Integrand size = 28, antiderivative size = 381 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {35 a^4 x}{128}+\frac {15}{64} a^2 b^2 x+\frac {3 b^4 x}{128}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {3 b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \]
35/128*a^4*x+15/64*a^2*b^2*x+3/128*b^4*x-2/3*a*b^3*cos(d*x+c)^6/d-1/2*a^3* b*cos(d*x+c)^8/d+1/2*a*b^3*cos(d*x+c)^8/d+35/128*a^4*cos(d*x+c)*sin(d*x+c) /d+15/64*a^2*b^2*cos(d*x+c)*sin(d*x+c)/d+3/128*b^4*cos(d*x+c)*sin(d*x+c)/d +35/192*a^4*cos(d*x+c)^3*sin(d*x+c)/d+5/32*a^2*b^2*cos(d*x+c)^3*sin(d*x+c) /d+1/64*b^4*cos(d*x+c)^3*sin(d*x+c)/d+7/48*a^4*cos(d*x+c)^5*sin(d*x+c)/d+1 /8*a^2*b^2*cos(d*x+c)^5*sin(d*x+c)/d-1/16*b^4*cos(d*x+c)^5*sin(d*x+c)/d+1/ 8*a^4*cos(d*x+c)^7*sin(d*x+c)/d-3/4*a^2*b^2*cos(d*x+c)^7*sin(d*x+c)/d-1/8* b^4*cos(d*x+c)^5*sin(d*x+c)^3/d
Time = 3.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.58 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {24 \left (35 a^4+30 a^2 b^2+3 b^4\right ) (c+d x)-96 a b \left (7 a^2+3 b^2\right ) \cos (2 (c+d x))-48 a b \left (7 a^2+b^2\right ) \cos (4 (c+d x))-32 a b \left (3 a^2-b^2\right ) \cos (6 (c+d x))-12 a b \left (a^2-b^2\right ) \cos (8 (c+d x))+96 a^2 \left (7 a^2+3 b^2\right ) \sin (2 (c+d x))+24 \left (7 a^4-6 a^2 b^2-b^4\right ) \sin (4 (c+d x))+32 a^2 \left (a^2-3 b^2\right ) \sin (6 (c+d x))+3 \left (a^4-6 a^2 b^2+b^4\right ) \sin (8 (c+d x))}{3072 d} \]
(24*(35*a^4 + 30*a^2*b^2 + 3*b^4)*(c + d*x) - 96*a*b*(7*a^2 + 3*b^2)*Cos[2 *(c + d*x)] - 48*a*b*(7*a^2 + b^2)*Cos[4*(c + d*x)] - 32*a*b*(3*a^2 - b^2) *Cos[6*(c + d*x)] - 12*a*b*(a^2 - b^2)*Cos[8*(c + d*x)] + 96*a^2*(7*a^2 + 3*b^2)*Sin[2*(c + d*x)] + 24*(7*a^4 - 6*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + 32*a^2*(a^2 - 3*b^2)*Sin[6*(c + d*x)] + 3*(a^4 - 6*a^2*b^2 + b^4)*Sin[8*(c + d*x)])/(3072*d)
Time = 0.61 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3042, 3569, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^4 (a \cos (c+d x)+b \sin (c+d x))^4dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^4 \cos ^8(c+d x)+4 a^3 b \sin (c+d x) \cos ^7(c+d x)+6 a^2 b^2 \sin ^2(c+d x) \cos ^6(c+d x)+4 a b^3 \sin ^3(c+d x) \cos ^5(c+d x)+b^4 \sin ^4(c+d x) \cos ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^4 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^4 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^4 x}{128}-\frac {a^3 b \cos ^8(c+d x)}{2 d}-\frac {3 a^2 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac {5 a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac {15 a^2 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {15}{64} a^2 b^2 x+\frac {a b^3 \cos ^8(c+d x)}{2 d}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 b^4 x}{128}\) |
(35*a^4*x)/128 + (15*a^2*b^2*x)/64 + (3*b^4*x)/128 - (2*a*b^3*Cos[c + d*x] ^6)/(3*d) - (a^3*b*Cos[c + d*x]^8)/(2*d) + (a*b^3*Cos[c + d*x]^8)/(2*d) + (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*b^2*Cos[c + d*x]*Sin[ c + d*x])/(64*d) + (3*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^4*Cos [c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a^2*b^2*Cos[c + d*x]^3*Sin[c + d*x] )/(32*d) + (b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^4*Cos[c + d*x]^ 5*Sin[c + d*x])/(48*d) + (a^2*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) - (b^ 4*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^4*Cos[c + d*x]^7*Sin[c + d*x])/ (8*d) - (3*a^2*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(4*d) - (b^4*Cos[c + d*x]^ 5*Sin[c + d*x]^3)/(8*d)
3.1.75.3.1 Defintions of rubi rules used
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*(a *cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte gerQ[m] && IGtQ[n, 0]
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}}\) | \(731\) |
1/3072*(24*(7*a^4-6*a^2*b^2-b^4)*sin(4*d*x+4*c)+3*(a^4-6*a^2*b^2+b^4)*sin( 8*d*x+8*c)+96*(-7*a^3*b-3*a*b^3)*cos(2*d*x+2*c)+48*(-7*a^3*b-a*b^3)*cos(4* d*x+4*c)+32*(-3*a^3*b+a*b^3)*cos(6*d*x+6*c)+12*(-a^3*b+a*b^3)*cos(8*d*x+8* c)+96*(7*a^4+3*a^2*b^2)*sin(2*d*x+2*c)+32*(a^4-3*a^2*b^2)*sin(6*d*x+6*c)+8 40*a^4*d*x+720*a^2*b^2*d*x+72*b^4*d*x+1116*a^3*b+292*a*b^3)/d
Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.48 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {256 \, a b^{3} \cos \left (d x + c\right )^{6} + 192 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{8} - 3 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x - {\left (48 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{4} + 6 \, a^{2} b^{2} - 9 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]
-1/384*(256*a*b^3*cos(d*x + c)^6 + 192*(a^3*b - a*b^3)*cos(d*x + c)^8 - 3* (35*a^4 + 30*a^2*b^2 + 3*b^4)*d*x - (48*(a^4 - 6*a^2*b^2 + b^4)*cos(d*x + c)^7 + 8*(7*a^4 + 6*a^2*b^2 - 9*b^4)*cos(d*x + c)^5 + 2*(35*a^4 + 30*a^2*b ^2 + 3*b^4)*cos(d*x + c)^3 + 3*(35*a^4 + 30*a^2*b^2 + 3*b^4)*cos(d*x + c)) *sin(d*x + c))/d
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} \]
Piecewise((35*a**4*x*sin(c + d*x)**8/128 + 35*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**4*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*a**4*x* sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**4*x*cos(c + d*x)**8/128 + 35*a* *4*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**4*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*a **4*sin(c + d*x)*cos(c + d*x)**7/(128*d) - a**3*b*cos(c + d*x)**8/(2*d) + 15*a**2*b**2*x*sin(c + d*x)**8/64 + 15*a**2*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 45*a**2*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 15*a**2* b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 15*a**2*b**2*x*cos(c + d*x)**8 /64 + 15*a**2*b**2*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 55*a**2*b**2*sin( c + d*x)**5*cos(c + d*x)**3/(64*d) + 73*a**2*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) - 15*a**2*b**2*sin(c + d*x)*cos(c + d*x)**7/(64*d) + a*b**3 *sin(c + d*x)**8/(6*d) + 2*a*b**3*sin(c + d*x)**6*cos(c + d*x)**2/(3*d) + a*b**3*sin(c + d*x)**4*cos(c + d*x)**4/d + 3*b**4*x*sin(c + d*x)**8/128 + 3*b**4*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 9*b**4*x*sin(c + d*x)**4*cos (c + d*x)**4/64 + 3*b**4*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*b**4*x*c os(c + d*x)**8/128 + 3*b**4*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*b**4 *sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 11*b**4*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 3*b**4*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**4*cos(c)**4, True))
Time = 0.23 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.52 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {1536 \, a^{3} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 6 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{2} - 512 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a b^{3} - 3 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{4}}{3072 \, d} \]
-1/3072*(1536*a^3*b*cos(d*x + c)^8 + (128*sin(2*d*x + 2*c)^3 - 840*d*x - 8 40*c - 3*sin(8*d*x + 8*c) - 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a ^4 - 6*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 24* sin(4*d*x + 4*c))*a^2*b^2 - 512*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6*s in(d*x + c)^4)*a*b^3 - 3*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*b^4)/d
Time = 0.49 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {1}{128} \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} x - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{256 \, d} - \frac {{\left (3 \, a^{3} b - a b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {{\left (7 \, a^{3} b + a b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {{\left (7 \, a^{4} - 6 \, a^{2} b^{2} - b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (7 \, a^{4} + 3 \, a^{2} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
1/128*(35*a^4 + 30*a^2*b^2 + 3*b^4)*x - 1/256*(a^3*b - a*b^3)*cos(8*d*x + 8*c)/d - 1/96*(3*a^3*b - a*b^3)*cos(6*d*x + 6*c)/d - 1/64*(7*a^3*b + a*b^3 )*cos(4*d*x + 4*c)/d - 1/32*(7*a^3*b + 3*a*b^3)*cos(2*d*x + 2*c)/d + 1/102 4*(a^4 - 6*a^2*b^2 + b^4)*sin(8*d*x + 8*c)/d + 1/96*(a^4 - 3*a^2*b^2)*sin( 6*d*x + 6*c)/d + 1/128*(7*a^4 - 6*a^2*b^2 - b^4)*sin(4*d*x + 4*c)/d + 1/32 *(7*a^4 + 3*a^2*b^2)*sin(2*d*x + 2*c)/d
Time = 23.87 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.90 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {35\,a^4\,x}{128}+\frac {3\,b^4\,x}{128}+\frac {15\,a^2\,b^2\,x}{64}-\frac {2\,a\,b^3\,{\cos \left (c+d\,x\right )}^6}{3\,d}+\frac {a\,b^3\,{\cos \left (c+d\,x\right )}^8}{2\,d}-\frac {a^3\,b\,{\cos \left (c+d\,x\right )}^8}{2\,d}+\frac {35\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{192\,d}+\frac {7\,a^4\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{48\,d}+\frac {a^4\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}+\frac {b^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{64\,d}-\frac {3\,b^4\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{16\,d}+\frac {b^4\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}+\frac {35\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d}+\frac {3\,b^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d}+\frac {15\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{64\,d}+\frac {5\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{32\,d}+\frac {a^2\,b^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{8\,d}-\frac {3\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{4\,d} \]
(35*a^4*x)/128 + (3*b^4*x)/128 + (15*a^2*b^2*x)/64 - (2*a*b^3*cos(c + d*x) ^6)/(3*d) + (a*b^3*cos(c + d*x)^8)/(2*d) - (a^3*b*cos(c + d*x)^8)/(2*d) + (35*a^4*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (7*a^4*cos(c + d*x)^5*sin(c + d*x))/(48*d) + (a^4*cos(c + d*x)^7*sin(c + d*x))/(8*d) + (b^4*cos(c + d *x)^3*sin(c + d*x))/(64*d) - (3*b^4*cos(c + d*x)^5*sin(c + d*x))/(16*d) + (b^4*cos(c + d*x)^7*sin(c + d*x))/(8*d) + (35*a^4*cos(c + d*x)*sin(c + d*x ))/(128*d) + (3*b^4*cos(c + d*x)*sin(c + d*x))/(128*d) + (15*a^2*b^2*cos(c + d*x)*sin(c + d*x))/(64*d) + (5*a^2*b^2*cos(c + d*x)^3*sin(c + d*x))/(32 *d) + (a^2*b^2*cos(c + d*x)^5*sin(c + d*x))/(8*d) - (3*a^2*b^2*cos(c + d*x )^7*sin(c + d*x))/(4*d)