Integrand size = 26, antiderivative size = 252 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {x}{a^3 c^6}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f} \]
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Time = 0.37 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3989, 3971, 3554, 8, 2686, 200, 2687, 30, 276} \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}+\frac {x}{a^3 c^6} \]
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Rule 8
Rule 30
Rule 200
Rule 276
Rule 2686
Rule 2687
Rule 3554
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cot ^{12}(e+f x) (a+a \sec (e+f x))^3 \, dx}{a^6 c^6} \\ & = \frac {\int \left (a^3 \cot ^{12}(e+f x)+3 a^3 \cot ^{11}(e+f x) \csc (e+f x)+3 a^3 \cot ^{10}(e+f x) \csc ^2(e+f x)+a^3 \cot ^9(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^6 c^6} \\ & = \frac {\int \cot ^{12}(e+f x) \, dx}{a^3 c^6}+\frac {\int \cot ^9(e+f x) \csc ^3(e+f x) \, dx}{a^3 c^6}+\frac {3 \int \cot ^{11}(e+f x) \csc (e+f x) \, dx}{a^3 c^6}+\frac {3 \int \cot ^{10}(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^6} \\ & = -\frac {\cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\int \cot ^{10}(e+f x) \, dx}{a^3 c^6}-\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}+\frac {3 \text {Subst}\left (\int x^{10} \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac {3 \text {Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f} \\ & = \frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\int \cot ^8(e+f x) \, dx}{a^3 c^6}-\frac {\text {Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}-\frac {3 \text {Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f} \\ & = -\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c^6} \\ & = \frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c^6} \\ & = -\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c^6} \\ & = \frac {\cot (e+f x)}{a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\int 1 \, dx}{a^3 c^6} \\ & = \frac {x}{a^3 c^6}+\frac {\cot (e+f x)}{a^3 c^6 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {3 \csc (e+f x)}{a^3 c^6 f}-\frac {16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac {19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 11.12 (sec) , antiderivative size = 787, normalized size of antiderivative = 3.12 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {\cot ^9(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {11}{2},1,-\frac {9}{2},-\tan ^2(e+f x)\right )}{11 a^2 c^5 f (a+a \sec (e+f x)) (c-c \sec (e+f x))}+\frac {16 \tan (e+f x)}{55 a^3 f (c-c \sec (e+f x))^6}-\frac {2 a^3 \tan (e+f x)}{11 f (a+a \sec (e+f x))^6 (c-c \sec (e+f x))^6}-\frac {3 a^3 \sec (e+f x) \tan (e+f x)}{11 f (a+a \sec (e+f x))^6 (c-c \sec (e+f x))^6}-\frac {a^2 \tan (e+f x)}{9 f (a+a \sec (e+f x))^5 (c-c \sec (e+f x))^6}-\frac {a \tan (e+f x)}{63 f (a+a \sec (e+f x))^4 (c-c \sec (e+f x))^6}-\frac {\tan (e+f x)}{35 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6}-\frac {8 \tan (e+f x)}{105 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^6}-\frac {8 \tan (e+f x)}{15 a^2 f (a+a \sec (e+f x)) (c-c \sec (e+f x))^6}+\frac {16 \tan (e+f x)}{99 a^3 c f (c-c \sec (e+f x))^5}-\frac {10 a^2 \sec (e+f x) \tan (e+f x)}{33 c f (a+a \sec (e+f x))^5 (c-c \sec (e+f x))^5}+\frac {64 \tan (e+f x)}{693 a^3 c^2 f (c-c \sec (e+f x))^4}-\frac {80 a \sec (e+f x) \tan (e+f x)}{231 c^2 f (a+a \sec (e+f x))^4 (c-c \sec (e+f x))^4}+\frac {64 \tan (e+f x)}{1155 a^3 c^3 f (c-c \sec (e+f x))^3}-\frac {32 \sec (e+f x) \tan (e+f x)}{77 c^3 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3}+\frac {128 \tan (e+f x)}{3465 a^3 c^4 f (c-c \sec (e+f x))^2}-\frac {128 \sec (e+f x) \tan (e+f x)}{231 a c^4 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2}+\frac {128 \tan (e+f x)}{3465 a^3 c^5 f (c-c \sec (e+f x))}-\frac {256 \sec (e+f x) \tan (e+f x)}{231 a^2 c^5 f (a+a \sec (e+f x)) (c-c \sec (e+f x))} \]
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Time = 0.69 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.52
method | result | size |
parallelrisch | \(\frac {-315 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+3850 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-22770 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-693 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+90090 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+11550 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-295680 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+887040 f x -159390 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1323630 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{887040 f \,a^{3} c^{6}}\) | \(130\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-46 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+512 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {10}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {46}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {26}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {256}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {382}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{256 f \,a^{3} c^{6}}\) | \(140\) |
default | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-46 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+512 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {10}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {46}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {26}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {256}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {382}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{256 f \,a^{3} c^{6}}\) | \(140\) |
risch | \(\frac {x}{a^{3} c^{6}}+\frac {2 i \left (10395 \,{\mathrm e}^{15 i \left (f x +e \right )}-31185 \,{\mathrm e}^{14 i \left (f x +e \right )}+1155 \,{\mathrm e}^{13 i \left (f x +e \right )}+148995 \,{\mathrm e}^{12 i \left (f x +e \right )}-190113 \,{\mathrm e}^{11 i \left (f x +e \right )}-117117 \,{\mathrm e}^{10 i \left (f x +e \right )}+434775 \,{\mathrm e}^{9 i \left (f x +e \right )}-138105 \,{\mathrm e}^{8 i \left (f x +e \right )}-385055 \,{\mathrm e}^{7 i \left (f x +e \right )}+374781 \,{\mathrm e}^{6 i \left (f x +e \right )}+63289 \,{\mathrm e}^{5 i \left (f x +e \right )}-223655 \,{\mathrm e}^{4 i \left (f x +e \right )}+75685 \,{\mathrm e}^{3 i \left (f x +e \right )}+43345 \,{\mathrm e}^{2 i \left (f x +e \right )}-34323 \,{\mathrm e}^{i \left (f x +e \right )}+7453\right )}{3465 f \,a^{3} c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{11} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\) | \(215\) |
norman | \(\frac {\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{c a}-\frac {1}{2816 a c f}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{1152 a c f}-\frac {23 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{896 a c f}+\frac {13 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{128 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 a c f}+\frac {191 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{128 a c f}-\frac {23 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{128 a c f}+\frac {5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{384 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{16}}{1280 a c f}}{c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}\) | \(226\) |
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Time = 0.28 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {7453 \, \cos \left (f x + e\right )^{8} - 11964 \, \cos \left (f x + e\right )^{7} - 11866 \, \cos \left (f x + e\right )^{6} + 30542 \, \cos \left (f x + e\right )^{5} + 90 \, \cos \left (f x + e\right )^{4} - 26438 \, \cos \left (f x + e\right )^{3} + 8539 \, \cos \left (f x + e\right )^{2} + 3465 \, {\left (f x \cos \left (f x + e\right )^{7} - 3 \, f x \cos \left (f x + e\right )^{6} + f x \cos \left (f x + e\right )^{5} + 5 \, f x \cos \left (f x + e\right )^{4} - 5 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} + 3 \, f x \cos \left (f x + e\right ) - f x\right )} \sin \left (f x + e\right ) + 7671 \, \cos \left (f x + e\right ) - 3712}{3465 \, {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{6} + a^{3} c^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{4} - 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{3} - a^{3} c^{6} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{6} f \cos \left (f x + e\right ) - a^{3} c^{6} f\right )} \sin \left (f x + e\right )} \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {\int \frac {1}{\sec ^{9}{\left (e + f x \right )} - 3 \sec ^{8}{\left (e + f x \right )} + 8 \sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{4}{\left (e + f x \right )} + 8 \sec ^{3}{\left (e + f x \right )} - 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=-\frac {\frac {231 \, {\left (\frac {690 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {50 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{6}} - \frac {1774080 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{6}} - \frac {5 \, {\left (\frac {770 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4554 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {18018 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {59136 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {264726 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{a^{3} c^{6} \sin \left (f x + e\right )^{11}}}{887040 \, f} \]
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Time = 0.43 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=\frac {\frac {887040 \, {\left (f x + e\right )}}{a^{3} c^{6}} + \frac {5 \, {\left (264726 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 59136 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 18018 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4554 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 770 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 63\right )}}{a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} - \frac {231 \, {\left (3 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 50 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 690 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{30}}}{887040 \, f} \]
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Time = 14.79 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx=-\frac {315\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+693\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}-11550\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+159390\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-1323630\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+295680\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-90090\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+22770\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-3850\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-887040\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (e+f\,x\right )}{887040\,a^3\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]
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