Integrand size = 25, antiderivative size = 161 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac {a \left (12 A b^2-a^2 C+8 b^2 C\right ) \sin (c+d x)}{6 b d}-\frac {\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {a C (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d} \]
1/8*(4*a^2*(2*A+C)+b^2*(4*A+3*C))*x+1/6*a*(12*A*b^2-C*a^2+8*C*b^2)*sin(d*x +c)/b/d-1/24*(2*a^2*C-3*b^2*(4*A+3*C))*cos(d*x+c)*sin(d*x+c)/d-1/12*a*C*(a +b*cos(d*x+c))^2*sin(d*x+c)/b/d+1/4*C*(a+b*cos(d*x+c))^3*sin(d*x+c)/b/d
Time = 1.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.66 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {12 \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) (c+d x)+48 a b (4 A+3 C) \sin (c+d x)+24 \left (A b^2+\left (a^2+b^2\right ) C\right ) \sin (2 (c+d x))+16 a b C \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d} \]
(12*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*(c + d*x) + 48*a*b*(4*A + 3*C)*Sin [c + d*x] + 24*(A*b^2 + (a^2 + b^2)*C)*Sin[2*(c + d*x)] + 16*a*b*C*Sin[3*( c + d*x)] + 3*b^2*C*Sin[4*(c + d*x)])/(96*d)
Time = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3503, 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.
| \(\displaystyle \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3503 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^2 (b (4 A+3 C)-a C \cos (c+d x))dx}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (4 A+3 C)-a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{3} \int (a+b \cos (c+d x)) \left (a b (12 A+7 C)-\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \cos (c+d x)\right )dx-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a b (12 A+7 C)+\left (3 b^2 (4 A+3 C)-2 a^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (12 A b^2-C \left (a^2-8 b^2\right )\right ) \sin (c+d x)}{d}-\frac {b \left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3}{2} b x \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )\right )-\frac {a C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}}{4 b}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}\) |
(C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d) + (-1/3*(a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/d + ((3*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x)/2 + (2*a*(12*A*b^2 - (a^2 - 8*b^2)*C)*Sin[c + d*x])/d - (b*(2*a^2*C - 3*b^2 *(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(2*d))/3)/(4*b)
3.6.33.3.1 Defintions of rubi rules used
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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^ (m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^ m*Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a , b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 3.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.61
| method | result | size |
| parallelrisch | \(\frac {24 \left (b^{2} \left (A +C \right )+a^{2} C \right ) \sin \left (2 d x +2 c \right )+16 a b \sin \left (3 d x +3 c \right ) C +3 C \sin \left (4 d x +4 c \right ) b^{2}+192 b \left (A +\frac {3 C}{4}\right ) a \sin \left (d x +c \right )+96 x \left (\frac {\left (A +\frac {3 C}{4}\right ) b^{2}}{2}+a^{2} \left (A +\frac {C}{2}\right )\right ) d}{96 d}\) | \(99\) |
| parts | \(a^{2} x A +\frac {\left (A \,b^{2}+a^{2} C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {b^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {2 \sin \left (d x +c \right ) A a b}{d}+\frac {2 C a b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(124\) |
| derivativedivides | \(\frac {b^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 C a b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \sin \left (d x +c \right ) a b +A \,a^{2} \left (d x +c \right )}{d}\) | \(140\) |
| default | \(\frac {b^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 C a b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \sin \left (d x +c \right ) a b +A \,a^{2} \left (d x +c \right )}{d}\) | \(140\) |
| risch | \(a^{2} x A +\frac {x A \,b^{2}}{2}+\frac {a^{2} C x}{2}+\frac {3 b^{2} C x}{8}+\frac {2 \sin \left (d x +c \right ) A a b}{d}+\frac {3 \sin \left (d x +c \right ) C a b}{2 d}+\frac {\sin \left (4 d x +4 c \right ) b^{2} C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) C a b}{6 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{4 d}+\frac {\sin \left (2 d x +2 c \right ) b^{2} C}{4 d}\) | \(146\) |
| norman | \(\frac {\left (A \,a^{2}+\frac {1}{2} A \,b^{2}+\frac {1}{2} a^{2} C +\frac {3}{8} b^{2} C \right ) x +\left (A \,a^{2}+\frac {1}{2} A \,b^{2}+\frac {1}{2} a^{2} C +\frac {3}{8} b^{2} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 A \,a^{2}+2 A \,b^{2}+2 a^{2} C +\frac {3}{2} b^{2} C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 A \,a^{2}+2 A \,b^{2}+2 a^{2} C +\frac {3}{2} b^{2} C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{2}+3 A \,b^{2}+3 a^{2} C +\frac {9}{4} b^{2} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (16 A a b -4 A \,b^{2}-4 a^{2} C +16 C a b -5 b^{2} C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (16 A a b +4 A \,b^{2}+4 a^{2} C +16 C a b +5 b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (144 A a b -12 A \,b^{2}-12 a^{2} C +80 C a b +9 b^{2} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {\left (144 A a b +12 A \,b^{2}+12 a^{2} C +80 C a b -9 b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(373\) |
1/96*(24*(b^2*(A+C)+a^2*C)*sin(2*d*x+2*c)+16*a*b*sin(3*d*x+3*c)*C+3*C*sin( 4*d*x+4*c)*b^2+192*b*(A+3/4*C)*a*sin(d*x+c)+96*x*(1/2*(A+3/4*C)*b^2+a^2*(A +1/2*C))*d)/d
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.65 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, {\left (2 \, A + C\right )} a^{2} + {\left (4 \, A + 3 \, C\right )} b^{2}\right )} d x + {\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 16 \, C a b \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, A + 2 \, C\right )} a b + 3 \, {\left (4 \, C a^{2} + {\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(3*(4*(2*A + C)*a^2 + (4*A + 3*C)*b^2)*d*x + (6*C*b^2*cos(d*x + c)^3 + 16*C*a*b*cos(d*x + c)^2 + 16*(3*A + 2*C)*a*b + 3*(4*C*a^2 + (4*A + 3*C)* b^2)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (141) = 282\).
Time = 0.18 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.92 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} A a^{2} x + \frac {2 A a b \sin {\left (c + d x \right )}}{d} + \frac {A b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 C a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a + b \cos {\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
Piecewise((A*a**2*x + 2*A*a*b*sin(c + d*x)/d + A*b**2*x*sin(c + d*x)**2/2 + A*b**2*x*cos(c + d*x)**2/2 + A*b**2*sin(c + d*x)*cos(c + d*x)/(2*d) + C* a**2*x*sin(c + d*x)**2/2 + C*a**2*x*cos(c + d*x)**2/2 + C*a**2*sin(c + d*x )*cos(c + d*x)/(2*d) + 4*C*a*b*sin(c + d*x)**3/(3*d) + 2*C*a*b*sin(c + d*x )*cos(c + d*x)**2/d + 3*C*b**2*x*sin(c + d*x)**4/8 + 3*C*b**2*x*sin(c + d* x)**2*cos(c + d*x)**2/4 + 3*C*b**2*x*cos(c + d*x)**4/8 + 3*C*b**2*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*C*b**2*sin(c + d*x)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(A + C*cos(c)**2)*(a + b*cos(c))**2, True))
Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.81 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {96 \, {\left (d x + c\right )} A a^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 192 \, A a b \sin \left (d x + c\right )}{96 \, d} \]
1/96*(96*(d*x + c)*A*a^2 + 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2 - 64* (sin(d*x + c)^3 - 3*sin(d*x + c))*C*a*b + 24*(2*d*x + 2*c + sin(2*d*x + 2* c))*A*b^2 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*b^ 2 + 192*A*a*b*sin(d*x + c))/d
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.72 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {C a b \sin \left (3 \, d x + 3 \, c\right )}{6 \, d} + \frac {1}{8} \, {\left (8 \, A a^{2} + 4 \, C a^{2} + 4 \, A b^{2} + 3 \, C b^{2}\right )} x + \frac {{\left (C a^{2} + A b^{2} + C b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a b + 3 \, C a b\right )} \sin \left (d x + c\right )}{2 \, d} \]
1/32*C*b^2*sin(4*d*x + 4*c)/d + 1/6*C*a*b*sin(3*d*x + 3*c)/d + 1/8*(8*A*a^ 2 + 4*C*a^2 + 4*A*b^2 + 3*C*b^2)*x + 1/4*(C*a^2 + A*b^2 + C*b^2)*sin(2*d*x + 2*c)/d + 1/2*(4*A*a*b + 3*C*a*b)*sin(d*x + c)/d
Time = 1.73 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx=A\,a^2\,x+\frac {A\,b^2\,x}{2}+\frac {C\,a^2\,x}{2}+\frac {3\,C\,b^2\,x}{8}+\frac {A\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {2\,A\,a\,b\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {C\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{6\,d} \]