Optimal. Leaf size=25 \[ \frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0192775, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3879, 43} \[ \frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int (a+a \sec (c+d x)) \tan (c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a+a x}{x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}+\frac{a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \log (\cos (c+d x))}{d}+\frac{a \sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0178748, size = 25, normalized size = 1. \[ \frac{a \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 25, normalized size = 1. \begin{align*}{\frac{a\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}}+{\frac{a\sec \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14846, size = 35, normalized size = 1.4 \begin{align*} -\frac{a \log \left (\cos \left (d x + c\right )\right ) - \frac{a}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.697311, size = 80, normalized size = 3.2 \begin{align*} -\frac{a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - a}{d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.423997, size = 37, normalized size = 1.48 \begin{align*} \begin{cases} \frac{a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right ) \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.41025, size = 143, normalized size = 5.72 \begin{align*} \frac{a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{3 \, a + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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