Integrand size = 33, antiderivative size = 200 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {7}{16} a^4 (10 A+8 B+7 C) x+\frac {4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d} \]
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Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3102, 2830, 2724, 2717, 2715, 8, 2713} \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (10 A+8 B+7 C)+\frac {(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 2724
Rule 2830
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {\int (a+a \cos (c+d x))^4 (a (6 A+5 C)+a (6 B-C) \cos (c+d x)) \, dx}{6 a} \\ & = \frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{10} (10 A+8 B+7 C) \int (a+a \cos (c+d x))^4 \, dx \\ & = \frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{10} (10 A+8 B+7 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx \\ & = \frac {1}{10} a^4 (10 A+8 B+7 C) x+\frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{10} \left (a^4 (10 A+8 B+7 C)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+8 B+7 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+8 B+7 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{10} a^4 (10 A+8 B+7 C) x+\frac {2 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {3 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{10 d}+\frac {a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{40} \left (3 a^4 (10 A+8 B+7 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (10 A+8 B+7 C)\right ) \int 1 \, dx-\frac {\left (2 a^4 (10 A+8 B+7 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {2}{5} a^4 (10 A+8 B+7 C) x+\frac {4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (10 A+8 B+7 C)\right ) \int 1 \, dx \\ & = \frac {7}{16} a^4 (10 A+8 B+7 C) x+\frac {4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (10 A+8 B+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (10 A+8 B+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 B-C) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 1.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.84 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 \sin (c+d x) \left (210 (10 A+8 B+7 C) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (16 (100 A+83 B+72 C)+15 (54 A+56 B+49 C) \cos (c+d x)+32 (10 A+17 B+18 C) \cos ^2(c+d x)+10 (6 A+24 B+41 C) \cos ^3(c+d x)+48 (B+4 C) \cos ^4(c+d x)+40 C \cos ^5(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{240 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 7.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {\left (3 \left (\frac {7 A}{4}+2 B +\frac {127 C}{64}\right ) \sin \left (2 d x +2 c \right )+\left (A +\frac {29 B}{16}+\frac {9 C}{4}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (\frac {A}{4}+B +\frac {15 C}{8}\right ) \sin \left (4 d x +4 c \right )}{8}+\frac {3 \left (\frac {B}{4}+C \right ) \sin \left (5 d x +5 c \right )}{20}+\frac {\sin \left (6 d x +6 c \right ) C}{64}+3 \left (7 A +\frac {49 B}{8}+\frac {11 C}{2}\right ) \sin \left (d x +c \right )+\frac {105 x \left (A +\frac {4 B}{5}+\frac {7 C}{10}\right ) d}{8}\right ) a^{4}}{3 d}\) | \(126\) |
parts | \(a^{4} x A +\frac {\left (4 a^{4} A +B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(263\) |
risch | \(\frac {35 a^{4} x A}{8}+\frac {7 a^{4} B x}{2}+\frac {49 a^{4} C x}{16}+\frac {7 \sin \left (d x +c \right ) a^{4} A}{d}+\frac {49 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {11 \sin \left (d x +c \right ) C \,a^{4}}{2 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{192 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {\sin \left (5 d x +5 c \right ) C \,a^{4}}{20 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{8 d}+\frac {15 \sin \left (4 d x +4 c \right ) C \,a^{4}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{3 d}+\frac {29 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {3 \sin \left (3 d x +3 c \right ) C \,a^{4}}{4 d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} A}{4 d}+\frac {2 \sin \left (2 d x +2 c \right ) B \,a^{4}}{d}+\frac {127 \sin \left (2 d x +2 c \right ) C \,a^{4}}{64 d}\) | \(284\) |
norman | \(\frac {\frac {7 a^{4} \left (10 A +8 B +7 C \right ) x}{16}+\frac {281 a^{4} \left (10 A +8 B +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {231 a^{4} \left (10 A +8 B +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {119 a^{4} \left (10 A +8 B +7 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {7 a^{4} \left (10 A +8 B +7 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a^{4} \left (10 A +8 B +7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{4} \left (10 A +8 B +7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (10 A +8 B +7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {105 a^{4} \left (10 A +8 B +7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (10 A +8 B +7 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{4} \left (10 A +8 B +7 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (186 A +200 B +207 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (2138 A +1864 B +1471 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(368\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )+C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(416\) |
default | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )+C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(416\) |
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.72 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} d x + {\left (40 \, C a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 24 \, B + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \, {\left (10 \, A + 17 \, B + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (54 \, A + 56 \, B + 49 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (100 \, A + 83 \, B + 72 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (187) = 374\).
Time = 0.46 (sec) , antiderivative size = 1005, normalized size of antiderivative = 5.02 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (186) = 372\).
Time = 0.23 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.00 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 960 \, {\left (d x + c\right )} A a^{4} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3840 \, A a^{4} \sin \left (d x + c\right ) - 960 \, B a^{4} \sin \left (d x + c\right )}{960 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {7}{16} \, {\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (2 \, A a^{4} + 8 \, B a^{4} + 15 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, A a^{4} + 29 \, B a^{4} + 36 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (112 \, A a^{4} + 128 \, B a^{4} + 127 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Time = 4.62 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.67 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {119\,B\,a^4}{3}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {231\,A\,a^4}{2}+\frac {462\,B\,a^4}{5}+\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {562\,B\,a^4}{5}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1069\,A\,a^4}{12}+\frac {233\,B\,a^4}{3}+\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+25\,B\,a^4+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,A+8\,B+7\,C\right )}{8\,\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4+\frac {49\,C\,a^4}{8}\right )}\right )\,\left (10\,A+8\,B+7\,C\right )}{8\,d} \]
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