Optimal. Leaf size=17 \[ -\frac{\log (\cos (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.0257337, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 31} \[ -\frac{\log (\cos (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a+a x} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\log (1+\cos (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0177193, size = 19, normalized size = 1.12 \[ -\frac{2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 33, normalized size = 1.9 \begin{align*} -{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{da}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15333, size = 23, normalized size = 1.35 \begin{align*} -\frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11244, size = 49, normalized size = 2.88 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.0738, size = 41, normalized size = 2.41 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} - \frac{\log{\left (\sec{\left (c + d x \right )} + 1 \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x \tan{\left (c \right )}}{a \sec{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32052, size = 42, normalized size = 2.47 \begin{align*} \frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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