Optimal. Leaf size=27 \[ -\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}} \]
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Rubi [A] time = 0.0298956, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3792} \[ -\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3792
Rubi steps
\begin{align*} \int \sec (c+d x) \sqrt{a-a \sec (c+d x)} \, dx &=-\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.110675, size = 30, normalized size = 1.11 \[ \frac{2 \cot \left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.15, size = 42, normalized size = 1.6 \begin{align*} -2\,{\frac{\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{{\frac{a \left ( -1+\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.935, size = 107, normalized size = 3.96 \begin{align*} \frac{2 \, \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}{\left (\cos \left (d x + c\right ) + 1\right )}}{d \sin \left (d x + c\right )} \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*} \int \sqrt{- a \left (\sec{\left (c + d x \right )} - 1\right )} \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5077, size = 77, normalized size = 2.85 \begin{align*} -\frac{2 \, \sqrt{2} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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