Optimal. Leaf size=133 \[ \frac {2 e^{2 (d+e x)} F^{c (a+b x)} (2 e-b c \log (F)) \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )}{3 e^2}+\frac {b c \log (F) \text {sech}^2(d+e x) F^{c (a+b x)}}{6 e^2}+\frac {\tanh (d+e x) \text {sech}^2(d+e x) F^{c (a+b x)}}{3 e} \]
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Rubi [A] time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5490, 5492} \[ \frac {2 e^{2 (d+e x)} F^{c (a+b x)} (2 e-b c \log (F)) \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )}{3 e^2}+\frac {b c \log (F) \text {sech}^2(d+e x) F^{c (a+b x)}}{6 e^2}+\frac {\tanh (d+e x) \text {sech}^2(d+e x) F^{c (a+b x)}}{3 e} \]
Antiderivative was successfully verified.
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Rule 5490
Rule 5492
Rubi steps
\begin {align*} \int F^{c (a+b x)} \text {sech}^4(d+e x) \, dx &=\frac {b c F^{c (a+b x)} \log (F) \text {sech}^2(d+e x)}{6 e^2}+\frac {F^{c (a+b x)} \text {sech}^2(d+e x) \tanh (d+e x)}{3 e}+\frac {1}{6} \left (4-\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \text {sech}^2(d+e x) \, dx\\ &=\frac {2 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1+\frac {b c \log (F)}{2 e};2+\frac {b c \log (F)}{2 e};-e^{2 (d+e x)}\right ) (2 e-b c \log (F))}{3 e^2}+\frac {b c F^{c (a+b x)} \log (F) \text {sech}^2(d+e x)}{6 e^2}+\frac {F^{c (a+b x)} \text {sech}^2(d+e x) \tanh (d+e x)}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 101, normalized size = 0.76 \[ \frac {F^{c (a+b x)} \left (4 e^{2 (d+e x)} (2 e-b c \log (F)) \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;-e^{2 (d+e x)}\right )+\text {sech}^2(d+e x) (b c \log (F)+2 e \tanh (d+e x))\right )}{6 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \operatorname {sech}\left (e x + d\right )^{4}, x\right ) \]
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 \[ \int F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \mathrm {sech}\left (e x +d \right )^{4}\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\mathrm {cosh}\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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