Optimal. Leaf size=174 \[ \frac {(1-n) \sin (e+f x) \sec ^{n-2}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(e+f x)\right )}{a f (2-n) \sqrt {\sin ^2(e+f x)}}-\frac {\sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{a f \sqrt {\sin ^2(e+f x)}}+\frac {\sin (e+f x) \sec ^n(e+f x)}{f (a \sec (e+f x)+a)} \]
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Rubi [A] time = 0.16, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3820, 3787, 3772, 2643} \[ \frac {(1-n) \sin (e+f x) \sec ^{n-2}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(e+f x)\right )}{a f (2-n) \sqrt {\sin ^2(e+f x)}}-\frac {\sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{a f \sqrt {\sin ^2(e+f x)}}+\frac {\sin (e+f x) \sec ^n(e+f x)}{f (a \sec (e+f x)+a)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rule 3787
Rule 3820
Rubi steps
\begin {align*} \int \frac {\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac {(1-n) \int \sec ^{-1+n}(e+f x) (a-a \sec (e+f x)) \, dx}{a^2}\\ &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac {(1-n) \int \sec ^{-1+n}(e+f x) \, dx}{a}+\frac {(1-n) \int \sec ^n(e+f x) \, dx}{a}\\ &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}-\frac {\left ((1-n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{1-n}(e+f x) \, dx}{a}+\frac {\left ((1-n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \, dx}{a}\\ &=\frac {\sec ^n(e+f x) \sin (e+f x)}{f (a+a \sec (e+f x))}+\frac {(1-n) \, _2F_1\left (\frac {1}{2},\frac {2-n}{2};\frac {4-n}{2};\cos ^2(e+f x)\right ) \sec ^{-2+n}(e+f x) \sin (e+f x)}{a f (2-n) \sqrt {\sin ^2(e+f x)}}-\frac {\, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{a f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [F] time = 1.05, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (f x + e\right )^{n}}{a \sec \left (f x + e\right ) + a}, x\right ) \]
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 \[ \int \frac {\sec \left (f x + e\right )^{n}}{a \sec \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.70, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{n}\left (f x +e \right )}{a +a \sec \left (f x +e \right )}\, dx \]
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 \[ \int \frac {\sec \left (f x + e\right )^{n}}{a \sec \left (f x + e\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n}{a+\frac {a}{\cos \left (e+f\,x\right )}} \,d x \]
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 \[ \frac {\int \frac {\sec ^{n}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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