Optimal. Leaf size=69 \[ \frac {2 a \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\sec (e+f x)+1\right )}{f \sqrt {a-a \sec (e+f x)}} \]
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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 = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3806, 67, 65} \[ \frac {2 a \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};\sec (e+f x)+1\right )}{f \sqrt {a-a \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 3806
Rubi steps
\begin {align*} \int (d \sec (e+f x))^n \sqrt {a-a \sec (e+f x)} \, dx &=-\frac {\left (a^2 d \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(d 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} (d \sec (e+f x))^n \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 {2 a \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1+\sec (e+f x)\right ) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 213, normalized size = 3.09 \[ \frac {2^{n-\frac {1}{2}} e^{\frac {1}{2} i (e+f (1-2 n) x)} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{n-\frac {1}{2}} \csc \left (\frac {e}{2}+\frac {f x}{2}\right ) \sqrt {a-a \sec (e+f x)} \left ((n+1) e^{i f n x} \, _2F_1\left (1,\frac {1-n}{2};\frac {n+2}{2};-e^{2 i (e+f x)}\right )-n e^{i (e+f (n+1) x)} \, _2F_1\left (1,1-\frac {n}{2};\frac {n+3}{2};-e^{2 i (e+f x)}\right )\right ) \sec ^{-n-\frac {1}{2}}(e+f x) (d \sec (e+f x))^n}{f n (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int \sqrt {-a \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.17, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{n} \sqrt {a -a \sec \left (f x +e \right )}\, dx \]
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 \[ \int \sqrt {-a \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int \sqrt {a-\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{n} \sqrt {- a \left (\sec {\left (e + f x \right )} - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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