Optimal. Leaf size=29 \[ \frac{x}{a}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)} \]
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Rubi [A] time = 0.0135317, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3777, 8} \[ \frac{x}{a}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3777
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{a+a \sec (c+d x)} \, dx &=-\frac{\tan (c+d x)}{d (a+a \sec (c+d x))}+\frac{\int a \, dx}{a^2}\\ &=\frac{x}{a}-\frac{\tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.122824, size = 58, normalized size = 2. \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (d x \cos \left (c+\frac{d x}{2}\right )-2 \sin \left (\frac{d x}{2}\right )+d x \cos \left (\frac{d x}{2}\right )\right )}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 37, normalized size = 1.3 \begin{align*} -{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6481, size = 66, normalized size = 2.28 \begin{align*} \frac{\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59883, size = 89, normalized size = 3.07 \begin{align*} \frac{d x \cos \left (d x + c\right ) + d x - \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29995, size = 38, normalized size = 1.31 \begin{align*} \frac{\frac{d x + c}{a} - \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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