Optimal. Leaf size=52 \[ \frac {2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5492} \[ \frac {2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \]
Antiderivative was successfully verified.
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Rule 5492
Rubi steps
\begin {align*} \int e^{a+b x} \text {sech}(c+d x) \, dx &=\frac {2 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 )}{b+d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.98 \[ \frac {2 e^{a+x (b+d)+c} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right ), x\right ) \]
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 \[ \int e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b x +a} \mathrm {sech}\left (d x +c \right )\, dx \]
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 \[ \int e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{\mathrm {cosh}\left (c+d\,x\right )} \,d x \]
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 \[ e^{a} \int e^{b x} \operatorname {sech}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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