Optimal. Leaf size=17 \[ \frac {\log \left (e^{2 a+2 b x}+1\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2282, 12, 260} \[ \frac {\log \left (e^{2 a+2 b x}+1\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \text {sech}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {2 x}{1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\log \left (1+e^{2 a+2 b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ \frac {\log \left (e^{2 a+2 b x}+1\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 30, normalized size = 1.76 \[ \frac {\log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 16, normalized size = 0.94 \[ \frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 1.12 \[ x +\frac {\ln \left (\cosh \left (b x +a \right )\right )}{b}+\frac {a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 16, normalized size = 0.94 \[ \frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 16, normalized size = 0.94 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{b} \]
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 \[ e^{a} \int e^{b x} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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