\right )^{4} + 264 \, \cos \left (x\right )^{2} + 433\right )} \sqrt {4 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) + \frac {625}{64} \, \arctan \left (\frac {4 \, {\left (8 \, \cos \left (x\right )^{2} - 3\right )} \sqrt {4 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) - 25 \, \cos \left (x\right ) \sin \left (x\right )}{64 \, \cos \left (x\right )^{4} - 23 \, \cos \left (x\right )^{2} - 16}\right ) + \frac {625}{64} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) \]
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Timed out. \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\frac {1}{6} \, {\left (-4 \, \sin \left (x\right )^{2} + 5\right )}^{\frac {5}{2}} \sin \left (x\right ) + \frac {25}{24} \, {\left (-4 \, \sin \left (x\right )^{2} + 5\right )}^{\frac {3}{2}} \sin \left (x\right ) + \frac {125}{16} \, \sqrt {-4 \, \sin \left (x\right )^{2} + 5} \sin \left (x\right ) + \frac {625}{32} \, \arcsin \left (\frac {2}{5} \, \sqrt {5} \sin \left (x\right )\right ) \]
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.59 \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\frac {1}{48} \, {\left (8 \, {\left (16 \, \sin \left (x\right )^{2} - 65\right )} \sin \left (x\right )^{2} + 825\right )} \sqrt {-4 \, \sin \left (x\right )^{2} + 5} \sin \left (x\right ) + \frac {625}{32} \, \arcsin \left (\frac {2}{5} \, \sqrt {5} \sin \left (x\right )\right ) \]
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Timed out. \[ \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx=\int \cos \left (x\right )\,{\left (5\,{\cos \left (x\right )}^2+{\sin \left (x\right )}^2\right )}^{5/2} \,d x \]
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