Integrand size = 22, antiderivative size = 48 \[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d} \]
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Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3880, 209} \[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d} \]
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Rule 209
Rule 3880
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.29 \[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right ) \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt {a-a \sec (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(39)=78\).
Time = 0.94 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {\sqrt {2}\, \sin \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\) | \(83\) |
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none
Time = 0.32 (sec) , antiderivative size = 161, normalized size of antiderivative = 3.35 \[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=\left [\frac {\sqrt {2} \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \sqrt {-\frac {1}{a}} - {\left (3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right )}{2 \, d}, \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right )}{\sqrt {a} d}\right ] \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\sqrt {- a \left (\sec {\left (c + d x \right )} - 1\right )}}\, dx \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\sqrt {-a \sec \left (d x + c\right ) + a}} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40 \[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{\sqrt {a} d \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]
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Timed out. \[ \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx=\int \frac {1}{\cos \left (c+d\,x\right )\,\sqrt {a-\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
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