Integrand size = 18, antiderivative size = 102 \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=-\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2+n+\frac {i b c \log (F)}{e}\right ),e^{-2 i (d+e x)}\right )}{i e n-b c \log (F)} \]
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Time = 0.26 (sec) , antiderivative size = 102, 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 = {4540, 2291} \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=-\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (n+\frac {i b c \log (F)}{e}+2\right ),e^{-2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]
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Rule 2291
Rule 4540
Rubi steps \begin{align*} \text {integral}& = \left (e^{i n (d+e x)} \left (1-e^{-2 i (d+e x)}\right )^n \csc ^n(d+e x)\right ) \int e^{-i d n-i e n x} \left (1-e^{-2 i (d+e x)}\right )^{-n} F^{a c+b c x} \, dx \\ & = -\frac {\left (1-e^{-2 i (d+e x)}\right )^n F^{a c+b c x} \csc ^n(d+e 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 )}{i e n-b c \log (F)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\frac {i \left (1-e^{-2 i (d+e x)}\right )^n F^{c (a+b x)} \csc ^n(d+e x) \operatorname {Hypergeometric2F1}\left (n,\frac {e n+i b c \log (F)}{2 e},\frac {1}{2} \left (2+n+\frac {i b c \log (F)}{e}\right ),e^{-2 i (d+e x)}\right )}{e n+i b c \log (F)} \]
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\[\int F^{c \left (x b +a \right )} \csc \left (e x +d \right )^{n}d x\]
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\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n} \,d x } \]
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\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int F^{c \left (a + b x\right )} \csc ^{n}{\left (d + e x \right )}\, dx \]
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\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n} \,d x } \]
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\[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{n} \,d x } \]
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Timed out. \[ \int F^{c (a+b x)} \csc ^n(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (\frac {1}{\sin \left (d+e\,x\right )}\right )}^n \,d x \]
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