Optimal. Leaf size=87 \[ \frac {2^{m+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{-m-\frac {1}{2}} (a-a \sec (e+f x))^m F_1\left (\frac {1}{2};1-n,\frac {1}{2}-m;\frac {3}{2};\sec (e+f x)+1,\frac {1}{2} (\sec (e+f x)+1)\right )}{f} \]
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Rubi [A] time = 0.12, antiderivative size = 87, 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 = {3828, 3825, 133} \[ \frac {2^{m+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{-m-\frac {1}{2}} (a-a \sec (e+f x))^m F_1\left (\frac {1}{2};1-n,\frac {1}{2}-m;\frac {3}{2};\sec (e+f x)+1,\frac {1}{2} (\sec (e+f x)+1)\right )}{f} \]
Antiderivative was successfully verified.
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Rule 133
Rule 3825
Rule 3828
Rubi steps
\begin {align*} \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx &=\left ((1-\sec (e+f x))^{-m} (a-a \sec (e+f x))^m\right ) \int (1-\sec (e+f x))^m (-\sec (e+f x))^n \, dx\\ &=\frac {\left ((1-\sec (e+f x))^{-\frac {1}{2}-m} (a-a \sec (e+f x))^m \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(1-x)^{-1+n} (2-x)^{-\frac {1}{2}+m}}{\sqrt {x}} \, dx,x,1+\sec (e+f x)\right )}{f \sqrt {1+\sec (e+f x)}}\\ &=\frac {2^{\frac {1}{2}+m} F_1\left (\frac {1}{2};1-n,\frac {1}{2}-m;\frac {3}{2};1+\sec (e+f x),\frac {1}{2} (1+\sec (e+f x))\right ) (1-\sec (e+f x))^{-\frac {1}{2}-m} (a-a \sec (e+f x))^m \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (-\sec \left (f x + e\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (-\sec \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.89, size = 0, normalized size = 0.00 \[ \int \left (-\sec \left (f x +e \right )\right )^{n} \left (a -a \sec \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (-\sec \left (f x + e\right )\right )^{n}\,{d x} \]
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 \[ \int {\left (a-\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (-\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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 \[ \int \left (- \sec {\left (e + f x \right )}\right )^{n} \left (- a \left (\sec {\left (e + f x \right )} - 1\right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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