Optimal. Leaf size=85 \[ \frac {2 e^{\frac {1}{2} (2 d+i \pi +2 e x)} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{e};2+\frac {b c \log (F)}{e};-e^{\frac {1}{2} (2 d+i \pi +2 e x)}\right )}{f (e+b c \log (F))} \]
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Rubi [A]
time = 0.06, 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 = {5604, 5600}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 5600
Rule 5604
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 = 2.10, size = 104, normalized size = 1.22 \begin {gather*} \frac {2 F^{c (a+b x)} \left (\, _2F_1\left (1,\frac {b c \log (F)}{e};1+\frac {b c \log (F)}{e};-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 ) \cosh \left (\frac {e x}{2}\right )+\left (1-i e^d\right ) \sinh \left (\frac {e x}{2}\right )}\right )}{e f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.76, 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 \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 {i \int \frac {F^{a c} F^{b c x}}{\sinh {\left (d + e x \right )} - i}\, dx}{f} \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^{c\,\left (a+b\,x\right )}}{f+f\,\mathrm {sinh}\left (d+e\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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