Integrand size = 26, antiderivative size = 210 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {x}{a^3 c^5}+\frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f} \]
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Time = 0.29 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3989, 3971, 3554, 8, 2686, 200, 2687, 30} \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot (e+f x)}{a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}+\frac {x}{a^3 c^5} \]
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Rule 8
Rule 30
Rule 200
Rule 2686
Rule 2687
Rule 3554
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^{10}(e+f x) (a+a \sec (e+f x))^2 \, dx}{a^5 c^5} \\ & = -\frac {\int \left (a^2 \cot ^{10}(e+f x)+2 a^2 \cot ^9(e+f x) \csc (e+f x)+a^2 \cot ^8(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^5 c^5} \\ & = -\frac {\int \cot ^{10}(e+f x) \, dx}{a^3 c^5}-\frac {\int \cot ^8(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^5}-\frac {2 \int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^3 c^5} \\ & = \frac {\cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int \cot ^8(e+f x) \, dx}{a^3 c^5}-\frac {\text {Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^3 c^5 f}+\frac {2 \text {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f} \\ & = -\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c^5}+\frac {2 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^5 f} \\ & = \frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c^5} \\ & = -\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c^5} \\ & = \frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f}+\frac {\int 1 \, dx}{a^3 c^5} \\ & = \frac {x}{a^3 c^5}+\frac {\cot (e+f x)}{a^3 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^3 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^3 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \cot ^9(e+f x)}{9 a^3 c^5 f}+\frac {2 \csc (e+f x)}{a^3 c^5 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^5 f}+\frac {12 \csc ^5(e+f x)}{5 a^3 c^5 f}-\frac {8 \csc ^7(e+f x)}{7 a^3 c^5 f}+\frac {2 \csc ^9(e+f x)}{9 a^3 c^5 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {\cot ^9(e+f x) \left (35+35 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},-\tan ^2(e+f x)\right )+630 \sec (e+f x)-1680 \sec ^3(e+f x)+2016 \sec ^5(e+f x)-1152 \sec ^7(e+f x)+256 \sec ^9(e+f x)\right )}{315 a^3 c^5 f} \]
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Time = 0.74 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(\frac {35 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-405 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-63 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+2331 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+945 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-9765 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+40320 f x -11655 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+51345 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{40320 f \,a^{3} c^{5}}\) | \(117\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+256 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {9}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {37}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {31}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {163}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{128 f \,c^{5} a^{3}}\) | \(127\) |
default | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+256 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {9}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {37}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {31}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {163}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{128 f \,c^{5} a^{3}}\) | \(127\) |
risch | \(\frac {x}{a^{3} c^{5}}+\frac {4 i \left (315 \,{\mathrm e}^{13 i \left (f x +e \right )}-315 \,{\mathrm e}^{12 i \left (f x +e \right )}-1470 \,{\mathrm e}^{11 i \left (f x +e \right )}+3360 \,{\mathrm e}^{10 i \left (f x +e \right )}+1113 \,{\mathrm e}^{9 i \left (f x +e \right )}-6447 \,{\mathrm e}^{8 i \left (f x +e \right )}+2028 \,{\mathrm e}^{7 i \left (f x +e \right )}+7008 \,{\mathrm e}^{6 i \left (f x +e \right )}-4867 \,{\mathrm e}^{5 i \left (f x +e \right )}-2321 \,{\mathrm e}^{4 i \left (f x +e \right )}+3314 \,{\mathrm e}^{3 i \left (f x +e \right )}-16 \,{\mathrm e}^{2 i \left (f x +e \right )}-881 \,{\mathrm e}^{i \left (f x +e \right )}+299\right )}{315 f \,c^{5} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\) | \(193\) |
norman | \(\frac {\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{c a}+\frac {1}{1152 a c f}-\frac {9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{896 a c f}+\frac {37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{640 a c f}-\frac {31 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{128 a c f}+\frac {163 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{128 a c f}-\frac {37 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{128 a c f}+\frac {3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{128 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{640 a c f}}{c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\) | \(204\) |
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Time = 0.26 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {598 \, \cos \left (f x + e\right )^{7} - 566 \, \cos \left (f x + e\right )^{6} - 1212 \, \cos \left (f x + e\right )^{5} + 1310 \, \cos \left (f x + e\right )^{4} + 860 \, \cos \left (f x + e\right )^{3} - 1014 \, \cos \left (f x + e\right )^{2} + 315 \, {\left (f x \cos \left (f x + e\right )^{6} - 2 \, f x \cos \left (f x + e\right )^{5} - f x \cos \left (f x + e\right )^{4} + 4 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} - 2 \, f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) - 197 \, \cos \left (f x + e\right ) + 256}{315 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{6} - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} - a^{3} c^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{5} f \cos \left (f x + e\right )^{3} - a^{3} c^{5} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} c^{5} f \cos \left (f x + e\right ) + a^{3} c^{5} f\right )} \sin \left (f x + e\right )} \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=- \frac {\int \frac {1}{\sec ^{8}{\left (e + f x \right )} - 2 \sec ^{7}{\left (e + f x \right )} - 2 \sec ^{6}{\left (e + f x \right )} + 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} - 1}\, dx}{a^{3} c^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=-\frac {\frac {63 \, {\left (\frac {185 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {15 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{5}} - \frac {80640 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{5}} + \frac {{\left (\frac {405 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2331 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {9765 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {51345 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{3} c^{5} \sin \left (f x + e\right )^{9}}}{40320 \, f} \]
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Time = 0.44 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {\frac {40320 \, {\left (f x + e\right )}}{a^{3} c^{5}} + \frac {51345 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 9765 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 2331 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35}{a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} - \frac {63 \, {\left (a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 185 \, a^{12} c^{20} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{25}}}{40320 \, f} \]
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Time = 14.61 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5} \, dx=\frac {35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-63\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-11655\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+51345\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-9765\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+2331\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-405\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+40320\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (e+f\,x\right )}{40320\,a^3\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]
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