Integrand size = 40, antiderivative size = 185 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3 (13 B+15 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^3 (38 B+45 C) \tan (c+d x)}{15 d}+\frac {a^3 (13 B+15 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^3 (43 B+45 C) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(7 B+5 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {a B (a+a \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d} \]
1/8*a^3*(13*B+15*C)*arctanh(sin(d*x+c))/d+1/15*a^3*(38*B+45*C)*tan(d*x+c)/ d+1/8*a^3*(13*B+15*C)*sec(d*x+c)*tan(d*x+c)/d+1/60*a^3*(43*B+45*C)*sec(d*x +c)^2*tan(d*x+c)/d+1/20*(7*B+5*C)*(a^3+a^3*cos(d*x+c))*sec(d*x+c)^3*tan(d* x+c)/d+1/5*a*B*(a+a*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/d
Time = 2.82 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.55 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3 \left (15 (13 B+15 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 (13 B+15 C) \sec (c+d x)+30 (3 B+C) \sec ^3(c+d x)+8 \left (60 (B+C)+5 (5 B+3 C) \tan ^2(c+d x)+3 B \tan ^4(c+d x)\right )\right )\right )}{120 d} \]
(a^3*(15*(13*B + 15*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(13*B + 15 *C)*Sec[c + d*x] + 30*(3*B + C)*Sec[c + d*x]^3 + 8*(60*(B + C) + 5*(5*B + 3*C)*Tan[c + d*x]^2 + 3*B*Tan[c + d*x]^4))))/(120*d)
Time = 1.45 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.03, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 3508, 3042, 3454, 3042, 3454, 3042, 3447, 3042, 3500, 3042, 3227, 3042, 4254, 24, 4255, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a \cos (c+d x)+a)^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \sec ^6(c+d x) (a \cos (c+d x)+a)^3 (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{5} \int (\cos (c+d x) a+a)^2 (a (7 B+5 C)+a (2 B+5 C) \cos (c+d x)) \sec ^5(c+d x)dx+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (7 B+5 C)+a (2 B+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int (\cos (c+d x) a+a) \left ((43 B+45 C) a^2+2 (11 B+15 C) \cos (c+d x) a^2\right ) \sec ^4(c+d x)dx+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((43 B+45 C) a^2+2 (11 B+15 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \left (2 (11 B+15 C) \cos ^2(c+d x) a^3+(43 B+45 C) a^3+\left (2 (11 B+15 C) a^3+(43 B+45 C) a^3\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {2 (11 B+15 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+(43 B+45 C) a^3+\left (2 (11 B+15 C) a^3+(43 B+45 C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \left (15 (13 B+15 C) a^3+4 (38 B+45 C) \cos (c+d x) a^3\right ) \sec ^3(c+d x)dx+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {15 (13 B+15 C) a^3+4 (38 B+45 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \int \sec ^3(c+d x)dx+4 a^3 (38 B+45 C) \int \sec ^2(c+d x)dx\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 B+45 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+15 a^3 (13 B+15 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 a^3 (38 B+45 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {4 a^3 (38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {4 a^3 (38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {4 a^3 (38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {4 a^3 (38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\) |
(a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (((7*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((a^3*( 43*B + 45*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((4*a^3*(38*B + 45*C)*Ta n[c + d*x])/d + 15*a^3*(13*B + 15*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/3)/4)/5
3.3.52.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, 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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
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_)]^(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_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 10.96 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.06
| method | result | size |
| parts | \(\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 B \,a^{3}+C \,a^{3}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 B \,a^{3}+3 C \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}-\frac {B \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,a^{3} \tan \left (d x +c \right )}{d}\) | \(196\) |
| parallelrisch | \(\frac {40 \left (-\frac {39 \left (B +\frac {15 C}{13}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32}+\frac {39 \left (B +\frac {15 C}{13}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{10}+\frac {\cos \left (3 d x +3 c \right )}{2}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32}+\left (\frac {15 B}{16}+\frac {57 C}{80}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {19 B}{20}+\frac {39 C}{40}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {39 B}{160}+\frac {9 C}{32}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {19 B}{100}+\frac {9 C}{40}\right ) \sin \left (5 d x +5 c \right )+\sin \left (d x +c \right ) \left (B +\frac {3 C}{4}\right )\right ) a^{3}}{3 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(217\) |
| derivativedivides | \(\frac {C \,a^{3} \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-3 C \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(271\) |
| default | \(\frac {C \,a^{3} \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-3 C \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(271\) |
| risch | \(-\frac {i a^{3} \left (195 B \,{\mathrm e}^{9 i \left (d x +c \right )}+225 C \,{\mathrm e}^{9 i \left (d x +c \right )}-120 C \,{\mathrm e}^{8 i \left (d x +c \right )}+750 B \,{\mathrm e}^{7 i \left (d x +c \right )}+570 C \,{\mathrm e}^{7 i \left (d x +c \right )}-720 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1200 C \,{\mathrm e}^{6 i \left (d x +c \right )}-2320 B \,{\mathrm e}^{4 i \left (d x +c \right )}-2400 C \,{\mathrm e}^{4 i \left (d x +c \right )}-750 B \,{\mathrm e}^{3 i \left (d x +c \right )}-570 C \,{\mathrm e}^{3 i \left (d x +c \right )}-1520 B \,{\mathrm e}^{2 i \left (d x +c \right )}-1680 C \,{\mathrm e}^{2 i \left (d x +c \right )}-195 B \,{\mathrm e}^{i \left (d x +c \right )}-225 C \,{\mathrm e}^{i \left (d x +c \right )}-304 B -360 C \right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {13 B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {13 B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\) | \(299\) |
(B*a^3+3*C*a^3)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)) )+(3*B*a^3+C*a^3)/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln (sec(d*x+c)+tan(d*x+c)))-(3*B*a^3+3*C*a^3)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d *x+c)-B*a^3/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+C*a^3/ d*tan(d*x+c)
Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.89 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left (13 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (13 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (38 \, B + 45 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (13 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (19 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) + 24 \, B a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
integrate((a+a*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="fricas")
1/240*(15*(13*B + 15*C)*a^3*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(13* B + 15*C)*a^3*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(8*(38*B + 45*C)*a ^3*cos(d*x + c)^4 + 15*(13*B + 15*C)*a^3*cos(d*x + c)^3 + 8*(19*B + 15*C)* a^3*cos(d*x + c)^2 + 30*(3*B + C)*a^3*cos(d*x + c) + 24*B*a^3)*sin(d*x + c ))/(d*cos(d*x + c)^5)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.82 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 45 \, B a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, C a^{3} \tan \left (d x + c\right )}{240 \, d} \]
integrate((a+a*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="maxima")
1/240*(16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^3 + 240*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3 + 240*(tan(d*x + c)^3 + 3*tan (d*x + c))*C*a^3 - 45*B*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d* x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 15*C*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c )^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 60*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)) - 180*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)) + 240*C*a^3*tan(d*x + c))/d
Time = 0.35 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.33 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left (13 \, B a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (13 \, B a^{3} + 15 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (195 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 225 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 910 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1050 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1664 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1920 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1330 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1830 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 765 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
1/120*(15*(13*B*a^3 + 15*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(1 3*B*a^3 + 15*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(195*B*a^3*tan( 1/2*d*x + 1/2*c)^9 + 225*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 910*B*a^3*tan(1/2* d*x + 1/2*c)^7 - 1050*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 1664*B*a^3*tan(1/2*d* x + 1/2*c)^5 + 1920*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 1330*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 1830*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 765*B*a^3*tan(1/2*d*x + 1 /2*c) + 735*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d
Time = 4.06 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.21 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (13\,B+15\,C\right )}{4\,d}-\frac {\left (\frac {13\,B\,a^3}{4}+\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {91\,B\,a^3}{6}-\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {416\,B\,a^3}{15}+32\,C\,a^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {133\,B\,a^3}{6}-\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,B\,a^3}{4}+\frac {49\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
(a^3*atanh(tan(c/2 + (d*x)/2))*(13*B + 15*C))/(4*d) - (tan(c/2 + (d*x)/2)* ((51*B*a^3)/4 + (49*C*a^3)/4) + tan(c/2 + (d*x)/2)^9*((13*B*a^3)/4 + (15*C *a^3)/4) - tan(c/2 + (d*x)/2)^7*((91*B*a^3)/6 + (35*C*a^3)/2) - tan(c/2 + (d*x)/2)^3*((133*B*a^3)/6 + (61*C*a^3)/2) + tan(c/2 + (d*x)/2)^5*((416*B*a ^3)/15 + 32*C*a^3))/(d*(5*tan(c/2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 - 1))