Optimal. Leaf size=28 \[ \text {Int}\left (\sqrt {a+b \sec (e+f x)} (d \sec (e+f x))^n,x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (d \sec (e+f x))^n \sqrt {a+b \sec (e+f x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (d \sec (e+f x))^n \sqrt {a+b \sec (e+f x)} \, dx &=\int (d \sec (e+f x))^n \sqrt {a+b \sec (e+f x)} \, dx\\ \end {align*}
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Mathematica [A] time = 0.54, size = 0, normalized size = 0.00 \[ \int (d \sec (e+f x))^n \sqrt {a+b \sec (e+f x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.25, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{n} \sqrt {a +b \sec \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right ) + a} \left (d \sec \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{n} \sqrt {a + b \sec {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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