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 = 0.0752858, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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 5496
Rule 5492
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*}
Mathematica [A] time = 3.36968, 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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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{f+if\sinh \left ( ex+d \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -4 \, F^{a c} b c e \int \frac{F^{b c x}}{i \, b^{2} c^{2} f \log \left (F\right )^{2} - 3 i \, b c e f \log \left (F\right ) + 2 i \, e^{2} f +{\left (b^{2} c^{2} f e^{\left (3 \, d\right )} \log \left (F\right )^{2} - 3 \, b c e f e^{\left (3 \, d\right )} \log \left (F\right ) + 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 \left (F\right )^{2} + 9 i \, b c e f e^{\left (2 \, d\right )} \log \left (F\right ) - 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 \left (F\right )^{2} - 3 \, b c e f e^{d} \log \left (F\right ) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}}\,{d x} \log \left (F\right ) + \frac{{\left (4 i \, F^{a c} e + 2 \,{\left (F^{a c} b c e^{d} \log \left (F\right ) - 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 \left (F\right )^{2} + 3 i \, b c e f \log \left (F\right ) - 2 i \, e^{2} f +{\left (i \, b^{2} c^{2} f e^{\left (2 \, d\right )} \log \left (F\right )^{2} - 3 i \, b c e f e^{\left (2 \, d\right )} \log \left (F\right ) + 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 \left (F\right )^{2} - 3 \, b c e f e^{d} \log \left (F\right ) + 2 \, e^{2} f e^{d}\right )} e^{\left (e x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \left (F\right )}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{F^{a c} F^{b c x}}{i \sinh{\left (d + e x \right )} + 1}\, dx}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{i \, f \sinh \left (e x + d\right ) + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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