Integrand size = 20, antiderivative size = 84 \[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(a+b x)\right ) \sin ^2(a+b x) \sin ^m(2 a+2 b x) \tan (a+b x)}{b (3+m)} \]
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Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4395, 2657} \[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\frac {\sin ^2(a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {m+3}{2},\frac {m+5}{2},\sin ^2(a+b x)\right )}{b (m+3)} \]
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Rule 2657
Rule 4395
Rubi steps \begin{align*} \text {integral}& = \left (\cos ^{-m}(a+b x) \sin ^{-m}(a+b x) \sin ^m(2 a+2 b x)\right ) \int \cos ^m(a+b x) \sin ^{2+m}(a+b x) \, dx \\ & = \frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(a+b x)\right ) \sin ^2(a+b x) \sin ^m(2 a+2 b x) \tan (a+b x)}{b (3+m)} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.77 \[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\frac {\operatorname {Hypergeometric2F1}\left (2+m,\frac {3+m}{2},\frac {5+m}{2},-\tan ^2(a+b x)\right ) \sec ^2(a+b x)^m \sin ^m(2 (a+b x)) \tan ^3(a+b x)}{b (3+m)} \]
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\[\int \sin \left (x b +a \right )^{2} \sin \left (2 x b +2 a \right )^{m}d x\]
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\[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\text {Timed out} \]
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\[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )^{2} \,d x } \]
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\[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\int { \sin \left (2 \, b x + 2 \, a\right )^{m} \sin \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^m \,d x \]
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