Integrand size = 16, antiderivative size = 103 \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=-\frac {(b-d) e^{a+c+b x+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
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Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5598, 5600} \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=-\frac {(b-d) e^{a+b x+c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (\frac {b}{d}+3\right ),-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
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Rule 5598
Rule 5600
Rubi steps \begin{align*} \text {integral}& = \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} \operatorname {Hypergeometric2F1}\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*}
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\frac {e^{a+b x} \left (-2 (b-d) e^{c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {b+d}{2 d},\frac {1}{2} \left (3+\frac {b}{d}\right ),-e^{2 (c+d x)}\right )+\text {sech}(c+d x) (b+d \tanh (c+d x))\right )}{2 d^2} \]
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\[\int {\mathrm e}^{b x +a} \operatorname {sech}\left (d x +c \right )^{3}d x\]
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\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3} \,d x } \]
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\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=e^{a} \int e^{b x} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]
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\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3} \,d x } \]
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\[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int e^{a+b x} \text {sech}^3(c+d x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^3} \,d x \]
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