Optimal. Leaf size=82 \[ \frac {2 i F^{a+b x} \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};-i e^{i (c+d x)}\right )}{b e \log (F)}-\frac {i F^{a+b x}}{b e \log (F)} \]
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Rubi [A] time = 0.13, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4459, 4442, 2194, 2251} \[ \frac {2 i F^{a+b x} \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};-i e^{i (c+d x)}\right )}{b e \log (F)}-\frac {i F^{a+b x}}{b e \log (F)} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 4442
Rule 4459
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 = 2.64, size = 64, normalized size = 0.78 \[ \frac {i F^{a+b x} \left (-1+2 \, _2F_1\left (1,-\frac {i b \log (F)}{d};1-\frac {i b \log (F)}{d};-i e^{i (c+d x)}\right )\right )}{b e \log (F)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {F^{b x + a} \cos \left (d x + c\right )}{e \sin \left (d x + c\right ) - e}, x\right ) \]
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 \[ \int -\frac {F^{b x + a} \cos \left (d x + c\right )}{e \sin \left (d x + c\right ) - e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, 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 \[ -\frac {2 \, F^{b x} F^{a} b d \cos \left (d x + c\right ) \log \relax (F) + 2 \, F^{b x} F^{a} d^{2} \sin \left (d x + c\right ) - {\left (F^{a} b^{2} \log \relax (F)^{2} + F^{a} d^{2}\right )} F^{b x} \cos \left (d x + c\right )^{2} - {\left (F^{a} b^{2} \log \relax (F)^{2} + F^{a} d^{2}\right )} F^{b x} \sin \left (d x + c\right )^{2} + {\left (F^{a} b^{2} \log \relax (F)^{2} - F^{a} d^{2}\right )} F^{b x} + \frac {{\left (F^{b x} d \cos \left (d x + c\right )^{2} - 2 \, F^{b x} b \cos \left (d x + c\right ) \log \relax (F) + F^{b x} d \sin \left (d x + c\right )^{2} - 2 \, F^{b x} d \sin \left (d x + c\right ) + F^{b x} d\right )} {\left ({\left (F^{a} b^{3} d \log \relax (F)^{3} + F^{a} b d^{3} \log \relax (F)\right )} e \cos \left (d x + c\right )^{2} + {\left (F^{a} b^{3} d \log \relax (F)^{3} + F^{a} b d^{3} \log \relax (F)\right )} e \sin \left (d x + c\right )^{2} - 2 \, {\left (F^{a} b^{3} d \log \relax (F)^{3} + F^{a} b d^{3} \log \relax (F)\right )} e \sin \left (d x + c\right ) + {\left (F^{a} b^{3} d \log \relax (F)^{3} + F^{a} b d^{3} \log \relax (F)\right )} e\right )}}{{\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e \cos \left (d x + c\right )^{2} + {\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e \sin \left (d x + c\right )^{2} - 2 \, {\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e \sin \left (d x + c\right ) + {\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e}}{{\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e \cos \left (d x + c\right )^{2} + {\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e \sin \left (d x + c\right )^{2} - 2 \, {\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e \sin \left (d x + c\right ) + {\left (b^{3} \log \relax (F)^{3} + b d^{2} \log \relax (F)\right )} e} \]
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 \[ \int \frac {F^{a+b\,x}\,\cos \left (c+d\,x\right )}{e-e\,\sin \left (c+d\,x\right )} \,d x \]
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 \[ - \frac {\int \frac {F^{a} F^{b x} \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} - 1}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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