Optimal. Leaf size=16 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
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Rubi [A] time = 0.0201203, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2735, 3770} \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \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 = 0.0072224, 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 = 0.055, size = 30, normalized size = 1.9 \begin{align*} ax+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{ca}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19116, size = 38, normalized size = 2.38 \begin{align*} \frac{{\left (d x + c\right )} a + 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 = 1.72294, 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 = 4.94182, size = 49, normalized size = 3.06 \begin{align*} a x + a \left (\begin{cases} \frac{x \tan{\left (c \right )} \sec{\left (c \right )}}{\tan{\left (c \right )} + \sec{\left (c \right )}} + \frac{x \sec ^{2}{\left (c \right )}}{\tan{\left (c \right )} + \sec{\left (c \right )}} & \text{for}\: d = 0 \\\frac{\log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )}}{d} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.46839, size = 58, normalized size = 3.62 \begin{align*} \frac{{\left (d x + c\right )} a + a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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