Integrand size = 19, antiderivative size = 128 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\frac {5 a x}{16}+\frac {b \sin (c+d x)}{d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {b \sin ^3(c+d x)}{d}+\frac {3 b \sin ^5(c+d x)}{5 d}-\frac {b \sin ^7(c+d x)}{7 d} \] Output:
5/16*a*x+b*sin(d*x+c)/d+5/16*a*cos(d*x+c)*sin(d*x+c)/d+5/24*a*cos(d*x+c)^3 *sin(d*x+c)/d+1/6*a*cos(d*x+c)^5*sin(d*x+c)/d-b*sin(d*x+c)^3/d+3/5*b*sin(d *x+c)^5/d-1/7*b*sin(d*x+c)^7/d
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\frac {6720 b \sin (c+d x)-6720 b \sin ^3(c+d x)+4032 b \sin ^5(c+d x)-960 b \sin ^7(c+d x)+35 a (60 c+60 d x+45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x)))}{6720 d} \] Input:
Integrate[Cos[c + d*x]^6*(a + b*Cos[c + d*x]),x]
Output:
(6720*b*Sin[c + d*x] - 6720*b*Sin[c + d*x]^3 + 4032*b*Sin[c + d*x]^5 - 960 *b*Sin[c + d*x]^7 + 35*a*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(6720*d)
Time = 0.49 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {3042, 3227, 3042, 3113, 2009, 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.
| \(\displaystyle \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^6 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle a \int \cos ^6(c+d x)dx+b \int \cos ^7(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx+b \int \sin \left (c+d x+\frac {\pi }{2}\right )^7dx\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {b \int \left (-\sin ^6(c+d x)+3 \sin ^4(c+d x)-3 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {b \left (\frac {1}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^6*(a + b*Cos[c + d*x]),x]
Output:
-((b*(-Sin[c + d*x] + Sin[c + d*x]^3 - (3*Sin[c + d*x]^5)/5 + Sin[c + d*x] ^7/7))/d) + a*((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*S in[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)
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 = 14.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(90\) |
| default | \(\frac {\frac {b \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(90\) |
| parts | \(\frac {a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {b \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7 d}\) | \(92\) |
| parallelrisch | \(\frac {2100 a x d +3675 b \sin \left (d x +c \right )+15 b \sin \left (7 d x +7 c \right )+35 a \sin \left (6 d x +6 c \right )+147 b \sin \left (5 d x +5 c \right )+315 a \sin \left (4 d x +4 c \right )+735 b \sin \left (3 d x +3 c \right )+1575 a \sin \left (2 d x +2 c \right )}{6720 d}\) | \(93\) |
| risch | \(\frac {5 a x}{16}+\frac {35 b \sin \left (d x +c \right )}{64 d}+\frac {b \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \sin \left (6 d x +6 c \right )}{192 d}+\frac {7 b \sin \left (5 d x +5 c \right )}{320 d}+\frac {3 a \sin \left (4 d x +4 c \right )}{64 d}+\frac {7 b \sin \left (3 d x +3 c \right )}{64 d}+\frac {15 a \sin \left (2 d x +2 c \right )}{64 d}\) | \(108\) |
| norman | \(\frac {\frac {5 a x}{16}+\frac {35 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {105 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {175 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}+\frac {175 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {105 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}+\frac {35 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}+\frac {5 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}+\frac {424 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {\left (7 a -24 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}+\frac {\left (7 a +24 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {\left (11 a -16 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}+\frac {\left (11 a +16 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (425 a -2064 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {\left (425 a +2064 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(280\) |
| orering | \(\text {Expression too large to display}\) | \(2244\) |
Input:
int(cos(d*x+c)^6*(a+cos(d*x+c)*b),x,method=_RETURNVERBOSE)
Output:
1/d*(1/7*b*(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 )+a*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d *x+5/16*c))
Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\frac {525 \, a d x + {\left (240 \, b \cos \left (d x + c\right )^{6} + 280 \, a \cos \left (d x + c\right )^{5} + 288 \, b \cos \left (d x + c\right )^{4} + 350 \, a \cos \left (d x + c\right )^{3} + 384 \, b \cos \left (d x + c\right )^{2} + 525 \, a \cos \left (d x + c\right ) + 768 \, b\right )} \sin \left (d x + c\right )}{1680 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+b*cos(d*x+c)),x, algorithm="fricas")
Output:
1/1680*(525*a*d*x + (240*b*cos(d*x + c)^6 + 280*a*cos(d*x + c)^5 + 288*b*c os(d*x + c)^4 + 350*a*cos(d*x + c)^3 + 384*b*cos(d*x + c)^2 + 525*a*cos(d* x + c) + 768*b)*sin(d*x + c))/d
Time = 0.45 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.86 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\begin {cases} \frac {5 a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {16 b \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {b \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right ) \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*(a+b*cos(d*x+c)),x)
Output:
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) + 16*b*sin(c + d *x)**7/(35*d) + 8*b*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*b*sin(c + d* x)**3*cos(c + d*x)**4/d + b*sin(c + d*x)*cos(c + d*x)**6/d, Ne(d, 0)), (x* (a + b*cos(c))*cos(c)**6, True))
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=-\frac {35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a + 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} b}{6720 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+b*cos(d*x+c)),x, algorithm="maxima")
Output:
-1/6720*(35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 4 8*sin(2*d*x + 2*c))*a + 192*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin (d*x + c)^3 - 35*sin(d*x + c))*b)/d
Time = 0.52 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\frac {5}{16} \, a x + \frac {b \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {7 \, b \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {3 \, a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {7 \, b \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {15 \, a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {35 \, b \sin \left (d x + c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+b*cos(d*x+c)),x, algorithm="giac")
Output:
5/16*a*x + 1/448*b*sin(7*d*x + 7*c)/d + 1/192*a*sin(6*d*x + 6*c)/d + 7/320 *b*sin(5*d*x + 5*c)/d + 3/64*a*sin(4*d*x + 4*c)/d + 7/64*b*sin(3*d*x + 3*c )/d + 15/64*a*sin(2*d*x + 2*c)/d + 35/64*b*sin(d*x + c)/d
Time = 42.56 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.20 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\frac {5\,a\,x}{16}+\frac {\left (2\,b-\frac {11\,a}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (4\,b-\frac {7\,a}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {86\,b}{5}-\frac {85\,a}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {424\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{35}+\left (\frac {85\,a}{24}+\frac {86\,b}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {7\,a}{6}+4\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,a}{8}+2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \] Input:
int(cos(c + d*x)^6*(a + b*cos(c + d*x)),x)
Output:
(5*a*x)/16 + (tan(c/2 + (d*x)/2)*((11*a)/8 + 2*b) + tan(c/2 + (d*x)/2)^3*( (7*a)/6 + 4*b) - tan(c/2 + (d*x)/2)^11*((7*a)/6 - 4*b) - tan(c/2 + (d*x)/2 )^13*((11*a)/8 - 2*b) + tan(c/2 + (d*x)/2)^5*((85*a)/24 + (86*b)/5) - tan( c/2 + (d*x)/2)^9*((85*a)/24 - (86*b)/5) + (424*b*tan(c/2 + (d*x)/2)^7)/35) /(d*(tan(c/2 + (d*x)/2)^2 + 1)^7)
Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx=\frac {280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -910 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +1155 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a -240 \sin \left (d x +c \right )^{7} b +1008 \sin \left (d x +c \right )^{5} b -1680 \sin \left (d x +c \right )^{3} b +1680 \sin \left (d x +c \right ) b +525 a d x}{1680 d} \] Input:
int(cos(d*x+c)^6*(a+b*cos(d*x+c)),x)
Output:
(280*cos(c + d*x)*sin(c + d*x)**5*a - 910*cos(c + d*x)*sin(c + d*x)**3*a + 1155*cos(c + d*x)*sin(c + d*x)*a - 240*sin(c + d*x)**7*b + 1008*sin(c + d *x)**5*b - 1680*sin(c + d*x)**3*b + 1680*sin(c + d*x)*b + 525*a*d*x)/(1680 *d)