Integrand size = 19, antiderivative size = 170 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {1}{2} a b \left (4 a^2+3 b^2\right ) x+\frac {2 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \sin (c+d x)}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac {\left (3 a^2+4 b^2\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac {a (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac {(a+b \cos (c+d x))^4 \sin (c+d x)}{5 d} \] Output:
1/2*a*b*(4*a^2+3*b^2)*x+2/15*(3*a^4+28*a^2*b^2+4*b^4)*sin(d*x+c)/d+1/30*a* b*(6*a^2+29*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/15*(3*a^2+4*b^2)*(a+b*cos(d*x+c ))^2*sin(d*x+c)/d+1/5*a*(a+b*cos(d*x+c))^3*sin(d*x+c)/d+1/5*(a+b*cos(d*x+c ))^4*sin(d*x+c)/d
Time = 1.00 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {30 \left (8 a^4+36 a^2 b^2+5 b^4\right ) \sin (c+d x)+b \left (480 a^3 c+360 a b^2 c+480 a^3 d x+360 a b^2 d x+240 a \left (a^2+b^2\right ) \sin (2 (c+d x))+5 \left (24 a^2 b+5 b^3\right ) \sin (3 (c+d x))+30 a b^2 \sin (4 (c+d x))+3 b^3 \sin (5 (c+d x))\right )}{240 d} \] Input:
Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4,x]
Output:
(30*(8*a^4 + 36*a^2*b^2 + 5*b^4)*Sin[c + d*x] + b*(480*a^3*c + 360*a*b^2*c + 480*a^3*d*x + 360*a*b^2*d*x + 240*a*(a^2 + b^2)*Sin[2*(c + d*x)] + 5*(2 4*a^2*b + 5*b^3)*Sin[3*(c + d*x)] + 30*a*b^2*Sin[4*(c + d*x)] + 3*b^3*Sin[ 5*(c + d*x)]))/(240*d)
Time = 0.67 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 3232, 27, 3042, 3232, 3042, 3232, 3042, 3213}
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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{5} \int 4 (b+a \cos (c+d x)) (a+b \cos (c+d x))^3dx+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} \int (b+a \cos (c+d x)) (a+b \cos (c+d x))^3dx+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \int \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3dx+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {4}{5} \left (\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (7 a b+\left (3 a^2+4 b^2\right ) \cos (c+d x)\right )dx+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \left (\frac {1}{4} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (7 a b+\left (3 a^2+4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {4}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (b \left (27 a^2+8 b^2\right )+a \left (6 a^2+29 b^2\right ) \cos (c+d x)\right )dx+\frac {\left (3 a^2+4 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (27 a^2+8 b^2\right )+a \left (6 a^2+29 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\left (3 a^2+4 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {4}{5} \left (\frac {1}{4} \left (\frac {\left (3 a^2+4 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {a b \left (6 a^2+29 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {15}{2} a b x \left (4 a^2+3 b^2\right )+\frac {2 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \sin (c+d x)}{d}\right )\right )+\frac {a \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
Input:
Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4,x]
Output:
((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + (4*((a*(a + b*Cos[c + d*x])^ 3*Sin[c + d*x])/(4*d) + (((3*a^2 + 4*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d *x])/(3*d) + ((15*a*b*(4*a^2 + 3*b^2)*x)/2 + (2*(3*a^4 + 28*a^2*b^2 + 4*b^ 4)*Sin[c + d*x])/d + (a*b*(6*a^2 + 29*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*d ))/3)/4))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Time = 34.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74
| method | result | size |
| parallelrisch | \(\frac {240 \left (a^{3} b +a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+5 \left (24 a^{2} b^{2}+5 b^{4}\right ) \sin \left (3 d x +3 c \right )+30 \sin \left (4 d x +4 c \right ) a \,b^{3}+3 \sin \left (5 d x +5 c \right ) b^{4}+30 \left (8 a^{4}+36 a^{2} b^{2}+5 b^{4}\right ) \sin \left (d x +c \right )+480 a d b x \left (a^{2}+\frac {3 b^{2}}{4}\right )}{240 d}\) | \(125\) |
| derivativedivides | \(\frac {a^{4} \sin \left (d x +c \right )+4 a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} b^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+4 a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(138\) |
| default | \(\frac {a^{4} \sin \left (d x +c \right )+4 a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} b^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )+4 a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(138\) |
| parts | \(\frac {a^{4} \sin \left (d x +c \right )}{d}+\frac {b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {4 a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {2 a^{2} b^{2} \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{d}+\frac {4 a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(149\) |
| risch | \(2 a^{3} b x +\frac {3 a \,b^{3} x}{2}+\frac {a^{4} \sin \left (d x +c \right )}{d}+\frac {9 \sin \left (d x +c \right ) a^{2} b^{2}}{2 d}+\frac {5 \sin \left (d x +c \right ) b^{4}}{8 d}+\frac {b^{4} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a \,b^{3} \sin \left (4 d x +4 c \right )}{8 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} b^{2}}{2 d}+\frac {5 \sin \left (3 d x +3 c \right ) b^{4}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) a \,b^{3}}{d}\) | \(166\) |
| norman | \(\frac {\left (2 a^{3} b +\frac {3}{2} a \,b^{3}\right ) x +\left (2 a^{3} b +\frac {3}{2} a \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (10 a^{3} b +\frac {15}{2} a \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (10 a^{3} b +\frac {15}{2} a \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (20 a^{3} b +15 a \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (20 a^{3} b +15 a \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {\left (2 a^{4}-4 a^{3} b +12 a^{2} b^{2}-5 a \,b^{3}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {\left (2 a^{4}+4 a^{3} b +12 a^{2} b^{2}+5 a \,b^{3}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 \left (45 a^{4}+150 a^{2} b^{2}+29 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {2 \left (12 a^{4}-12 a^{3} b +48 a^{2} b^{2}-3 a \,b^{3}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {2 \left (12 a^{4}+12 a^{3} b +48 a^{2} b^{2}+3 a \,b^{3}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(382\) |
| orering | \(\text {Expression too large to display}\) | \(2147\) |
Input:
int(cos(d*x+c)*(a+cos(d*x+c)*b)^4,x,method=_RETURNVERBOSE)
Output:
1/240*(240*(a^3*b+a*b^3)*sin(2*d*x+2*c)+5*(24*a^2*b^2+5*b^4)*sin(3*d*x+3*c )+30*sin(4*d*x+4*c)*a*b^3+3*sin(5*d*x+5*c)*b^4+30*(8*a^4+36*a^2*b^2+5*b^4) *sin(d*x+c)+480*a*d*b*x*(a^2+3/4*b^2))/d
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.71 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} d x + {\left (6 \, b^{4} \cos \left (d x + c\right )^{4} + 30 \, a b^{3} \cos \left (d x + c\right )^{3} + 30 \, a^{4} + 120 \, a^{2} b^{2} + 16 \, b^{4} + 4 \, {\left (15 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4,x, algorithm="fricas")
Output:
1/30*(15*(4*a^3*b + 3*a*b^3)*d*x + (6*b^4*cos(d*x + c)^4 + 30*a*b^3*cos(d* x + c)^3 + 30*a^4 + 120*a^2*b^2 + 16*b^4 + 4*(15*a^2*b^2 + 2*b^4)*cos(d*x + c)^2 + 15*(4*a^3*b + 3*a*b^3)*cos(d*x + c))*sin(d*x + c))/d
Time = 0.25 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.77 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\begin {cases} \frac {a^{4} \sin {\left (c + d x \right )}}{d} + 2 a^{3} b x \sin ^{2}{\left (c + d x \right )} + 2 a^{3} b x \cos ^{2}{\left (c + d x \right )} + \frac {2 a^{3} b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {6 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a b^{3} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 a b^{3} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 a b^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 a b^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {8 b^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {b^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{4} \cos {\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))**4,x)
Output:
Piecewise((a**4*sin(c + d*x)/d + 2*a**3*b*x*sin(c + d*x)**2 + 2*a**3*b*x*c os(c + d*x)**2 + 2*a**3*b*sin(c + d*x)*cos(c + d*x)/d + 4*a**2*b**2*sin(c + d*x)**3/d + 6*a**2*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3*a*b**3*x*sin( c + d*x)**4/2 + 3*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*a*b**3*x*co s(c + d*x)**4/2 + 3*a*b**3*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 5*a*b**3*s in(c + d*x)*cos(c + d*x)**3/(2*d) + 8*b**4*sin(c + d*x)**5/(15*d) + 4*b**4 *sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + b**4*sin(c + d*x)*cos(c + d*x)**4 /d, Ne(d, 0)), (x*(a + b*cos(c))**4*cos(c), True))
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} b - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} b^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} + 8 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} b^{4} + 120 \, a^{4} \sin \left (d x + c\right )}{120 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4,x, algorithm="maxima")
Output:
1/120*(120*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^3*b - 240*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^2*b^2 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d *x + 2*c))*a*b^3 + 8*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*b^4 + 120*a^4*sin(d*x + c))/d
Time = 0.46 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {b^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {a b^{3} \sin \left (4 \, d x + 4 \, c\right )}{8 \, d} + \frac {1}{2} \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} x + \frac {{\left (24 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (a^{3} b + a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac {{\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4,x, algorithm="giac")
Output:
1/80*b^4*sin(5*d*x + 5*c)/d + 1/8*a*b^3*sin(4*d*x + 4*c)/d + 1/2*(4*a^3*b + 3*a*b^3)*x + 1/48*(24*a^2*b^2 + 5*b^4)*sin(3*d*x + 3*c)/d + (a^3*b + a*b ^3)*sin(2*d*x + 2*c)/d + 1/8*(8*a^4 + 36*a^2*b^2 + 5*b^4)*sin(d*x + c)/d
Time = 41.69 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.14 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {\left (2\,a^4-4\,a^3\,b+12\,a^2\,b^2-5\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (8\,a^4-8\,a^3\,b+32\,a^2\,b^2-2\,a\,b^3+\frac {8\,b^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,a^4+40\,a^2\,b^2+\frac {116\,b^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (8\,a^4+8\,a^3\,b+32\,a^2\,b^2+2\,a\,b^3+\frac {8\,b^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^4+4\,a^3\,b+12\,a^2\,b^2+5\,a\,b^3+2\,b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^2+3\,b^2\right )}{4\,a^3\,b+3\,a\,b^3}\right )\,\left (4\,a^2+3\,b^2\right )}{d}-\frac {a\,b\,\left (4\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{d} \] Input:
int(cos(c + d*x)*(a + b*cos(c + d*x))^4,x)
Output:
(tan(c/2 + (d*x)/2)^5*(12*a^4 + (116*b^4)/15 + 40*a^2*b^2) + tan(c/2 + (d* x)/2)^9*(2*a^4 - 4*a^3*b - 5*a*b^3 + 2*b^4 + 12*a^2*b^2) + tan(c/2 + (d*x) /2)^3*(2*a*b^3 + 8*a^3*b + 8*a^4 + (8*b^4)/3 + 32*a^2*b^2) + tan(c/2 + (d* x)/2)^7*(8*a^4 - 8*a^3*b - 2*a*b^3 + (8*b^4)/3 + 32*a^2*b^2) + tan(c/2 + ( d*x)/2)*(5*a*b^3 + 4*a^3*b + 2*a^4 + 2*b^4 + 12*a^2*b^2))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (a*b*atan((a*b*tan(c/2 + (d* x)/2)*(4*a^2 + 3*b^2))/(3*a*b^3 + 4*a^3*b))*(4*a^2 + 3*b^2))/d - (a*b*(4*a ^2 + 3*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/d
Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {-30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3}+60 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b +75 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3}+6 \sin \left (d x +c \right )^{5} b^{4}-60 \sin \left (d x +c \right )^{3} a^{2} b^{2}-20 \sin \left (d x +c \right )^{3} b^{4}+30 \sin \left (d x +c \right ) a^{4}+180 \sin \left (d x +c \right ) a^{2} b^{2}+30 \sin \left (d x +c \right ) b^{4}+60 a^{3} b d x +45 a \,b^{3} d x}{30 d} \] Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))^4,x)
Output:
( - 30*cos(c + d*x)*sin(c + d*x)**3*a*b**3 + 60*cos(c + d*x)*sin(c + d*x)* a**3*b + 75*cos(c + d*x)*sin(c + d*x)*a*b**3 + 6*sin(c + d*x)**5*b**4 - 60 *sin(c + d*x)**3*a**2*b**2 - 20*sin(c + d*x)**3*b**4 + 30*sin(c + d*x)*a** 4 + 180*sin(c + d*x)*a**2*b**2 + 30*sin(c + d*x)*b**4 + 60*a**3*b*d*x + 45 *a*b**3*d*x)/(30*d)