Optimal. Leaf size=22 \[ \frac{\tan (c+d x)}{d (a \sec (c+d x)+a)} \]
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Rubi [A] time = 0.0238029, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3794} \[ \frac{\tan (c+d x)}{d (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{\tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0257344, size = 17, normalized size = 0.77 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 17, normalized size = 0.8 \begin{align*}{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08566, size = 31, normalized size = 1.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52776, size = 53, normalized size = 2.41 \begin{align*} \frac{\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{\sec{\left (c + d x \right )}}{\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 = 1.26239, size = 22, normalized size = 1. \begin{align*} \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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