Optimal. Leaf size=85 \[ \frac {2 e^{\frac {1}{2} (2 d+2 e x+i \pi )} F^{c (a+b x)} \, _2F_1\left (2,\frac {b c \log (F)}{e}+1;\frac {b c \log (F)}{e}+2;-e^{\frac {1}{2} (2 d+2 e x+i \pi )}\right )}{f (b c \log (F)+e)} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5496, 5492} \[ \frac {2 e^{\frac {1}{2} (2 d+2 e x+i \pi )} F^{c (a+b x)} \, _2F_1\left (2,\frac {b c \log (F)}{e}+1;\frac {b c \log (F)}{e}+2;-e^{\frac {1}{2} (2 d+2 e x+i \pi )}\right )}{f (b c \log (F)+e)} \]
Antiderivative was successfully verified.
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Rule 5492
Rule 5496
Rubi steps
\begin {align*} \int \frac {F^{c (a+b x)}}{f+i f \sinh (d+e x)} \, dx &=\frac {\int F^{c (a+b x)} \text {sech}^2\left (\frac {d}{2}+\frac {i \pi }{4}+\frac {e x}{2}\right ) \, dx}{2 f}\\ &=\frac {2 e^{\frac {1}{2} (2 d+i \pi +2 e x)} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{e};2+\frac {b c \log (F)}{e};-e^{\frac {1}{2} (2 d+i \pi +2 e x)}\right )}{f (e+b c \log (F))}\\ \end {align*}
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Mathematica [A] time = 3.74, size = 104, normalized size = 1.22 \[ \frac {2 F^{c (a+b x)} \left (\, _2F_1\left (1,\frac {b c \log (F)}{e};\frac {b c \log (F)}{e}+1;-i e^{d+e x}\right )+\frac {\cosh \left (\frac {e x}{2}\right )-\sinh \left (\frac {e x}{2}\right )}{\left (1-i e^d\right ) \sinh \left (\frac {e x}{2}\right )+\left (-1-i e^d\right ) \cosh \left (\frac {e x}{2}\right )}\right )}{e f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ \frac {{\left (e f e^{\left (e x + d\right )} - i \, e f\right )} {\rm integral}\left (-\frac {2 i \, F^{b c x + a c} b c \log \relax (F)}{e f e^{\left (e x + d\right )} - i \, e f}, x\right ) + 2 i \, F^{b c x + a c}}{e f e^{\left (e x + d\right )} - i \, e f} \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{i \, f \sinh \left (e x + d\right ) + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {F^{c \left (b x +a \right )}}{f +i f \sinh \left (e x +d \right )}\, dx \]
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 \[ -4 \, F^{a c} b c e \int \frac {F^{b c x}}{i \, b^{2} c^{2} f \log \relax (F)^{2} - 3 i \, b c e f \log \relax (F) + 2 i \, e^{2} f + {\left (b^{2} c^{2} f e^{\left (3 \, d\right )} \log \relax (F)^{2} - 3 \, b c e f e^{\left (3 \, d\right )} \log \relax (F) + 2 \, e^{2} f e^{\left (3 \, d\right )}\right )} e^{\left (3 \, e x\right )} + {\left (-3 i \, b^{2} c^{2} f e^{\left (2 \, d\right )} \log \relax (F)^{2} + 9 i \, b c e f e^{\left (2 \, d\right )} \log \relax (F) - 6 i \, e^{2} f e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} - 3 \, {\left (b^{2} c^{2} f e^{d} \log \relax (F)^{2} - 3 \, b c e f e^{d} \log \relax (F) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}}\,{d x} \log \relax (F) - \frac {2 \, {\left (-2 i \, F^{a c} e - {\left (F^{a c} b c e^{d} \log \relax (F) - 2 \, F^{a c} e e^{d}\right )} e^{\left (e x\right )}\right )} F^{b c x}}{-i \, b^{2} c^{2} f \log \relax (F)^{2} + 3 i \, b c e f \log \relax (F) - 2 i \, e^{2} f + {\left (i \, b^{2} c^{2} f e^{\left (2 \, d\right )} \log \relax (F)^{2} - 3 i \, b c e f e^{\left (2 \, d\right )} \log \relax (F) + 2 i \, e^{2} f e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} + 2 \, {\left (b^{2} c^{2} f e^{d} \log \relax (F)^{2} - 3 \, b c e f e^{d} \log \relax (F) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}} \]
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 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{f+f\,\mathrm {sinh}\left (d+e\,x\right )\,1{}\mathrm {i}} \,d x \]
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 \[ - \frac {i \int \frac {F^{a c} F^{b c x}}{\sinh {\left (d + e x \right )} - i}\, dx}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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