Optimal. Leaf size=11 \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0050338, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3770} \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \text{sech}(a+b x) \, dx &=\frac{\tan ^{-1}(\sinh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0020975, size = 11, normalized size = 1. \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 12, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \sinh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0037, size = 15, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\sinh \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00671, size = 58, normalized size = 5.27 \begin{align*} \frac{2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09724, size = 16, normalized size = 1.45 \begin{align*} \frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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