Integrand size = 20, antiderivative size = 110 \[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=-\frac {3 \arcsin (\cos (a+b x)-\sin (a+b x))}{16 b}-\frac {3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{16 b}+\frac {3 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{8 b}-\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{4 b} \]
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Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4387, 4386, 4391} \[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=-\frac {3 \arcsin (\cos (a+b x)-\sin (a+b x))}{16 b}+\frac {3 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{8 b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (a+b x)}{4 b}-\frac {3 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{16 b} \]
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Rule 4386
Rule 4387
Rule 4391
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{4 b}+\frac {3}{4} \int \cos (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx \\ & = \frac {3 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{8 b}-\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{4 b}+\frac {3}{8} \int \frac {\sin (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = -\frac {3 \arcsin (\cos (a+b x)-\sin (a+b x))}{16 b}-\frac {3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{16 b}+\frac {3 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{8 b}-\frac {\cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{4 b} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=\frac {-3 \left (\arcsin (\cos (a+b x)-\sin (a+b x))+\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )\right )+2 \sqrt {\sin (2 (a+b x))} (2 \sin (a+b x)-\sin (3 (a+b x)))}{16 b} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 24.15 (sec) , antiderivative size = 73677072, normalized size of antiderivative = 669791.56
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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 \[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=-\frac {8 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right ) - 6 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) + 6 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 3 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{64 \, b} \]
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Timed out. \[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=\text {Timed out} \]
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\[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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\[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx=\int \sin \left (a+b\,x\right )\,{\sin \left (2\,a+2\,b\,x\right )}^{3/2} \,d x \]
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