Integrand size = 28, antiderivative size = 265 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {35 a^3 x}{128}+\frac {15}{128} a b^2 x-\frac {b^3 \cos ^6(c+d x)}{6 d}-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}+\frac {b^3 \cos ^8(c+d x)}{8 d}+\frac {35 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d} \]
35/128*a^3*x+15/128*a*b^2*x-1/6*b^3*cos(d*x+c)^6/d-3/8*a^2*b*cos(d*x+c)^8/ d+1/8*b^3*cos(d*x+c)^8/d+35/128*a^3*cos(d*x+c)*sin(d*x+c)/d+15/128*a*b^2*c os(d*x+c)*sin(d*x+c)/d+35/192*a^3*cos(d*x+c)^3*sin(d*x+c)/d+5/64*a*b^2*cos (d*x+c)^3*sin(d*x+c)/d+7/48*a^3*cos(d*x+c)^5*sin(d*x+c)/d+1/16*a*b^2*cos(d *x+c)^5*sin(d*x+c)/d+1/8*a^3*cos(d*x+c)^7*sin(d*x+c)/d-3/8*a*b^2*cos(d*x+c )^7*sin(d*x+c)/d
Time = 1.35 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {5 a \left (7 a^2+3 b^2\right ) (c+d x)}{128 d}-\frac {3 b \left (7 a^2+b^2\right ) \cos (2 (c+d x))}{128 d}-\frac {b \left (21 a^2+b^2\right ) \cos (4 (c+d x))}{256 d}-\frac {b \left (9 a^2-b^2\right ) \cos (6 (c+d x))}{384 d}-\frac {b \left (3 a^2-b^2\right ) \cos (8 (c+d x))}{1024 d}+\frac {a \left (14 a^2+3 b^2\right ) \sin (2 (c+d x))}{64 d}+\frac {a \left (7 a^2-3 b^2\right ) \sin (4 (c+d x))}{128 d}+\frac {a \left (2 a^2-3 b^2\right ) \sin (6 (c+d x))}{192 d}+\frac {a \left (a^2-3 b^2\right ) \sin (8 (c+d x))}{1024 d} \]
(5*a*(7*a^2 + 3*b^2)*(c + d*x))/(128*d) - (3*b*(7*a^2 + b^2)*Cos[2*(c + d* x)])/(128*d) - (b*(21*a^2 + b^2)*Cos[4*(c + d*x)])/(256*d) - (b*(9*a^2 - b ^2)*Cos[6*(c + d*x)])/(384*d) - (b*(3*a^2 - b^2)*Cos[8*(c + d*x)])/(1024*d ) + (a*(14*a^2 + 3*b^2)*Sin[2*(c + d*x)])/(64*d) + (a*(7*a^2 - 3*b^2)*Sin[ 4*(c + d*x)])/(128*d) + (a*(2*a^2 - 3*b^2)*Sin[6*(c + d*x)])/(192*d) + (a* (a^2 - 3*b^2)*Sin[8*(c + d*x)])/(1024*d)
Time = 0.50 (sec) , antiderivative size = 265, 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.
| \(\displaystyle \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^5 (a \cos (c+d x)+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^3 \cos ^8(c+d x)+3 a^2 b \sin (c+d x) \cos ^7(c+d x)+3 a b^2 \sin ^2(c+d x) \cos ^6(c+d x)+b^3 \sin ^3(c+d x) \cos ^5(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^3 x}{128}-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}-\frac {3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^2 x+\frac {b^3 \cos ^8(c+d x)}{8 d}-\frac {b^3 \cos ^6(c+d x)}{6 d}\) |
(35*a^3*x)/128 + (15*a*b^2*x)/128 - (b^3*Cos[c + d*x]^6)/(6*d) - (3*a^2*b* Cos[c + d*x]^8)/(8*d) + (b^3*Cos[c + d*x]^8)/(8*d) + (35*a^3*Cos[c + d*x]* Sin[c + d*x])/(128*d) + (15*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35 *a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)
3.1.57.3.1 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 = 1.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.66
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {3 a^{2} b \cos \left (d x +c \right )^{8}}{8}+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )}{d}\) | \(175\) |
| default | \(\frac {a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {3 a^{2} b \cos \left (d x +c \right )^{8}}{8}+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )}{d}\) | \(175\) |
| parts | \(\frac {a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {b^{3} \left (\frac {\cos \left (d x +c \right )^{8}}{8}-\frac {\cos \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}-\frac {3 a^{2} b \cos \left (d x +c \right )^{8}}{8 d}\) | \(175\) |
| parallelrisch | \(\frac {\left (-504 a^{2} b -72 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-252 a^{2} b -12 b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (-72 a^{2} b +8 b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (-9 a^{2} b +3 b^{3}\right ) \cos \left (8 d x +8 c \right )+\left (672 a^{3}+144 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (168 a^{3}-72 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (32 a^{3}-48 a \,b^{2}\right ) \sin \left (6 d x +6 c \right )+\left (3 a^{3}-9 a \,b^{2}\right ) \sin \left (8 d x +8 c \right )+840 a^{3} x d +360 a \,b^{2} d x +837 a^{2} b +73 b^{3}}{3072 d}\) | \(209\) |
| risch | \(\frac {35 a^{3} x}{128}+\frac {15 a \,b^{2} x}{128}-\frac {3 b \cos \left (8 d x +8 c \right ) a^{2}}{1024 d}+\frac {b^{3} \cos \left (8 d x +8 c \right )}{1024 d}+\frac {a^{3} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {3 a \sin \left (8 d x +8 c \right ) b^{2}}{1024 d}-\frac {3 b \cos \left (6 d x +6 c \right ) a^{2}}{128 d}+\frac {b^{3} \cos \left (6 d x +6 c \right )}{384 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{96 d}-\frac {a \sin \left (6 d x +6 c \right ) b^{2}}{64 d}-\frac {21 b \cos \left (4 d x +4 c \right ) a^{2}}{256 d}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{256 d}+\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{128 d}-\frac {3 a \sin \left (4 d x +4 c \right ) b^{2}}{128 d}-\frac {21 b \cos \left (2 d x +2 c \right ) a^{2}}{128 d}-\frac {3 b^{3} \cos \left (2 d x +2 c \right )}{128 d}+\frac {7 a^{3} \sin \left (2 d x +2 c \right )}{32 d}+\frac {3 a \sin \left (2 d x +2 c \right ) b^{2}}{64 d}\) | \(295\) |
| norman | \(\frac {\left (\frac {35}{128} a^{3}+\frac {15}{128} a \,b^{2}\right ) x +\left (\frac {35}{16} a^{3}+\frac {15}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {35}{16} a^{3}+\frac {15}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {35}{128} a^{3}+\frac {15}{128} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {245}{16} a^{3}+\frac {105}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {245}{16} a^{3}+\frac {105}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {245}{32} a^{3}+\frac {105}{32} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {245}{32} a^{3}+\frac {105}{32} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {1225}{64} a^{3}+\frac {525}{64} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {4 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {4 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {40 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}+\frac {6 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {6 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {2 \left (63 a^{2} b -8 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {2 \left (63 a^{2} b -8 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}+\frac {3 a \left (31 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {3 a \left (31 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}+\frac {a \left (91 a^{2}+1191 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d}-\frac {a \left (91 a^{2}+1191 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}-\frac {5 a \left (217 a^{2}-1059 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{192 d}+\frac {5 a \left (217 a^{2}-1059 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192 d}+\frac {a \left (1799 a^{2}-2685 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 d}-\frac {a \left (1799 a^{2}-2685 b^{2}\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}}\) | \(607\) |
1/d*(a^3*(1/8*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos( d*x+c))*sin(d*x+c)+35/128*d*x+35/128*c)-3/8*a^2*b*cos(d*x+c)^8+3*a*b^2*(-1 /8*cos(d*x+c)^7*sin(d*x+c)+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d* x+c))*sin(d*x+c)+5/128*d*x+5/128*c)+b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/ 24*cos(d*x+c)^6))
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.57 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=-\frac {64 \, b^{3} \cos \left (d x + c\right )^{6} + 48 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{8} - 15 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} d x - {\left (48 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]
-1/384*(64*b^3*cos(d*x + c)^6 + 48*(3*a^2*b - b^3)*cos(d*x + c)^8 - 15*(7* a^3 + 3*a*b^2)*d*x - (48*(a^3 - 3*a*b^2)*cos(d*x + c)^7 + 8*(7*a^3 + 3*a*b ^2)*cos(d*x + c)^5 + 10*(7*a^3 + 3*a*b^2)*cos(d*x + c)^3 + 15*(7*a^3 + 3*a *b^2)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (257) = 514\).
Time = 0.78 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.01 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {35 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 a^{2} b \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {15 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {15 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {73 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {15 a b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {b^{3} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {b^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{3} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((35*a**3*x*sin(c + d*x)**8/128 + 35*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*a**3*x* sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**3*x*cos(c + d*x)**8/128 + 35*a* *3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*a **3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 3*a**2*b*cos(c + d*x)**8/(8*d) + 15*a*b**2*x*sin(c + d*x)**8/128 + 15*a*b**2*x*sin(c + d*x)**6*cos(c + d* x)**2/32 + 45*a*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a*b**2*x*si n(c + d*x)**2*cos(c + d*x)**6/32 + 15*a*b**2*x*cos(c + d*x)**8/128 + 15*a* b**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*a*b**2*sin(c + d*x)**5*cos( c + d*x)**3/(128*d) + 73*a*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 15*a*b**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) + b**3*sin(c + d*x)**8/(24* d) + b**3*sin(c + d*x)**6*cos(c + d*x)**2/(6*d) + b**3*sin(c + d*x)**4*cos (c + d*x)**4/(4*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**3*cos(c)**5, True ))
Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.62 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=-\frac {1152 \, a^{2} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 3 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 128 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} b^{3}}{3072 \, d} \]
-1/3072*(1152*a^2*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 ^3 - 3*(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*b^2 - 128*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6*sin (d*x + c)^4)*b^3)/d
Time = 0.42 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.82 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {5}{128} \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (9 \, a^{2} b - b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (21 \, a^{2} b + b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, {\left (7 \, a^{2} b + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (7 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (14 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
5/128*(7*a^3 + 3*a*b^2)*x - 1/1024*(3*a^2*b - b^3)*cos(8*d*x + 8*c)/d - 1/ 384*(9*a^2*b - b^3)*cos(6*d*x + 6*c)/d - 1/256*(21*a^2*b + b^3)*cos(4*d*x + 4*c)/d - 3/128*(7*a^2*b + b^3)*cos(2*d*x + 2*c)/d + 1/1024*(a^3 - 3*a*b^ 2)*sin(8*d*x + 8*c)/d + 1/192*(2*a^3 - 3*a*b^2)*sin(6*d*x + 6*c)/d + 1/128 *(7*a^3 - 3*a*b^2)*sin(4*d*x + 4*c)/d + 1/64*(14*a^3 + 3*a*b^2)*sin(2*d*x + 2*c)/d
Time = 24.22 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.97 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,a\,b^2}{64}-\frac {93\,a^3}{64}\right )+\frac {40\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {15\,a\,b^2}{64}-\frac {93\,a^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {91\,a^3}{192}+\frac {397\,a\,b^2}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {91\,a^3}{192}+\frac {397\,a\,b^2}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {895\,a\,b^2}{64}-\frac {1799\,a^3}{192}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {895\,a\,b^2}{64}-\frac {1799\,a^3}{192}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1765\,a\,b^2}{64}-\frac {1085\,a^3}{192}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1765\,a\,b^2}{64}-\frac {1085\,a^3}{192}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (42\,a^2\,b-\frac {16\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (42\,a^2\,b-\frac {16\,b^3}{3}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {5\,a\,\mathrm {atan}\left (\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,a^2+3\,b^2\right )}{64\,\left (\frac {35\,a^3}{64}+\frac {15\,a\,b^2}{64}\right )}\right )\,\left (7\,a^2+3\,b^2\right )}{64\,d}-\frac {5\,a\,\left (7\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d} \]
(4*b^3*tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)*((15*a*b^2)/64 - (93*a^3) /64) + (40*b^3*tan(c/2 + (d*x)/2)^8)/3 + 4*b^3*tan(c/2 + (d*x)/2)^12 + tan (c/2 + (d*x)/2)^15*((15*a*b^2)/64 - (93*a^3)/64) + tan(c/2 + (d*x)/2)^3*(( 397*a*b^2)/64 + (91*a^3)/192) - tan(c/2 + (d*x)/2)^13*((397*a*b^2)/64 + (9 1*a^3)/192) - tan(c/2 + (d*x)/2)^5*((895*a*b^2)/64 - (1799*a^3)/192) + tan (c/2 + (d*x)/2)^11*((895*a*b^2)/64 - (1799*a^3)/192) + tan(c/2 + (d*x)/2)^ 7*((1765*a*b^2)/64 - (1085*a^3)/192) - tan(c/2 + (d*x)/2)^9*((1765*a*b^2)/ 64 - (1085*a^3)/192) + tan(c/2 + (d*x)/2)^6*(42*a^2*b - (16*b^3)/3) + tan( c/2 + (d*x)/2)^10*(42*a^2*b - (16*b^3)/3) + 6*a^2*b*tan(c/2 + (d*x)/2)^2 + 6*a^2*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*tan(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((5*a*tan(c/2 + (d*x)/2)*(7*a^2 + 3*b^ 2))/(64*((15*a*b^2)/64 + (35*a^3)/64)))*(7*a^2 + 3*b^2))/(64*d) - (5*a*(7* a^2 + 3*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64*d)