Optimal. Leaf size=56 \[ \frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d} \]
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Rubi [A] time = 0.026573, antiderivative size = 56, normalized size of antiderivative = 1., 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 = {5492} \[ \frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d} \]
Antiderivative was successfully verified.
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Rule 5492
Rubi steps
\begin{align*} \int e^{a+b x} \text{sech}^2(c+d x) \, dx &=\frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,1+\frac{b}{2 d};2+\frac{b}{2 d};-e^{2 (c+d x)}\right )}{b+2 d}\\ \end{align*}
Mathematica [A] time = 0.0167047, size = 56, normalized size = 1. \[ \frac{4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}} \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b \int \frac{e^{\left (b x + a\right )}}{2 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} + d\right )}}\,{d x} - \frac{2 \, e^{\left (b x + a\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\left (b x + a\right )} \operatorname{sech}\left (d x + c\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int e^{b x} \operatorname{sech}^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (b x + a\right )} \operatorname{sech}\left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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