Integrand size = 26, antiderivative size = 98 \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {x}{a^2 c^3}+\frac {\cot ^5(e+f x) (1+\sec (e+f x))}{5 a^2 c^3 f}-\frac {\cot ^3(e+f x) (5+4 \sec (e+f x))}{15 a^2 c^3 f}+\frac {\cot (e+f x) (15+8 \sec (e+f x))}{15 a^2 c^3 f} \]
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Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3967, 8} \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\cot ^5(e+f x) (\sec (e+f x)+1)}{5 a^2 c^3 f}-\frac {\cot ^3(e+f x) (4 \sec (e+f x)+5)}{15 a^2 c^3 f}+\frac {\cot (e+f x) (8 \sec (e+f x)+15)}{15 a^2 c^3 f}+\frac {x}{a^2 c^3} \]
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Rule 8
Rule 3967
Rule 3989
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^6(e+f x) (a+a \sec (e+f x)) \, dx}{a^3 c^3} \\ & = \frac {\cot ^5(e+f x) (1+\sec (e+f x))}{5 a^2 c^3 f}-\frac {\int \cot ^4(e+f x) (-5 a-4 a \sec (e+f x)) \, dx}{5 a^3 c^3} \\ & = \frac {\cot ^5(e+f x) (1+\sec (e+f x))}{5 a^2 c^3 f}-\frac {\cot ^3(e+f x) (5+4 \sec (e+f x))}{15 a^2 c^3 f}-\frac {\int \cot ^2(e+f x) (15 a+8 a \sec (e+f x)) \, dx}{15 a^3 c^3} \\ & = \frac {\cot ^5(e+f x) (1+\sec (e+f x))}{5 a^2 c^3 f}-\frac {\cot ^3(e+f x) (5+4 \sec (e+f x))}{15 a^2 c^3 f}+\frac {\cot (e+f x) (15+8 \sec (e+f x))}{15 a^2 c^3 f}-\frac {\int -15 a \, dx}{15 a^3 c^3} \\ & = \frac {x}{a^2 c^3}+\frac {\cot ^5(e+f x) (1+\sec (e+f x))}{5 a^2 c^3 f}-\frac {\cot ^3(e+f x) (5+4 \sec (e+f x))}{15 a^2 c^3 f}+\frac {\cot (e+f x) (15+8 \sec (e+f x))}{15 a^2 c^3 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\cot ^5(e+f x) \left (3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(e+f x)\right )+15 \sec (e+f x)-20 \sec ^3(e+f x)+8 \sec ^5(e+f x)\right )}{15 a^2 c^3 f} \]
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Time = 0.58 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {3 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-30 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+240 f x +240 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )-90 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{240 f \,a^{2} c^{3}}\) | \(78\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+32 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 f \,a^{2} c^{3}}\) | \(88\) |
default | \(\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {16}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+32 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 f \,a^{2} c^{3}}\) | \(88\) |
risch | \(\frac {x}{a^{2} c^{3}}+\frac {2 i \left (15 \,{\mathrm e}^{7 i \left (f x +e \right )}+15 \,{\mathrm e}^{6 i \left (f x +e \right )}-65 \,{\mathrm e}^{5 i \left (f x +e \right )}+25 \,{\mathrm e}^{4 i \left (f x +e \right )}+73 \,{\mathrm e}^{3 i \left (f x +e \right )}-31 \,{\mathrm e}^{2 i \left (f x +e \right )}-31 \,{\mathrm e}^{i \left (f x +e \right )}+23\right )}{15 f \,a^{2} c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}\) | \(127\) |
norman | \(\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{a c f}+\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{c a}+\frac {1}{80 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{8 a c f}-\frac {3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{8 a c f}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{48 a c f}}{a \,c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(137\) |
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Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {23 \, \cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{3} - 27 \, \cos \left (f x + e\right )^{2} + 15 \, {\left (f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} - f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) + 7 \, \cos \left (f x + e\right ) + 8}{15 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )} \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=- \frac {\int \frac {1}{\sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} + 2 \sec ^{2}{\left (e + f x \right )} + \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=-\frac {\frac {5 \, {\left (\frac {18 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{3}} - \frac {480 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c^{3}} + \frac {3 \, {\left (\frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {80 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{a^{2} c^{3} \sin \left (f x + e\right )^{5}}}{240 \, f} \]
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Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {\frac {240 \, {\left (f x + e\right )}}{a^{2} c^{3}} + \frac {3 \, {\left (80 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 10 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}}{a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} + \frac {5 \, {\left (a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 18 \, a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{9}}}{240 \, f} \]
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Time = 14.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3} \, dx=\frac {3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-90\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+240\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-30\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+240\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (e+f\,x\right )}{240\,a^2\,c^3\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]
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