Integrand size = 29, antiderivative size = 198 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {a+b} c f} \]
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Time = 0.13 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {4021} \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{c f \sqrt {a+b}} \]
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Rule 4021
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {a+b} c f} \\ \end{align*}
Time = 5.31 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\frac {4 \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(a+b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{-b c+a d}} \csc (e+f x) \left ((a+b) c \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{-b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right )-a (c+d) \operatorname {EllipticPi}\left (\frac {b c-a d}{a c+b c},\arcsin \left (\frac {\sqrt {\frac {(a+b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{-b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right )\right ) \sqrt {a+b \sec (e+f x)} \sin ^2\left (\frac {1}{2} (e+f x)\right )}{(a+b) c f \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {c+d \sec (e+f x)}} \]
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Time = 10.63 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.63
method | result | size |
default | \(-\frac {2 \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) a -\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) b -2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), -\frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) a \right ) \left (\cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{f \sqrt {\frac {a -b}{a +b}}\, \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right )}\) | \(322\) |
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{\sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {\sqrt {a + b \sec {\left (e + f x \right )}}}{\sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{\sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{\sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}}{\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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