Integrand size = 40, antiderivative size = 125 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=a^3 C x+\frac {a^3 (5 B+7 C) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac {(5 B+3 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {a B (a+a \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
a^3*C*x+1/2*a^3*(5*B+7*C)*arctanh(sin(d*x+c))/d+5/2*a^3*(B+C)*tan(d*x+c)/d +1/6*(5*B+3*C)*(a^3+a^3*cos(d*x+c))*sec(d*x+c)*tan(d*x+c)/d+1/3*a*B*(a+a*c os(d*x+c))^2*sec(d*x+c)^2*tan(d*x+c)/d
Time = 1.74 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {a^3 \left (6 C d x+3 (5 B+7 C) \text {arctanh}(\sin (c+d x))+3 (8 B+6 C+(3 B+C) \sec (c+d x)) \tan (c+d x)+2 B \tan ^3(c+d x)\right )}{6 d} \]
(a^3*(6*C*d*x + 3*(5*B + 7*C)*ArcTanh[Sin[c + d*x]] + 3*(8*B + 6*C + (3*B + C)*Sec[c + d*x])*Tan[c + d*x] + 2*B*Tan[c + d*x]^3))/(6*d)
Time = 1.13 (sec) , antiderivative size = 131, 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, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a \cos (c+d x)+a)^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \sec ^4(c+d x) (a \cos (c+d x)+a)^3 (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{3} \int (\cos (c+d x) a+a)^2 (a (5 B+3 C)+3 a C \cos (c+d x)) \sec ^3(c+d x)dx+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (5 B+3 C)+3 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int 3 (\cos (c+d x) a+a) \left (5 (B+C) a^2+2 C \cos (c+d x) a^2\right ) \sec ^2(c+d x)dx+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left (5 (B+C) a^2+2 C \cos (c+d x) a^2\right ) \sec ^2(c+d x)dx+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (B+C) a^2+2 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \left (2 C \cos ^2(c+d x) a^3+5 (B+C) a^3+\left (2 C a^3+5 (B+C) a^3\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+5 (B+C) a^3+\left (2 C a^3+5 (B+C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\int \left ((5 B+7 C) a^3+2 C \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {5 a^3 (B+C) \tan (c+d x)}{d}\right )+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\int \frac {(5 B+7 C) a^3+2 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {5 a^3 (B+C) \tan (c+d x)}{d}\right )+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (a^3 (5 B+7 C) \int \sec (c+d x)dx+\frac {5 a^3 (B+C) \tan (c+d x)}{d}+2 a^3 C x\right )+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (a^3 (5 B+7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {5 a^3 (B+C) \tan (c+d x)}{d}+2 a^3 C x\right )+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {3}{2} \left (\frac {a^3 (5 B+7 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {5 a^3 (B+C) \tan (c+d x)}{d}+2 a^3 C x\right )+\frac {(5 B+3 C) \tan (c+d x) \sec (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{2 d}\right )+\frac {a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
(a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (((5*B + 3*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(2*a^3 *C*x + (a^3*(5*B + 7*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Tan[c + d*x])/d))/2)/3
3.3.50.3.1 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^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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 7.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14
| method | result | size |
| parts | \(\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 B \,a^{3}+C \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 B \,a^{3}+3 C \,a^{3}\right ) \tan \left (d x +c \right )}{d}-\frac {B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,a^{3} \left (d x +c \right )}{d}\) | \(142\) |
| parallelrisch | \(\frac {3 a^{3} \left (-\frac {5 \left (B +\frac {7 C}{5}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {5 \left (B +\frac {7 C}{5}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}+\frac {d x C \cos \left (3 d x +3 c \right )}{3}+\left (B +\frac {C}{3}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {11 B}{9}+C \right ) \sin \left (3 d x +3 c \right )+d x C \cos \left (d x +c \right )+\frac {5 \left (B +\frac {3 C}{5}\right ) \sin \left (d x +c \right )}{3}\right )}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(171\) |
| derivativedivides | \(\frac {C \,a^{3} \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+3 C \,a^{3} \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(176\) |
| default | \(\frac {C \,a^{3} \left (d x +c \right )+B \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{3} \tan \left (d x +c \right )+3 C \,a^{3} \tan \left (d x +c \right )+3 B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(176\) |
| risch | \(a^{3} C x -\frac {i a^{3} \left (9 B \,{\mathrm e}^{5 i \left (d x +c \right )}+3 C \,{\mathrm e}^{5 i \left (d x +c \right )}-18 B \,{\mathrm e}^{4 i \left (d x +c \right )}-18 C \,{\mathrm e}^{4 i \left (d x +c \right )}-48 B \,{\mathrm e}^{2 i \left (d x +c \right )}-36 C \,{\mathrm e}^{2 i \left (d x +c \right )}-9 B \,{\mathrm e}^{i \left (d x +c \right )}-3 C \,{\mathrm e}^{i \left (d x +c \right )}-22 B -18 C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {5 B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}-\frac {5 B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}\) | \(221\) |
(B*a^3+3*C*a^3)/d*ln(sec(d*x+c)+tan(d*x+c))+(3*B*a^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+3*C*a^3)/d*tan(d*x+ c)-B*a^3/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+C*a^3/d*(d*x+c)
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.13 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {12 \, C a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (11 \, B + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, B a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
integrate((a+a*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="fricas")
1/12*(12*C*a^3*d*x*cos(d*x + c)^3 + 3*(5*B + 7*C)*a^3*cos(d*x + c)^3*log(s in(d*x + c) + 1) - 3*(5*B + 7*C)*a^3*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(2*(11*B + 9*C)*a^3*cos(d*x + c)^2 + 3*(3*B + C)*a^3*cos(d*x + c) + 2 *B*a^3)*sin(d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.70 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 12 \, {\left (d x + c\right )} C a^{3} - 9 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{3} \tan \left (d x + c\right ) + 36 \, C a^{3} \tan \left (d x + c\right )}{12 \, d} \]
integrate((a+a*cos(d*x+c))^3*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="maxima")
1/12*(4*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3 + 12*(d*x + c)*C*a^3 - 9*B *a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(si n(d*x + c) - 1)) - 3*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)) + 6*B*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*C*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c ) - 1)) + 36*B*a^3*tan(d*x + c) + 36*C*a^3*tan(d*x + c))/d
Time = 0.35 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.51 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {6 \, {\left (d x + c\right )} C a^{3} + 3 \, {\left (5 \, B a^{3} + 7 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (5 \, B a^{3} + 7 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(6*(d*x + c)*C*a^3 + 3*(5*B*a^3 + 7*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c ) + 1)) - 3*(5*B*a^3 + 7*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 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 - 40*B*a^3 *tan(1/2*d*x + 1/2*c)^3 - 36*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 33*B*a^3*tan(1 /2*d*x + 1/2*c) + 21*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 = 209, normalized size of antiderivative = 1.67 \[ \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {5\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {7\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {11\,B\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {B\,a^3\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,a^3\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]
(5*B*a^3*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*C*a^3*atan(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (7*C*a^3*atanh(sin(c/2 + (d*x)/ 2)/cos(c/2 + (d*x)/2)))/d + (11*B*a^3*sin(c + d*x))/(3*d*cos(c + d*x)) + ( 3*B*a^3*sin(c + d*x))/(2*d*cos(c + d*x)^2) + (B*a^3*sin(c + d*x))/(3*d*cos (c + d*x)^3) + (3*C*a^3*sin(c + d*x))/(d*cos(c + d*x)) + (C*a^3*sin(c + d* x))/(2*d*cos(c + d*x)^2)