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 = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 1.00 \[ \frac {4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac {b}{2 d}+1;\frac {b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2}, x\right ) \]
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 \[ \int e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b x +a} \mathrm {sech}\left (d x +c \right )^{2}\, dx \]
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 \[ 4 \, b \int \frac {e^{\left (b x + a\right )}}{2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} + d\right )}}\,{d x} - \frac {2 \, e^{\left (b x + a\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \]
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 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^2} \,d x \]
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 \[ e^{a} \int e^{b x} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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