Integrand size = 31, antiderivative size = 103 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=a b (2 A+C) x+\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin (c+d x)}{3 d}+\frac {a b C \cos (c+d x) \sin (c+d x)}{3 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d} \]
a*b*(2*A+C)*x+a^2*A*arctanh(sin(d*x+c))/d+1/3*(3*A*b^2+2*(a^2+b^2)*C)*sin( d*x+c)/d+1/3*a*b*C*cos(d*x+c)*sin(d*x+c)/d+1/3*C*(a+b*cos(d*x+c))^2*sin(d* x+c)/d
Time = 1.79 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.41 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {24 a A b c+12 a b c C+24 a A b d x+12 a b C d x-12 a^2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^2 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \left (4 A b^2+4 a^2 C+3 b^2 C\right ) \sin (c+d x)+6 a b C \sin (2 (c+d x))+b^2 C \sin (3 (c+d x))}{12 d} \]
(24*a*A*b*c + 12*a*b*c*C + 24*a*A*b*d*x + 12*a*b*C*d*x - 12*a^2*A*Log[Cos[ (c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^2*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A*b^2 + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 6*a*b*C*Sin[2* (c + d*x)] + b^2*C*Sin[3*(c + d*x)])/(12*d)
Time = 0.83 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {3042, 3529, 3042, 3512, 27, 3042, 3502, 27, 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.
| \(\displaystyle \int \sec (c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {1}{3} \int (a+b \cos (c+d x)) \left (2 a C \cos ^2(c+d x)+b (3 A+2 C) \cos (c+d x)+3 a A\right ) \sec (c+d x)dx+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (2 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (3 A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int 2 \left (3 A a^2+3 b (2 A+C) \cos (c+d x) a+\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \left (3 A a^2+3 b (2 A+C) \cos (c+d x) a+\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {3 A a^2+3 b (2 A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (3 A b^2+2 \left (a^2+b^2\right ) C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\int 3 \left (A a^2+b (2 A+C) \cos (c+d x) a\right ) \sec (c+d x)dx+\frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{d}+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (3 \int \left (A a^2+b (2 A+C) \cos (c+d x) a\right ) \sec (c+d x)dx+\frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{d}+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (3 \int \frac {A a^2+b (2 A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{d}+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a^2 A \int \sec (c+d x)dx+a b x (2 A+C)\right )+\frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{d}+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a^2 A \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+a b x (2 A+C)\right )+\frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{d}+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (3 \left (\frac {a^2 A \text {arctanh}(\sin (c+d x))}{d}+a b x (2 A+C)\right )+\frac {\left (2 C \left (a^2+b^2\right )+3 A b^2\right ) \sin (c+d x)}{d}+\frac {a b C \sin (c+d x) \cos (c+d x)}{d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
(C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (3*(a*b*(2*A + C)*x + (a^2 *A*ArcTanh[Sin[c + d*x]])/d) + ((3*A*b^2 + 2*(a^2 + b^2)*C)*Sin[c + d*x])/ d + (a*b*C*Cos[c + d*x]*Sin[c + d*x])/d)/3
3.6.34.3.1 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_.)*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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !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/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 4.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02
| method | result | size |
| parallelrisch | \(\frac {-12 A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+12 A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 a b \sin \left (2 d x +2 c \right ) C +C \sin \left (3 d x +3 c \right ) b^{2}+12 \left (a^{2} C +\left (A +\frac {3 C}{4}\right ) b^{2}\right ) \sin \left (d x +c \right )+24 x \left (A +\frac {C}{2}\right ) b d a}{12 d}\) | \(105\) |
| derivativedivides | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \sin \left (d x +c \right )+2 A a b \left (d x +c \right )+2 C a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \sin \left (d x +c \right ) b^{2}+\frac {b^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(106\) |
| default | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{2} C \sin \left (d x +c \right )+2 A a b \left (d x +c \right )+2 C a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \sin \left (d x +c \right ) b^{2}+\frac {b^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(106\) |
| parts | \(\frac {A \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,b^{2}+a^{2} C \right ) \sin \left (d x +c \right )}{d}+\frac {b^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {2 A a b \left (d x +c \right )}{d}+\frac {2 C a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(113\) |
| risch | \(2 a b x A +a b C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{2}}{2 d}-\frac {i a^{2} C \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} b^{2} C}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{2}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{2} C}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} b^{2} C}{8 d}+\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (3 d x +3 c \right ) b^{2} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) C a b}{2 d}\) | \(205\) |
| norman | \(\frac {\left (2 A a b +C a b \right ) x +\left (2 A a b +C a b \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8 A a b +4 C a b \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8 A a b +4 C a b \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12 A a b +6 C a b \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 \left (A \,b^{2}+a^{2} C -C a b +b^{2} C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (A \,b^{2}+a^{2} C +C a b +b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (9 A \,b^{2}+9 a^{2} C -3 C a b +5 b^{2} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (9 A \,b^{2}+9 a^{2} C +3 C a b +5 b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(318\) |
1/12*(-12*A*a^2*ln(tan(1/2*d*x+1/2*c)-1)+12*A*a^2*ln(tan(1/2*d*x+1/2*c)+1) +6*a*b*sin(2*d*x+2*c)*C+C*sin(3*d*x+3*c)*b^2+12*(a^2*C+(A+3/4*C)*b^2)*sin( d*x+c)+24*x*(A+1/2*C)*b*d*a)/d
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {6 \, {\left (2 \, A + C\right )} a b d x + 3 \, A a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C b^{2} \cos \left (d x + c\right )^{2} + 3 \, C a b \cos \left (d x + c\right ) + 3 \, C a^{2} + {\left (3 \, A + 2 \, C\right )} b^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
1/6*(6*(2*A + C)*a*b*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*b^2*cos(d*x + c)^2 + 3*C*a*b*cos(d*x + c) + 3*C*a^2 + (3*A + 2*C)*b^2)*sin(d*x + c))/d
\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec {\left (c + d x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {12 \, {\left (d x + c\right )} A a b + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b - 2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{2} + 6 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 6 \, C a^{2} \sin \left (d x + c\right ) + 6 \, A b^{2} \sin \left (d x + c\right )}{6 \, d} \]
1/6*(12*(d*x + c)*A*a*b + 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a*b - 2*(si n(d*x + c)^3 - 3*sin(d*x + c))*C*b^2 + 6*A*a^2*log(sec(d*x + c) + tan(d*x + c)) + 6*C*a^2*sin(d*x + c) + 6*A*b^2*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (97) = 194\).
Time = 0.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.49 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {3 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, A a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (2 \, A a b + C a b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
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*b + C*a*b)*(d*x + c) + 2*(3*C*a^2*tan(1/2*d* x + 1/2*c)^5 - 3*C*a*b*tan(1/2*d*x + 1/2*c)^5 + 3*A*b^2*tan(1/2*d*x + 1/2* c)^5 + 3*C*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*tan(1/2*d*x + 1/2*c)^3 + 6 *A*b^2*tan(1/2*d*x + 1/2*c)^3 + 2*C*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*C*a^2*t an(1/2*d*x + 1/2*c) + 3*C*a*b*tan(1/2*d*x + 1/2*c) + 3*A*b^2*tan(1/2*d*x + 1/2*c) + 3*C*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 2.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.65 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {A\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {4\,A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2\,d} \]
(A*b^2*sin(c + d*x))/d + (C*a^2*sin(c + d*x))/d + (3*C*b^2*sin(c + d*x))/( 4*d) + (2*A*a^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (C*b^2*s in(3*c + 3*d*x))/(12*d) + (4*A*a*b*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x) /2)))/d + (2*C*a*b*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (C*a*b *sin(2*c + 2*d*x))/(2*d)