Optimal. Leaf size=103 \[ -\frac{(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );-e^{2 (c+d x)}\right )}{d^2}+\frac{b e^{a+b x} \text{sech}(c+d x)}{2 d^2}+\frac{e^{a+b x} \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.0500383, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5490, 5492} \[ -\frac{(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );-e^{2 (c+d x)}\right )}{d^2}+\frac{b e^{a+b x} \text{sech}(c+d x)}{2 d^2}+\frac{e^{a+b x} \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 5490
Rule 5492
Rubi steps
\begin{align*} \int e^{a+b x} \text{sech}^3(c+d x) \, dx &=\frac{b e^{a+b x} \text{sech}(c+d x)}{2 d^2}+\frac{e^{a+b x} \text{sech}(c+d x) \tanh (c+d x)}{2 d}+\frac{1}{2} \left (1-\frac{b^2}{d^2}\right ) \int e^{a+b x} \text{sech}(c+d x) \, dx\\ &=-\frac{(b-d) e^{a+c+b x+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (3+\frac{b}{d}\right );-e^{2 (c+d x)}\right )}{d^2}+\frac{b e^{a+b x} \text{sech}(c+d x)}{2 d^2}+\frac{e^{a+b x} \text{sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.146712, size = 80, normalized size = 0.78 \[ \frac{e^{a+b x} \left (\text{sech}(c+d x) (b+d \tanh (c+d x))-2 (b-d) e^{c+d x} \, _2F_1\left (1,\frac{b+d}{2 d};\frac{1}{2} \left (\frac{b}{d}+3\right );-e^{2 (c+d x)}\right )\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}} \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}\, 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*} -8 \,{\left (b^{2} e^{c} - d^{2} e^{c}\right )} \int \frac{e^{\left (b x + d x + a\right )}}{8 \,{\left (d^{2} e^{\left (2 \, d x + 2 \, c\right )} + d^{2}\right )}}\,{d x} + \frac{{\left (b e^{\left (3 \, c\right )} + d e^{\left (3 \, c\right )}\right )} e^{\left (b x + 3 \, d x + a\right )} +{\left (b e^{c} - d e^{c}\right )} e^{\left (b x + d x + a\right )}}{d^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d^{2} e^{\left (2 \, d x + 2 \, c\right )} + d^{2}} \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 )^{3}, 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}^{3}{\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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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