Integrand size = 31, antiderivative size = 96 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=a^2 (2 A+C) x+\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 (A+C) \sin (c+d x)}{d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {C \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{3 d} \]
a^2*(2*A+C)*x+a^2*A*arctanh(sin(d*x+c))/d+a^2*(A+C)*sin(d*x+c)/d+1/3*C*(a+ a*cos(d*x+c))^2*sin(d*x+c)/d+1/3*C*(a^2+a^2*cos(d*x+c))*sin(d*x+c)/d
Time = 1.51 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.14 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {a^2 \left (24 A d x+12 C d x-12 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 (4 A+7 C) \sin (c+d x)+6 C \sin (2 (c+d x))+C \sin (3 (c+d x))\right )}{12 d} \]
(a^2*(24*A*d*x + 12*C*d*x - 12*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A + 7*C)*Sin[c + d* x] + 6*C*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)
Time = 0.87 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 3525, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3214, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+a)^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^2 (3 a A+2 a C \cos (c+d x)) \sec (c+d x)dx}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a A+2 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{2} \int 6 (\cos (c+d x) a+a) \left (A a^2+(A+C) \cos (c+d x) a^2\right ) \sec (c+d x)dx+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int (\cos (c+d x) a+a) \left (A a^2+(A+C) \cos (c+d x) a^2\right ) \sec (c+d x)dx+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (A a^2+(A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {3 \int \left ((A+C) \cos ^2(c+d x) a^3+A a^3+\left (A a^3+(A+C) a^3\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {(A+C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+A a^3+\left (A a^3+(A+C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {3 \left (\int \left (A a^3+(2 A+C) \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {a^3 (A+C) \sin (c+d x)}{d}\right )+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\int \frac {A a^3+(2 A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^3 (A+C) \sin (c+d x)}{d}\right )+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {3 \left (a^3 A \int \sec (c+d x)dx+\frac {a^3 (A+C) \sin (c+d x)}{d}+a^3 x (2 A+C)\right )+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (a^3 A \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {a^3 (A+C) \sin (c+d x)}{d}+a^3 x (2 A+C)\right )+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {3 \left (\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^3 (A+C) \sin (c+d x)}{d}+a^3 x (2 A+C)\right )+\frac {C \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{d}}{3 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\) |
(C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((C*(a^3 + a^3*Cos[c + d*x ])*Sin[c + d*x])/d + 3*(a^3*(2*A + C)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (a^3*(A + C)*Sin[c + d*x])/d))/(3*a)
3.1.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 4.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90
| method | result | size |
| parallelrisch | \(-\frac {\left (A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\sin \left (2 d x +2 c \right ) C}{2}-\frac {\sin \left (3 d x +3 c \right ) C}{12}+\left (-A -\frac {7 C}{4}\right ) \sin \left (d x +c \right )-2 d x \left (A +\frac {C}{2}\right )\right ) a^{2}}{d}\) | \(86\) |
| derivativedivides | \(\frac {A \,a^{2} \sin \left (d x +c \right )+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 A \,a^{2} \left (d x +c \right )+2 a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \sin \left (d x +c \right )}{d}\) | \(108\) |
| default | \(\frac {A \,a^{2} \sin \left (d x +c \right )+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 A \,a^{2} \left (d x +c \right )+2 a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \sin \left (d x +c \right )}{d}\) | \(108\) |
| parts | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,a^{2}+a^{2} C \right ) \sin \left (d x +c \right )}{d}+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {2 A \,a^{2} \left (d x +c \right )}{d}+\frac {2 a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(115\) |
| risch | \(2 a^{2} x A +a^{2} C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2}}{2 d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} a^{2} C}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2}}{2 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} C}{8 d}+\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(170\) |
| norman | \(\frac {\left (2 A \,a^{2}+a^{2} C \right ) x +\left (2 A \,a^{2}+a^{2} C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8 A \,a^{2}+4 a^{2} C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8 A \,a^{2}+4 a^{2} C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12 A \,a^{2}+6 a^{2} C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 a^{2} \left (A +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \left (A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{2} \left (9 A +11 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{2} \left (9 A +17 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(273\) |
-(A*ln(tan(1/2*d*x+1/2*c)-1)-A*ln(tan(1/2*d*x+1/2*c)+1)-1/2*sin(2*d*x+2*c) *C-1/12*sin(3*d*x+3*c)*C+(-A-7/4*C)*sin(d*x+c)-2*d*x*(A+1/2*C))*a^2/d
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.99 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {6 \, {\left (2 \, A + C\right )} a^{2} d x + 3 \, A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C a^{2} \cos \left (d x + c\right )^{2} + 3 \, C a^{2} \cos \left (d x + c\right ) + {\left (3 \, A + 5 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
1/6*(6*(2*A + C)*a^2*d*x + 3*A*a^2*log(sin(d*x + c) + 1) - 3*A*a^2*log(-si n(d*x + c) + 1) + 2*(C*a^2*cos(d*x + c)^2 + 3*C*a^2*cos(d*x + c) + (3*A + 5*C)*a^2)*sin(d*x + c))/d
\[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=a^{2} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 2 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 2 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
a**2*(Integral(A*sec(c + d*x), x) + Integral(2*A*cos(c + d*x)*sec(c + d*x) , x) + Integral(A*cos(c + d*x)**2*sec(c + d*x), x) + Integral(C*cos(c + d* x)**2*sec(c + d*x), x) + Integral(2*C*cos(c + d*x)**3*sec(c + d*x), x) + I ntegral(C*cos(c + d*x)**4*sec(c + d*x), x))
Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.11 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {12 \, {\left (d x + c\right )} A a^{2} - 2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 6 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 6 \, A a^{2} \sin \left (d x + c\right ) + 6 \, C a^{2} \sin \left (d x + c\right )}{6 \, d} \]
1/6*(12*(d*x + c)*A*a^2 - 2*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2 + 3*(2 *d*x + 2*c + sin(2*d*x + 2*c))*C*a^2 + 6*A*a^2*log(sec(d*x + c) + tan(d*x + c)) + 6*A*a^2*sin(d*x + c) + 6*C*a^2*sin(d*x + c))/d
Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.86 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {3 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (2 \, A a^{2} + C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
1/3*(3*A*a^2*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*A*a^2*log(abs(tan(1/2* d*x + 1/2*c) - 1)) + 3*(2*A*a^2 + C*a^2)*(d*x + c) + 2*(3*A*a^2*tan(1/2*d* x + 1/2*c)^5 + 3*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^2*tan(1/2*d*x + 1/2* c)^3 + 8*C*a^2*tan(1/2*d*x + 1/2*c)^3 + 3*A*a^2*tan(1/2*d*x + 1/2*c) + 9*C *a^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 1.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.66 \[ \int (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {7\,C\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {4\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
(A*a^2*sin(c + d*x))/d + (7*C*a^2*sin(c + d*x))/(4*d) + (4*A*a^2*atan(sin( c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*A*a^2*atanh(sin(c/2 + (d*x)/2)/ cos(c/2 + (d*x)/2)))/d + (2*C*a^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/ 2)))/d + (C*a^2*sin(2*c + 2*d*x))/(2*d) + (C*a^2*sin(3*c + 3*d*x))/(12*d)