Optimal. Leaf size=90 \[ \frac {\left (1+e^{2 (d+e x)}\right )^n F^{a c+b c x} \, _2F_1\left (n,\frac {e n+b c \log (F)}{2 e};1+\frac {e n+b c \log (F)}{2 e};-e^{2 (d+e x)}\right ) \text {sech}^n(d+e x)}{e n+b c \log (F)} \]
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Rubi [A]
time = 0.10, 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 = {5602, 2291}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2291
Rule 5602
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 = 0.05, size = 89, normalized size = 0.99 \begin {gather*} \frac {\left (1+e^{2 (d+e x)}\right )^n F^{c (a+b 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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.41, 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 \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}^{n}{\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 F^{c\,\left (a+b\,x\right )}\,{\left (\frac {1}{\mathrm {cosh}\left (d+e\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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