Integrand size = 11, antiderivative size = 16 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4416} \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \]
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Rule 4416
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
Time = 0.58 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44
method | result | size |
default | \(\frac {\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right ) \left (4 \left (\sin ^{2}\left (x \right )\right )-1\right )}{12 \left (\sin ^{4}\left (x \right )\right )-12 \left (\sin ^{2}\left (x \right )\right )+3}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {{\left (4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1}}{3 \, {\left (4 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 1\right )}} \]
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\[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=\int \frac {\sin {\left (x \right )}}{\cos ^{\frac {5}{2}}{\left (2 x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (12) = 24\).
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 5.62 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {\sqrt {2} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + {\left (\sqrt {2} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + \sqrt {2}\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )}{3 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {3}{4}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \cos \left (x\right )}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {3}{2}}} \]
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Time = 0.44 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx=-\frac {\cos \left (3\,x\right )}{3\,{\cos \left (2\,x\right )}^{3/2}} \]
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