Optimal. Leaf size=107 \[ -\frac {\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \, _2F_1\left (-n,-\frac {e n+i b c \log (F)}{2 e};\frac {1}{2} \left (2-n-\frac {i b c \log (F)}{e}\right );-e^{2 i (d+e x)}\right )}{i e n-b c \log (F)} \]
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Rubi [A]
time = 0.08, 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 = {4526, 2291}
\begin {gather*} -\frac {\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2291
Rule 4526
Rubi steps
\begin {align*} \int F^{c (a+b x)} \cos ^n(d+e x) \, dx &=\left (e^{i n (d+e x)} \left (1+e^{2 i (d+e x)}\right )^{-n} \cos ^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)} \cos ^n(d+e 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 )}{i e n-b c \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 110, normalized size = 1.03 \begin {gather*} \frac {\left (1+e^{2 i (d+e x)}\right )^{-n} F^{c (a+b x)} \cos ^n(d+e x) \, _2F_1\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 )}{-i e n+b c \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (\cos ^{n}\left (e x +d \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int F^{c \left (a + b x\right )} \cos ^{n}{\left (d + e x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,{\cos \left (d+e\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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