Integrand size = 38, antiderivative size = 116 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {5}{8} a^3 (4 B+3 C) x+\frac {a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac {3 a^3 (4 B+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d} \]
5/8*a^3*(4*B+3*C)*x+a^3*(4*B+3*C)*sin(d*x+c)/d+3/8*a^3*(4*B+3*C)*cos(d*x+c )*sin(d*x+c)/d+1/4*C*(a+a*cos(d*x+c))^3*sin(d*x+c)/d-1/12*a^3*(4*B+3*C)*si n(d*x+c)^3/d
Time = 0.61 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {a^3 \sin (c+d x) \left (30 (4 B+3 C) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (88 B+72 C+9 (4 B+5 C) \cos (c+d x)+8 (B+3 C) \cos ^2(c+d x)+6 C \cos ^3(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{24 d \sqrt {\sin ^2(c+d x)}} \]
(a^3*Sin[c + d*x]*(30*(4*B + 3*C)*ArcSin[Sqrt[Sin[(c + d*x)/2]^2]] + (88*B + 72*C + 9*(4*B + 5*C)*Cos[c + d*x] + 8*(B + 3*C)*Cos[c + d*x]^2 + 6*C*Co s[c + d*x]^3)*Sqrt[Sin[c + d*x]^2]))/(24*d*Sqrt[Sin[c + d*x]^2])
Time = 0.46 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3042, 3508, 3042, 3230, 3042, 3124, 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.
| \(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+a)^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int (a \cos (c+d x)+a)^3 (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{4} (4 B+3 C) \int (\cos (c+d x) a+a)^3dx+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} (4 B+3 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3dx+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {1}{4} (4 B+3 C) \int \left (\cos ^3(c+d x) a^3+3 \cos ^2(c+d x) a^3+3 \cos (c+d x) a^3+a^3\right )dx+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} (4 B+3 C) \left (-\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {5 a^3 x}{2}\right )+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}\) |
(C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + ((4*B + 3*C)*((5*a^3*x)/2 + (4*a^3*Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3* Sin[c + d*x]^3)/(3*d)))/4
3.3.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 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[(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_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Time = 5.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68
| method | result | size |
| parallelrisch | \(\frac {5 \left (\frac {\left (\frac {3 B}{2}+2 C \right ) \sin \left (2 d x +2 c \right )}{5}+\frac {\left (\frac {B}{3}+C \right ) \sin \left (3 d x +3 c \right )}{10}+\frac {\sin \left (4 d x +4 c \right ) C}{80}+\frac {\left (3 B +\frac {13 C}{5}\right ) \sin \left (d x +c \right )}{2}+d x \left (B +\frac {3 C}{4}\right )\right ) a^{3}}{2 d}\) | \(79\) |
| risch | \(\frac {5 a^{3} B x}{2}+\frac {15 a^{3} C x}{8}+\frac {15 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {13 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (4 d x +4 c \right ) C \,a^{3}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{3}}{d}\) | \(135\) |
| parts | \(\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (3 B \,a^{3}+C \,a^{3}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (3 B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \,a^{3} \left (d x +c \right )}{d}+\frac {C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(150\) |
| derivativedivides | \(\frac {C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \sin \left (d x +c \right ) a^{3}+C \,a^{3} \sin \left (d x +c \right )+B \,a^{3} \left (d x +c \right )}{d}\) | \(176\) |
| default | \(\frac {C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \sin \left (d x +c \right ) a^{3}+C \,a^{3} \sin \left (d x +c \right )+B \,a^{3} \left (d x +c \right )}{d}\) | \(176\) |
| norman | \(\frac {\frac {5 a^{3} \left (4 B +3 C \right ) x}{8}+\frac {32 a^{3} \left (4 B +3 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {35 a^{3} \left (4 B +3 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {5 a^{3} \left (4 B +3 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {25 a^{3} \left (4 B +3 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {25 a^{3} \left (4 B +3 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {25 a^{3} \left (4 B +3 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {25 a^{3} \left (4 B +3 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a^{3} \left (4 B +3 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (44 B +49 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (212 B +183 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(279\) |
5/2*(1/5*(3/2*B+2*C)*sin(2*d*x+2*c)+1/10*(1/3*B+C)*sin(3*d*x+3*c)+1/80*sin (4*d*x+4*c)*C+1/2*(3*B+13/5*C)*sin(d*x+c)+d*x*(B+3/4*C))*a^3/d
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {15 \, {\left (4 \, B + 3 \, C\right )} a^{3} d x + {\left (6 \, C a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 9 \, {\left (4 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (11 \, B + 9 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{24 \, d} \]
1/24*(15*(4*B + 3*C)*a^3*d*x + (6*C*a^3*cos(d*x + c)^3 + 8*(B + 3*C)*a^3*c os(d*x + c)^2 + 9*(4*B + 5*C)*a^3*cos(d*x + c) + 8*(11*B + 9*C)*a^3)*sin(d *x + c))/d
\[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=a^{3} \left (\int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(B*cos(c + d*x)*sec(c + d*x), x) + Integral(3*B*cos(c + d*x) **2*sec(c + d*x), x) + Integral(3*B*cos(c + d*x)**3*sec(c + d*x), x) + Int egral(B*cos(c + d*x)**4*sec(c + d*x), x) + Integral(C*cos(c + d*x)**2*sec( c + d*x), x) + Integral(3*C*cos(c + d*x)**3*sec(c + d*x), x) + Integral(3* C*cos(c + d*x)**4*sec(c + d*x), x) + Integral(C*cos(c + d*x)**5*sec(c + d* x), x))
Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.44 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 96 \, {\left (d x + c\right )} B a^{3} + 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 288 \, B a^{3} \sin \left (d x + c\right ) - 96 \, C a^{3} \sin \left (d x + c\right )}{96 \, d} \]
-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 - 72*(2*d*x + 2*c + sin( 2*d*x + 2*c))*B*a^3 - 96*(d*x + c)*B*a^3 + 96*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C* a^3 - 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 - 288*B*a^3*sin(d*x + c) - 96*C*a^3*sin(d*x + c))/d
Time = 0.31 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.52 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {15 \, {\left (4 \, B a^{3} + 3 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (60 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 220 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 165 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 292 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 219 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 132 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 147 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
1/24*(15*(4*B*a^3 + 3*C*a^3)*(d*x + c) + 2*(60*B*a^3*tan(1/2*d*x + 1/2*c)^ 7 + 45*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 220*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 1 65*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 292*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 219*C *a^3*tan(1/2*d*x + 1/2*c)^3 + 132*B*a^3*tan(1/2*d*x + 1/2*c) + 147*C*a^3*t an(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 1.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {5\,B\,a^3\,x}{2}+\frac {15\,C\,a^3\,x}{8}+\frac {15\,B\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {13\,C\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]