Optimal. Leaf size=125 \[ -\frac {2 \cot (e+f x) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right )|\frac {a-b}{a+b}\right )}{f \sqrt {a+b}} \]
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Rubi [A] time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3780} \[ -\frac {2 \cot (e+f x) \sqrt {-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt {\frac {b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left (\frac {a}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b}}{\sqrt {a+b \sec (e+f x)}}\right )|\frac {a-b}{a+b}\right )}{f \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 3780
Rubi steps
\begin {align*} \int \sqrt {a+b \sec (e+f x)} \, dx &=-\frac {2 \cot (e+f x) \Pi \left (\frac {a}{a+b};\sin ^{-1}\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*}
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Mathematica [A] time = 0.27, size = 151, normalized size = 1.21 \[ \frac {4 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{\cos (e+f x)+1}} \sqrt {\frac {a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt {a+b \sec (e+f x)} \left ((b-a) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )+2 a \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )\right )}{f (a \cos (e+f x)+b)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.34, size = 215, normalized size = 1.72 \[ -\frac {2 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \left (\EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) a -\EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) b -2 a \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {a -b}{a +b}}\right )\right ) \left (-1+\cos \left (f x +e \right )\right )}{f \left (b +a \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sec {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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