Integrand size = 40, antiderivative size = 144 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^2 (7 B+8 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 (4 B+5 C) \tan (c+d x)}{3 d}+\frac {a^2 (7 B+8 C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^2 (5 B+4 C) \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {B \left (a^2+a^2 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Output:
1/8*a^2*(7*B+8*C)*arctanh(sin(d*x+c))/d+1/3*a^2*(4*B+5*C)*tan(d*x+c)/d+1/8 *a^2*(7*B+8*C)*sec(d*x+c)*tan(d*x+c)/d+1/12*a^2*(5*B+4*C)*sec(d*x+c)^2*tan (d*x+c)/d+1/4*B*(a^2+a^2*cos(d*x+c))*sec(d*x+c)^3*tan(d*x+c)/d
Time = 1.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.56 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^2 \left (3 (7 B+8 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (48 (B+C)+3 (7 B+8 C) \sec (c+d x)+6 B \sec ^3(c+d x)+8 (2 B+C) \tan ^2(c+d x)\right )\right )}{24 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]
Output:
(a^2*(3*(7*B + 8*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(48*(B + C) + 3*( 7*B + 8*C)*Sec[c + d*x] + 6*B*Sec[c + d*x]^3 + 8*(2*B + C)*Tan[c + d*x]^2) ))/(24*d)
Time = 1.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3508, 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.
| \(\displaystyle \int \sec ^6(c+d x) (a \cos (c+d x)+a)^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \sec ^5(c+d x) (a \cos (c+d x)+a)^2 (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{4} \int (\cos (c+d x) a+a) (a (5 B+4 C)+2 a (B+2 C) \cos (c+d x)) \sec ^4(c+d x)dx+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a (5 B+4 C)+2 a (B+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{4} \int \left (2 (B+2 C) \cos ^2(c+d x) a^2+(5 B+4 C) a^2+\left (2 (B+2 C) a^2+(5 B+4 C) a^2\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {2 (B+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+(5 B+4 C) a^2+\left (2 (B+2 C) a^2+(5 B+4 C) a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \left (3 (7 B+8 C) a^2+4 (4 B+5 C) \cos (c+d x) a^2\right ) \sec ^3(c+d x)dx+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {3 (7 B+8 C) a^2+4 (4 B+5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (3 a^2 (7 B+8 C) \int \sec ^3(c+d x)dx+4 a^2 (4 B+5 C) \int \sec ^2(c+d x)dx\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (4 a^2 (4 B+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+3 a^2 (7 B+8 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (3 a^2 (7 B+8 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 a^2 (4 B+5 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (3 a^2 (7 B+8 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {4 a^2 (4 B+5 C) \tan (c+d x)}{d}\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (3 a^2 (7 B+8 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {4 a^2 (4 B+5 C) \tan (c+d x)}{d}\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (3 a^2 (7 B+8 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {4 a^2 (4 B+5 C) \tan (c+d x)}{d}\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (3 a^2 (7 B+8 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {4 a^2 (4 B+5 C) \tan (c+d x)}{d}\right )+\frac {a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {B \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{4 d}\) |
Input:
Int[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x ]^6,x]
Output:
(B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((a^2*(5* B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((4*a^2*(4*B + 5*C)*Tan[c + d*x])/d + 3*a^2*(7*B + 8*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*T an[c + d*x])/(2*d)))/3)/4
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 = 3.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(\frac {\left (B \,a^{2}+2 a^{2} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (2 B \,a^{2}+a^{2} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {a^{2} C \tan \left (d x +c \right )}{d}\) | \(150\) |
| parallelrisch | \(\frac {16 a^{2} \left (-\frac {21 \left (B +\frac {8 C}{7}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32}+\frac {21 \left (B +\frac {8 C}{7}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32}+\left (B +\frac {7 C}{8}\right ) \sin \left (2 d x +2 c \right )+\frac {3 \left (\frac {7 B}{8}+C \right ) \sin \left (3 d x +3 c \right )}{8}+\frac {\left (B +\frac {5 C}{4}\right ) \sin \left (4 d x +4 c \right )}{4}+\frac {45 \left (B +\frac {8 C}{15}\right ) \sin \left (d x +c \right )}{64}\right )}{3 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(176\) |
| derivativedivides | \(\frac {a^{2} C \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 a^{2} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-2 B \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{2} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(187\) |
| default | \(\frac {a^{2} C \tan \left (d x +c \right )+B \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 a^{2} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-2 B \,a^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-a^{2} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(187\) |
| risch | \(-\frac {i a^{2} \left (21 B \,{\mathrm e}^{7 i \left (d x +c \right )}+24 C \,{\mathrm e}^{7 i \left (d x +c \right )}-24 C \,{\mathrm e}^{6 i \left (d x +c \right )}+45 B \,{\mathrm e}^{5 i \left (d x +c \right )}+24 C \,{\mathrm e}^{5 i \left (d x +c \right )}-96 B \,{\mathrm e}^{4 i \left (d x +c \right )}-120 C \,{\mathrm e}^{4 i \left (d x +c \right )}-45 B \,{\mathrm e}^{3 i \left (d x +c \right )}-24 C \,{\mathrm e}^{3 i \left (d x +c \right )}-128 B \,{\mathrm e}^{2 i \left (d x +c \right )}-136 C \,{\mathrm e}^{2 i \left (d x +c \right )}-21 B \,{\mathrm e}^{i \left (d x +c \right )}-24 C \,{\mathrm e}^{i \left (d x +c \right )}-32 B -40 C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {7 B \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {7 B \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(274\) |
Input:
int((a+a*cos(d*x+c))^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x,method =_RETURNVERBOSE)
Output:
(B*a^2+2*C*a^2)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)) )-(2*B*a^2+C*a^2)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+B*a^2/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)))+a^2*C/ d*tan(d*x+c)
Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 \, {\left (7 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (7 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (4 \, B + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (7 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right ) + 6 \, B a^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate((a+a*cos(d*x+c))^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="fricas")
Output:
1/48*(3*(7*B + 8*C)*a^2*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(7*B + 8* C)*a^2*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(8*(4*B + 5*C)*a^2*cos(d* x + c)^3 + 3*(7*B + 8*C)*a^2*cos(d*x + c)^2 + 8*(2*B + C)*a^2*cos(d*x + c) + 6*B*a^2)*sin(d*x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**2*(B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**6 ,x)
Output:
Timed out
Time = 0.07 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.60 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {32 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} - 3 \, B a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, C a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C a^{2} \tan \left (d x + c\right )}{48 \, d} \] Input:
integrate((a+a*cos(d*x+c))^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="maxima")
Output:
1/48*(32*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^2 + 16*(tan(d*x + c)^3 + 3* tan(d*x + c))*C*a^2 - 3*B*a^2*(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)) - 12*B*a^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d* x + c) + 1) + log(sin(d*x + c) - 1)) - 24*C*a^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 48*C*a^2*tan (d*x + c))/d
Time = 0.15 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.47 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 \, {\left (7 \, B a^{2} + 8 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (7 \, B a^{2} + 8 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (21 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 77 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 88 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 83 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 136 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \] Input:
integrate((a+a*cos(d*x+c))^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="giac")
Output:
1/24*(3*(7*B*a^2 + 8*C*a^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(7*B*a^ 2 + 8*C*a^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(21*B*a^2*tan(1/2*d*x + 1/2*c)^7 + 24*C*a^2*tan(1/2*d*x + 1/2*c)^7 - 77*B*a^2*tan(1/2*d*x + 1/2* c)^5 - 88*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 83*B*a^2*tan(1/2*d*x + 1/2*c)^3 + 136*C*a^2*tan(1/2*d*x + 1/2*c)^3 - 75*B*a^2*tan(1/2*d*x + 1/2*c) - 72*C*a ^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d
Time = 1.74 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.27 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\left (-\frac {7\,B\,a^2}{4}-2\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {77\,B\,a^2}{12}+\frac {22\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {83\,B\,a^2}{12}-\frac {34\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {25\,B\,a^2}{4}+6\,C\,a^2\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {7\,B}{8}+C\right )}{d} \] Input:
int(((B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2)/cos(c + d *x)^6,x)
Output:
(tan(c/2 + (d*x)/2)*((25*B*a^2)/4 + 6*C*a^2) - tan(c/2 + (d*x)/2)^7*((7*B* a^2)/4 + 2*C*a^2) + tan(c/2 + (d*x)/2)^5*((77*B*a^2)/12 + (22*C*a^2)/3) - tan(c/2 + (d*x)/2)^3*((83*B*a^2)/12 + (34*C*a^2)/3))/(d*(6*tan(c/2 + (d*x) /2)^4 - 4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/ 2)^8 + 1)) + (2*a^2*atanh(tan(c/2 + (d*x)/2))*((7*B)/8 + C))/d
Time = 0.16 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.63 \[ \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {a^{2} \left (-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +48 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +48 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c -21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} b -24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} c +42 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} b +48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} c -21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b -24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) c +21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} b +24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} c -42 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} b -48 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} c +21 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b +24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) c -21 \sin \left (d x +c \right )^{3} b -24 \sin \left (d x +c \right )^{3} c +27 \sin \left (d x +c \right ) b +24 \sin \left (d x +c \right ) c \right )}{24 d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:
int((a+a*cos(d*x+c))^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x)
Output:
(a**2*( - 32*cos(c + d*x)*sin(c + d*x)**3*b - 40*cos(c + d*x)*sin(c + d*x) **3*c + 48*cos(c + d*x)*sin(c + d*x)*b + 48*cos(c + d*x)*sin(c + d*x)*c - 21*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*b - 24*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*c + 42*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b + 4 8*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*c - 21*log(tan((c + d*x)/2) - 1)*b - 24*log(tan((c + d*x)/2) - 1)*c + 21*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b + 24*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*c - 42*log(tan ((c + d*x)/2) + 1)*sin(c + d*x)**2*b - 48*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*c + 21*log(tan((c + d*x)/2) + 1)*b + 24*log(tan((c + d*x)/2) + 1 )*c - 21*sin(c + d*x)**3*b - 24*sin(c + d*x)**3*c + 27*sin(c + d*x)*b + 24 *sin(c + d*x)*c))/(24*d*(sin(c + d*x)**4 - 2*sin(c + d*x)**2 + 1))