Integrand size = 40, antiderivative size = 111 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{2} a^3 (7 B+5 C) x+\frac {a^3 B \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac {a C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {(3 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{6 d} \]
1/2*a^3*(7*B+5*C)*x+a^3*B*arctanh(sin(d*x+c))/d+5/2*a^3*(B+C)*sin(d*x+c)/d +1/3*a*C*(a+a*cos(d*x+c))^2*sin(d*x+c)/d+1/6*(3*B+5*C)*(a^3+a^3*cos(d*x+c) )*sin(d*x+c)/d
Time = 1.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {a^3 \left (42 B d x+30 C d x-12 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 (4 B+5 C) \sin (c+d x)+3 (B+3 C) \sin (2 (c+d x))+C \sin (3 (c+d x))\right )}{12 d} \]
(a^3*(42*B*d*x + 30*C*d*x - 12*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*(4*B + 5*C)*Sin[c + d* x] + 3*(B + 3*C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)
Time = 1.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3508, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3214, 3042, 4257}
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.
| \(\displaystyle \int \sec ^2(c+d x) (a \cos (c+d x)+a)^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+a)^3 (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{3} \int (\cos (c+d x) a+a)^2 (3 a B+a (3 B+5 C) \cos (c+d x)) \sec (c+d x)dx+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a B+a (3 B+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int 3 (\cos (c+d x) a+a) \left (2 B a^2+5 (B+C) \cos (c+d x) a^2\right ) \sec (c+d x)dx+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left (2 B a^2+5 (B+C) \cos (c+d x) a^2\right ) \sec (c+d x)dx+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 B a^2+5 (B+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \left (5 (B+C) \cos ^2(c+d x) a^3+2 B a^3+\left (2 B a^3+5 (B+C) a^3\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {5 (B+C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+2 B a^3+\left (2 B a^3+5 (B+C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\int \left (2 B a^3+(7 B+5 C) \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {5 a^3 (B+C) \sin (c+d x)}{d}\right )+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\int \frac {2 B a^3+(7 B+5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 a^3 (B+C) \sin (c+d x)}{d}\right )+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (2 a^3 B \int \sec (c+d x)dx+\frac {5 a^3 (B+C) \sin (c+d x)}{d}+a^3 x (7 B+5 C)\right )+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (2 a^3 B \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^3 (B+C) \sin (c+d x)}{d}+a^3 x (7 B+5 C)\right )+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\frac {2 a^3 B \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (B+C) \sin (c+d x)}{d}+a^3 x (7 B+5 C)\right )+\frac {(3 B+5 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
(a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (((3*B + 5*C)*(a^3 + a^3 *Cos[c + d*x])*Sin[c + d*x])/(2*d) + (3*(a^3*(7*B + 5*C)*x + (2*a^3*B*ArcT anh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/d))/2)/3
3.3.47.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
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]
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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 5.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(-\frac {\left (B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-B \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {\left (-B -3 C \right ) \sin \left (2 d x +2 c \right )}{4}-\frac {\sin \left (3 d x +3 c \right ) C}{12}+3 \left (-B -\frac {5 C}{4}\right ) \sin \left (d x +c \right )-\frac {7 \left (B +\frac {5 C}{7}\right ) x d}{2}\right ) a^{3}}{d}\) | \(93\) |
| parts | \(\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 B \,a^{3}+C \,a^{3}\right ) \left (d x +c \right )}{d}+\frac {\left (3 B \,a^{3}+3 C \,a^{3}\right ) \sin \left (d x +c \right )}{d}+\frac {B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(131\) |
| derivativedivides | \(\frac {\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \sin \left (d x +c \right ) a^{3}+3 C \,a^{3} \sin \left (d x +c \right )+3 B \,a^{3} \left (d x +c \right )+C \,a^{3} \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(147\) |
| default | \(\frac {\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \sin \left (d x +c \right ) a^{3}+3 C \,a^{3} \sin \left (d x +c \right )+3 B \,a^{3} \left (d x +c \right )+C \,a^{3} \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(147\) |
| risch | \(\frac {7 a^{3} B x}{2}+\frac {5 a^{3} C x}{2}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3}}{2 d}-\frac {15 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{3}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3}}{2 d}+\frac {15 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{3}}{8 d}+\frac {B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C \,a^{3}}{4 d}\) | \(189\) |
| norman | \(\frac {\left (-\frac {7}{2} B \,a^{3}-\frac {5}{2} C \,a^{3}\right ) x +\left (-\frac {35}{2} B \,a^{3}-\frac {25}{2} C \,a^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7}{2} B \,a^{3}+\frac {5}{2} C \,a^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35}{2} B \,a^{3}+\frac {25}{2} C \,a^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-14 B \,a^{3}-10 C \,a^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (14 B \,a^{3}+10 C \,a^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 a^{3} \left (B +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3} \left (7 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{3} \left (15 B +11 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{3} \left (21 B +29 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{3} \left (51 B +55 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{3} \left (57 B +73 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {B \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {B \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(370\) |
-(B*ln(tan(1/2*d*x+1/2*c)-1)-B*ln(tan(1/2*d*x+1/2*c)+1)+1/4*(-B-3*C)*sin(2 *d*x+2*c)-1/12*sin(3*d*x+3*c)*C+3*(-B-5/4*C)*sin(d*x+c)-7/2*(B+5/7*C)*x*d) *a^3/d
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (7 \, B + 5 \, C\right )} a^{3} d x + 3 \, B a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, B a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (2 \, C a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, {\left (9 \, B + 11 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
integrate((a+a*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2,x, algorithm="fricas")
1/6*(3*(7*B + 5*C)*a^3*d*x + 3*B*a^3*log(sin(d*x + c) + 1) - 3*B*a^3*log(- sin(d*x + c) + 1) + (2*C*a^3*cos(d*x + c)^2 + 3*(B + 3*C)*a^3*cos(d*x + c) + 2*(9*B + 11*C)*a^3)*sin(d*x + c))/d
\[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=a^{3} \left (\int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(B*cos(c + d*x)*sec(c + d*x)**2, x) + Integral(3*B*cos(c + d *x)**2*sec(c + d*x)**2, x) + Integral(3*B*cos(c + d*x)**3*sec(c + d*x)**2, x) + Integral(B*cos(c + d*x)**4*sec(c + d*x)**2, x) + Integral(C*cos(c + d*x)**2*sec(c + d*x)**2, x) + Integral(3*C*cos(c + d*x)**3*sec(c + d*x)**2 , x) + Integral(3*C*cos(c + d*x)**4*sec(c + d*x)**2, x) + Integral(C*cos(c + d*x)**5*sec(c + d*x)**2, x))
Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 36 \, {\left (d x + c\right )} B a^{3} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 12 \, {\left (d x + c\right )} C a^{3} + 6 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{3} \sin \left (d x + c\right ) + 36 \, C a^{3} \sin \left (d x + c\right )}{12 \, d} \]
integrate((a+a*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2,x, algorithm="maxima")
1/12*(3*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3 + 36*(d*x + c)*B*a^3 - 4*(s in(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 + 9*(2*d*x + 2*c + sin(2*d*x + 2*c)) *C*a^3 + 12*(d*x + c)*C*a^3 + 6*B*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*B*a^3*sin(d*x + c) + 36*C*a^3*sin(d*x + c))/d
Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.62 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {6 \, B a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, B a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (7 \, B a^{3} + 5 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
1/6*(6*B*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 6*B*a^3*log(abs(tan(1/2* d*x + 1/2*c) - 1)) + 3*(7*B*a^3 + 5*C*a^3)*(d*x + c) + 2*(15*B*a^3*tan(1/2 *d*x + 1/2*c)^5 + 15*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 40*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 21*B*a^3*tan(1/2*d*x + 1/2*c ) + 33*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 1.42 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.60 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{d}+\frac {15\,C\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {7\,B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {5\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
(3*B*a^3*sin(c + d*x))/d + (15*C*a^3*sin(c + d*x))/(4*d) + (7*B*a^3*atan(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*B*a^3*atanh(sin(c/2 + (d*x)/ 2)/cos(c/2 + (d*x)/2)))/d + (5*C*a^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d* x)/2)))/d + (B*a^3*sin(2*c + 2*d*x))/(4*d) + (3*C*a^3*sin(2*c + 2*d*x))/(4 *d) + (C*a^3*sin(3*c + 3*d*x))/(12*d)