Integrand size = 22, antiderivative size = 69 \[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{10 b}-\frac {\cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{10 b}-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b} \]
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Time = 0.06 (sec) , antiderivative size = 69, 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, 2715, 2719} \[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}+\frac {3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{10 b}-\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{10 b} \]
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Rule 2715
Rule 2719
Rule 4383
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}+\frac {1}{2} \int \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx \\ & = -\frac {\cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{10 b}-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b}+\frac {3}{10} \int \sqrt {\sin (2 a+2 b x)} \, dx \\ & = \frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{10 b}-\frac {\cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{10 b}-\frac {\sin ^{\frac {7}{2}}(2 a+2 b x)}{14 b} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\frac {84 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\sqrt {\sin (2 (a+b x))} (-15 \sin (2 (a+b x))-14 \sin (4 (a+b x))+5 \sin (6 (a+b x)))}{280 b} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 176.98 (sec) , antiderivative size = 312731247, normalized size of antiderivative = 4532336.91
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\[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}} \sin \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\text {Timed out} \]
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\[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}} \sin \left (b x + a\right )^{2} \,d x } \]
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\[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}} \sin \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \sin ^2(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{5/2} \,d x \]
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