Integrand size = 21, antiderivative size = 89 \[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,1-n,\frac {1}{2},\frac {3}{2}+m,1-\sec (e+f x),\frac {1}{2} (1-\sec (e+f x))\right ) (1-\sec (e+f x))^m \tan (e+f x)}{f (1+2 m) \sqrt {1+\sec (e+f x)}} \]
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Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3911, 138} \[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\frac {\sqrt {2} \tan (e+f x) (1-\sec (e+f x))^m \operatorname {AppellF1}\left (m+\frac {1}{2},1-n,\frac {1}{2},m+\frac {3}{2},1-\sec (e+f x),\frac {1}{2} (1-\sec (e+f x))\right )}{f (2 m+1) \sqrt {\sec (e+f x)+1}} \]
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Rule 138
Rule 3911
Rubi steps \begin{align*} \text {integral}& = \frac {\tan (e+f x) \text {Subst}\left (\int \frac {(1-x)^{-1+n} x^{-\frac {1}{2}+m}}{\sqrt {2-x}} \, dx,x,1-\sec (e+f x)\right )}{f \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}} \\ & = \frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,1-n,\frac {1}{2},\frac {3}{2}+m,1-\sec (e+f x),\frac {1}{2} (1-\sec (e+f x))\right ) (1-\sec (e+f x))^m \tan (e+f x)}{f (1+2 m) \sqrt {1+\sec (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(89)=178\).
Time = 3.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.87 \[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\frac {(3+2 m) \operatorname {AppellF1}\left (\frac {1}{2}+m,m+n,1-n,\frac {3}{2}+m,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (1-\sec (e+f x))^m \sec ^n(e+f x) \sin (e+f x)}{f (1+2 m) \left ((3+2 m) \operatorname {AppellF1}\left (\frac {1}{2}+m,m+n,1-n,\frac {3}{2}+m,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 \left ((-1+n) \operatorname {AppellF1}\left (\frac {3}{2}+m,m+n,2-n,\frac {5}{2}+m,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(m+n) \operatorname {AppellF1}\left (\frac {3}{2}+m,1+m+n,1-n,\frac {5}{2}+m,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (1-\sec \left (f x +e \right )\right )^{m} \sec \left (f x +e \right )^{n}d x\]
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\[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\int { {\left (-\sec \left (f x + e\right ) + 1\right )}^{m} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\int \left (1 - \sec {\left (e + f x \right )}\right )^{m} \sec ^{n}{\left (e + f x \right )}\, dx \]
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\[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\int { {\left (-\sec \left (f x + e\right ) + 1\right )}^{m} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\int { {\left (-\sec \left (f x + e\right ) + 1\right )}^{m} \sec \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx=\int {\left (1-\frac {1}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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