Integrand size = 28, antiderivative size = 38 \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a-a \sec (e+f x)}}\right )}{f} \]
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Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3886, 221} \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a-a \sec (e+f x)}}\right )}{f} \]
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Rule 221
Rule 3886
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,\frac {a \tan (e+f x)}{\sqrt {a-a \sec (e+f x)}}\right )}{f} \\ & = \frac {2 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a-a \sec (e+f x)}}\right )}{f} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45 \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\frac {2 \arcsin \left (\sqrt {-\sec (e+f x)}\right ) \cot \left (\frac {1}{2} (e+f x)\right ) \sqrt {a-a \sec (e+f x)}}{f \sqrt {1+\sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(32)=64\).
Time = 1.72 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.42
method | result | size |
default | \(-\frac {\sqrt {-a \left (\sec \left (f x +e \right )-1\right )}\, \sqrt {-\sec \left (f x +e \right )}\, \left (\operatorname {arctanh}\left (\frac {-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\right )-\operatorname {arctanh}\left (\frac {\cos \left (f x +e \right )+\sin \left (f x +e \right )+1}{2 \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\right )\right ) \cot \left (f x +e \right )}{f \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}}\) | \(130\) |
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (32) = 64\).
Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 5.66 \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {4 \, {\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{\cos \left (f x + e\right )}} + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) + 8 \, a\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{2} \sin \left (f x + e\right )}\right )}{2 \, f}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{\cos \left (f x + e\right )}}}{{\left (a \cos \left (f x + e\right ) + 2 \, a\right )} \sin \left (f x + e\right )}\right )}{f}\right ] \]
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\[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\int \sqrt {- \sec {\left (e + f x \right )}} \sqrt {- a \left (\sec {\left (e + f x \right )} - 1\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (32) = 64\).
Time = 0.41 (sec) , antiderivative size = 353, normalized size of antiderivative = 9.29 \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=-\frac {\sqrt {a} {\left (\log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right ) + \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) - 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} - 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right )^{2} - 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) - 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right )\right )\right ) + 2\right )\right )}}{2 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (32) = 64\).
Time = 0.66 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.58 \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\frac {\sqrt {2} {\left (\frac {\sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {2} a^{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right )}{\sqrt {-a}}\right )} {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a^{2} f} \]
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Timed out. \[ \int \sqrt {-\sec (e+f x)} \sqrt {a-a \sec (e+f x)} \, dx=\int \sqrt {a-\frac {a}{\cos \left (e+f\,x\right )}}\,\sqrt {-\frac {1}{\cos \left (e+f\,x\right )}} \,d x \]
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