Optimal. Leaf size=90 \[ \frac {\left (e^{2 (d+e x)}+1\right )^n F^{a c+b c x} \text {sech}^n(d+e x) \, _2F_1\left (n,\frac {e n+b c \log (F)}{2 e};\frac {e n+b c \log (F)}{2 e}+1;-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
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Rubi [A] time = 0.14, antiderivative size = 90, 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 = {5494, 2259} \[ \frac {\left (e^{2 (d+e x)}+1\right )^n F^{a c+b c x} \text {sech}^n(d+e x) \, _2F_1\left (n,\frac {e n+b c \log (F)}{2 e};\frac {e n+b c \log (F)}{2 e}+1;-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
Antiderivative was successfully verified.
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Rule 2259
Rule 5494
Rubi steps
\begin {align*} \int F^{c (a+b x)} \text {sech}^n(d+e x) \, dx &=\left (e^{-n (d+e x)} \left (1+e^{2 (d+e x)}\right )^n \text {sech}^n(d+e x)\right ) \int e^{d n+e n x} \left (1+e^{2 (d+e x)}\right )^{-n} F^{a c+b c x} \, dx\\ &=\frac {\left (1+e^{2 (d+e x)}\right )^n F^{a c+b c x} \, _2F_1\left (n,\frac {e n+b c \log (F)}{2 e};1+\frac {e n+b c \log (F)}{2 e};-e^{2 (d+e x)}\right ) \text {sech}^n(d+e x)}{e n+b c \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 89, normalized size = 0.99 \[ \frac {\left (e^{2 (d+e x)}+1\right )^n F^{c (a+b x)} \text {sech}^n(d+e x) \, _2F_1\left (n,\frac {e n+b c \log (F)}{2 e};\frac {e n+b c \log (F)}{2 e}+1;-e^{2 (d+e x)}\right )}{b c \log (F)+e n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \operatorname {sech}\left (e x + d\right )^{n}, x\right ) \]
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 \[ \int F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \mathrm {sech}\left (e x +d \right )^{n}\, dx \]
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 \[ \int F^{{\left (b x + a\right )} c} \operatorname {sech}\left (e x + d\right )^{n}\,{d x} \]
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 \[ \int F^{c\,\left (a+b\,x\right )}\,{\left (\frac {1}{\mathrm {cosh}\left (d+e\,x\right )}\right )}^n \,d x \]
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 \[ \int F^{c \left (a + b x\right )} \operatorname {sech}^{n}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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