Integrand size = 18, antiderivative size = 107 \[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=-\frac {\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2-n-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right ) \sin ^n(d+e x)}{i e n-b c \log (F)} \]
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Time = 0.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4525, 2291} \[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=-\frac {\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \sin ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (-n-\frac {i b c \log (F)}{e}+2\right ),e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]
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Rule 2291
Rule 4525
Rubi steps \begin{align*} \text {integral}& = \left (e^{i n (d+e x)} \left (-1+e^{2 i (d+e x)}\right )^{-n} \sin ^n(d+e x)\right ) \int e^{-i n (d+e x)} \left (-1+e^{2 i (d+e x)}\right )^n F^{c (a+b x)} \, dx \\ & = -\frac {\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2-n-\frac {i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right ) \sin ^n(d+e x)}{i e n-b c \log (F)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03 \[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=\frac {\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (-n,-\frac {i (-i e n+b c \log (F))}{2 e},1-\frac {i (-i e n+b c \log (F))}{2 e},e^{2 i (d+e x)}\right ) \sin ^n(d+e x)}{-i e n+b c \log (F)} \]
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\[\int F^{c \left (x b +a \right )} \sin \left (e x +d \right )^{n}d x\]
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\[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n} \,d x } \]
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\[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=\int F^{c \left (a + b x\right )} \sin ^{n}{\left (d + e x \right )}\, dx \]
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\[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n} \,d x } \]
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\[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n} \,d x } \]
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Timed out. \[ \int F^{c (a+b x)} \sin ^n(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\sin \left (d+e\,x\right )}^n \,d x \]
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