Optimal. Leaf size=11 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]
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Rubi [A] time = 0.0042149, 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} \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \sec (a+b x) \, dx &=\frac{\tanh ^{-1}(\sin (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0022212, size = 11, normalized size = 1. \[ \frac{\tanh ^{-1}(\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \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 = 1.10381, size = 24, normalized size = 2.18 \begin{align*} \frac{\log \left (\sec \left (b x + a\right ) + \tan \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52565, size = 76, normalized size = 6.91 \begin{align*} \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.91471, size = 36, normalized size = 3.27 \begin{align*} \begin{cases} \frac{\log{\left (\tan{\left (a + b x \right )} + \sec{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x \left (\tan{\left (a \right )} \sec{\left (a \right )} + \sec ^{2}{\left (a \right )}\right )}{\tan{\left (a \right )} + \sec{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31224, size = 59, normalized size = 5.36 \begin{align*} \frac{\log \left ({\left | \frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) + 2 \right |}\right ) - \log \left ({\left | \frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) - 2 \right |}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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