Optimal. Leaf size=107 \[ -\frac {\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \sin ^n(d+e x) \, _2F_1\left (-n,-\frac {e n+i b c \log (F)}{2 e};\frac {1}{2} \left (-n-\frac {i b c \log (F)}{e}+2\right );e^{2 i (d+e x)}\right )}{-b c \log (F)+i e n} \]
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Rubi [A] time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.00, 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 2259
Rule 4440
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*}
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Mathematica [A] time = 0.06, size = 110, normalized size = 1.03 \[ \frac {\left (1-e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \sin ^n(d+e x) \, _2F_1\left (-n,-\frac {i (b c \log (F)-i e n)}{2 e};1-\frac {i (b c \log (F)-i e n)}{2 e};e^{2 i (d+e x)}\right )}{b c \log (F)-i e n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \sin \left (e x + d\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \left (\sin ^{n}\left (e x +d \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \sin \left (e x + d\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{c\,\left (a+b\,x\right )}\,{\sin \left (d+e\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \sin ^{n}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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