Integrand size = 21, antiderivative size = 247 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {1}{4} a b \left (6 a^2+5 b^2\right ) x+\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d} \]
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Time = 0.46 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2872, 3112, 3102, 2827, 2713, 2715, 8} \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {b^2 \left (37 a^2+6 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac {a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x \left (6 a^2+5 b^2\right )-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 2872
Rule 3102
Rule 3112
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^3(c+d x) (a+b \cos (c+d x)) \left (a \left (7 a^2+4 b^2\right )+3 b \left (7 a^2+2 b^2\right ) \cos (c+d x)+16 a b^2 \cos ^2(c+d x)\right ) \, dx \\ & = \frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{42} \int \cos ^3(c+d x) \left (6 a^2 \left (7 a^2+4 b^2\right )+28 a b \left (6 a^2+5 b^2\right ) \cos (c+d x)+6 b^2 \left (37 a^2+6 b^2\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{210} \int \cos ^3(c+d x) \left (6 \left (35 a^4+168 a^2 b^2+24 b^4\right )+140 a b \left (6 a^2+5 b^2\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{3} \left (2 a b \left (6 a^2+5 b^2\right )\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{35} \left (35 a^4+168 a^2 b^2+24 b^4\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{2} \left (a b \left (6 a^2+5 b^2\right )\right ) \int \cos ^2(c+d x) \, dx-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac {1}{4} \left (a b \left (6 a^2+5 b^2\right )\right ) \int 1 \, dx \\ & = \frac {1}{4} a b \left (6 a^2+5 b^2\right ) x+\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.73 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {1680 a b \left (6 a^2+5 b^2\right ) (c+d x)+105 \left (48 a^4+240 a^2 b^2+35 b^4\right ) \sin (c+d x)+420 a b \left (16 a^2+15 b^2\right ) \sin (2 (c+d x))+35 \left (16 a^4+120 a^2 b^2+21 b^4\right ) \sin (3 (c+d x))+420 a b \left (2 a^2+3 b^2\right ) \sin (4 (c+d x))+21 b^2 \left (24 a^2+7 b^2\right ) \sin (5 (c+d x))+140 a b^3 \sin (6 (c+d x))+15 b^4 \sin (7 (c+d x))}{6720 d} \]
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Time = 5.95 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.71
method | result | size |
parallelrisch | \(\frac {\left (560 a^{4}+4200 a^{2} b^{2}+735 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (6720 a^{3} b +6300 a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (840 a^{3} b +1260 a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+\left (504 a^{2} b^{2}+147 b^{4}\right ) \sin \left (5 d x +5 c \right )+140 a \,b^{3} \sin \left (6 d x +6 c \right )+15 b^{4} \sin \left (7 d x +7 c \right )+\left (5040 a^{4}+25200 a^{2} b^{2}+3675 b^{4}\right ) \sin \left (d x +c \right )+10080 b \left (a^{2}+\frac {5 b^{2}}{6}\right ) d x a}{6720 d}\) | \(176\) |
derivativedivides | \(\frac {\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(190\) |
default | \(\frac {\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(190\) |
parts | \(\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {4 a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {4 a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(201\) |
risch | \(\frac {3 a^{3} b x}{2}+\frac {5 a \,b^{3} x}{4}+\frac {3 a^{4} \sin \left (d x +c \right )}{4 d}+\frac {15 \sin \left (d x +c \right ) a^{2} b^{2}}{4 d}+\frac {35 \sin \left (d x +c \right ) b^{4}}{64 d}+\frac {b^{4} \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \,b^{3} \sin \left (6 d x +6 c \right )}{48 d}+\frac {3 \sin \left (5 d x +5 c \right ) a^{2} b^{2}}{40 d}+\frac {7 \sin \left (5 d x +5 c \right ) b^{4}}{320 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} b}{8 d}+\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{3}}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4}}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{2} b^{2}}{8 d}+\frac {7 \sin \left (3 d x +3 c \right ) b^{4}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b}{d}+\frac {15 \sin \left (2 d x +2 c \right ) a \,b^{3}}{16 d}\) | \(258\) |
norman | \(\frac {\left (\frac {3}{2} a^{3} b +\frac {5}{4} a \,b^{3}\right ) x +\left (\frac {3}{2} a^{3} b +\frac {5}{4} a \,b^{3}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{2} a^{3} b +\frac {35}{4} a \,b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{2} a^{3} b +\frac {35}{4} a \,b^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{2} a^{3} b +\frac {105}{4} a \,b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{2} a^{3} b +\frac {105}{4} a \,b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{3} b +\frac {175}{4} a \,b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{3} b +\frac {175}{4} a \,b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8 \left (105 a^{4}+546 a^{2} b^{2}+53 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {\left (4 a^{4}-10 a^{3} b +24 a^{2} b^{2}-11 a \,b^{3}+4 b^{4}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {\left (4 a^{4}+10 a^{3} b +24 a^{2} b^{2}+11 a \,b^{3}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {2 \left (14 a^{4}-18 a^{3} b +60 a^{2} b^{2}-7 a \,b^{3}+6 b^{4}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (14 a^{4}+18 a^{3} b +60 a^{2} b^{2}+7 a \,b^{3}+6 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {\left (580 a^{4}-270 a^{3} b +2712 a^{2} b^{2}-425 a \,b^{3}+516 b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}+\frac {\left (580 a^{4}+270 a^{3} b +2712 a^{2} b^{2}+425 a \,b^{3}+516 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(530\) |
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Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.69 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {105 \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} d x + {\left (60 \, b^{4} \cos \left (d x + c\right )^{6} + 280 \, a b^{3} \cos \left (d x + c\right )^{5} + 72 \, {\left (7 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 280 \, a^{4} + 1344 \, a^{2} b^{2} + 192 \, b^{4} + 70 \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (35 \, a^{4} + 168 \, a^{2} b^{2} + 24 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (231) = 462\).
Time = 0.52 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.00 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\begin {cases} \frac {2 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a^{3} b x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a^{3} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 a^{3} b x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 a^{3} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 a^{3} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {16 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {8 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {6 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a b^{3} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {15 a b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {15 a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + \frac {5 a b^{3} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {5 a b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {10 a b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 a b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {16 b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {b^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{4} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=-\frac {560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} b - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} b^{2} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} + 48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} b^{4}}{1680 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a b^{3} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac {1}{4} \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} x + \frac {{\left (24 \, a^{2} b^{2} + 7 \, b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {{\left (16 \, a^{4} + 120 \, a^{2} b^{2} + 21 \, b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (16 \, a^{3} b + 15 \, a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (48 \, a^{4} + 240 \, a^{2} b^{2} + 35 \, b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
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Time = 15.71 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.93 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {\left (2\,a^4-5\,a^3\,b+12\,a^2\,b^2-\frac {11\,a\,b^3}{2}+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {28\,a^4}{3}-12\,a^3\,b+40\,a^2\,b^2-\frac {14\,a\,b^3}{3}+4\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {58\,a^4}{3}-9\,a^3\,b+\frac {452\,a^2\,b^2}{5}-\frac {85\,a\,b^3}{6}+\frac {86\,b^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (24\,a^4+\frac {624\,a^2\,b^2}{5}+\frac {424\,b^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {58\,a^4}{3}+9\,a^3\,b+\frac {452\,a^2\,b^2}{5}+\frac {85\,a\,b^3}{6}+\frac {86\,b^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {28\,a^4}{3}+12\,a^3\,b+40\,a^2\,b^2+\frac {14\,a\,b^3}{3}+4\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^4+5\,a^3\,b+12\,a^2\,b^2+\frac {11\,a\,b^3}{2}+2\,b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2+5\,b^2\right )}{2\,\left (3\,a^3\,b+\frac {5\,a\,b^3}{2}\right )}\right )\,\left (6\,a^2+5\,b^2\right )}{2\,d}-\frac {a\,b\,\left (6\,a^2+5\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{2\,d} \]
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