Integrand size = 26, antiderivative size = 101 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\frac {2^{\frac {1}{2}+n} c \operatorname {AppellF1}\left (\frac {7}{2},\frac {1}{2}-n,1,\frac {9}{2},\frac {1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (1-\sec (e+f x))^{\frac {1}{2}-n} (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{-1+n} \tan (e+f x)}{7 f} \]
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Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3997, 142, 141} \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\frac {c 2^{n+\frac {1}{2}} \tan (e+f x) (a \sec (e+f x)+a)^3 (1-\sec (e+f x))^{\frac {1}{2}-n} \operatorname {AppellF1}\left (\frac {7}{2},\frac {1}{2}-n,1,\frac {9}{2},\frac {1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right ) (c-c \sec (e+f x))^{n-1}}{7 f} \]
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Rule 141
Rule 142
Rule 3997
Rubi steps \begin{align*} \text {integral}& = -\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^{5/2} (c-c x)^{-\frac {1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = -\frac {\left (2^{-\frac {1}{2}+n} a c (c-c \sec (e+f x))^{-1+n} \left (\frac {c-c \sec (e+f x)}{c}\right )^{\frac {1}{2}-n} \tan (e+f x)\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}-\frac {x}{2}\right )^{-\frac {1}{2}+n} (a+a x)^{5/2}}{x} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2^{\frac {1}{2}+n} c \operatorname {AppellF1}\left (\frac {7}{2},\frac {1}{2}-n,1,\frac {9}{2},\frac {1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (1-\sec (e+f x))^{\frac {1}{2}-n} (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{-1+n} \tan (e+f x)}{7 f} \\ \end{align*}
\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx \]
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\[\int \left (a +a \sec \left (f x +e \right )\right )^{3} \left (c -c \sec \left (f x +e \right )\right )^{n}d x\]
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\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=a^{3} \left (\int 3 \left (- c \sec {\left (e + f x \right )} + c\right )^{n} \sec {\left (e + f x \right )}\, dx + \int 3 \left (- c \sec {\left (e + f x \right )} + c\right )^{n} \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- c \sec {\left (e + f x \right )} + c\right )^{n} \sec ^{3}{\left (e + f x \right )}\, dx + \int \left (- c \sec {\left (e + f x \right )} + c\right )^{n}\, dx\right ) \]
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\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} {\left (-c \sec \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^n \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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