Optimal. Leaf size=69 \[ \frac {2 a \sin (e+f x) (-\sec (e+f x))^{-n} \sec ^{n+1}(e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\sec (e+f x)+1\right )}{f \sqrt {a-a \sec (e+f x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3806, 67, 65} \[ \frac {2 a \sin (e+f x) (-\sec (e+f x))^{-n} \sec ^{n+1}(e+f x) \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\sec (e+f x)+1\right )}{f \sqrt {a-a \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 3806
Rubi steps
\begin {align*} \int \sec ^n(e+f x) \sqrt {a-a \sec (e+f x)} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {x^{-1+n}}{\sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {\left (a^2 (-\sec (e+f x))^{-n} \sec ^{1+n}(e+f x) \sin (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(-x)^{-1+n}}{\sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1+\sec (e+f x)\right ) (-\sec (e+f x))^{-n} \sec ^{1+n}(e+f x) \sin (e+f x)}{f \sqrt {a-a \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 185, normalized size = 2.68 \[ -\frac {2^n e^{\frac {1}{2} i (e+f (1-2 n) x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^n \cos (e+f x) \csc \left (\frac {1}{2} (e+f x)\right ) \sqrt {a-a \sec (e+f x)} \left (n e^{i (e+f (n+1) x)} \, _2F_1\left (1,1-\frac {n}{2};\frac {n+3}{2};-e^{2 i (e+f x)}\right )-(n+1) e^{i f n x} \, _2F_1\left (1,\frac {1-n}{2};\frac {n+2}{2};-e^{2 i (e+f x)}\right )\right )}{f n (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )^{n}, x\right ) \]
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 \[ \int \sqrt {-a \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.26, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{n}\left (f x +e \right )\right ) \sqrt {a -a \sec \left (f x +e \right )}\, dx \]
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 \[ \int \sqrt {-a \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )^{n}\,{d x} \]
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 \[ \int \sqrt {a-\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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 \[ \int \sqrt {- a \left (\sec {\left (e + f x \right )} - 1\right )} \sec ^{n}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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