Integrand size = 22, antiderivative size = 110 \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=-\frac {5 \arcsin (\cos (a+b x)-\sin (a+b x))}{32 b}+\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{32 b}-\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b} \]
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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 = {4383, 4387, 4390} \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=-\frac {5 \arcsin (\cos (a+b x)-\sin (a+b x))}{32 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}-\frac {5 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{16 b}+\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{32 b} \]
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Rule 4383
Rule 4387
Rule 4390
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{8} \int \sin (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx \\ & = -\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{16} \int \frac {\cos (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 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\frac {5 \left (-\arcsin (\cos (a+b x)-\sin (a+b x))+\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )\right )+2 (-6 \cos (a+b x)+\cos (3 (a+b x))) \sqrt {\sin (2 (a+b x))}}{32 b} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 16.22 (sec) , antiderivative size = 57690707, normalized size of antiderivative = 524460.97
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (96) = 192\).
Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.55 \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\frac {8 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - 9 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{128 \, b} \]
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Timed out. \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\text {Timed out} \]
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\[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\int { \sqrt {\sin \left (2 \, b x + 2 \, a\right )} \sin \left (b x + a\right )^{3} \,d x } \]
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Exception generated. \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx=\int {\sin \left (a+b\,x\right )}^3\,\sqrt {\sin \left (2\,a+2\,b\,x\right )} \,d x \]
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