Integrand size = 22, antiderivative size = 110 \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=-\frac {5 \arcsin (\cos (a+b x)-\sin (a+b x))}{32 b}-\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{32 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}+\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b} \]
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Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4382, 4386, 4391} \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=-\frac {5 \arcsin (\cos (a+b x)-\sin (a+b x))}{32 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}+\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (a+b x)}{8 b}-\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{32 b} \]
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Rule 4382
Rule 4386
Rule 4391
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{8} \int \cos (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx \\ & = \frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}+\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{16} \int \frac {\sin (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = -\frac {5 \arcsin (\cos (a+b x)-\sin (a+b x))}{32 b}-\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{32 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}+\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\frac {-5 \left (\arcsin (\cos (a+b x)-\sin (a+b x))+\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )\right )+2 \sqrt {\sin (2 (a+b x))} (6 \sin (a+b x)+\sin (3 (a+b x)))}{32 b} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 18.15 (sec) , antiderivative size = 103475646, normalized size of antiderivative = 940687.69
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (96) = 192\).
Time = 0.28 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.55 \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\frac {8 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} + 5\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right ) + 10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{128 \, b} \]
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Timed out. \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\text {Timed out} \]
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\[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\int { \cos \left (b x + a\right )^{3} \sqrt {\sin \left (2 \, b x + 2 \, a\right )} \,d x } \]
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Exception generated. \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \cos ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\int {\cos \left (a+b\,x\right )}^3\,\sqrt {\sin \left (2\,a+2\,b\,x\right )} \,d x \]
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