Optimal. Leaf size=68 \[ \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {e+b c \log (F)}{2 e};\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right );-e^{2 (d+e x)}\right )}{e+b c \log (F)} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5600}
\begin {gather*} \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {e+b c \log (F)}{2 e};\frac {1}{2} \left (\frac {b c \log (F)}{e}+3\right );-e^{2 (d+e x)}\right )}{b c \log (F)+e} \end {gather*}
Antiderivative was successfully verified.
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Rule 5600
Rubi steps
\begin {align*} \int F^{c (a+b x)} \text {sech}(d+e x) \, dx &=\frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {e+b c \log (F)}{2 e};\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right );-e^{2 (d+e x)}\right )}{e+b c \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.03 \begin {gather*} \frac {2 e^{d+e x} F^{c (a+b x)} \, _2F_1\left (1,\frac {1}{2}+\frac {b c \log (F)}{2 e};\frac {3}{2}+\frac {b c \log (F)}{2 e};-e^{2 (d+e x)}\right )}{e+b c \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \mathrm {sech}\left (e x +d \right )\, dx\]
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*} \int F^{c \left (a + b x\right )} \operatorname {sech}{\left (d + e x \right )}\, dx \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 )}}{\mathrm {cosh}\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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