Integrand size = 32, antiderivative size = 129 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^2 (8 B+7 C) x+\frac {a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 B+7 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 B-C) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d} \]
1/8*a^2*(8*B+7*C)*x+1/6*a^2*(8*B+7*C)*sin(d*x+c)/d+1/24*a^2*(8*B+7*C)*cos( d*x+c)*sin(d*x+c)/d+1/12*(4*B-C)*(a+a*cos(d*x+c))^2*sin(d*x+c)/d+1/4*C*(a+ a*cos(d*x+c))^3*sin(d*x+c)/a/d
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.67 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 (84 c C+96 B d x+84 C d x+24 (7 B+6 C) \sin (c+d x)+48 (B+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x)))}{96 d} \]
(a^2*(84*c*C + 96*B*d*x + 84*C*d*x + 24*(7*B + 6*C)*Sin[c + d*x] + 48*(B + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 16*C*Sin[3*(c + d*x)] + 3*C* Sin[4*(c + d*x)]))/(96*d)
Time = 0.46 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3502, 3042, 3230, 3042, 3123}
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.
| \(\displaystyle \int (a \cos (c+d x)+a)^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^2 (3 a C+a (4 B-C) \cos (c+d x))dx}{4 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a C+a (4 B-C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{4 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{3} a (8 B+7 C) \int (\cos (c+d x) a+a)^2dx+\frac {a (4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}}{4 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} a (8 B+7 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2dx+\frac {a (4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}}{4 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\) |
| \(\Big \downarrow \) 3123 |
| \(\displaystyle \frac {\frac {1}{3} a (8 B+7 C) \left (\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3 a^2 x}{2}\right )+\frac {a (4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}}{4 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}\) |
(C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d) + ((a*(4*B - C)*(a + a*Cos [c + d*x])^2*Sin[c + d*x])/(3*d) + (a*(8*B + 7*C)*((3*a^2*x)/2 + (2*a^2*Si n[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)))/3)/(4*a)
3.3.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(2*a^2 + b^ 2)*(x/2), x] + (-Simp[2*a*b*(Cos[c + d*x]/d), x] - Simp[b^2*Cos[c + d*x]*(S in[c + d*x]/(2*d)), x]) /; FreeQ[{a, b, c, d}, x]
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[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[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) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 4.73 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.57
| method | result | size |
| parallelrisch | \(\frac {\left (\frac {\left (B +C \right ) \sin \left (2 d x +2 c \right )}{2}+\frac {\left (\frac {B}{2}+C \right ) \sin \left (3 d x +3 c \right )}{6}+\frac {\sin \left (4 d x +4 c \right ) C}{32}+\frac {\left (\frac {7 B}{2}+3 C \right ) \sin \left (d x +c \right )}{2}+d x \left (B +\frac {7 C}{8}\right )\right ) a^{2}}{d}\) | \(74\) |
| parts | \(\frac {\left (B \,a^{2}+2 a^{2} C \right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (2 B \,a^{2}+a^{2} C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\sin \left (d x +c \right ) B \,a^{2}}{d}\) | \(128\) |
| risch | \(a^{2} B x +\frac {7 a^{2} C x}{8}+\frac {7 \sin \left (d x +c \right ) B \,a^{2}}{4 d}+\frac {3 \sin \left (d x +c \right ) a^{2} C}{2 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} C}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{6 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(135\) |
| derivativedivides | \(\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) a^{2}}{d}\) | \(154\) |
| default | \(\frac {a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) a^{2}}{d}\) | \(154\) |
| norman | \(\frac {\frac {a^{2} \left (8 B +7 C \right ) x}{8}+\frac {11 a^{2} \left (8 B +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{2} \left (8 B +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (8 B +7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{2} \left (8 B +7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {a^{2} \left (8 B +7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a^{2} \left (8 B +7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} \left (24 B +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (136 B +83 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(229\) |
(1/2*(B+C)*sin(2*d*x+2*c)+1/6*(1/2*B+C)*sin(3*d*x+3*c)+1/32*sin(4*d*x+4*c) *C+1/2*(7/2*B+3*C)*sin(d*x+c)+d*x*(B+7/8*C))*a^2/d
Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (8 \, B + 7 \, C\right )} a^{2} d x + {\left (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (5 \, B + 4 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(3*(8*B + 7*C)*a^2*d*x + (6*C*a^2*cos(d*x + c)^3 + 8*(B + 2*C)*a^2*co s(d*x + c)^2 + 3*(8*B + 7*C)*a^2*cos(d*x + c) + 8*(5*B + 4*C)*a^2)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (112) = 224\).
Time = 0.19 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.64 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} B a^{2} x \sin ^{2}{\left (c + d x \right )} + B a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 C a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Piecewise((B*a**2*x*sin(c + d*x)**2 + B*a**2*x*cos(c + d*x)**2 + 2*B*a**2* sin(c + d*x)**3/(3*d) + B*a**2*sin(c + d*x)*cos(c + d*x)**2/d + B*a**2*sin (c + d*x)*cos(c + d*x)/d + B*a**2*sin(c + d*x)/d + 3*C*a**2*x*sin(c + d*x) **4/8 + 3*C*a**2*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + C*a**2*x*sin(c + d* x)**2/2 + 3*C*a**2*x*cos(c + d*x)**4/8 + C*a**2*x*cos(c + d*x)**2/2 + 3*C* a**2*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 4*C*a**2*sin(c + d*x)**3/(3*d) + 5*C*a**2*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 2*C*a**2*sin(c + d*x)*cos(c + d*x)**2/d + C*a**2*sin(c + d*x)*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(B*co s(c) + C*cos(c)**2)*(a*cos(c) + a)**2, True))
Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \]
-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2 - 48*(2*d*x + 2*c + sin( 2*d*x + 2*c))*B*a^2 + 64*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2 - 3*(12*d *x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^2 - 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2 - 96*B*a^2*sin(d*x + c))/d
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.85 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (8 \, B a^{2} + 7 \, C a^{2}\right )} x + \frac {{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (B a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (7 \, B a^{2} + 6 \, C a^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
1/32*C*a^2*sin(4*d*x + 4*c)/d + 1/8*(8*B*a^2 + 7*C*a^2)*x + 1/12*(B*a^2 + 2*C*a^2)*sin(3*d*x + 3*c)/d + 1/2*(B*a^2 + C*a^2)*sin(2*d*x + 2*c)/d + 1/4 *(7*B*a^2 + 6*C*a^2)*sin(d*x + c)/d
Time = 1.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=B\,a^2\,x+\frac {7\,C\,a^2\,x}{8}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {C\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]