Integrand size = 19, antiderivative size = 161 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=\frac {5}{16} \left (a^2+b^2\right )^3 x-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d} \]
5/16*(a^2+b^2)^3*x-5/16*(a^2+b^2)^2*(b*cos(d*x+c)-a*sin(d*x+c))*(a*cos(d*x +c)+b*sin(d*x+c))/d-5/24*(a^2+b^2)*(b*cos(d*x+c)-a*sin(d*x+c))*(a*cos(d*x+ c)+b*sin(d*x+c))^3/d-1/6*(b*cos(d*x+c)-a*sin(d*x+c))*(a*cos(d*x+c)+b*sin(d *x+c))^5/d
Time = 2.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=\frac {60 \left (a^2+b^2\right )^3 (c+d x)-90 a b \left (a^2+b^2\right )^2 \cos (2 (c+d x))-36 a b \left (a^4-b^4\right ) \cos (4 (c+d x))-2 a b \left (3 a^4-10 a^2 b^2+3 b^4\right ) \cos (6 (c+d x))+45 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \sin (2 (c+d x))+9 \left (a^6-5 a^4 b^2-5 a^2 b^4+b^6\right ) \sin (4 (c+d x))+\left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \sin (6 (c+d x))}{192 d} \]
(60*(a^2 + b^2)^3*(c + d*x) - 90*a*b*(a^2 + b^2)^2*Cos[2*(c + d*x)] - 36*a *b*(a^4 - b^4)*Cos[4*(c + d*x)] - 2*a*b*(3*a^4 - 10*a^2*b^2 + 3*b^4)*Cos[6 *(c + d*x)] + 45*(a^2 - b^2)*(a^2 + b^2)^2*Sin[2*(c + d*x)] + 9*(a^6 - 5*a ^4*b^2 - 5*a^2*b^4 + b^6)*Sin[4*(c + d*x)] + (a^6 - 15*a^4*b^2 + 15*a^2*b^ 4 - b^6)*Sin[6*(c + d*x)])/(192*d)
Time = 0.43 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3042, 3552, 3042, 3552, 3042, 3552, 24}
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.
| \(\displaystyle \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \cos (c+d x)+b \sin (c+d x))^6dx\) |
| \(\Big \downarrow \) 3552 |
| \(\displaystyle \frac {5}{6} \left (a^2+b^2\right ) \int (a \cos (c+d x)+b \sin (c+d x))^4dx-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \left (a^2+b^2\right ) \int (a \cos (c+d x)+b \sin (c+d x))^4dx-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\) |
| \(\Big \downarrow \) 3552 |
| \(\displaystyle \frac {5}{6} \left (a^2+b^2\right ) \left (\frac {3}{4} \left (a^2+b^2\right ) \int (a \cos (c+d x)+b \sin (c+d x))^2dx-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{4 d}\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} \left (a^2+b^2\right ) \left (\frac {3}{4} \left (a^2+b^2\right ) \int (a \cos (c+d x)+b \sin (c+d x))^2dx-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{4 d}\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\) |
| \(\Big \downarrow \) 3552 |
| \(\displaystyle \frac {5}{6} \left (a^2+b^2\right ) \left (\frac {3}{4} \left (a^2+b^2\right ) \left (\frac {1}{2} \left (a^2+b^2\right ) \int 1dx-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{2 d}\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{4 d}\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {5}{6} \left (a^2+b^2\right ) \left (\frac {3}{4} \left (a^2+b^2\right ) \left (\frac {1}{2} x \left (a^2+b^2\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{2 d}\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{4 d}\right )-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\) |
-1/6*((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^ 5)/d + (5*(a^2 + b^2)*(-1/4*((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/d + (3*(a^2 + b^2)*(((a^2 + b^2)*x)/2 - ((b*Cos[ c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(2*d)))/4))/ 6
3.3.20.3.1 Defintions of rubi rules used
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[(-(b*Cos[c + d*x] - a*Sin[c + d*x]))*((a*Cos[c + d*x] + b* Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[(n - 1)*((a^2 + b^2)/n) Int[(a*Co s[c + d*x] + b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d}, x] && N eQ[a^2 + b^2, 0] && !IntegerQ[(n - 1)/2] && GtQ[n, 1]
Time = 2.36 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.46
| method | result | size |
| parallelrisch | \(\frac {45 \left (a -b \right ) \left (a +b \right ) \left (a^{2}+b^{2}\right )^{2} \sin \left (2 d x +2 c \right )+9 \left (a^{6}-5 a^{4} b^{2}-5 a^{2} b^{4}+b^{6}\right ) \sin \left (4 d x +4 c \right )+\left (a^{6}-15 a^{4} b^{2}+15 a^{2} b^{4}-b^{6}\right ) \sin \left (6 d x +6 c \right )-90 b a \left (a^{2}+b^{2}\right )^{2} \cos \left (2 d x +2 c \right )+2 \left (-3 a^{5} b +10 a^{3} b^{3}-3 a \,b^{5}\right ) \cos \left (6 d x +6 c \right )+36 \left (-a^{5} b +a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+60 a^{6} d x +180 a^{4} b^{2} d x +180 a^{2} b^{4} d x +60 b^{6} d x +132 a^{5} b +160 a^{3} b^{3}+60 a \,b^{5}}{192 d}\) | \(235\) |
| derivativedivides | \(\frac {a^{6} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-a^{5} b \cos \left (d x +c \right )^{6}+15 a^{4} b^{2} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+20 a^{3} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{6}-\frac {\cos \left (d x +c \right )^{4}}{12}\right )+15 a^{2} b^{4} \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a \,b^{5} \sin \left (d x +c \right )^{6}+b^{6} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(285\) |
| default | \(\frac {a^{6} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-a^{5} b \cos \left (d x +c \right )^{6}+15 a^{4} b^{2} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+20 a^{3} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{6}-\frac {\cos \left (d x +c \right )^{4}}{12}\right )+15 a^{2} b^{4} \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a \,b^{5} \sin \left (d x +c \right )^{6}+b^{6} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(285\) |
| parts | \(\frac {a^{6} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {b^{6} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {20 a^{3} b^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}+\frac {\sin \left (d x +c \right )^{4}}{4}\right )}{d}+\frac {a \,b^{5} \sin \left (d x +c \right )^{6}}{d}-\frac {a^{5} b \cos \left (d x +c \right )^{6}}{d}+\frac {15 a^{4} b^{2} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}+\frac {15 a^{2} b^{4} \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(294\) |
| risch | \(\frac {5 x \,a^{6}}{16}+\frac {15 x \,a^{4} b^{2}}{16}+\frac {15 x \,a^{2} b^{4}}{16}+\frac {5 b^{6} x}{16}-\frac {a^{5} b \cos \left (6 d x +6 c \right )}{32 d}+\frac {5 a^{3} b^{3} \cos \left (6 d x +6 c \right )}{48 d}-\frac {a \,b^{5} \cos \left (6 d x +6 c \right )}{32 d}+\frac {\sin \left (6 d x +6 c \right ) a^{6}}{192 d}-\frac {5 \sin \left (6 d x +6 c \right ) a^{4} b^{2}}{64 d}+\frac {5 \sin \left (6 d x +6 c \right ) a^{2} b^{4}}{64 d}-\frac {\sin \left (6 d x +6 c \right ) b^{6}}{192 d}-\frac {3 a^{5} b \cos \left (4 d x +4 c \right )}{16 d}+\frac {3 a \,b^{5} \cos \left (4 d x +4 c \right )}{16 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{6}}{64 d}-\frac {15 \sin \left (4 d x +4 c \right ) a^{4} b^{2}}{64 d}-\frac {15 \sin \left (4 d x +4 c \right ) a^{2} b^{4}}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) b^{6}}{64 d}-\frac {15 a^{5} b \cos \left (2 d x +2 c \right )}{32 d}-\frac {15 a^{3} b^{3} \cos \left (2 d x +2 c \right )}{16 d}-\frac {15 a \,b^{5} \cos \left (2 d x +2 c \right )}{32 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{6}}{64 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{4} b^{2}}{64 d}-\frac {15 \sin \left (2 d x +2 c \right ) a^{2} b^{4}}{64 d}-\frac {15 \sin \left (2 d x +2 c \right ) b^{6}}{64 d}\) | \(402\) |
| norman | \(\frac {\left (\frac {5}{16} a^{6}+\frac {15}{16} a^{4} b^{2}+\frac {15}{16} a^{2} b^{4}+\frac {5}{16} b^{6}\right ) x +\left (\frac {5}{16} a^{6}+\frac {15}{16} a^{4} b^{2}+\frac {15}{16} a^{2} b^{4}+\frac {5}{16} b^{6}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {15}{8} a^{6}+\frac {45}{8} a^{4} b^{2}+\frac {45}{8} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {15}{8} a^{6}+\frac {45}{8} a^{4} b^{2}+\frac {45}{8} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {25}{4} a^{6}+\frac {75}{4} a^{4} b^{2}+\frac {75}{4} a^{2} b^{4}+\frac {25}{4} b^{6}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {75}{16} a^{6}+\frac {225}{16} a^{4} b^{2}+\frac {225}{16} a^{2} b^{4}+\frac {75}{16} b^{6}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {75}{16} a^{6}+\frac {225}{16} a^{4} b^{2}+\frac {225}{16} a^{2} b^{4}+\frac {75}{16} b^{6}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {80 a^{3} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {80 a^{3} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {12 a^{5} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {12 a^{5} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {5 \left (a^{6}-141 a^{4} b^{2}+51 a^{2} b^{4}+17 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {5 \left (a^{6}-141 a^{4} b^{2}+51 a^{2} b^{4}+17 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {3 \left (5 a^{6}-65 a^{4} b^{2}+95 a^{2} b^{4}-11 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {3 \left (5 a^{6}-65 a^{4} b^{2}+95 a^{2} b^{4}-11 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (11 a^{6}-15 a^{4} b^{2}-15 a^{2} b^{4}-5 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (11 a^{6}-15 a^{4} b^{2}-15 a^{2} b^{4}-5 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {4 \left (30 a^{5} b -40 a^{3} b^{3}+48 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(660\) |
1/192*(45*(a-b)*(a+b)*(a^2+b^2)^2*sin(2*d*x+2*c)+9*(a^6-5*a^4*b^2-5*a^2*b^ 4+b^6)*sin(4*d*x+4*c)+(a^6-15*a^4*b^2+15*a^2*b^4-b^6)*sin(6*d*x+6*c)-90*b* a*(a^2+b^2)^2*cos(2*d*x+2*c)+2*(-3*a^5*b+10*a^3*b^3-3*a*b^5)*cos(6*d*x+6*c )+36*(-a^5*b+a*b^5)*cos(4*d*x+4*c)+60*a^6*d*x+180*a^4*b^2*d*x+180*a^2*b^4* d*x+60*b^6*d*x+132*a^5*b+160*a^3*b^3+60*a*b^5)/d
Time = 0.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.36 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=-\frac {144 \, a b^{5} \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{6} + 48 \, {\left (5 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d x - {\left (8 \, {\left (a^{6} - 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, a^{6} + 15 \, a^{4} b^{2} - 105 \, a^{2} b^{4} + 13 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{6} + 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 11 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]
-1/48*(144*a*b^5*cos(d*x + c)^2 + 16*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*cos( d*x + c)^6 + 48*(5*a^3*b^3 - 3*a*b^5)*cos(d*x + c)^4 - 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*x - (8*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*cos(d*x + c)^5 + 2*(5*a^6 + 15*a^4*b^2 - 105*a^2*b^4 + 13*b^6)*cos(d*x + c)^3 + 3 *(5*a^6 + 15*a^4*b^2 + 15*a^2*b^4 - 11*b^6)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (151) = 302\).
Time = 0.41 (sec) , antiderivative size = 770, normalized size of antiderivative = 4.78 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=\begin {cases} \frac {5 a^{6} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{6} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{6} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{6} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{6} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{6} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{6} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{5} b \cos ^{6}{\left (c + d x \right )}}{d} + \frac {15 a^{4} b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{4} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{4} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{4} b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{4} b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{4} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {15 a^{4} b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 a^{3} b^{3} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {5 a^{3} b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {15 a^{2} b^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 a^{2} b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {15 a^{2} b^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {a b^{5} \sin ^{6}{\left (c + d x \right )}}{d} + \frac {5 b^{6} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{6} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{6} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{6} x \cos ^{6}{\left (c + d x \right )}}{16} - \frac {11 b^{6} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 b^{6} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {5 b^{6} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{6} & \text {otherwise} \end {cases} \]
Piecewise((5*a**6*x*sin(c + d*x)**6/16 + 15*a**6*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**6*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*a**6*x*cos( c + d*x)**6/16 + 5*a**6*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a**6*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*a**6*sin(c + d*x)*cos(c + d*x)**5/(1 6*d) - a**5*b*cos(c + d*x)**6/d + 15*a**4*b**2*x*sin(c + d*x)**6/16 + 45*a **4*b**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 45*a**4*b**2*x*sin(c + d*x )**2*cos(c + d*x)**4/16 + 15*a**4*b**2*x*cos(c + d*x)**6/16 + 15*a**4*b**2 *sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a**4*b**2*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) - 15*a**4*b**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 5*a**3 *b**3*sin(c + d*x)**6/(3*d) + 5*a**3*b**3*sin(c + d*x)**4*cos(c + d*x)**2/ d + 15*a**2*b**4*x*sin(c + d*x)**6/16 + 45*a**2*b**4*x*sin(c + d*x)**4*cos (c + d*x)**2/16 + 45*a**2*b**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 15*a **2*b**4*x*cos(c + d*x)**6/16 + 15*a**2*b**4*sin(c + d*x)**5*cos(c + d*x)/ (16*d) - 5*a**2*b**4*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) - 15*a**2*b**4* sin(c + d*x)*cos(c + d*x)**5/(16*d) + a*b**5*sin(c + d*x)**6/d + 5*b**6*x* sin(c + d*x)**6/16 + 15*b**6*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*b** 6*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*b**6*x*cos(c + d*x)**6/16 - 11* b**6*sin(c + d*x)**5*cos(c + d*x)/(16*d) - 5*b**6*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 5*b**6*sin(c + d*x)*cos(c + d*x)**5/(16*d), Ne(d, 0)), (x* (a*cos(c) + b*sin(c))**6, True))
Time = 0.22 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.48 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=-\frac {192 \, a^{5} b \cos \left (d x + c\right )^{6} - 192 \, a b^{5} \sin \left (d x + c\right )^{6} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{6} - 15 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{2} + 320 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{3} b^{3} + 15 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{4} - {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{6}}{192 \, d} \]
-1/192*(192*a^5*b*cos(d*x + c)^6 - 192*a*b^5*sin(d*x + c)^6 + (4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^6 - 15*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^4*b^2 + 320*(2*sin(d*x + c)^6 - 3*sin(d*x + c)^4)*a^3*b^3 + 15*(4*sin(2*d*x + 2*c )^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*a^2*b^4 - (4*sin(2*d*x + 2*c)^3 + 60*d*x + 60*c + 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*b^6)/d
Time = 0.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.46 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=\frac {5}{16} \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} x - \frac {{\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {3 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac {15 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (a^{6} - 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - b^{6}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, {\left (a^{6} - 5 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
5/16*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x - 1/96*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*cos(6*d*x + 6*c)/d - 3/16*(a^5*b - a*b^5)*cos(4*d*x + 4*c)/d - 15 /32*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2*d*x + 2*c)/d + 1/192*(a^6 - 15*a^4*b ^2 + 15*a^2*b^4 - b^6)*sin(6*d*x + 6*c)/d + 3/64*(a^6 - 5*a^4*b^2 - 5*a^2* b^4 + b^6)*sin(4*d*x + 4*c)/d + 15/64*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*sin( 2*d*x + 2*c)/d
Time = 28.25 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.22 \[ \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx=\frac {5\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\left (a^2+b^2\right )}^3}{8\,\left (\frac {5\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )}\right )\,{\left (a^2+b^2\right )}^3}{8\,d}-\frac {5\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,{\left (a^2+b^2\right )}^3}{8\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (40\,a^5\,b-\frac {160\,a^3\,b^3}{3}+64\,a\,b^5\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-\frac {11\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {11\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a^6}{24}-\frac {235\,a^4\,b^2}{8}+\frac {85\,a^2\,b^4}{8}+\frac {85\,b^6}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {5\,a^6}{24}-\frac {235\,a^4\,b^2}{8}+\frac {85\,a^2\,b^4}{8}+\frac {85\,b^6}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {15\,a^6}{4}-\frac {195\,a^4\,b^2}{4}+\frac {285\,a^2\,b^4}{4}-\frac {33\,b^6}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {15\,a^6}{4}-\frac {195\,a^4\,b^2}{4}+\frac {285\,a^2\,b^4}{4}-\frac {33\,b^6}{4}\right )+80\,a^3\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+80\,a^3\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+12\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(5*atan((5*tan(c/2 + (d*x)/2)*(a^2 + b^2)^3)/(8*((5*a^6)/8 + (5*b^6)/8 + ( 15*a^2*b^4)/8 + (15*a^4*b^2)/8)))*(a^2 + b^2)^3)/(8*d) - (5*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(a^2 + b^2)^3)/(8*d) + (tan(c/2 + (d*x)/2)^6*(64*a* b^5 + 40*a^5*b - (160*a^3*b^3)/3) - tan(c/2 + (d*x)/2)*((5*b^6)/8 - (11*a^ 6)/8 + (15*a^2*b^4)/8 + (15*a^4*b^2)/8) + tan(c/2 + (d*x)/2)^11*((5*b^6)/8 - (11*a^6)/8 + (15*a^2*b^4)/8 + (15*a^4*b^2)/8) - tan(c/2 + (d*x)/2)^3*(( 5*a^6)/24 + (85*b^6)/24 + (85*a^2*b^4)/8 - (235*a^4*b^2)/8) + tan(c/2 + (d *x)/2)^9*((5*a^6)/24 + (85*b^6)/24 + (85*a^2*b^4)/8 - (235*a^4*b^2)/8) + t an(c/2 + (d*x)/2)^5*((15*a^6)/4 - (33*b^6)/4 + (285*a^2*b^4)/4 - (195*a^4* b^2)/4) - tan(c/2 + (d*x)/2)^7*((15*a^6)/4 - (33*b^6)/4 + (285*a^2*b^4)/4 - (195*a^4*b^2)/4) + 80*a^3*b^3*tan(c/2 + (d*x)/2)^4 + 80*a^3*b^3*tan(c/2 + (d*x)/2)^8 + 12*a^5*b*tan(c/2 + (d*x)/2)^2 + 12*a^5*b*tan(c/2 + (d*x)/2) ^10)/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + ( d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))