Integrand size = 19, antiderivative size = 76 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\frac {3 b x}{8}+\frac {a \sin (c+d x)}{d}+\frac {3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a \sin ^3(c+d x)}{3 d} \] Output:
3/8*b*x+a*sin(d*x+c)/d+3/8*b*cos(d*x+c)*sin(d*x+c)/d+1/4*b*cos(d*x+c)^3*si n(d*x+c)/d-1/3*a*sin(d*x+c)^3/d
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\frac {3 b (c+d x)}{8 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{3 d}+\frac {b \sin (2 (c+d x))}{4 d}+\frac {b \sin (4 (c+d x))}{32 d} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x]),x]
Output:
(3*b*(c + d*x))/(8*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (b *Sin[2*(c + d*x)])/(4*d) + (b*Sin[4*(c + d*x)])/(32*d)
Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 3227, 3042, 3113, 2009, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle a \int \cos ^3(c+d x)dx+b \int \cos ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx+b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle b \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle b \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {a \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x]),x]
Output:
-((a*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d) + b*((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)
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 = 5.71 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
| method | result | size |
| parallelrisch | \(\frac {36 d x b +72 a \sin \left (d x +c \right )+3 b \sin \left (4 d x +4 c \right )+8 a \sin \left (3 d x +3 c \right )+24 b \sin \left (2 d x +2 c \right )}{96 d}\) | \(57\) |
| derivativedivides | \(\frac {b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(60\) |
| default | \(\frac {b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(60\) |
| parts | \(\frac {a \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}+\frac {b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(62\) |
| risch | \(\frac {3 b x}{8}+\frac {3 a \sin \left (d x +c \right )}{4 d}+\frac {b \sin \left (4 d x +4 c \right )}{32 d}+\frac {a \sin \left (3 d x +3 c \right )}{12 d}+\frac {b \sin \left (2 d x +2 c \right )}{4 d}\) | \(63\) |
| norman | \(\frac {\frac {3 b x}{8}+\frac {3 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {9 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {3 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {3 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {\left (8 a -5 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (8 a +5 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (40 a -9 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {\left (40 a +9 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(172\) |
| orering | \(x \cos \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right )-\frac {205 \left (-3 \cos \left (d x +c \right )^{2} \left (a +\cos \left (d x +c \right ) b \right ) d \sin \left (d x +c \right )-b \cos \left (d x +c \right )^{3} d \sin \left (d x +c \right )\right )}{144 d^{2}}+\frac {205 x \left (6 \cos \left (d x +c \right ) \left (a +\cos \left (d x +c \right ) b \right ) d^{2} \sin \left (d x +c \right )^{2}+6 \cos \left (d x +c \right )^{2} d^{2} \sin \left (d x +c \right )^{2} b -3 \cos \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right ) d^{2}-b \cos \left (d x +c \right )^{4} d^{2}\right )}{144 d^{2}}-\frac {91 \left (-6 d^{3} \sin \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right )-18 \cos \left (d x +c \right ) d^{3} \sin \left (d x +c \right )^{3} b +21 \cos \left (d x +c \right )^{2} \left (a +\cos \left (d x +c \right ) b \right ) d^{3} \sin \left (d x +c \right )+19 \cos \left (d x +c \right )^{3} d^{3} \sin \left (d x +c \right ) b \right )}{192 d^{4}}+\frac {91 x \left (-60 d^{4} \sin \left (d x +c \right )^{2} \left (a +\cos \left (d x +c \right ) b \right ) \cos \left (d x +c \right )+24 \sin \left (d x +c \right )^{4} b \,d^{4}-132 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2} b \,d^{4}+21 \cos \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right ) d^{4}+19 \cos \left (d x +c \right )^{4} b \,d^{4}\right )}{192 d^{4}}-\frac {5 \left (-183 d^{5} \sin \left (d x +c \right ) \left (a +\cos \left (d x +c \right ) b \right ) \cos \left (d x +c \right )^{2}+420 d^{5} \sin \left (d x +c \right )^{3} b \cos \left (d x +c \right )+60 d^{5} \sin \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right )-361 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3} b \,d^{5}\right )}{96 d^{6}}+\frac {5 x \left (-183 d^{6} \cos \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right )+2526 d^{6} \sin \left (d x +c \right )^{2} b \cos \left (d x +c \right )^{2}+546 d^{6} \sin \left (d x +c \right )^{2} \left (a +\cos \left (d x +c \right ) b \right ) \cos \left (d x +c \right )-480 d^{6} \sin \left (d x +c \right )^{4} b -361 d^{6} \cos \left (d x +c \right )^{4} b \right )}{96 d^{6}}-\frac {1641 d^{7} \cos \left (d x +c \right )^{2} \left (a +\cos \left (d x +c \right ) b \right ) \sin \left (d x +c \right )+6679 d^{7} \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) b -7518 d^{7} \sin \left (d x +c \right )^{3} b \cos \left (d x +c \right )-546 d^{7} \sin \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right )}{576 d^{8}}+\frac {x \left (-4920 d^{8} \cos \left (d x +c \right ) \left (a +\cos \left (d x +c \right ) b \right ) \sin \left (d x +c \right )^{2}-44232 d^{8} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} b +1641 d^{8} \cos \left (d x +c \right )^{3} \left (a +\cos \left (d x +c \right ) b \right )+6679 d^{8} \cos \left (d x +c \right )^{4} b +8064 d^{8} \sin \left (d x +c \right )^{4} b \right )}{576 d^{8}}\) | \(789\) |
Input:
int(cos(d*x+c)^3*(a+cos(d*x+c)*b),x,method=_RETURNVERBOSE)
Output:
1/96*(36*d*x*b+72*a*sin(d*x+c)+3*b*sin(4*d*x+4*c)+8*a*sin(3*d*x+3*c)+24*b* sin(2*d*x+2*c))/d
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\frac {9 \, b d x + {\left (6 \, b \cos \left (d x + c\right )^{3} + 8 \, a \cos \left (d x + c\right )^{2} + 9 \, b \cos \left (d x + c\right ) + 16 \, a\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c)),x, algorithm="fricas")
Output:
1/24*(9*b*d*x + (6*b*cos(d*x + c)^3 + 8*a*cos(d*x + c)^2 + 9*b*cos(d*x + c ) + 16*a)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (70) = 140\).
Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.89 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\begin {cases} \frac {2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right ) \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**3*(a+b*cos(d*x+c)),x)
Output:
Piecewise((2*a*sin(c + d*x)**3/(3*d) + a*sin(c + d*x)*cos(c + d*x)**2/d + 3*b*x*sin(c + d*x)**4/8 + 3*b*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*b*x* cos(c + d*x)**4/8 + 3*b*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*b*sin(c + d *x)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a + b*cos(c))*cos(c)**3, True))
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{96 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c)),x, algorithm="maxima")
Output:
-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*a - 3*(12*d*x + 12*c + sin(4*d *x + 4*c) + 8*sin(2*d*x + 2*c))*b)/d
Time = 0.50 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\frac {3}{8} \, b x + \frac {b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {b \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{4 \, d} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c)),x, algorithm="giac")
Output:
3/8*b*x + 1/32*b*sin(4*d*x + 4*c)/d + 1/12*a*sin(3*d*x + 3*c)/d + 1/4*b*si n(2*d*x + 2*c)/d + 3/4*a*sin(d*x + c)/d
Time = 39.83 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\frac {3\,b\,x}{8}+\frac {2\,a\,\sin \left (c+d\,x\right )}{3\,d}+\frac {3\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {b\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d} \] Input:
int(cos(c + d*x)^3*(a + b*cos(c + d*x)),x)
Output:
(3*b*x)/8 + (2*a*sin(c + d*x))/(3*d) + (3*b*cos(c + d*x)*sin(c + d*x))/(8* d) + (a*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (b*cos(c + d*x)^3*sin(c + d*x ))/(4*d)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -8 \sin \left (d x +c \right )^{3} a +24 \sin \left (d x +c \right ) a +9 b d x}{24 d} \] Input:
int(cos(d*x+c)^3*(a+b*cos(d*x+c)),x)
Output:
( - 6*cos(c + d*x)*sin(c + d*x)**3*b + 15*cos(c + d*x)*sin(c + d*x)*b - 8* sin(c + d*x)**3*a + 24*sin(c + d*x)*a + 9*b*d*x)/(24*d)