Integrand size = 21, antiderivative size = 61 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} (4 A+3 C) x+\frac {(4 A+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d} \] Output:
1/8*(4*A+3*C)*x+1/8*(4*A+3*C)*cos(d*x+c)*sin(d*x+c)/d+1/4*C*cos(d*x+c)^3*s in(d*x+c)/d
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x))}{32 d} \] Input:
Integrate[Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2),x]
Output:
(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)] )/(32*d)
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3493, 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.
\(\displaystyle \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {1}{4} (4 A+3 C) \int \cos ^2(c+d x)dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} (4 A+3 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{4} (4 A+3 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{4} (4 A+3 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}\) |
Input:
Int[Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2),x]
Output:
(C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + ((4*A + 3*C)*(x/2 + (Cos[c + d*x]* Sin[c + d*x])/(2*d)))/4
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_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 0.72 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {\left (8 A +8 C \right ) \sin \left (2 d x +2 c \right )+\sin \left (4 d x +4 c \right ) C +16 x \left (A +\frac {3 C}{4}\right ) d}{32 d}\) | \(44\) |
risch | \(\frac {x A}{2}+\frac {3 C x}{8}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 d}\) | \(55\) |
derivativedivides | \(\frac {C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(65\) |
default | \(\frac {C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(65\) |
parts | \(\frac {A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(67\) |
norman | \(\frac {\left (\frac {A}{2}+\frac {3 C}{8}\right ) x +\left (2 A +\frac {3 C}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (2 A +\frac {3 C}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (3 A +\frac {9 C}{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {A}{2}+\frac {3 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {\left (4 A -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}-\frac {\left (4 A -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}+\frac {\left (4 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (4 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\) | \(197\) |
orering | \(x \cos \left (d x +c \right )^{2} \left (A +C \cos \left (d x +c \right )^{2}\right )-\frac {5 \left (-2 \cos \left (d x +c \right ) \left (A +C \cos \left (d x +c \right )^{2}\right ) d \sin \left (d x +c \right )-2 \cos \left (d x +c \right )^{3} C d \sin \left (d x +c \right )\right )}{16 d^{2}}+\frac {5 x \left (2 d^{2} \sin \left (d x +c \right )^{2} \left (A +C \cos \left (d x +c \right )^{2}\right )+10 C \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} d^{2}-2 \cos \left (d x +c \right )^{2} \left (A +C \cos \left (d x +c \right )^{2}\right ) d^{2}-2 C \cos \left (d x +c \right )^{4} d^{2}\right )}{16 d^{2}}-\frac {8 d^{3} \sin \left (d x +c \right ) \left (A +C \cos \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )-24 C \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} d^{3}+32 C \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) d^{3}}{64 d^{4}}+\frac {x \left (8 d^{4} \cos \left (d x +c \right )^{2} \left (A +C \cos \left (d x +c \right )^{2}\right )-184 d^{4} \sin \left (d x +c \right )^{2} C \cos \left (d x +c \right )^{2}-8 d^{4} \sin \left (d x +c \right )^{2} \left (A +C \cos \left (d x +c \right )^{2}\right )+24 C \,d^{4} \sin \left (d x +c \right )^{4}+32 C \cos \left (d x +c \right )^{4} d^{4}\right )}{64 d^{4}}\) | \(350\) |
Input:
int(cos(d*x+c)^2*(A+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/32*((8*A+8*C)*sin(2*d*x+2*c)+sin(4*d*x+4*c)*C+16*x*(A+3/4*C)*d)/d
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {{\left (4 \, A + 3 \, C\right )} d x + {\left (2 \, C \cos \left (d x + c\right )^{3} + {\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+C*cos(d*x+c)^2),x, algorithm="fricas")
Output:
1/8*((4*A + 3*C)*d*x + (2*C*cos(d*x + c)^3 + (4*A + 3*C)*cos(d*x + c))*sin (d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (53) = 106\).
Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.59 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 C x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*(A+C*cos(d*x+c)**2),x)
Output:
Piecewise((A*x*sin(c + d*x)**2/2 + A*x*cos(c + d*x)**2/2 + A*sin(c + d*x)* cos(c + d*x)/(2*d) + 3*C*x*sin(c + d*x)**4/8 + 3*C*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*C*x*cos(c + d*x)**4/8 + 3*C*sin(c + d*x)**3*cos(c + d*x)/ (8*d) + 5*C*sin(c + d*x)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(A + C*cos(c )**2)*cos(c)**2, True))
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {{\left (d x + c\right )} {\left (4 \, A + 3 \, C\right )} + \frac {{\left (4 \, A + 3 \, C\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, A + 5 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+C*cos(d*x+c)^2),x, algorithm="maxima")
Output:
1/8*((d*x + c)*(4*A + 3*C) + ((4*A + 3*C)*tan(d*x + c)^3 + (4*A + 5*C)*tan (d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1))/d
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \, {\left (4 \, A + 3 \, C\right )} x + \frac {C \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (A + C\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+C*cos(d*x+c)^2),x, algorithm="giac")
Output:
1/8*(4*A + 3*C)*x + 1/32*C*sin(4*d*x + 4*c)/d + 1/4*(A + C)*sin(2*d*x + 2* c)/d
Time = 41.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=x\,\left (\frac {A}{2}+\frac {3\,C}{8}\right )+\frac {\left (\frac {A}{2}+\frac {3\,C}{8}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {A}{2}+\frac {5\,C}{8}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \] Input:
int(cos(c + d*x)^2*(A + C*cos(c + d*x)^2),x)
Output:
x*(A/2 + (3*C)/8) + (tan(c + d*x)*(A/2 + (5*C)/8) + tan(c + d*x)^3*(A/2 + (3*C)/8))/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1))
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03 \[ \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +4 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +5 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +4 a d x +3 c d x}{8 d} \] Input:
int(cos(d*x+c)^2*(A+C*cos(d*x+c)^2),x)
Output:
( - 2*cos(c + d*x)*sin(c + d*x)**3*c + 4*cos(c + d*x)*sin(c + d*x)*a + 5*c os(c + d*x)*sin(c + d*x)*c + 4*a*d*x + 3*c*d*x)/(8*d)