Integrand size = 41, antiderivative size = 185 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {(4 A-B) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {2 (332 A-80 B+3 C) \tan (c+d x)}{105 a^4 d}-\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(4 A-B) \tan (c+d x)}{a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
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Time = 0.80 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3120, 3057, 2827, 3852, 8, 3855} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {(4 A-B) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {2 (332 A-80 B+3 C) \tan (c+d x)}{105 a^4 d}-\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rule 8
Rule 2827
Rule 3057
Rule 3120
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {(a (8 A-B+C)-a (4 A-4 B-3 C) \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (a^2 (52 A-10 B+3 C)-3 a^2 (12 A-5 B-2 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (a^3 (244 A-55 B+6 C)-2 a^3 (88 A-25 B-3 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \left (2 a^4 (332 A-80 B+3 C)-105 a^4 (4 A-B) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{105 a^8} \\ & = -\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(4 A-B) \int \sec (c+d x) \, dx}{a^4}+\frac {(2 (332 A-80 B+3 C)) \int \sec ^2(c+d x) \, dx}{105 a^4} \\ & = -\frac {(4 A-B) \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(2 (332 A-80 B+3 C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = -\frac {(4 A-B) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {2 (332 A-80 B+3 C) \tan (c+d x)}{105 a^4 d}-\frac {(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 A-B) \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1190\) vs. \(2(185)=370\).
Time = 8.74 (sec) , antiderivative size = 1190, normalized size of antiderivative = 6.43 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {32 (4 A-B) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right )}{d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac {32 (4 A-B) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right )}{d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {4 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (A \sin \left (\frac {c}{2}\right )-B \sin \left (\frac {c}{2}\right )+C \sin \left (\frac {c}{2}\right )\right )}{7 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {8 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (17 A \sin \left (\frac {c}{2}\right )-10 B \sin \left (\frac {c}{2}\right )+3 C \sin \left (\frac {c}{2}\right )\right )}{35 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {16 \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (139 A \sin \left (\frac {c}{2}\right )-55 B \sin \left (\frac {c}{2}\right )+6 C \sin \left (\frac {c}{2}\right )\right )}{105 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {4 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{7 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {8 \cos ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (17 A \sin \left (\frac {d x}{2}\right )-10 B \sin \left (\frac {d x}{2}\right )+3 C \sin \left (\frac {d x}{2}\right )\right )}{35 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {32 \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (559 A \sin \left (\frac {d x}{2}\right )-160 B \sin \left (\frac {d x}{2}\right )+6 C \sin \left (\frac {d x}{2}\right )\right )}{105 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {16 \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^2(c+d x) \sec \left (\frac {c}{2}\right ) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (139 A \sin \left (\frac {d x}{2}\right )-55 B \sin \left (\frac {d x}{2}\right )+6 C \sin \left (\frac {d x}{2}\right )\right )}{105 d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac {32 A \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos (c+d x) \sec (c) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \sin (d x)}{d (1+\cos (c+d x))^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}}{a^4} \]
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Time = 2.82 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {3360 \cos \left (d x +c \right ) \left (A -\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3360 \cos \left (d x +c \right ) \left (A -\frac {B}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+559 \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\frac {15 \left (\frac {110 A}{13}-2 B +\frac {3 C}{26}\right ) \cos \left (2 d x +2 c \right )}{43}+\left (A -\frac {535 B}{2236}+\frac {6 C}{559}\right ) \cos \left (3 d x +3 c \right )+\frac {\left (83 A -20 B +\frac {3 C}{4}\right ) \cos \left (4 d x +4 c \right )}{559}+\frac {\left (2861 A -\frac {2645 B}{4}+54 C \right ) \cos \left (d x +c \right )}{559}+\frac {1672 A}{559}-\frac {370 B}{559}+\frac {87 C}{2236}\right )}{840 d \,a^{4} \cos \left (d x +c \right )}\) | \(178\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}+\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (32 A -8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-32 A +8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(242\) |
default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}+\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (32 A -8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-32 A +8 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(242\) |
norman | \(\frac {\frac {\left (A -B +C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (27 A -20 B +13 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}-\frac {\left (65 A -15 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (133 A -28 B +3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}+\frac {\left (359 A -155 B -9 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 a d}+\frac {\left (937 A -475 B +153 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{840 a d}+\frac {\left (1447 A -460 B +33 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{210 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{3}}+\frac {\left (4 A -B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4} d}-\frac {\left (4 A -B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4} d}\) | \(285\) |
risch | \(\frac {2 i \left (420 A \,{\mathrm e}^{8 i \left (d x +c \right )}-105 B \,{\mathrm e}^{8 i \left (d x +c \right )}+2940 A \,{\mathrm e}^{7 i \left (d x +c \right )}-735 B \,{\mathrm e}^{7 i \left (d x +c \right )}+9100 A \,{\mathrm e}^{6 i \left (d x +c \right )}-2275 B \,{\mathrm e}^{6 i \left (d x +c \right )}+16660 A \,{\mathrm e}^{5 i \left (d x +c \right )}-4165 B \,{\mathrm e}^{5 i \left (d x +c \right )}+210 C \,{\mathrm e}^{5 i \left (d x +c \right )}+20524 A \,{\mathrm e}^{4 i \left (d x +c \right )}-4795 B \,{\mathrm e}^{4 i \left (d x +c \right )}+126 C \,{\mathrm e}^{4 i \left (d x +c \right )}+18788 A \,{\mathrm e}^{3 i \left (d x +c \right )}-4445 B \,{\mathrm e}^{3 i \left (d x +c \right )}+252 C \,{\mathrm e}^{3 i \left (d x +c \right )}+11668 A \,{\mathrm e}^{2 i \left (d x +c \right )}-2785 B \,{\mathrm e}^{2 i \left (d x +c \right )}+132 C \,{\mathrm e}^{2 i \left (d x +c \right )}+4228 A \,{\mathrm e}^{i \left (d x +c \right )}-1015 B \,{\mathrm e}^{i \left (d x +c \right )}+42 C \,{\mathrm e}^{i \left (d x +c \right )}+664 A -160 B +6 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{4} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}-\frac {4 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}\) | \(386\) |
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Time = 0.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.88 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {105 \, {\left ({\left (4 \, A - B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, A - B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (4 \, A - B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, A - B\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, A - B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, {\left (332 \, A - 80 \, B + 3 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (2236 \, A - 535 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2636 \, A - 620 \, B + 39 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (296 \, A - 65 \, B + 9 \, C\right )} \cos \left (d x + c\right ) + 105 \, A\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{2}{\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\cos ^{4}{\left (c + d x \right )} + 4 \cos ^{3}{\left (c + d x \right )} + 6 \cos ^{2}{\left (c + d x \right )} + 4 \cos {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (177) = 354\).
Time = 0.23 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.22 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {A {\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 {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + \frac {3 \, C {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.57 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {840 \, {\left (4 \, A - B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, {\left (4 \, A - B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {1680 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 1.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.36 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {5\,A-3\,B+C}{40\,a^4}+\frac {A-B+C}{20\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,A-3\,B+C}{12\,a^4}-\frac {2\,B-10\,A+2\,C}{24\,a^4}+\frac {A-B+C}{8\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (5\,A-3\,B+C\right )}{8\,a^4}-\frac {2\,B-10\,A+2\,C}{4\,a^4}+\frac {10\,A+2\,B-2\,C}{8\,a^4}+\frac {A-B+C}{2\,a^4}\right )}{d}-\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,A-B\right )}{a^4\,d} \]
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