Optimal. Leaf size=16 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
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Rubi [A] time = 0.0072979, 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} \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \, dx &=a x+a \int \sec (c+d x) \, dx\\ &=a x+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0015236, size = 16, normalized size = 1. \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 24, normalized size = 1.5 \begin{align*} ax+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \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 = 1.05706, size = 31, normalized size = 1.94 \begin{align*} a x + \frac{a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76666, size = 95, normalized size = 5.94 \begin{align*} \frac{2 \, a d x + a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.73496, size = 41, normalized size = 2.56 \begin{align*} a x + a \left (\begin{cases} \frac{\log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \left (\tan{\left (c \right )} \sec{\left (c \right )} + \sec ^{2}{\left (c \right )}\right )}{\tan{\left (c \right )} + \sec{\left (c \right )}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35175, size = 66, normalized size = 4.12 \begin{align*} a x + \frac{a{\left (\log \left ({\left | \frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac{1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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