Optimal. Leaf size=16 \[ a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]
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Rubi [A] time = 0.009225, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3770} \[ a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int (a+b \text{sech}(c+d x)) \, dx &=a x+b \int \text{sech}(c+d x) \, dx\\ &=a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0017844, size = 16, normalized size = 1. \[ a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 1.1 \begin{align*} ax+{\frac{b\arctan \left ( \sinh \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21941, size = 22, normalized size = 1.38 \begin{align*} a x + \frac{b \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31228, size = 74, normalized size = 4.62 \begin{align*} \frac{a d x + 2 \, b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d} \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 \left (a + b \operatorname{sech}{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13199, size = 23, normalized size = 1.44 \begin{align*} a x + \frac{2 \, b \arctan \left (e^{\left (d x + c\right )}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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