Optimal. Leaf size=107 \[ -\frac{\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \sin ^n(d+e x) \text{Hypergeometric2F1}\left (-n,-\frac{e n+i b c \log (F)}{2 e},\frac{1}{2} \left (-\frac{i b c \log (F)}{e}-n+2\right ),e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]
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Rubi [A] time = 0.13809, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4440, 2259} \[ -\frac{\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \sin ^n(d+e x) \, _2F_1\left (-n,-\frac{e n+i b c \log (F)}{2 e};\frac{1}{2} \left (-n-\frac{i b c \log (F)}{e}+2\right );e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]
Antiderivative was successfully verified.
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Rule 4440
Rule 2259
Rubi steps
\begin{align*} \int F^{c (a+b x)} \sin ^n(d+e x) \, dx &=\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)} \, _2F_1\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*}
Mathematica [A] time = 0.0633474, size = 110, normalized size = 1.03 \[ \frac{\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \sin ^n(d+e x) \text{Hypergeometric2F1}\left (-n,-\frac{i (b c \log (F)-i e n)}{2 e},1-\frac{i (b c \log (F)-i e n)}{2 e},e^{2 i (d+e x)}\right )}{b c \log (F)-i e n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.889, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \sin \left ( ex+d \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b c x + a c} \sin \left (e x + d\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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