Integrand size = 29, antiderivative size = 61 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {c} f} \]
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Time = 0.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4019, 209} \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {c} f} \]
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Rule 209
Rule 4019
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 a) \text {Subst}\left (\int \frac {1}{1+a c x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \\ & = \frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {c} f} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {d+c \cos (e+f x)}}\right ) \sqrt {d+c \cos (e+f x)} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{\sqrt {c} f \sqrt {c+d \sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(165\) vs. \(2(49)=98\).
Time = 2.90 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.72
method | result | size |
default | \(-\frac {2 \sqrt {2}\, \sqrt {-\left (c -d \right )^{4} c}\, \arctan \left (\frac {\left (c -d \right )^{2} c \sqrt {2}\, \sin \left (f x +e \right )}{\sqrt {-\left (c -d \right )^{4} c}\, \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \cos \left (f x +e \right )}{f \left (c^{2}-2 c d +d^{2}\right ) c \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )+1}}}\) | \(166\) |
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Time = 0.34 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.38 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\left [\frac {\sqrt {-\frac {a}{c}} \log \left (-\frac {2 \, c \sqrt {-\frac {a}{c}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a c \cos \left (f x + e\right )^{2} + a c - a d - {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{f}, -\frac {2 \, \sqrt {\frac {a}{c}} \arctan \left (\frac {\sqrt {\frac {a}{c}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a \sin \left (f x + e\right )}\right )}{f}\right ] \]
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\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}}{\sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sqrt {a \sec \left (f x + e\right ) + a}}{\sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}}{\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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