Integrand size = 41, antiderivative size = 304 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{128} a^4 (392 A+352 B+323 C) x+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d} \]
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Time = 0.93 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3124, 3055, 3047, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d}+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac {7 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac {a^4 (392 A+352 B+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a^4 x (392 A+352 B+323 C)+\frac {(56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{336 d}+\frac {a (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rule 3124
Rubi steps \begin{align*} \text {integral}& = \frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (a (8 A+3 C)+4 a (2 B+C) \cos (c+d x)) \, dx}{8 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (a^2 (56 A+24 B+33 C)+a^2 (56 A+80 B+61 C) \cos (c+d x)\right ) \, dx}{56 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^3 (168 A+128 B+127 C)+98 a^3 (8 A+8 B+7 C) \cos (c+d x)\right ) \, dx}{336 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^4 (1624 A+1424 B+1321 C)+3 a^4 (2408 A+2208 B+2007 C) \cos (c+d x)\right ) \, dx}{1680 a} \\ & = \frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) \left (3 a^5 (1624 A+1424 B+1321 C)+\left (3 a^5 (1624 A+1424 B+1321 C)+3 a^5 (2408 A+2208 B+2007 C)\right ) \cos (c+d x)+3 a^5 (2408 A+2208 B+2007 C) \cos ^2(c+d x)\right ) \, dx}{1680 a} \\ & = \frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) \left (105 a^5 (392 A+352 B+323 C)+192 a^5 (252 A+227 B+208 C) \cos (c+d x)\right ) \, dx}{6720 a} \\ & = \frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{35} \left (a^4 (252 A+227 B+208 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{64} \left (a^4 (392 A+352 B+323 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{128} \left (a^4 (392 A+352 B+323 C)\right ) \int 1 \, dx-\frac {\left (a^4 (252 A+227 B+208 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {1}{128} a^4 (392 A+352 B+323 C) x+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d} \\ \end{align*}
Time = 1.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (295680 B c+164640 c C+329280 A d x+295680 B d x+271320 C d x+1680 (352 A+323 B+300 C) \sin (c+d x)+1680 (127 A+124 B+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+87920 B \sin (3 (c+d x))+91840 C \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+33600 B \sin (4 (c+d x))+39480 C \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+10416 B \sin (5 (c+d x))+14784 C \sin (5 (c+d x))+560 A \sin (6 (c+d x))+2240 B \sin (6 (c+d x))+4480 C \sin (6 (c+d x))+240 B \sin (7 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x)))}{107520 d} \]
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Time = 12.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.54
method | result | size |
parallelrisch | \(\frac {3 \left (\frac {\left (\frac {127 A}{24}+\frac {31 B}{6}+5 C \right ) \sin \left (2 d x +2 c \right )}{2}+\left (A +\frac {157 B}{144}+\frac {41 C}{36}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (\frac {5 A}{4}+\frac {5 B}{3}+\frac {47 C}{24}\right ) \sin \left (4 d x +4 c \right )}{4}+\frac {\left (A +\frac {31 B}{16}+\frac {11 C}{4}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {\left (\frac {A}{8}+\frac {B}{2}+C \right ) \sin \left (6 d x +6 c \right )}{18}+\frac {\left (\frac {B}{4}+C \right ) \sin \left (7 d x +7 c \right )}{84}+\frac {C \sin \left (8 d x +8 c \right )}{768}+\left (\frac {22 A}{3}+\frac {323 B}{48}+\frac {25 C}{4}\right ) \sin \left (d x +c \right )+\frac {49 x \left (A +\frac {44 B}{49}+\frac {323 C}{392}\right ) d}{12}\right ) a^{4}}{4 d}\) | \(165\) |
parts | \(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(349\) |
risch | \(\frac {49 a^{4} x A}{16}+\frac {11 a^{4} B x}{4}+\frac {323 a^{4} C x}{128}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {323 \sin \left (d x +c \right ) B \,a^{4}}{64 d}+\frac {75 \sin \left (d x +c \right ) C \,a^{4}}{16 d}+\frac {C \,a^{4} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {\sin \left (7 d x +7 c \right ) B \,a^{4}}{448 d}+\frac {\sin \left (7 d x +7 c \right ) C \,a^{4}}{112 d}+\frac {\sin \left (6 d x +6 c \right ) a^{4} A}{192 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{24 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{20 d}+\frac {31 \sin \left (5 d x +5 c \right ) B \,a^{4}}{320 d}+\frac {11 \sin \left (5 d x +5 c \right ) C \,a^{4}}{80 d}+\frac {15 \sin \left (4 d x +4 c \right ) a^{4} A}{64 d}+\frac {5 \sin \left (4 d x +4 c \right ) B \,a^{4}}{16 d}+\frac {47 \sin \left (4 d x +4 c \right ) C \,a^{4}}{128 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{4} A}{4 d}+\frac {157 \sin \left (3 d x +3 c \right ) B \,a^{4}}{192 d}+\frac {41 \sin \left (3 d x +3 c \right ) C \,a^{4}}{48 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {31 \sin \left (2 d x +2 c \right ) B \,a^{4}}{16 d}+\frac {15 \sin \left (2 d x +2 c \right ) C \,a^{4}}{8 d}\) | \(392\) |
norman | \(\frac {\frac {a^{4} \left (392 A +352 B +323 C \right ) x}{128}+\frac {5053 a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d}+\frac {383 a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {23 a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {a^{4} \left (1656 A +1696 B +1725 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {a^{4} \left (21704 A +18528 B +15099 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (236936 A +211296 B +206019 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {a^{4} \left (2277016 A +2090016 B +1872009 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(480\) |
derivativedivides | \(\frac {a^{4} A \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 {B \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 a^{4} A \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}+4 B \,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 C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+6 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 {6 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}+6 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}+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 )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+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 )}{d}\) | \(577\) |
default | \(\frac {a^{4} A \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 {B \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 a^{4} A \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}+4 B \,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 C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+6 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 {6 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}+6 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}+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 )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+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 )}{d}\) | \(577\) |
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Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.63 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (392 \, A + 352 \, B + 323 \, C\right )} a^{4} d x + {\left (1680 \, C a^{4} \cos \left (d x + c\right )^{7} + 1920 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (8 \, A + 32 \, B + 55 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \, {\left (7 \, A + 12 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (328 \, A + 352 \, B + 323 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 128 \, {\left (252 \, A + 227 \, B + 208 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (392 \, A + 352 \, B + 323 \, C\right )} a^{4} \cos \left (d x + c\right ) + 256 \, {\left (252 \, A + 227 \, B + 208 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1640 vs. \(2 (289) = 578\).
Time = 0.86 (sec) , antiderivative size = 1640, normalized size of antiderivative = 5.39 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (286) = 572\).
Time = 0.22 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.90 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {28672 \, {\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 )} A a^{4} - 560 \, {\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 )} A a^{4} - 143360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 20160 \, {\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} + 26880 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 3072 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} B a^{4} + 43008 \, {\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} - 2240 \, {\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 )} B a^{4} - 35840 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 13440 \, {\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} - 12288 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 28672 \, {\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} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3360 \, {\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} + 3360 \, {\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}}{107520 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {1}{128} \, {\left (392 \, A a^{4} + 352 \, B a^{4} + 323 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (A a^{4} + 4 \, B a^{4} + 8 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{4} + 31 \, B a^{4} + 44 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (30 \, A a^{4} + 40 \, B a^{4} + 47 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (144 \, A a^{4} + 157 \, B a^{4} + 164 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (127 \, A a^{4} + 124 \, B a^{4} + 120 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (352 \, A a^{4} + 323 \, B a^{4} + 300 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
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Time = 3.61 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.49 \[ \int \cos ^2(c+d x) (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 {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}+\frac {323\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (\frac {1127\,A\,a^4}{24}+\frac {253\,B\,a^4}{6}+\frac {7429\,C\,a^4}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {18767\,A\,a^4}{120}+\frac {4213\,B\,a^4}{30}+\frac {123709\,C\,a^4}{960}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {35371\,A\,a^4}{120}+\frac {55583\,B\,a^4}{210}+\frac {1632119\,C\,a^4}{6720}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {40661\,A\,a^4}{120}+\frac {21771\,B\,a^4}{70}+\frac {624003\,C\,a^4}{2240}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {29617\,A\,a^4}{120}+\frac {2201\,B\,a^4}{10}+\frac {68673\,C\,a^4}{320}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {2713\,A\,a^4}{24}+\frac {193\,B\,a^4}{2}+\frac {5033\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {53\,B\,a^4}{2}+\frac {1725\,C\,a^4}{64}\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 )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}+\frac {323\,C\,a^4}{64}\right )}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,d}-\frac {a^4\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,d} \]
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