Integrand size = 33, antiderivative size = 248 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {2 (11 A+2 C) x}{a^4}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \cos (c+d x) \sin (c+d x)}{a^4 d}-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^4 d (1+\sec (c+d x))}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d} \]
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Time = 0.79 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4170, 4105, 3872, 2713, 2715, 8} \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \sin (c+d x) \cos (c+d x)}{a^4 d}-\frac {4 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {2 x (11 A+2 C)}{a^4}-\frac {2 (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4105
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {\int \frac {\cos ^3(c+d x) (-a (10 A+3 C)+a (6 A-C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (-7 a^2 (14 A+3 C)+10 a^2 (8 A+C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (-12 a^3 (69 A+13 C)+4 a^3 (178 A+31 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {\int \cos ^3(c+d x) \left (-12 a^4 (454 A+83 C)+420 a^4 (11 A+2 C) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(4 (11 A+2 C)) \int \cos ^2(c+d x) \, dx}{a^4}+\frac {(4 (454 A+83 C)) \int \cos ^3(c+d x) \, dx}{35 a^4} \\ & = -\frac {2 (11 A+2 C) \cos (c+d x) \sin (c+d x)}{a^4 d}-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(2 (11 A+2 C)) \int 1 \, dx}{a^4}-\frac {(4 (454 A+83 C)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 a^4 d} \\ & = -\frac {2 (11 A+2 C) x}{a^4}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \cos (c+d x) \sin (c+d x)}{a^4 d}-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(575\) vs. \(2(248)=496\).
Time = 8.87 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.32 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (58800 (11 A+2 C) d x \cos \left (\frac {d x}{2}\right )+58800 (11 A+2 C) d x \cos \left (c+\frac {d x}{2}\right )+388080 A d x \cos \left (c+\frac {3 d x}{2}\right )+70560 C d x \cos \left (c+\frac {3 d x}{2}\right )+388080 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+70560 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+129360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+23520 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+129360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+23520 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+18480 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+3360 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+18480 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+3360 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-1010660 A \sin \left (\frac {d x}{2}\right )-243320 C \sin \left (\frac {d x}{2}\right )+687260 A \sin \left (c+\frac {d x}{2}\right )+184520 C \sin \left (c+\frac {d x}{2}\right )-814107 A \sin \left (c+\frac {3 d x}{2}\right )-184464 C \sin \left (c+\frac {3 d x}{2}\right )+204645 A \sin \left (2 c+\frac {3 d x}{2}\right )+72240 C \sin \left (2 c+\frac {3 d x}{2}\right )-357609 A \sin \left (2 c+\frac {5 d x}{2}\right )-77168 C \sin \left (2 c+\frac {5 d x}{2}\right )-18025 A \sin \left (3 c+\frac {5 d x}{2}\right )+8400 C \sin \left (3 c+\frac {5 d x}{2}\right )-72522 A \sin \left (3 c+\frac {7 d x}{2}\right )-15164 C \sin \left (3 c+\frac {7 d x}{2}\right )-24010 A \sin \left (4 c+\frac {7 d x}{2}\right )-2940 C \sin \left (4 c+\frac {7 d x}{2}\right )-2310 A \sin \left (4 c+\frac {9 d x}{2}\right )-420 C \sin \left (4 c+\frac {9 d x}{2}\right )-2310 A \sin \left (5 c+\frac {9 d x}{2}\right )-420 C \sin \left (5 c+\frac {9 d x}{2}\right )+175 A \sin \left (5 c+\frac {11 d x}{2}\right )+175 A \sin \left (6 c+\frac {11 d x}{2}\right )-35 A \sin \left (6 c+\frac {13 d x}{2}\right )-35 A \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{107520 a^4 d} \]
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Time = 0.48 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(\frac {31 \left (\frac {\left (\frac {238253 A}{2}+21928 C \right ) \cos \left (2 d x +2 c \right )}{1085}+\frac {2 \left (12609 A +2368 C \right ) \cos \left (3 d x +3 c \right )}{1085}+\left (A +\frac {6 C}{31}\right ) \cos \left (4 d x +4 c \right )-\frac {2 A \cos \left (5 d x +5 c \right )}{31}+\frac {A \cos \left (6 d x +6 c \right )}{62}+\frac {4 \left (67873 A +12496 C \right ) \cos \left (d x +c \right )}{1085}+\frac {176171 A}{1085}+\frac {32342 C}{1085}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-8448 x d \left (A +\frac {2 C}{11}\right )}{384 a^{4} d}\) | \(138\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+209 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-13 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {62 A}{3}-2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-9 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-32 \left (11 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(210\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+209 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-13 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {62 A}{3}-2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-9 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-32 \left (11 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(210\) |
norman | \(\frac {\frac {2 \left (11 A +2 C \right ) x}{a}-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{56 a d}+\frac {4 \left (11 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {4 \left (11 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {2 \left (11 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}+\frac {\left (67 A +39 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{280 a d}-\frac {\left (229 A +73 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{120 a d}-\frac {\left (353 A +65 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (1755 A +319 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}+\frac {\left (2397 A +449 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 a d}+\frac {\left (3569 A +713 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{168 a d}+\frac {\left (21761 A +3897 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{280 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\) | \(323\) |
risch | \(-\frac {22 A x}{a^{4}}-\frac {4 x C}{a^{4}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 a^{4} d}-\frac {43 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{4} d}+\frac {43 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{4} d}-\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 a^{4} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {2 i \left (3675 A \,{\mathrm e}^{6 i \left (d x +c \right )}+1050 C \,{\mathrm e}^{6 i \left (d x +c \right )}+19845 A \,{\mathrm e}^{5 i \left (d x +c \right )}+5250 C \,{\mathrm e}^{5 i \left (d x +c \right )}+46550 A \,{\mathrm e}^{4 i \left (d x +c \right )}+11900 C \,{\mathrm e}^{4 i \left (d x +c \right )}+59570 A \,{\mathrm e}^{3 i \left (d x +c \right )}+14840 C \,{\mathrm e}^{3 i \left (d x +c \right )}+43827 A \,{\mathrm e}^{2 i \left (d x +c \right )}+10794 C \,{\mathrm e}^{2 i \left (d x +c \right )}+17549 A \,{\mathrm e}^{i \left (d x +c \right )}+4298 C \,{\mathrm e}^{i \left (d x +c \right )}+3032 A +764 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(341\) |
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Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {210 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{4} + 840 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{3} + 1260 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 840 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 210 \, {\left (11 \, A + 2 \, C\right )} d x - {\left (35 \, A \cos \left (d x + c\right )^{6} - 70 \, A \cos \left (d x + c\right )^{5} + 35 \, {\left (14 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (799 \, A + 148 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (3592 \, A + 659 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6109 \, A + 1118 \, C\right )} \cos \left (d x + c\right ) + 3632 \, A + 664 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {36960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {1680 \, {\left (d x + c\right )} {\left (11 \, A + 2 \, C\right )}}{a^{4}} - \frac {560 \, {\left (39 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 15.84 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {209\,A}{8}+\frac {49\,C}{8}\right )}{a^4\,d}-\frac {22\,A\,d\,x+4\,C\,d\,x}{a^4\,d}+\frac {\left (26\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {124\,A}{3}+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {11\,A}{40}+\frac {7\,C}{40}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {A}{56}+\frac {C}{56}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {59\,A}{24}+\frac {23\,C}{24}\right )}{a^4\,d} \]
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