Integrand size = 41, antiderivative size = 195 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {(B-4 C) x}{a^4}+\frac {(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(B-4 C) \sin (c+d x)}{a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
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Time = 0.74 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3120, 3056, 3047, 3102, 12, 2814, 2727} \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(B-4 C) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac {x (B-4 C)}{a^4}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(2 A+5 B-12 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
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Rule 12
Rule 2727
Rule 2814
Rule 3047
Rule 3056
Rule 3102
Rule 3120
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {\cos ^3(c+d x) (a (3 A+4 B-4 C)+a (A-B+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) \left (3 a^2 (2 A+5 B-12 C)+a^2 (3 A-10 B+52 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (2 a^3 (3 A+25 B-88 C)+a^3 (6 A-55 B+244 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = \frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {2 a^3 (3 A+25 B-88 C) \cos (c+d x)+a^3 (6 A-55 B+244 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = \frac {(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {105 a^4 (B-4 C) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^7} \\ & = \frac {(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(B-4 C) \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^3} \\ & = \frac {(B-4 C) x}{a^4}+\frac {(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(B-4 C) \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^3} \\ & = \frac {(B-4 C) x}{a^4}+\frac {(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(B-4 C) \sin (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. \(571\) vs. \(2(195)=390\).
Time = 7.18 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.93 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (7350 (B-4 C) d x \cos \left (\frac {d x}{2}\right )+7350 (B-4 C) d x \cos \left (c+\frac {d x}{2}\right )+4410 B d x \cos \left (c+\frac {3 d x}{2}\right )-17640 C d x \cos \left (c+\frac {3 d x}{2}\right )+4410 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-17640 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+1470 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-5880 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+1470 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-5880 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+210 B d x \cos \left (3 c+\frac {7 d x}{2}\right )-840 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+210 B d x \cos \left (4 c+\frac {7 d x}{2}\right )-840 C d x \cos \left (4 c+\frac {7 d x}{2}\right )+2520 A \sin \left (\frac {d x}{2}\right )-19880 B \sin \left (\frac {d x}{2}\right )+60830 C \sin \left (\frac {d x}{2}\right )-2520 A \sin \left (c+\frac {d x}{2}\right )+16520 B \sin \left (c+\frac {d x}{2}\right )-46130 C \sin \left (c+\frac {d x}{2}\right )+1764 A \sin \left (c+\frac {3 d x}{2}\right )-14280 B \sin \left (c+\frac {3 d x}{2}\right )+46116 C \sin \left (c+\frac {3 d x}{2}\right )-1260 A \sin \left (2 c+\frac {3 d x}{2}\right )+7560 B \sin \left (2 c+\frac {3 d x}{2}\right )-18060 C \sin \left (2 c+\frac {3 d x}{2}\right )+588 A \sin \left (2 c+\frac {5 d x}{2}\right )-5600 B \sin \left (2 c+\frac {5 d x}{2}\right )+19292 C \sin \left (2 c+\frac {5 d x}{2}\right )-420 A \sin \left (3 c+\frac {5 d x}{2}\right )+1680 B \sin \left (3 c+\frac {5 d x}{2}\right )-2100 C \sin \left (3 c+\frac {5 d x}{2}\right )+144 A \sin \left (3 c+\frac {7 d x}{2}\right )-1040 B \sin \left (3 c+\frac {7 d x}{2}\right )+3791 C \sin \left (3 c+\frac {7 d x}{2}\right )+735 C \sin \left (4 c+\frac {7 d x}{2}\right )+105 C \sin \left (4 c+\frac {9 d x}{2}\right )+105 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{1680 a^4 d (1+\cos (c+d x))^4} \]
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Time = 1.91 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {13 A}{6}-\frac {310 B}{9}+\frac {2741 C}{18}\right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {65 B}{9}+\frac {296 C}{9}\right ) \cos \left (3 d x +3 c \right )+\frac {35 C \cos \left (4 d x +4 c \right )}{24}+\left (\frac {17 A}{3}-\frac {730 B}{9}+\frac {3124 C}{9}\right ) \cos \left (d x +c \right )+\frac {17 A}{6}-\frac {470 B}{9}+\frac {16171 C}{72}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6720 d x \left (B -4 C \right )}{6720 a^{4} d}\) | \(118\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {3 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {16 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+16 \left (B -4 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(215\) |
default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {3 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {16 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+16 \left (B -4 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(215\) |
risch | \(\frac {B x}{a^{4}}-\frac {4 C x}{a^{4}}-\frac {i C \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{4} d}+\frac {2 i \left (105 A \,{\mathrm e}^{6 i \left (d x +c \right )}-420 B \,{\mathrm e}^{6 i \left (d x +c \right )}+1050 C \,{\mathrm e}^{6 i \left (d x +c \right )}+315 A \,{\mathrm e}^{5 i \left (d x +c \right )}-1890 B \,{\mathrm e}^{5 i \left (d x +c \right )}+5250 C \,{\mathrm e}^{5 i \left (d x +c \right )}+630 A \,{\mathrm e}^{4 i \left (d x +c \right )}-4130 B \,{\mathrm e}^{4 i \left (d x +c \right )}+11900 C \,{\mathrm e}^{4 i \left (d x +c \right )}+630 A \,{\mathrm e}^{3 i \left (d x +c \right )}-4970 B \,{\mathrm e}^{3 i \left (d x +c \right )}+14840 C \,{\mathrm e}^{3 i \left (d x +c \right )}+441 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3570 B \,{\mathrm e}^{2 i \left (d x +c \right )}+10794 C \,{\mathrm e}^{2 i \left (d x +c \right )}+147 A \,{\mathrm e}^{i \left (d x +c \right )}-1400 B \,{\mathrm e}^{i \left (d x +c \right )}+4298 C \,{\mathrm e}^{i \left (d x +c \right )}+36 A -260 B +764 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(301\) |
norman | \(\frac {\frac {\left (B -4 C \right ) x}{a}+\frac {\left (B -4 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 \left (B -4 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 \left (B -4 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 \left (B -4 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 \left (B -4 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (A -15 B +65 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (A -B +C \right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {\left (2 A +5 B -12 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}+\frac {\left (3 A -500 B +2075 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}+\frac {\left (6 A -107 B +452 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}+\frac {\left (6 A -55 B +244 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}+\frac {\left (6 A +B -22 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}+\frac {\left (30 A -1241 B +5084 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}+\frac {\left (42 A -995 B +4118 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{3}}\) | \(391\) |
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Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left (B - 4 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (B - 4 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (B - 4 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (B - 4 \, C\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (B - 4 \, C\right )} d x + {\left (105 \, C \cos \left (d x + c\right )^{4} + 4 \, {\left (9 \, A - 65 \, B + 296 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A - 620 \, B + 2636 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (24 \, A - 535 \, B + 2236 \, C\right )} \cos \left (d x + c\right ) + 6 \, A - 160 \, B + 664 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (189) = 378\).
Time = 6.70 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.83 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} - \frac {15 A \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {48 A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {42 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {105 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {840 B d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {840 B d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {15 B \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {90 B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {280 B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {1190 B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {1575 B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 C d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {15 C \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {132 C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {658 C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {4340 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {6825 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos ^{3}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.83 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (d x + c\right )} {\left (B - 4 \, C\right )}}{a^{4}} + \frac {1680 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 1.47 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {B\,d\,x-4\,C\,d\,x}{a^4\,d}+\frac {\left (\frac {12\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {52\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}+\frac {764\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {16\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}-\frac {23\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}-\frac {143\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {9\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}-\frac {5\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}+\frac {8\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}-\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}+\frac {2\,C\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d} \]
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