Optimal. Leaf size=82 \[ \frac {i F^{a+b x}}{b e \log (F)}-\frac {2 i F^{a+b x} \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};i e^{i (c+d x)}\right )}{b e \log (F)} \]
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Rubi [A]
time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4547, 4527,
2225, 2283} \begin {gather*} \frac {i F^{a+b x}}{b e \log (F)}-\frac {2 i F^{a+b x} \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};i e^{i (c+d x)}\right )}{b e \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 4527
Rule 4547
Rubi steps
\begin {align*} \int \frac {F^{a+b x} \cos (c+d x)}{e+e \sin (c+d x)} \, dx &=-\frac {\int F^{a+b x} \tan \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{e}\\ &=-\frac {i \int \left (-F^{a+b x}+\frac {2 F^{a+b x}}{1+e^{2 i \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right )}}\right ) \, dx}{e}\\ &=\frac {i \int F^{a+b x} \, dx}{e}-\frac {(2 i) \int \frac {F^{a+b x}}{1+e^{2 i \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right )}} \, dx}{e}\\ &=\frac {i F^{a+b x}}{b e \log (F)}-\frac {2 i F^{a+b x} \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};i e^{i (c+d x)}\right )}{b e \log (F)}\\ \end {align*}
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Mathematica [A]
time = 1.61, size = 64, normalized size = 0.78 \begin {gather*} -\frac {i F^{a+b x} \left (-1+2 \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};i e^{i (c+d x)}\right )\right )}{b e \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {F^{b x +a} \cos \left (d x +c \right )}{e +e \sin \left (d x +c \right )}\, dx\]
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*} \frac {\int \frac {F^{a} F^{b x} \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{e} \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 \frac {F^{a+b\,x}\,\cos \left (c+d\,x\right )}{e+e\,\sin \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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