Integrand size = 14, antiderivative size = 125 \[ \int \sqrt {a+b \sec (e+f x)} \, dx=-\frac {2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (1+\sec (e+f x))}{a+b \sec (e+f x)}} (a+b \sec (e+f x))}{\sqrt {a+b} f} \]
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Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3865} \[ \int \sqrt {a+b \sec (e+f x)} \, dx=-\frac {2 \cot (e+f x) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {a-b}{a+b}\right )}{f \sqrt {a+b}} \]
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Rule 3865
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (1+\sec (e+f x))}{a+b \sec (e+f x)}} (a+b \sec (e+f x))}{\sqrt {a+b} f} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.21 \[ \int \sqrt {a+b \sec (e+f x)} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left ((-a+b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )+2 a \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sqrt {a+b \sec (e+f x)}}{f (b+a \cos (e+f x))} \]
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Time = 6.61 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {2 \left (\cos \left (f x +e \right )+1\right ) \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a -\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b -2 a \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a +b \sec \left (f x +e \right )}}{f \left (b +a \cos \left (f x +e \right )\right )}\) | \(182\) |
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\[ \int \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \,d x } \]
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\[ \int \sqrt {a+b \sec (e+f x)} \, dx=\int \sqrt {a + b \sec {\left (e + f x \right )}}\, dx \]
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\[ \int \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \,d x } \]
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\[ \int \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \sec (e+f x)} \, dx=\int \sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
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