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 = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.12, 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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fricas [A] time = 0.96, size = 44, normalized size = 1.63 \[ \frac {2 \, \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) + 1\right )}}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.44, size = 57, normalized size = 2.11 \[ -\frac {2 \, \sqrt {2} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 42, normalized size = 1.56 \[ -\frac {2 \sqrt {\frac {a \left (-1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )}{d \left (-1+\cos \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 36, normalized size = 1.33 \[ \frac {\sin \left (c+d\,x\right )\,\sqrt {a-\frac {a}{\cos \left (c+d\,x\right )}}}{d\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \left (\sec {\left (c + d x \right )} - 1\right )} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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