Integrand size = 21, antiderivative size = 129 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {11 a^2 x}{16}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {11 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d} \] Output:
11/16*a^2*x+2*a^2*sin(d*x+c)/d+11/16*a^2*cos(d*x+c)*sin(d*x+c)/d+11/24*a^2 *cos(d*x+c)^3*sin(d*x+c)/d+1/6*a^2*cos(d*x+c)^5*sin(d*x+c)/d-4/3*a^2*sin(d *x+c)^3/d+2/5*a^2*sin(d*x+c)^5/d
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^2 (660 d x+1200 \sin (c+d x)+465 \sin (2 (c+d x))+200 \sin (3 (c+d x))+75 \sin (4 (c+d x))+24 \sin (5 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \] Input:
Integrate[Cos[c + d*x]^4*(a + a*Cos[c + d*x])^2,x]
Output:
(a^2*(660*d*x + 1200*Sin[c + d*x] + 465*Sin[2*(c + d*x)] + 200*Sin[3*(c + d*x)] + 75*Sin[4*(c + d*x)] + 24*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]))/( 960*d)
Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3236, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a \cos (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2dx\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \int \left (a^2 \cos ^6(c+d x)+2 a^2 \cos ^5(c+d x)+a^2 \cos ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {11 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {11 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {11 a^2 x}{16}\) |
Input:
Int[Cos[c + d*x]^4*(a + a*Cos[c + d*x])^2,x]
Output:
(11*a^2*x)/16 + (2*a^2*Sin[c + d*x])/d + (11*a^2*Cos[c + d*x]*Sin[c + d*x] )/(16*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x] ^5*Sin[c + d*x])/(6*d) - (4*a^2*Sin[c + d*x]^3)/(3*d) + (2*a^2*Sin[c + d*x ]^5)/(5*d)
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Time = 19.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.58
| method | result | size |
| parallelrisch | \(\frac {a^{2} \left (132 d x +\sin \left (6 d x +6 c \right )+240 \sin \left (d x +c \right )+93 \sin \left (2 d x +2 c \right )+40 \sin \left (3 d x +3 c \right )+15 \sin \left (4 d x +4 c \right )+\frac {24 \sin \left (5 d x +5 c \right )}{5}\right )}{192 d}\) | \(75\) |
| risch | \(\frac {11 a^{2} x}{16}+\frac {5 a^{2} \sin \left (d x +c \right )}{4 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{2} \sin \left (5 d x +5 c \right )}{40 d}+\frac {5 a^{2} \sin \left (4 d x +4 c \right )}{64 d}+\frac {5 a^{2} \sin \left (3 d x +3 c \right )}{24 d}+\frac {31 a^{2} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {2 a^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(121\) |
| default | \(\frac {a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {2 a^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(121\) |
| parts | \(\frac {a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {2 a^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(126\) |
| norman | \(\frac {\frac {11 a^{2} x}{16}+\frac {53 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {87 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 d}+\frac {501 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}+\frac {331 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {187 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {11 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {33 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {165 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {55 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {165 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {33 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {11 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(238\) |
| orering | \(\text {Expression too large to display}\) | \(2560\) |
Input:
int(cos(d*x+c)^4*(a+a*cos(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/192*a^2*(132*d*x+sin(6*d*x+6*c)+240*sin(d*x+c)+93*sin(2*d*x+2*c)+40*sin( 3*d*x+3*c)+15*sin(4*d*x+4*c)+24/5*sin(5*d*x+5*c))/d
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.69 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {165 \, a^{2} d x + {\left (40 \, a^{2} \cos \left (d x + c\right )^{5} + 96 \, a^{2} \cos \left (d x + c\right )^{4} + 110 \, a^{2} \cos \left (d x + c\right )^{3} + 128 \, a^{2} \cos \left (d x + c\right )^{2} + 165 \, a^{2} \cos \left (d x + c\right ) + 256 \, a^{2}\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+a*cos(d*x+c))^2,x, algorithm="fricas")
Output:
1/240*(165*a^2*d*x + (40*a^2*cos(d*x + c)^5 + 96*a^2*cos(d*x + c)^4 + 110* a^2*cos(d*x + c)^3 + 128*a^2*cos(d*x + c)^2 + 165*a^2*cos(d*x + c) + 256*a ^2)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (122) = 244\).
Time = 0.34 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.66 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\begin {cases} \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {16 a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {8 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {11 a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {2 a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right )^{2} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**4*(a+a*cos(d*x+c))**2,x)
Output:
Piecewise((5*a**2*x*sin(c + d*x)**6/16 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*a**2*x*sin(c + d*x)**4/8 + 15*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*a**2*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 5*a**2*x*cos( c + d*x)**6/16 + 3*a**2*x*cos(c + d*x)**4/8 + 5*a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 16*a**2*sin(c + d*x)**5/(15*d) + 5*a**2*sin(c + d*x)**3*c os(c + d*x)**3/(6*d) + 8*a**2*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*a* *2*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 11*a**2*sin(c + d*x)*cos(c + d*x)* *5/(16*d) + 2*a**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*a**2*sin(c + d*x)*co s(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a*cos(c) + a)**2*cos(c)**4, True))
Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {128 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{960 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+a*cos(d*x+c))^2,x, algorithm="maxima")
Output:
1/960*(128*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^2 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d* x + 2*c))*a^2 + 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c)) *a^2)/d
Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {11}{16} \, a^{2} x + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a^{2} \sin \left (5 \, d x + 5 \, c\right )}{40 \, d} + \frac {5 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {5 \, a^{2} \sin \left (3 \, d x + 3 \, c\right )}{24 \, d} + \frac {31 \, a^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {5 \, a^{2} \sin \left (d x + c\right )}{4 \, d} \] Input:
integrate(cos(d*x+c)^4*(a+a*cos(d*x+c))^2,x, algorithm="giac")
Output:
11/16*a^2*x + 1/192*a^2*sin(6*d*x + 6*c)/d + 1/40*a^2*sin(5*d*x + 5*c)/d + 5/64*a^2*sin(4*d*x + 4*c)/d + 5/24*a^2*sin(3*d*x + 3*c)/d + 31/64*a^2*sin (2*d*x + 2*c)/d + 5/4*a^2*sin(d*x + c)/d
Time = 45.84 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {11\,a^2\,x}{16}+\frac {\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {187\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {331\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {501\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {87\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {53\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int(cos(c + d*x)^4*(a + a*cos(c + d*x))^2,x)
Output:
(11*a^2*x)/16 + ((87*a^2*tan(c/2 + (d*x)/2)^3)/8 + (501*a^2*tan(c/2 + (d*x )/2)^5)/20 + (331*a^2*tan(c/2 + (d*x)/2)^7)/20 + (187*a^2*tan(c/2 + (d*x)/ 2)^9)/24 + (11*a^2*tan(c/2 + (d*x)/2)^11)/8 + (53*a^2*tan(c/2 + (d*x)/2))/ 8)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^6)
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.67 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^{2} \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-190 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+315 \cos \left (d x +c \right ) \sin \left (d x +c \right )+96 \sin \left (d x +c \right )^{5}-320 \sin \left (d x +c \right )^{3}+480 \sin \left (d x +c \right )+165 d x \right )}{240 d} \] Input:
int(cos(d*x+c)^4*(a+a*cos(d*x+c))^2,x)
Output:
(a**2*(40*cos(c + d*x)*sin(c + d*x)**5 - 190*cos(c + d*x)*sin(c + d*x)**3 + 315*cos(c + d*x)*sin(c + d*x) + 96*sin(c + d*x)**5 - 320*sin(c + d*x)**3 + 480*sin(c + d*x) + 165*d*x))/(240*d)