Integrand size = 28, antiderivative size = 381 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {35 a^4 x}{128}+\frac {15}{64} a^2 b^2 x+\frac {3 b^4 x}{128}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {3 b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \]
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Time = 0.44 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 2715, 8, 2645, 30, 2648, 14} \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^4 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^4 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^4 x}{128}-\frac {a^3 b \cos ^8(c+d x)}{2 d}-\frac {3 a^2 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac {5 a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac {15 a^2 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {15}{64} a^2 b^2 x+\frac {a b^3 \cos ^8(c+d x)}{2 d}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 b^4 x}{128} \]
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Rule 8
Rule 14
Rule 30
Rule 2645
Rule 2648
Rule 2715
Rule 3169
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \cos ^8(c+d x)+4 a^3 b \cos ^7(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^6(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^5(c+d x) \sin ^3(c+d x)+b^4 \cos ^4(c+d x) \sin ^4(c+d x)\right ) \, dx \\ & = a^4 \int \cos ^8(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^7(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx \\ & = \frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{8} \left (7 a^4\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{4} \left (3 a^2 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (3 b^4\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{48} \left (35 a^4\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{8} \left (5 a^2 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} b^4 \int \cos ^4(c+d x) \, dx-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{64} \left (35 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (15 a^2 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{64} \left (3 b^4\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {3 b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{128} \left (35 a^4\right ) \int 1 \, dx+\frac {1}{64} \left (15 a^2 b^2\right ) \int 1 \, dx+\frac {1}{128} \left (3 b^4\right ) \int 1 \, dx \\ & = \frac {35 a^4 x}{128}+\frac {15}{64} a^2 b^2 x+\frac {3 b^4 x}{128}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {3 b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \\ \end{align*}
Time = 3.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.58 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {24 \left (35 a^4+30 a^2 b^2+3 b^4\right ) (c+d x)-96 a b \left (7 a^2+3 b^2\right ) \cos (2 (c+d x))-48 a b \left (7 a^2+b^2\right ) \cos (4 (c+d x))-32 a b \left (3 a^2-b^2\right ) \cos (6 (c+d x))-12 a b \left (a^2-b^2\right ) \cos (8 (c+d x))+96 a^2 \left (7 a^2+3 b^2\right ) \sin (2 (c+d x))+24 \left (7 a^4-6 a^2 b^2-b^4\right ) \sin (4 (c+d x))+32 a^2 \left (a^2-3 b^2\right ) \sin (6 (c+d x))+3 \left (a^4-6 a^2 b^2+b^4\right ) \sin (8 (c+d x))}{3072 d} \]
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Time = 1.62 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {24 \left (7 a^{4}-6 a^{2} b^{2}-b^{4}\right ) \sin \left (4 d x +4 c \right )+3 \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \sin \left (8 d x +8 c \right )+96 \left (-7 a^{3} b -3 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+48 \left (-7 a^{3} b -a \,b^{3}\right ) \cos \left (4 d x +4 c \right )+32 \left (-3 a^{3} b +a \,b^{3}\right ) \cos \left (6 d x +6 c \right )+12 \left (-a^{3} b +a \,b^{3}\right ) \cos \left (8 d x +8 c \right )+96 \left (7 a^{4}+3 a^{2} b^{2}\right ) \sin \left (2 d x +2 c \right )+32 \left (a^{4}-3 a^{2} b^{2}\right ) \sin \left (6 d x +6 c \right )+840 a^{4} d x +720 a^{2} b^{2} d x +72 b^{4} d x +1116 a^{3} b +292 a \,b^{3}}{3072 d}\) | \(241\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {a^{3} b \cos \left (d x +c \right )^{8}}{2}+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+4 a \,b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(250\) |
default | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {a^{3} b \cos \left (d x +c \right )^{8}}{2}+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+4 a \,b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(250\) |
parts | \(\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\cos \left (d x +c \right )^{8}}{8}-\frac {\cos \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}-\frac {a^{3} b \cos \left (d x +c \right )^{8}}{2 d}\) | \(253\) |
risch | \(\frac {35 a^{4} x}{128}+\frac {15 a^{2} b^{2} x}{64}+\frac {3 b^{4} x}{128}-\frac {a^{3} b \cos \left (8 d x +8 c \right )}{256 d}+\frac {a \,b^{3} \cos \left (8 d x +8 c \right )}{256 d}+\frac {\sin \left (8 d x +8 c \right ) a^{4}}{1024 d}-\frac {3 \sin \left (8 d x +8 c \right ) a^{2} b^{2}}{512 d}+\frac {\sin \left (8 d x +8 c \right ) b^{4}}{1024 d}-\frac {a^{3} b \cos \left (6 d x +6 c \right )}{32 d}+\frac {a \,b^{3} \cos \left (6 d x +6 c \right )}{96 d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{96 d}-\frac {a^{2} \sin \left (6 d x +6 c \right ) b^{2}}{32 d}-\frac {7 a^{3} b \cos \left (4 d x +4 c \right )}{64 d}-\frac {a \,b^{3} \cos \left (4 d x +4 c \right )}{64 d}+\frac {7 \sin \left (4 d x +4 c \right ) a^{4}}{128 d}-\frac {3 \sin \left (4 d x +4 c \right ) a^{2} b^{2}}{64 d}-\frac {\sin \left (4 d x +4 c \right ) b^{4}}{128 d}-\frac {7 a^{3} b \cos \left (2 d x +2 c \right )}{32 d}-\frac {3 a \,b^{3} \cos \left (2 d x +2 c \right )}{32 d}+\frac {7 a^{4} \sin \left (2 d x +2 c \right )}{32 d}+\frac {3 a^{2} \sin \left (2 d x +2 c \right ) b^{2}}{32 d}\) | \(349\) |
norman | \(\frac {\left (\frac {35}{128} a^{4}+\frac {15}{64} a^{2} b^{2}+\frac {3}{128} b^{4}\right ) x +\left (\frac {35}{16} a^{4}+\frac {15}{8} a^{2} b^{2}+\frac {3}{16} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {35}{16} a^{4}+\frac {15}{8} a^{2} b^{2}+\frac {3}{16} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {35}{128} a^{4}+\frac {15}{64} a^{2} b^{2}+\frac {3}{128} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {245}{16} a^{4}+\frac {105}{8} a^{2} b^{2}+\frac {21}{16} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {245}{16} a^{4}+\frac {105}{8} a^{2} b^{2}+\frac {21}{16} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {245}{32} a^{4}+\frac {105}{16} a^{2} b^{2}+\frac {21}{32} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {245}{32} a^{4}+\frac {105}{16} a^{2} b^{2}+\frac {21}{32} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {1225}{64} a^{4}+\frac {525}{32} a^{2} b^{2}+\frac {105}{64} b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {16 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {16 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {160 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}+\frac {8 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {8 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {3 \left (31 a^{4}-10 a^{2} b^{2}-b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {3 \left (31 a^{4}-10 a^{2} b^{2}-b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}+\frac {\left (91 a^{4}+2382 a^{2} b^{2}-69 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d}-\frac {\left (91 a^{4}+2382 a^{2} b^{2}-69 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}-\frac {\left (1085 a^{4}-10590 a^{2} b^{2}+2013 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{192 d}+\frac {\left (1085 a^{4}-10590 a^{2} b^{2}+2013 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192 d}+\frac {\left (1799 a^{4}-5370 a^{2} b^{2}+999 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 d}-\frac {\left (1799 a^{4}-5370 a^{2} b^{2}+999 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{192 d}+\frac {8 \left (21 a^{3} b -8 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {8 \left (21 a^{3} b -8 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(731\) |
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Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.48 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {256 \, a b^{3} \cos \left (d x + c\right )^{6} + 192 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{8} - 3 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x - {\left (48 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{4} + 6 \, a^{2} b^{2} - 9 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (367) = 734\).
Time = 0.81 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.99 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\begin {cases} \frac {35 a^{4} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 a^{4} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 a^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a^{3} b \cos ^{8}{\left (c + d x \right )}}{2 d} + \frac {15 a^{2} b^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {15 a^{2} b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {15 a^{2} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {15 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {55 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {73 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac {15 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} + \frac {a b^{3} \sin ^{8}{\left (c + d x \right )}}{6 d} + \frac {2 a b^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {3 b^{4} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 b^{4} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {11 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 b^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{4} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.52 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {1536 \, a^{3} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 6 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{2} - 512 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a b^{3} - 3 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{4}}{3072 \, d} \]
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Time = 0.49 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {1}{128} \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} x - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{256 \, d} - \frac {{\left (3 \, a^{3} b - a b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {{\left (7 \, a^{3} b + a b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {{\left (7 \, a^{4} - 6 \, a^{2} b^{2} - b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (7 \, a^{4} + 3 \, a^{2} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
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Time = 23.87 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.90 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {35\,a^4\,x}{128}+\frac {3\,b^4\,x}{128}+\frac {15\,a^2\,b^2\,x}{64}-\frac {2\,a\,b^3\,{\cos \left (c+d\,x\right )}^6}{3\,d}+\frac {a\,b^3\,{\cos \left (c+d\,x\right )}^8}{2\,d}-\frac {a^3\,b\,{\cos \left (c+d\,x\right )}^8}{2\,d}+\frac {35\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{192\,d}+\frac {7\,a^4\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{48\,d}+\frac {a^4\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}+\frac {b^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{64\,d}-\frac {3\,b^4\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{16\,d}+\frac {b^4\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}+\frac {35\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d}+\frac {3\,b^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d}+\frac {15\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{64\,d}+\frac {5\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{32\,d}+\frac {a^2\,b^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{8\,d}-\frac {3\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{4\,d} \]
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