Integrand size = 24, antiderivative size = 87 \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\frac {2^{\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1}{2},1-n,\frac {1}{2}-m,\frac {3}{2},1+\sec (e+f x),\frac {1}{2} (1+\sec (e+f x))\right ) (1-\sec (e+f x))^{-\frac {1}{2}-m} (a-a \sec (e+f x))^m \tan (e+f x)}{f} \]
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Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3913, 3910, 138} \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\frac {2^{m+\frac {1}{2}} \tan (e+f x) (1-\sec (e+f x))^{-m-\frac {1}{2}} (a-a \sec (e+f x))^m \operatorname {AppellF1}\left (\frac {1}{2},1-n,\frac {1}{2}-m,\frac {3}{2},\sec (e+f x)+1,\frac {1}{2} (\sec (e+f x)+1)\right )}{f} \]
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Rule 138
Rule 3910
Rule 3913
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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} \operatorname {AppellF1}\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*}
\[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx \]
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\[\int \left (-\sec \left (f x +e \right )\right )^{n} \left (a -a \sec \left (f x +e \right )\right )^{m}d x\]
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\[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int { {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (-\sec \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int \left (- \sec {\left (e + f x \right )}\right )^{n} \left (- a \left (\sec {\left (e + f x \right )} - 1\right )\right )^{m}\, dx \]
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\[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int { {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (-\sec \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int { {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (-\sec \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int {\left (a-\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (-\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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