Integrand size = 19, antiderivative size = 150 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\frac {35 b x}{128}+\frac {a \sin (c+d x)}{d}+\frac {35 b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d} \] Output:
35/128*b*x+a*sin(d*x+c)/d+35/128*b*cos(d*x+c)*sin(d*x+c)/d+35/192*b*cos(d* x+c)^3*sin(d*x+c)/d+7/48*b*cos(d*x+c)^5*sin(d*x+c)/d+1/8*b*cos(d*x+c)^7*si n(d*x+c)/d-a*sin(d*x+c)^3/d+3/5*a*sin(d*x+c)^5/d-1/7*a*sin(d*x+c)^7/d
Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.90 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\frac {35 b (c+d x)}{128 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d}+\frac {7 b \sin (2 (c+d x))}{32 d}+\frac {7 b \sin (4 (c+d x))}{128 d}+\frac {b \sin (6 (c+d x))}{96 d}+\frac {b \sin (8 (c+d x))}{1024 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + b*Cos[c + d*x]),x]
Output:
(35*b*(c + d*x))/(128*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3* a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d) + (7*b*Sin[2*(c + d*x)] )/(32*d) + (7*b*Sin[4*(c + d*x)])/(128*d) + (b*Sin[6*(c + d*x)])/(96*d) + (b*Sin[8*(c + d*x)])/(1024*d)
Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {3042, 3227, 3042, 3113, 2009, 3115, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^7 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle a \int \cos ^7(c+d x)dx+b \int \cos ^8(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^7dx+b \int \sin \left (c+d x+\frac {\pi }{2}\right )^8dx\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^8dx-\frac {a \int \left (-\sin ^6(c+d x)+3 \sin ^4(c+d x)-3 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^8dx-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {7}{8} \int \cos ^6(c+d x)dx+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b \left (\frac {7}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {7}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b \left (\frac {7}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )+\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle b \left (\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )\right )-\frac {a \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^7*(a + b*Cos[c + d*x]),x]
Output:
-((a*(-Sin[c + d*x] + Sin[c + d*x]^3 - (3*Sin[c + d*x]^5)/5 + Sin[c + d*x] ^7/7))/d) + b*((Cos[c + d*x]^7*Sin[c + d*x])/(8*d) + (7*((Cos[c + d*x]^5*S in[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)
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Time = 19.65 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(\frac {b \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(100\) |
| default | \(\frac {b \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(100\) |
| parts | \(\frac {a \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {b \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(102\) |
| parallelrisch | \(\frac {29400 d x b +105 b \sin \left (8 d x +8 c \right )+240 a \sin \left (7 d x +7 c \right )+1120 b \sin \left (6 d x +6 c \right )+2352 a \sin \left (5 d x +5 c \right )+5880 b \sin \left (4 d x +4 c \right )+11760 a \sin \left (3 d x +3 c \right )+23520 b \sin \left (2 d x +2 c \right )+58800 a \sin \left (d x +c \right )}{107520 d}\) | \(105\) |
| risch | \(\frac {35 b x}{128}+\frac {35 a \sin \left (d x +c \right )}{64 d}+\frac {b \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a \sin \left (7 d x +7 c \right )}{448 d}+\frac {b \sin \left (6 d x +6 c \right )}{96 d}+\frac {7 a \sin \left (5 d x +5 c \right )}{320 d}+\frac {7 b \sin \left (4 d x +4 c \right )}{128 d}+\frac {7 a \sin \left (3 d x +3 c \right )}{64 d}+\frac {7 b \sin \left (2 d x +2 c \right )}{32 d}\) | \(123\) |
| norman | \(\frac {\frac {35 b x}{128}+\frac {35 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {245 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32}+\frac {245 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}+\frac {1225 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64}+\frac {245 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}+\frac {245 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32}+\frac {35 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}+\frac {35 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128}+\frac {\left (128 a -93 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}+\frac {\left (128 a +93 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {\left (1152 a -91 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}+\frac {\left (1152 a +91 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d}+\frac {\left (20352 a -8995 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{960 d}+\frac {\left (20352 a +8995 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{960 d}+\frac {\left (196992 a -37975 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6720 d}+\frac {\left (196992 a +37975 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6720 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(324\) |
| orering | \(\text {Expression too large to display}\) | \(2885\) |
Input:
int(cos(d*x+c)^7*(a+cos(d*x+c)*b),x,method=_RETURNVERBOSE)
Output:
1/d*(b*(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)+1/7*a*(16/5+cos(d*x+c)^6+6/5*cos(d*x +c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.65 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\frac {3675 \, b d x + {\left (1680 \, b \cos \left (d x + c\right )^{7} + 1920 \, a \cos \left (d x + c\right )^{6} + 1960 \, b \cos \left (d x + c\right )^{5} + 2304 \, a \cos \left (d x + c\right )^{4} + 2450 \, b \cos \left (d x + c\right )^{3} + 3072 \, a \cos \left (d x + c\right )^{2} + 3675 \, b \cos \left (d x + c\right ) + 6144 \, a\right )} \sin \left (d x + c\right )}{13440 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*cos(d*x+c)),x, algorithm="fricas")
Output:
1/13440*(3675*b*d*x + (1680*b*cos(d*x + c)^7 + 1920*a*cos(d*x + c)^6 + 196 0*b*cos(d*x + c)^5 + 2304*a*cos(d*x + c)^4 + 2450*b*cos(d*x + c)^3 + 3072* a*cos(d*x + c)^2 + 3675*b*cos(d*x + c) + 6144*a)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (141) = 282\).
Time = 0.65 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.91 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {35 b x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 b x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 b x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 b x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 b x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 b \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 b \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 b \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 b \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**7*(a+b*cos(d*x+c)),x)
Output:
Piecewise((16*a*sin(c + d*x)**7/(35*d) + 8*a*sin(c + d*x)**5*cos(c + d*x)* *2/(5*d) + 2*a*sin(c + d*x)**3*cos(c + d*x)**4/d + a*sin(c + d*x)*cos(c + d*x)**6/d + 35*b*x*sin(c + d*x)**8/128 + 35*b*x*sin(c + d*x)**6*cos(c + d* x)**2/32 + 105*b*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*b*x*sin(c + d*x )**2*cos(c + d*x)**6/32 + 35*b*x*cos(c + d*x)**8/128 + 35*b*sin(c + d*x)** 7*cos(c + d*x)/(128*d) + 385*b*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 5 11*b*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*b*sin(c + d*x)*cos(c + d *x)**7/(128*d), Ne(d, 0)), (x*(a + b*cos(c))*cos(c)**7, True))
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=-\frac {3072 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a + 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{107520 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*cos(d*x+c)),x, algorithm="maxima")
Output:
-1/107520*(3072*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a + 35*(128*sin(2*d*x + 2*c)^3 - 840*d*x - 840*c - 3*si n(8*d*x + 8*c) - 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*b)/d
Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\frac {35}{128} \, b x + \frac {b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {7 \, b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {35 \, a \sin \left (d x + c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*cos(d*x+c)),x, algorithm="giac")
Output:
35/128*b*x + 1/1024*b*sin(8*d*x + 8*c)/d + 1/448*a*sin(7*d*x + 7*c)/d + 1/ 96*b*sin(6*d*x + 6*c)/d + 7/320*a*sin(5*d*x + 5*c)/d + 7/128*b*sin(4*d*x + 4*c)/d + 7/64*a*sin(3*d*x + 3*c)/d + 7/32*b*sin(2*d*x + 2*c)/d + 35/64*a* sin(d*x + c)/d
Time = 42.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.17 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\frac {35\,b\,x}{128}+\frac {\left (2\,a-\frac {93\,b}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (6\,a-\frac {91\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {106\,a}{5}-\frac {1799\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1026\,a}{35}+\frac {1085\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1026\,a}{35}-\frac {1085\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {106\,a}{5}+\frac {1799\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (6\,a+\frac {91\,b}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a+\frac {93\,b}{64}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \] Input:
int(cos(c + d*x)^7*(a + b*cos(c + d*x)),x)
Output:
(35*b*x)/128 + (tan(c/2 + (d*x)/2)*(2*a + (93*b)/64) + tan(c/2 + (d*x)/2)^ 15*(2*a - (93*b)/64) + tan(c/2 + (d*x)/2)^3*(6*a + (91*b)/192) + tan(c/2 + (d*x)/2)^13*(6*a - (91*b)/192) + tan(c/2 + (d*x)/2)^5*((106*a)/5 + (1799* b)/192) + tan(c/2 + (d*x)/2)^11*((106*a)/5 - (1799*b)/192) + tan(c/2 + (d* x)/2)^7*((1026*a)/35 - (1085*b)/192) + tan(c/2 + (d*x)/2)^9*((1026*a)/35 + (1085*b)/192))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^8)
Time = 0.16 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.79 \[ \int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx=\frac {-1680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b +7000 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b -11410 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +9765 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -1920 \sin \left (d x +c \right )^{7} a +8064 \sin \left (d x +c \right )^{5} a -13440 \sin \left (d x +c \right )^{3} a +13440 \sin \left (d x +c \right ) a +3675 b d x}{13440 d} \] Input:
int(cos(d*x+c)^7*(a+b*cos(d*x+c)),x)
Output:
( - 1680*cos(c + d*x)*sin(c + d*x)**7*b + 7000*cos(c + d*x)*sin(c + d*x)** 5*b - 11410*cos(c + d*x)*sin(c + d*x)**3*b + 9765*cos(c + d*x)*sin(c + d*x )*b - 1920*sin(c + d*x)**7*a + 8064*sin(c + d*x)**5*a - 13440*sin(c + d*x) **3*a + 13440*sin(c + d*x)*a + 3675*b*d*x)/(13440*d)