Integrand size = 28, antiderivative size = 426 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {35 a^5 x}{128}+\frac {25}{64} a^3 b^2 x+\frac {15}{128} a b^4 x-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d} \] Output:
35/128*a^5*x+25/64*a^3*b^2*x+15/128*a*b^4*x-5/3*a^2*b^3*cos(d*x+c)^6/d-5/8 *a^4*b*cos(d*x+c)^8/d+5/4*a^2*b^3*cos(d*x+c)^8/d+35/128*a^5*cos(d*x+c)*sin (d*x+c)/d+25/64*a^3*b^2*cos(d*x+c)*sin(d*x+c)/d+15/128*a*b^4*cos(d*x+c)*si n(d*x+c)/d+35/192*a^5*cos(d*x+c)^3*sin(d*x+c)/d+25/96*a^3*b^2*cos(d*x+c)^3 *sin(d*x+c)/d+5/64*a*b^4*cos(d*x+c)^3*sin(d*x+c)/d+7/48*a^5*cos(d*x+c)^5*s in(d*x+c)/d+5/24*a^3*b^2*cos(d*x+c)^5*sin(d*x+c)/d-5/16*a*b^4*cos(d*x+c)^5 *sin(d*x+c)/d+1/8*a^5*cos(d*x+c)^7*sin(d*x+c)/d-5/4*a^3*b^2*cos(d*x+c)^7*s in(d*x+c)/d-5/8*a*b^4*cos(d*x+c)^5*sin(d*x+c)^3/d+1/6*b^5*sin(d*x+c)^6/d-1 /8*b^5*sin(d*x+c)^8/d
Result contains complex when optimal does not.
Time = 2.55 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.61 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {120 a (a-i b) (a+i b) \left (7 a^2+3 b^2\right ) (c+d x)-24 b \left (35 a^4+30 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+12 b \left (-35 a^4-10 a^2 b^2+b^4\right ) \cos (4 (c+d x))+8 b \left (-15 a^4+10 a^2 b^2+b^4\right ) \cos (6 (c+d x))-3 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (8 (c+d x))+96 a^3 \left (7 a^2+5 b^2\right ) \sin (2 (c+d x))+24 a \left (7 a^4-10 a^2 b^2-5 b^4\right ) \sin (4 (c+d x))+32 a^3 \left (a^2-5 b^2\right ) \sin (6 (c+d x))+3 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (8 (c+d x))}{3072 d} \] Input:
Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]
Output:
(120*a*(a - I*b)*(a + I*b)*(7*a^2 + 3*b^2)*(c + d*x) - 24*b*(35*a^4 + 30*a ^2*b^2 + 3*b^4)*Cos[2*(c + d*x)] + 12*b*(-35*a^4 - 10*a^2*b^2 + b^4)*Cos[4 *(c + d*x)] + 8*b*(-15*a^4 + 10*a^2*b^2 + b^4)*Cos[6*(c + d*x)] - 3*b*(5*a ^4 - 10*a^2*b^2 + b^4)*Cos[8*(c + d*x)] + 96*a^3*(7*a^2 + 5*b^2)*Sin[2*(c + d*x)] + 24*a*(7*a^4 - 10*a^2*b^2 - 5*b^4)*Sin[4*(c + d*x)] + 32*a^3*(a^2 - 5*b^2)*Sin[6*(c + d*x)] + 3*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[8*(c + d*x )])/(3072*d)
Time = 0.69 (sec) , antiderivative size = 426, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^5dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^5 \cos ^8(c+d x)+5 a^4 b \sin (c+d x) \cos ^7(c+d x)+10 a^3 b^2 \sin ^2(c+d x) \cos ^6(c+d x)+10 a^2 b^3 \sin ^3(c+d x) \cos ^5(c+d x)+5 a b^4 \sin ^4(c+d x) \cos ^4(c+d x)+b^5 \sin ^5(c+d x) \cos ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^5 x}{128}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}-\frac {5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {25}{64} a^3 b^2 x+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^4 x-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}\) |
Input:
Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]
Output:
(35*a^5*x)/128 + (25*a^3*b^2*x)/64 + (15*a*b^4*x)/128 - (5*a^2*b^3*Cos[c + d*x]^6)/(3*d) - (5*a^4*b*Cos[c + d*x]^8)/(8*d) + (5*a^2*b^3*Cos[c + d*x]^ 8)/(4*d) + (35*a^5*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (25*a^3*b^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^5*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*b^2*Cos[c + d*x] ^3*Sin[c + d*x])/(96*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + ( 7*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (5*a^3*b^2*Cos[c + d*x]^5*Sin[ c + d*x])/(24*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^5*Cos [c + d*x]^7*Sin[c + d*x])/(8*d) - (5*a^3*b^2*Cos[c + d*x]^7*Sin[c + d*x])/ (4*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d) + (b^5*Sin[c + d*x]^ 6)/(6*d) - (b^5*Sin[c + d*x]^8)/(8*d)
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 = 7.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(\frac {24 \left (-35 a^{4} b -30 a^{2} b^{3}-3 b^{5}\right ) \cos \left (2 d x +2 c \right )+12 \left (-35 a^{4} b -10 a^{2} b^{3}+b^{5}\right ) \cos \left (4 d x +4 c \right )+8 \left (-15 a^{4} b +10 a^{2} b^{3}+b^{5}\right ) \cos \left (6 d x +6 c \right )+3 \left (-5 a^{4} b +10 a^{2} b^{3}-b^{5}\right ) \cos \left (8 d x +8 c \right )+24 \left (7 a^{5}-10 a^{3} b^{2}-5 b^{4} a \right ) \sin \left (4 d x +4 c \right )+3 \left (a^{5}-10 a^{3} b^{2}+5 b^{4} a \right ) \sin \left (8 d x +8 c \right )+96 \left (7 a^{5}+5 a^{3} b^{2}\right ) \sin \left (2 d x +2 c \right )+32 \left (a^{5}-5 a^{3} b^{2}\right ) \sin \left (6 d x +6 c \right )+840 a^{5} d x +1200 a^{3} b^{2} d x +360 a \,b^{4} d x +1395 a^{4} b +730 a^{2} b^{3}+55 b^{5}}{3072 d}\) | \(279\) |
| parts | \(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {b^{5} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}+\frac {\sin \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {10 a^{3} b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}+\frac {5 b^{4} a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}-\frac {5 a^{4} b \cos \left (d x +c \right )^{8}}{8 d}+\frac {10 a^{2} b^{3} \left (\frac {\cos \left (d x +c \right )^{8}}{8}-\frac {\cos \left (d x +c \right )^{6}}{6}\right )}{d}\) | \(285\) |
| derivativedivides | \(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {5 a^{4} b \cos \left (d x +c \right )^{8}}{8}+10 a^{3} b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+10 a^{2} b^{3} \left (-\frac {\cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{2}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )+5 b^{4} a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{4}}{8}-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{12}-\frac {\cos \left (d x +c \right )^{4}}{24}\right )}{d}\) | \(305\) |
| default | \(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {5 a^{4} b \cos \left (d x +c \right )^{8}}{8}+10 a^{3} b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+10 a^{2} b^{3} \left (-\frac {\cos \left (d x +c \right )^{6} \sin \left (d x +c \right )^{2}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )+5 b^{4} a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{4}}{8}-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{12}-\frac {\cos \left (d x +c \right )^{4}}{24}\right )}{d}\) | \(305\) |
| risch | \(-\frac {b^{5} \cos \left (8 d x +8 c \right )}{1024 d}+\frac {b^{5} \cos \left (4 d x +4 c \right )}{256 d}+\frac {35 a^{5} x}{128}+\frac {25 a^{3} b^{2} x}{64}+\frac {15 a \,b^{4} x}{128}-\frac {5 a^{4} b \cos \left (8 d x +8 c \right )}{1024 d}+\frac {5 a^{2} b^{3} \cos \left (8 d x +8 c \right )}{512 d}+\frac {a^{5} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {b^{5} \cos \left (6 d x +6 c \right )}{384 d}+\frac {a^{5} \sin \left (6 d x +6 c \right )}{96 d}+\frac {7 a^{5} \sin \left (4 d x +4 c \right )}{128 d}-\frac {3 b^{5} \cos \left (2 d x +2 c \right )}{128 d}+\frac {7 a^{5} \sin \left (2 d x +2 c \right )}{32 d}-\frac {5 a^{3} \sin \left (8 d x +8 c \right ) b^{2}}{512 d}+\frac {5 a \sin \left (8 d x +8 c \right ) b^{4}}{1024 d}-\frac {5 b \cos \left (6 d x +6 c \right ) a^{4}}{128 d}+\frac {5 b^{3} \cos \left (6 d x +6 c \right ) a^{2}}{192 d}-\frac {5 a^{3} \sin \left (6 d x +6 c \right ) b^{2}}{96 d}-\frac {35 a^{4} b \cos \left (4 d x +4 c \right )}{256 d}-\frac {5 a^{2} b^{3} \cos \left (4 d x +4 c \right )}{128 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right ) b^{2}}{64 d}-\frac {5 a \sin \left (4 d x +4 c \right ) b^{4}}{128 d}-\frac {35 b \cos \left (2 d x +2 c \right ) a^{4}}{128 d}-\frac {15 b^{3} \cos \left (2 d x +2 c \right ) a^{2}}{64 d}+\frac {5 a^{3} \sin \left (2 d x +2 c \right ) b^{2}}{32 d}\) | \(428\) |
| norman | \(\frac {\left (\frac {35}{128} a^{5}+\frac {25}{64} a^{3} b^{2}+\frac {15}{128} b^{4} a \right ) x +\left (\frac {35}{16} a^{5}+\frac {25}{8} a^{3} b^{2}+\frac {15}{16} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {35}{16} a^{5}+\frac {25}{8} a^{3} b^{2}+\frac {15}{16} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {35}{128} a^{5}+\frac {25}{64} a^{3} b^{2}+\frac {15}{128} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {245}{16} a^{5}+\frac {175}{8} a^{3} b^{2}+\frac {105}{16} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {245}{16} a^{5}+\frac {175}{8} a^{3} b^{2}+\frac {105}{16} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {245}{32} a^{5}+\frac {175}{16} a^{3} b^{2}+\frac {105}{32} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {245}{32} a^{5}+\frac {175}{16} a^{3} b^{2}+\frac {105}{32} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {1225}{64} a^{5}+\frac {875}{32} a^{3} b^{2}+\frac {525}{64} b^{4} a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {40 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {40 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {10 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {10 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {2 \left (200 a^{2} b^{3}-16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}+\frac {2 \left (105 a^{4} b -80 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {2 \left (105 a^{4} b -80 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}+\frac {a \left (91 a^{4}+3970 a^{2} b^{2}-345 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d}-\frac {a \left (91 a^{4}+3970 a^{2} b^{2}-345 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}+\frac {a \left (93 a^{4}-50 a^{2} b^{2}-15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {a \left (93 a^{4}-50 a^{2} b^{2}-15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}-\frac {5 a \left (217 a^{4}-3530 a^{2} b^{2}+2013 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{192 d}+\frac {5 a \left (217 a^{4}-3530 a^{2} b^{2}+2013 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192 d}+\frac {a \left (1799 a^{4}-8950 a^{2} b^{2}+4995 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 d}-\frac {a \left (1799 a^{4}-8950 a^{2} b^{2}+4995 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{192 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(776\) |
| orering | \(\text {Expression too large to display}\) | \(16696\) |
Input:
int(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)
Output:
1/3072*(24*(-35*a^4*b-30*a^2*b^3-3*b^5)*cos(2*d*x+2*c)+12*(-35*a^4*b-10*a^ 2*b^3+b^5)*cos(4*d*x+4*c)+8*(-15*a^4*b+10*a^2*b^3+b^5)*cos(6*d*x+6*c)+3*(- 5*a^4*b+10*a^2*b^3-b^5)*cos(8*d*x+8*c)+24*(7*a^5-10*a^3*b^2-5*a*b^4)*sin(4 *d*x+4*c)+3*(a^5-10*a^3*b^2+5*a*b^4)*sin(8*d*x+8*c)+96*(7*a^5+5*a^3*b^2)*s in(2*d*x+2*c)+32*(a^5-5*a^3*b^2)*sin(6*d*x+6*c)+840*a^5*d*x+1200*a^3*b^2*d *x+360*a*b^4*d*x+1395*a^4*b+730*a^2*b^3+55*b^5)/d
Time = 0.09 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.52 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {96 \, b^{5} \cos \left (d x + c\right )^{4} + 48 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{8} + 128 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{6} - 15 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x - {\left (48 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} - 45 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \] Input:
integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas" )
Output:
-1/384*(96*b^5*cos(d*x + c)^4 + 48*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^8 + 128*(5*a^2*b^3 - b^5)*cos(d*x + c)^6 - 15*(7*a^5 + 10*a^3*b^2 + 3*a *b^4)*d*x - (48*(a^5 - 10*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^7 + 8*(7*a^5 + 1 0*a^3*b^2 - 45*a*b^4)*cos(d*x + c)^5 + 10*(7*a^5 + 10*a^3*b^2 + 3*a*b^4)*c os(d*x + c)^3 + 15*(7*a^5 + 10*a^3*b^2 + 3*a*b^4)*cos(d*x + c))*sin(d*x + c))/d
Time = 0.85 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.94 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)**3*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)
Output:
Piecewise((35*a**5*x*sin(c + d*x)**8/128 + 35*a**5*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**5*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*a**5*x* sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**5*x*cos(c + d*x)**8/128 + 35*a* *5*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**5*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**5*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*a **5*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 5*a**4*b*cos(c + d*x)**8/(8*d) + 25*a**3*b**2*x*sin(c + d*x)**8/64 + 25*a**3*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 75*a**3*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 25*a** 3*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 25*a**3*b**2*x*cos(c + d*x)* *8/64 + 25*a**3*b**2*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 275*a**3*b**2*s in(c + d*x)**5*cos(c + d*x)**3/(192*d) + 365*a**3*b**2*sin(c + d*x)**3*cos (c + d*x)**5/(192*d) - 25*a**3*b**2*sin(c + d*x)*cos(c + d*x)**7/(64*d) + 5*a**2*b**3*sin(c + d*x)**8/(12*d) + 5*a**2*b**3*sin(c + d*x)**6*cos(c + d *x)**2/(3*d) + 5*a**2*b**3*sin(c + d*x)**4*cos(c + d*x)**4/(2*d) + 15*a*b* *4*x*sin(c + d*x)**8/128 + 15*a*b**4*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a*b**4*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a*b**4*x*sin(c + d*x )**2*cos(c + d*x)**6/32 + 15*a*b**4*x*cos(c + d*x)**8/128 + 15*a*b**4*sin( c + d*x)**7*cos(c + d*x)/(128*d) + 55*a*b**4*sin(c + d*x)**5*cos(c + d*x)* *3/(128*d) - 55*a*b**4*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 15*a*b**4 *sin(c + d*x)*cos(c + d*x)**7/(128*d) + b**5*sin(c + d*x)**8/(24*d) + b...
Time = 0.04 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.54 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {1920 \, a^{4} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} - 10 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} - 1280 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} + 128 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{3072 \, d} \] Input:
integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima" )
Output:
-1/3072*(1920*a^4*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 ^5 - 10*(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^3*b^2 - 1280*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6 *sin(d*x + c)^4)*a^2*b^3 - 15*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4* d*x + 4*c))*a*b^4 + 128*(3*sin(d*x + c)^8 - 4*sin(d*x + c)^6)*b^5)/d
Time = 0.35 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.65 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {5}{128} \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} x - \frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (15 \, a^{4} b - 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (35 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (35 \, a^{4} b + 30 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{5} - 5 \, a^{3} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {{\left (7 \, a^{5} - 10 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (7 \, a^{5} + 5 \, a^{3} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \] Input:
integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")
Output:
5/128*(7*a^5 + 10*a^3*b^2 + 3*a*b^4)*x - 1/1024*(5*a^4*b - 10*a^2*b^3 + b^ 5)*cos(8*d*x + 8*c)/d - 1/384*(15*a^4*b - 10*a^2*b^3 - b^5)*cos(6*d*x + 6* c)/d - 1/256*(35*a^4*b + 10*a^2*b^3 - b^5)*cos(4*d*x + 4*c)/d - 1/128*(35* a^4*b + 30*a^2*b^3 + 3*b^5)*cos(2*d*x + 2*c)/d + 1/1024*(a^5 - 10*a^3*b^2 + 5*a*b^4)*sin(8*d*x + 8*c)/d + 1/96*(a^5 - 5*a^3*b^2)*sin(6*d*x + 6*c)/d + 1/128*(7*a^5 - 10*a^3*b^2 - 5*a*b^4)*sin(4*d*x + 4*c)/d + 1/32*(7*a^5 + 5*a^3*b^2)*sin(2*d*x + 2*c)/d
Time = 18.18 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.53 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^3*(a*cos(c + d*x) + b*sin(c + d*x))^5,x)
Output:
(tan(c/2 + (d*x)/2)^15*((15*a*b^4)/64 - (93*a^5)/64 + (25*a^3*b^2)/32) - t an(c/2 + (d*x)/2)*((15*a*b^4)/64 - (93*a^5)/64 + (25*a^3*b^2)/32) + tan(c/ 2 + (d*x)/2)^3*((91*a^5)/192 - (115*a*b^4)/64 + (1985*a^3*b^2)/96) - tan(c /2 + (d*x)/2)^13*((91*a^5)/192 - (115*a*b^4)/64 + (1985*a^3*b^2)/96) + tan (c/2 + (d*x)/2)^5*((1665*a*b^4)/64 + (1799*a^5)/192 - (4475*a^3*b^2)/96) - tan(c/2 + (d*x)/2)^11*((1665*a*b^4)/64 + (1799*a^5)/192 - (4475*a^3*b^2)/ 96) - tan(c/2 + (d*x)/2)^7*((3355*a*b^4)/64 + (1085*a^5)/192 - (8825*a^3*b ^2)/96) + tan(c/2 + (d*x)/2)^9*((3355*a*b^4)/64 + (1085*a^5)/192 - (8825*a ^3*b^2)/96) + tan(c/2 + (d*x)/2)^6*(70*a^4*b + (32*b^5)/3 - (160*a^2*b^3)/ 3) + tan(c/2 + (d*x)/2)^10*(70*a^4*b + (32*b^5)/3 - (160*a^2*b^3)/3) - tan (c/2 + (d*x)/2)^8*((32*b^5)/3 - (400*a^2*b^3)/3) + 40*a^2*b^3*tan(c/2 + (d *x)/2)^4 + 40*a^2*b^3*tan(c/2 + (d*x)/2)^12 + 10*a^4*b*tan(c/2 + (d*x)/2)^ 2 + 10*a^4*b*tan(c/2 + (d*x)/2)^14)/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*tan(c/ 2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*ta n(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) - (5*a*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*( 7*a^4 + 3*b^4 + 10*a^2*b^2))/(64*d) + (5*a*atan((5*a*tan(c/2 + (d*x)/2)*(7 *a^2 + 3*b^2)*(a^2 + b^2))/(64*((15*a*b^4)/64 + (35*a^5)/64 + (25*a^3*b^2) /32)))*(7*a^2 + 3*b^2)*(a^2 + b^2))/(64*d)
Time = 0.16 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a^{5}+480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a^{3} b^{2}-240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a \,b^{4}+200 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{5}-1360 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{3} b^{2}+360 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a \,b^{4}-326 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{5}+1180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b^{2}-30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{4}+279 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{5}-150 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b^{2}-45 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{4}-240 \sin \left (d x +c \right )^{8} a^{4} b +480 \sin \left (d x +c \right )^{8} a^{2} b^{3}-48 \sin \left (d x +c \right )^{8} b^{5}+960 \sin \left (d x +c \right )^{6} a^{4} b -1280 \sin \left (d x +c \right )^{6} a^{2} b^{3}+64 \sin \left (d x +c \right )^{6} b^{5}-1440 \sin \left (d x +c \right )^{4} a^{4} b +960 \sin \left (d x +c \right )^{4} a^{2} b^{3}+960 \sin \left (d x +c \right )^{2} a^{4} b +105 a^{5} d x +150 a^{3} b^{2} d x +45 a \,b^{4} d x}{384 d} \] Input:
int(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)
Output:
( - 48*cos(c + d*x)*sin(c + d*x)**7*a**5 + 480*cos(c + d*x)*sin(c + d*x)** 7*a**3*b**2 - 240*cos(c + d*x)*sin(c + d*x)**7*a*b**4 + 200*cos(c + d*x)*s in(c + d*x)**5*a**5 - 1360*cos(c + d*x)*sin(c + d*x)**5*a**3*b**2 + 360*co s(c + d*x)*sin(c + d*x)**5*a*b**4 - 326*cos(c + d*x)*sin(c + d*x)**3*a**5 + 1180*cos(c + d*x)*sin(c + d*x)**3*a**3*b**2 - 30*cos(c + d*x)*sin(c + d* x)**3*a*b**4 + 279*cos(c + d*x)*sin(c + d*x)*a**5 - 150*cos(c + d*x)*sin(c + d*x)*a**3*b**2 - 45*cos(c + d*x)*sin(c + d*x)*a*b**4 - 240*sin(c + d*x) **8*a**4*b + 480*sin(c + d*x)**8*a**2*b**3 - 48*sin(c + d*x)**8*b**5 + 960 *sin(c + d*x)**6*a**4*b - 1280*sin(c + d*x)**6*a**2*b**3 + 64*sin(c + d*x) **6*b**5 - 1440*sin(c + d*x)**4*a**4*b + 960*sin(c + d*x)**4*a**2*b**3 + 9 60*sin(c + d*x)**2*a**4*b + 105*a**5*d*x + 150*a**3*b**2*d*x + 45*a*b**4*d *x)/(384*d)