Integrand size = 21, antiderivative size = 117 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {5}{128} (8 A+7 C) x+\frac {5 (8 A+7 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {C \cos ^7(c+d x) \sin (c+d x)}{8 d} \]
5/128*(8*A+7*C)*x+5/128*(8*A+7*C)*cos(d*x+c)*sin(d*x+c)/d+5/192*(8*A+7*C)* cos(d*x+c)^3*sin(d*x+c)/d+1/48*(8*A+7*C)*cos(d*x+c)^5*sin(d*x+c)/d+1/8*C*c os(d*x+c)^7*sin(d*x+c)/d
Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {960 A c+840 c C+960 A d x+840 C d x+48 (15 A+14 C) \sin (2 (c+d x))+24 (6 A+7 C) \sin (4 (c+d x))+16 A \sin (6 (c+d x))+32 C \sin (6 (c+d x))+3 C \sin (8 (c+d x))}{3072 d} \]
(960*A*c + 840*c*C + 960*A*d*x + 840*C*d*x + 48*(15*A + 14*C)*Sin[2*(c + d *x)] + 24*(6*A + 7*C)*Sin[4*(c + d*x)] + 16*A*Sin[6*(c + d*x)] + 32*C*Sin[ 6*(c + d*x)] + 3*C*Sin[8*(c + d*x)])/(3072*d)
Time = 0.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3493, 3042, 3115, 3042, 3115, 3042, 3115, 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.
| \(\displaystyle \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^6 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3493 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \int \cos ^6(c+d x)dx+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{8} (8 A+7 C) \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {C \sin (c+d x) \cos ^7(c+d x)}{8 d}\) |
(C*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) + ((8*A + 7*C)*((Cos[c + d*x]^5*Sin[ c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos [c + d*x]*Sin[c + d*x])/(2*d)))/4))/6))/8
3.1.9.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 3.96 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68
| method | result | size |
| parallelrisch | \(\frac {\left (720 A +672 C \right ) \sin \left (2 d x +2 c \right )+\left (144 A +168 C \right ) \sin \left (4 d x +4 c \right )+\left (16 A +32 C \right ) \sin \left (6 d x +6 c \right )+3 C \sin \left (8 d x +8 c \right )+960 d \left (A +\frac {7 C}{8}\right ) x}{3072 d}\) | \(79\) |
| derivativedivides | \(\frac {C \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(106\) |
| default | \(\frac {C \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(106\) |
| parts | \(\frac {A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {C \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(108\) |
| risch | \(\frac {5 x A}{16}+\frac {35 C x}{128}+\frac {C \sin \left (8 d x +8 c \right )}{1024 d}+\frac {\sin \left (6 d x +6 c \right ) A}{192 d}+\frac {\sin \left (6 d x +6 c \right ) C}{96 d}+\frac {3 \sin \left (4 d x +4 c \right ) A}{64 d}+\frac {7 \sin \left (4 d x +4 c \right ) C}{128 d}+\frac {15 \sin \left (2 d x +2 c \right ) A}{64 d}+\frac {7 \sin \left (2 d x +2 c \right ) C}{32 d}\) | \(115\) |
| norman | \(\frac {\left (\frac {5 A}{16}+\frac {35 C}{128}\right ) x +\left (\frac {5 A}{2}+\frac {35 C}{16}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5 A}{2}+\frac {35 C}{16}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5 A}{16}+\frac {35 C}{128}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 A}{2}+\frac {245 C}{16}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 A}{2}+\frac {245 C}{16}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 A}{4}+\frac {245 C}{32}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 A}{4}+\frac {245 C}{32}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {175 A}{8}+\frac {1225 C}{64}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (88 A +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {\left (88 A +93 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {5 \left (136 A -217 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {5 \left (136 A -217 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {\left (488 A +91 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {\left (488 A +91 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {\left (904 A +1799 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {\left (904 A +1799 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(369\) |
1/3072*((720*A+672*C)*sin(2*d*x+2*c)+(144*A+168*C)*sin(4*d*x+4*c)+(16*A+32 *C)*sin(6*d*x+6*c)+3*C*sin(8*d*x+8*c)+960*d*(A+7/8*C)*x)/d
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (8 \, A + 7 \, C\right )} d x + {\left (48 \, C \cos \left (d x + c\right )^{7} + 8 \, {\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]
1/384*(15*(8*A + 7*C)*d*x + (48*C*cos(d*x + c)^7 + 8*(8*A + 7*C)*cos(d*x + c)^5 + 10*(8*A + 7*C)*cos(d*x + c)^3 + 15*(8*A + 7*C)*cos(d*x + c))*sin(d *x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (109) = 218\).
Time = 0.72 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.03 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {5 A x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 A x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 A x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 A x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 A \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 A \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 A \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {35 C x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 C x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 C x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 C x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 C x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 C \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 C \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 C \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 C \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((5*A*x*sin(c + d*x)**6/16 + 15*A*x*sin(c + d*x)**4*cos(c + d*x)* *2/16 + 15*A*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*A*x*cos(c + d*x)**6/ 16 + 5*A*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*A*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*A*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 35*C*x*sin(c + d*x)**8/128 + 35*C*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*C*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*C*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*C*x*cos(c + d*x)**8/128 + 35*C*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 3 85*C*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*C*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*C*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (x*(A + C*cos(c)**2)*cos(c)**6, True))
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (d x + c\right )} {\left (8 \, A + 7 \, C\right )} + \frac {15 \, {\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{7} + 55 \, {\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{5} + 73 \, {\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (88 \, A + 93 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1}}{384 \, d} \]
1/384*(15*(d*x + c)*(8*A + 7*C) + (15*(8*A + 7*C)*tan(d*x + c)^7 + 55*(8*A + 7*C)*tan(d*x + c)^5 + 73*(8*A + 7*C)*tan(d*x + c)^3 + 3*(88*A + 93*C)*t an(d*x + c))/(tan(d*x + c)^8 + 4*tan(d*x + c)^6 + 6*tan(d*x + c)^4 + 4*tan (d*x + c)^2 + 1))/d
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {5}{128} \, {\left (8 \, A + 7 \, C\right )} x + \frac {C \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (A + 2 \, C\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, A + 7 \, C\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, A + 14 \, C\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
5/128*(8*A + 7*C)*x + 1/1024*C*sin(8*d*x + 8*c)/d + 1/192*(A + 2*C)*sin(6* d*x + 6*c)/d + 1/128*(6*A + 7*C)*sin(4*d*x + 4*c)/d + 1/64*(15*A + 14*C)*s in(2*d*x + 2*c)/d
Time = 2.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=x\,\left (\frac {5\,A}{16}+\frac {35\,C}{128}\right )+\frac {\left (\frac {5\,A}{16}+\frac {35\,C}{128}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^7+\left (\frac {55\,A}{48}+\frac {385\,C}{384}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (\frac {73\,A}{48}+\frac {511\,C}{384}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {11\,A}{16}+\frac {93\,C}{128}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^8+4\,{\mathrm {tan}\left (c+d\,x\right )}^6+6\,{\mathrm {tan}\left (c+d\,x\right )}^4+4\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \]