Integrand size = 16, antiderivative size = 56 \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 (c+d x)}\right )}{b+2 d} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5600} \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b}{2 d}+1,\frac {b}{2 d}+2,-e^{2 (c+d x)}\right )}{b+2 d} \]
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Rule 5600
Rubi steps \begin{align*} \text {integral}& = \frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 (c+d x)}\right )}{b+2 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\frac {4 e^{a+b x+2 (c+d x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b}{2 d},2+\frac {b}{2 d},-e^{2 (c+d x)}\right )}{b+2 d} \]
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\[\int {\mathrm e}^{b x +a} \operatorname {sech}\left (d x +c \right )^{2}d x\]
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\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2} \,d x } \]
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\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=e^{a} \int e^{b x} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \]
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\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2} \,d x } \]
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\[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int { e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int e^{a+b x} \text {sech}^2(c+d x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^2} \,d x \]
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