Integrand size = 13, antiderivative size = 49 \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=2 \sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )-\frac {5}{2} \arctan \left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x) \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {4449, 424, 537, 222, 385, 209} \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=2 \sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )-\frac {5}{2} \arctan \left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\frac {1}{2} \sqrt {\cos (2 x)} \tan (x) \sec (x) \]
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Rule 209
Rule 222
Rule 385
Rule 424
Rule 537
Rule 4449
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-2 x^2\right )^{3/2}}{\left (1-x^2\right )^2} \, dx,x,\sin (x)\right ) \\ & = -\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)-\frac {1}{2} \text {Subst}\left (\int \frac {-3+8 x^2}{\sqrt {1-2 x^2} \left (1-x^2\right )} \, dx,x,\sin (x)\right ) \\ & = -\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)-\frac {5}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2} \left (1-x^2\right )} \, dx,x,\sin (x)\right )+4 \text {Subst}\left (\int \frac {1}{\sqrt {1-2 x^2}} \, dx,x,\sin (x)\right ) \\ & = 2 \sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x)-\frac {5}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right ) \\ & = 2 \sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )-\frac {5}{2} \arctan \left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\frac {1}{2} \sqrt {\cos (2 x)} \sec (x) \tan (x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=\frac {1}{2} \left (4 \sqrt {2} \arcsin \left (\sqrt {2} \sin (x)\right )-5 \arctan \left (\frac {\sin (x)}{\sqrt {\cos (2 x)}}\right )-\sqrt {\cos (2 x)} \sec (x) \tan (x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(37)=74\).
Time = 0.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.04
method | result | size |
default | \(-\frac {\sqrt {\left (2 \left (\cos ^{2}\left (x \right )\right )-1\right ) \left (\sin ^{2}\left (x \right )\right )}\, \left (4 \sqrt {2}\, \arcsin \left (4 \left (\cos ^{2}\left (x \right )\right )-3\right ) \left (\cos ^{2}\left (x \right )\right )-5 \arctan \left (\frac {3 \left (\cos ^{2}\left (x \right )\right )-2}{2 \sqrt {-2 \left (\sin ^{4}\left (x \right )\right )+\sin ^{2}\left (x \right )}}\right ) \left (\cos ^{2}\left (x \right )\right )+2 \sqrt {-2 \left (\sin ^{4}\left (x \right )\right )+\sin ^{2}\left (x \right )}\right )}{4 \cos \left (x \right )^{2} \sin \left (x \right ) \sqrt {2 \left (\cos ^{2}\left (x \right )\right )-1}}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (37) = 74\).
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.41 \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {{\left (32 \, \sqrt {2} \cos \left (x\right )^{4} - 48 \, \sqrt {2} \cos \left (x\right )^{2} + 17 \, \sqrt {2}\right )} \sqrt {2 \, \cos \left (x\right )^{2} - 1}}{8 \, {\left (8 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}\right ) \cos \left (x\right )^{2} - 5 \, \arctan \left (\frac {3 \, \cos \left (x\right )^{2} - 2}{2 \, \sqrt {2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}\right ) \cos \left (x\right )^{2} + 2 \, \sqrt {2 \, \cos \left (x\right )^{2} - 1} \sin \left (x\right )}{4 \, \cos \left (x\right )^{2}} \]
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Timed out. \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=\int { \frac {\cos \left (2 \, x\right )^{\frac {3}{2}}}{\cos \left (x\right )^{3}} \,d x } \]
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\[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=\int { \frac {\cos \left (2 \, x\right )^{\frac {3}{2}}}{\cos \left (x\right )^{3}} \,d x } \]
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Timed out. \[ \int \cos ^{\frac {3}{2}}(2 x) \sec ^3(x) \, dx=\int \frac {{\cos \left (2\,x\right )}^{3/2}}{{\cos \left (x\right )}^3} \,d x \]
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