Integrand size = 39, antiderivative size = 156 \[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^3 (3 A+B) x+\frac {a^3 (6 A+7 B+5 C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {5 a^3 (B+C) \tan (c+d x)}{2 d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac {(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d} \]
a^3*(3*A+B)*x+1/2*a^3*(6*A+7*B+5*C)*arctanh(sin(d*x+c))/d+A*(a+a*sec(d*x+c ))^3*sin(d*x+c)/d+5/2*a^3*(B+C)*tan(d*x+c)/d-1/3*(3*A-C)*(a^2+a^2*sec(d*x+ c))^2*tan(d*x+c)/a/d-1/6*(6*A-3*B-5*C)*(a^3+a^3*sec(d*x+c))*tan(d*x+c)/d
Time = 5.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87 \[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (18 A d x+6 B d x+3 (6 A+7 B+5 C) \text {arctanh}(\sin (c+d x))+6 A \sin (c+d x)+18 B \tan (c+d x)+24 C \tan (c+d x)+3 B \sec (c+d x) \tan (c+d x)+9 C \sec (c+d x) \tan (c+d x)+6 A \sec ^2(c+d x) \tan (c+d x)-6 A \tan ^3(c+d x)+2 C \tan ^3(c+d x)\right )}{6 d} \]
(a^3*(18*A*d*x + 6*B*d*x + 3*(6*A + 7*B + 5*C)*ArcTanh[Sin[c + d*x]] + 6*A *Sin[c + d*x] + 18*B*Tan[c + d*x] + 24*C*Tan[c + d*x] + 3*B*Sec[c + d*x]*T an[c + d*x] + 9*C*Sec[c + d*x]*Tan[c + d*x] + 6*A*Sec[c + d*x]^2*Tan[c + d *x] - 6*A*Tan[c + d*x]^3 + 2*C*Tan[c + d*x]^3))/(6*d)
Time = 0.99 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4574, 3042, 4405, 3042, 4405, 27, 3042, 4402, 3042, 4254, 24, 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.
| \(\displaystyle \int \cos (c+d x) (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a)^3 (a (3 A+B)-a (3 A-C) \sec (c+d x))dx}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (3 A+B)-a (3 A-C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {1}{3} \int (\sec (c+d x) a+a)^2 \left (3 a^2 (3 A+B)-a^2 (6 A-3 B-5 C) \sec (c+d x)\right )dx-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a^2 (3 A+B)-a^2 (6 A-3 B-5 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 4405 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int 3 (\sec (c+d x) a+a) \left (2 (3 A+B) a^3+5 (B+C) \sec (c+d x) a^3\right )dx-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \int (\sec (c+d x) a+a) \left (2 (3 A+B) a^3+5 (B+C) \sec (c+d x) a^3\right )dx-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \int \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 (3 A+B) a^3+5 (B+C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 4402 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \left (a^4 (6 A+7 B+5 C) \int \sec (c+d x)dx+5 a^4 (B+C) \int \sec ^2(c+d x)dx+2 a^4 x (3 A+B)\right )-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \left (a^4 (6 A+7 B+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+5 a^4 (B+C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+2 a^4 x (3 A+B)\right )-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \left (a^4 (6 A+7 B+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {5 a^4 (B+C) \int 1d(-\tan (c+d x))}{d}+2 a^4 x (3 A+B)\right )-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \left (a^4 (6 A+7 B+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+2 a^4 x (3 A+B)+\frac {5 a^4 (B+C) \tan (c+d x)}{d}\right )-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} \left (\frac {a^4 (6 A+7 B+5 C) \text {arctanh}(\sin (c+d x))}{d}+2 a^4 x (3 A+B)+\frac {5 a^4 (B+C) \tan (c+d x)}{d}\right )-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{2 d}\right )-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 d}}{a}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}\) |
(A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + (-1/3*((3*A - C)*(a^2 + a^2*Se c[c + d*x])^2*Tan[c + d*x])/d + (-1/2*((6*A - 3*B - 5*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/d + (3*(2*a^4*(3*A + B)*x + (a^4*(6*A + 7*B + 5*C)*A rcTanh[Sin[c + d*x]])/d + (5*a^4*(B + C)*Tan[c + d*x])/d))/2)/3)/a
3.5.31.3.1 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[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[a*c*x, x] + (Simp[b*d Int[Csc[e + f*x]^2, x], x ] + Simp[(b*c + a*d) Int[Csc[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f }, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[1/m Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2 *m]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.59 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.30
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (-9 \left (A +\frac {7 B}{6}+\frac {5 C}{6}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (A +\frac {7 B}{6}+\frac {5 C}{6}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 x d \left (A +\frac {B}{3}\right ) \cos \left (3 d x +3 c \right )+\left (A +B +3 C \right ) \sin \left (2 d x +2 c \right )+\left (A +3 B +\frac {11 C}{3}\right ) \sin \left (3 d x +3 c \right )+\frac {A \sin \left (4 d x +4 c \right )}{2}+9 x d \left (A +\frac {B}{3}\right ) \cos \left (d x +c \right )+\sin \left (d x +c \right ) \left (A +3 B +5 C \right )\right )}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(203\) |
| derivativedivides | \(\frac {a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \left (d x +c \right )+3 B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \tan \left (d x +c \right )+a^{3} A \sin \left (d x +c \right )+B \,a^{3} \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(229\) |
| default | \(\frac {a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} A \left (d x +c \right )+3 B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \tan \left (d x +c \right )+a^{3} A \sin \left (d x +c \right )+B \,a^{3} \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(229\) |
| risch | \(3 a^{3} A x +a^{3} B x -\frac {i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} \left (3 B \,{\mathrm e}^{5 i \left (d x +c \right )}+9 C \,{\mathrm e}^{5 i \left (d x +c \right )}-6 A \,{\mathrm e}^{4 i \left (d x +c \right )}-18 B \,{\mathrm e}^{4 i \left (d x +c \right )}-18 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,{\mathrm e}^{2 i \left (d x +c \right )}-36 B \,{\mathrm e}^{2 i \left (d x +c \right )}-48 C \,{\mathrm e}^{2 i \left (d x +c \right )}-3 B \,{\mathrm e}^{i \left (d x +c \right )}-9 C \,{\mathrm e}^{i \left (d x +c \right )}-6 A -18 B -22 C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\) | \(337\) |
| norman | \(\frac {\left (3 a^{3} A +B \,a^{3}\right ) x +\left (-9 a^{3} A -3 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-9 a^{3} A -3 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (3 a^{3} A +B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (6 a^{3} A +2 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (6 a^{3} A +2 B \,a^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a^{3} \left (4 A +7 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{3} \left (B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {4 a^{3} \left (3 A -9 B -10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {2 a^{3} \left (6 A -B -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {4 a^{3} \left (9 B +9 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {a^{3} \left (6 A +7 B +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{3} \left (6 A +7 B +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(372\) |
a^3*(-9*(A+7/6*B+5/6*C)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x+1/2 *c)-1)+9*(A+7/6*B+5/6*C)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x+1/ 2*c)+1)+3*x*d*(A+1/3*B)*cos(3*d*x+3*c)+(A+B+3*C)*sin(2*d*x+2*c)+(A+3*B+11/ 3*C)*sin(3*d*x+3*c)+1/2*A*sin(4*d*x+4*c)+9*x*d*(A+1/3*B)*cos(d*x+c)+sin(d* x+c)*(A+3*B+5*C))/d/(cos(3*d*x+3*c)+3*cos(d*x+c))
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08 \[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (3 \, A + B\right )} a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (6 \, A + 7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (6 \, A + 7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A + 9 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
integrate(cos(d*x+c)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
1/12*(12*(3*A + B)*a^3*d*x*cos(d*x + c)^3 + 3*(6*A + 7*B + 5*C)*a^3*cos(d* x + c)^3*log(sin(d*x + c) + 1) - 3*(6*A + 7*B + 5*C)*a^3*cos(d*x + c)^3*lo g(-sin(d*x + c) + 1) + 2*(6*A*a^3*cos(d*x + c)^3 + 2*(3*A + 9*B + 11*C)*a^ 3*cos(d*x + c)^2 + 3*(B + 3*C)*a^3*cos(d*x + c) + 2*C*a^3)*sin(d*x + c))/( d*cos(d*x + c)^3)
\[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A \cos {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(A*cos(c + d*x), x) + Integral(3*A*cos(c + d*x)*sec(c + d*x) , x) + Integral(3*A*cos(c + d*x)*sec(c + d*x)**2, x) + Integral(A*cos(c + d*x)*sec(c + d*x)**3, x) + Integral(B*cos(c + d*x)*sec(c + d*x), x) + Inte gral(3*B*cos(c + d*x)*sec(c + d*x)**2, x) + Integral(3*B*cos(c + d*x)*sec( c + d*x)**3, x) + Integral(B*cos(c + d*x)*sec(c + d*x)**4, x) + Integral(C *cos(c + d*x)*sec(c + d*x)**2, x) + Integral(3*C*cos(c + d*x)*sec(c + d*x) **3, x) + Integral(3*C*cos(c + d*x)*sec(c + d*x)**4, x) + Integral(C*cos(c + d*x)*sec(c + d*x)**5, x))
Time = 0.23 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.76 \[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {36 \, {\left (d x + c\right )} A a^{3} + 12 \, {\left (d x + c\right )} B a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 3 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, A a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} \tan \left (d x + c\right ) + 36 \, B a^{3} \tan \left (d x + c\right ) + 36 \, C a^{3} \tan \left (d x + c\right )}{12 \, d} \]
integrate(cos(d*x+c)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
1/12*(36*(d*x + c)*A*a^3 + 12*(d*x + c)*B*a^3 + 4*(tan(d*x + c)^3 + 3*tan( d*x + c))*C*a^3 - 3*B*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d *x + c) + 1) + log(sin(d*x + c) - 1)) - 9*C*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 18*A*a^3*(lo g(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*B*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 6*C*a^3*(log(sin(d*x + c) + 1) - log(sin(d *x + c) - 1)) + 12*A*a^3*sin(d*x + c) + 12*A*a^3*tan(d*x + c) + 36*B*a^3*t an(d*x + c) + 36*C*a^3*tan(d*x + c))/d
Time = 0.37 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.85 \[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 6 \, {\left (3 \, A a^{3} + B a^{3}\right )} {\left (d x + c\right )} + 3 \, {\left (6 \, A a^{3} + 7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (6 \, A a^{3} + 7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(12*A*a^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 6*(3*A*a ^3 + B*a^3)*(d*x + c) + 3*(6*A*a^3 + 7*B*a^3 + 5*C*a^3)*log(abs(tan(1/2*d* x + 1/2*c) + 1)) - 3*(6*A*a^3 + 7*B*a^3 + 5*C*a^3)*log(abs(tan(1/2*d*x + 1 /2*c) - 1)) - 2*(6*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 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 - 12*A*a^3*tan(1/2*d*x + 1/2*c)^ 3 - 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 + 6* A*a^3*tan(1/2*d*x + 1/2*c) + 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 = 17.92 (sec) , antiderivative size = 523, normalized size of antiderivative = 3.35 \[ \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {3\,C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {11\,C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {5\,C\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {9\,A\,a^3\,\cos \left (c+d\,x\right )\,\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 )}{2}+\frac {9\,A\,a^3\,\cos \left (c+d\,x\right )\,\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 )}{2}+\frac {3\,B\,a^3\,\cos \left (c+d\,x\right )\,\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 )}{2}+\frac {21\,B\,a^3\,\cos \left (c+d\,x\right )\,\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 )}{4}+\frac {15\,C\,a^3\,\cos \left (c+d\,x\right )\,\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 )}{4}+\frac {3\,A\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {3\,A\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {7\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{4}+\frac {5\,C\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{4}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
((A*a^3*sin(2*c + 2*d*x))/4 + (A*a^3*sin(3*c + 3*d*x))/4 + (A*a^3*sin(4*c + 4*d*x))/8 + (B*a^3*sin(2*c + 2*d*x))/4 + (3*B*a^3*sin(3*c + 3*d*x))/4 + (3*C*a^3*sin(2*c + 2*d*x))/4 + (11*C*a^3*sin(3*c + 3*d*x))/12 + (A*a^3*sin (c + d*x))/4 + (3*B*a^3*sin(c + d*x))/4 + (5*C*a^3*sin(c + d*x))/4 + (9*A* a^3*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + (9*A*a^3 *cos(c + d*x)*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + (3*B*a^3*c os(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + (21*B*a^3*cos (c + d*x)*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/4 + (15*C*a^3*cos( c + d*x)*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/4 + (3*A*a^3*atan(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/2 + (3*A*a^3*atanh (sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/2 + (B*a^3*atan( sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/2 + (7*B*a^3*atan h(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/4 + (5*C*a^3*at anh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/4)/(d*((3*cos (c + d*x))/4 + cos(3*c + 3*d*x)/4))