Integrand size = 33, antiderivative size = 112 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=b^2 C x+\frac {a b (A+2 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (2 A b^2+a^2 (2 A+3 C)\right ) \tan (c+d x)}{3 d}+\frac {a A b \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
b^2*C*x+a*b*(A+2*C)*arctanh(sin(d*x+c))/d+1/3*(2*A*b^2+a^2*(2*A+3*C))*tan( d*x+c)/d+1/3*a*A*b*sec(d*x+c)*tan(d*x+c)/d+1/3*A*(a+b*cos(d*x+c))^2*sec(d* x+c)^2*tan(d*x+c)/d
Time = 0.47 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 b^2 C d x+3 a b (A+2 C) \text {arctanh}(\sin (c+d x))+3 \left (A b^2+a^2 (A+C)+a A b \sec (c+d x)\right ) \tan (c+d x)+a^2 A \tan ^3(c+d x)}{3 d} \]
(3*b^2*C*d*x + 3*a*b*(A + 2*C)*ArcTanh[Sin[c + d*x]] + 3*(A*b^2 + a^2*(A + C) + a*A*b*Sec[c + d*x])*Tan[c + d*x] + a^2*A*Tan[c + d*x]^3)/(3*d)
Time = 0.88 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3527, 3042, 3510, 27, 3042, 3500, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{3} \int (a+b \cos (c+d x)) \left (3 b C \cos ^2(c+d x)+a (2 A+3 C) \cos (c+d x)+2 A b\right ) \sec ^3(c+d x)dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 b C \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (2 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{3} \left (\frac {a A b \tan (c+d x) \sec (c+d x)}{d}-\frac {1}{2} \int -2 \left ((2 A+3 C) a^2+3 b (A+2 C) \cos (c+d x) a+2 A b^2+3 b^2 C \cos ^2(c+d x)\right ) \sec ^2(c+d x)dx\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \left ((2 A+3 C) a^2+3 b (A+2 C) \cos (c+d x) a+2 A b^2+3 b^2 C \cos ^2(c+d x)\right ) \sec ^2(c+d x)dx+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {(2 A+3 C) a^2+3 b (A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+2 A b^2+3 b^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{3} \left (\int 3 \left (C \cos (c+d x) b^2+a (A+2 C) b\right ) \sec (c+d x)dx+\frac {\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{d}+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (3 \int \left (C \cos (c+d x) b^2+a (A+2 C) b\right ) \sec (c+d x)dx+\frac {\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{d}+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (3 \int \frac {C \sin \left (c+d x+\frac {\pi }{2}\right ) b^2+a (A+2 C) b}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{d}+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a b (A+2 C) \int \sec (c+d x)dx+b^2 C x\right )+\frac {\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{d}+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a b (A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+b^2 C x\right )+\frac {\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{d}+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{3} \left (\frac {\left (a^2 (2 A+3 C)+2 A b^2\right ) \tan (c+d x)}{d}+3 \left (\frac {a b (A+2 C) \text {arctanh}(\sin (c+d x))}{d}+b^2 C x\right )+\frac {a A b \tan (c+d x) \sec (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\) |
(A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (3*(b^2*C*x + (a*b*(A + 2*C)*ArcTanh[Sin[c + d*x]])/d) + ((2*A*b^2 + a^2*(2*A + 3*C)) *Tan[c + d*x])/d + (a*A*b*Sec[c + d*x]*Tan[c + d*x])/d)/3
3.6.37.3.1 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[(-(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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, 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^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 7.77 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {-A \,a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{2} C \tan \left (d x +c \right )+2 A a b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{2} \tan \left (d x +c \right )+b^{2} C \left (d x +c \right )}{d}\) | \(117\) |
| default | \(\frac {-A \,a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{2} C \tan \left (d x +c \right )+2 A a b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+2 C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{2} \tan \left (d x +c \right )+b^{2} C \left (d x +c \right )}{d}\) | \(117\) |
| parts | \(-\frac {A \,a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{2}+a^{2} C \right ) \tan \left (d x +c \right )}{d}+\frac {b^{2} C \left (d x +c \right )}{d}+\frac {2 A a b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {2 C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(124\) |
| parallelrisch | \(\frac {-9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) b \left (A +2 C \right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) b \left (A +2 C \right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 C \,b^{2} d x \cos \left (3 d x +3 c \right )+\left (3 A \,b^{2}+a^{2} \left (2 A +3 C \right )\right ) \sin \left (3 d x +3 c \right )+6 A \sin \left (2 d x +2 c \right ) a b +9 C \,b^{2} d x \cos \left (d x +c \right )+6 \sin \left (d x +c \right ) \left (a^{2} \left (A +\frac {C}{2}\right )+\frac {A \,b^{2}}{2}\right )}{3 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(202\) |
| risch | \(b^{2} C x -\frac {2 i \left (3 A a b \,{\mathrm e}^{5 i \left (d x +c \right )}-3 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 A a b \,{\mathrm e}^{i \left (d x +c \right )}-2 A \,a^{2}-3 A \,b^{2}-3 a^{2} C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {A a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {A a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {2 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}\) | \(232\) |
| norman | \(\frac {b^{2} C x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b^{2} C x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b^{2} C x -\frac {8 \left (A \,a^{2}-A \,b^{2}-a^{2} C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (A \,a^{2}-A a b +A \,b^{2}+a^{2} C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (A \,a^{2}+A a b +A \,b^{2}+a^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 \left (5 A \,a^{2}-6 A a b +3 A \,b^{2}+3 a^{2} C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 \left (5 A \,a^{2}+6 A a b +3 A \,b^{2}+3 a^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 \left (13 A \,a^{2}-15 A a b -3 A \,b^{2}-3 a^{2} C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 \left (13 A \,a^{2}+15 A a b -3 A \,b^{2}-3 a^{2} C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-b^{2} C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b^{2} C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b^{2} C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{2} C x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 b^{2} C x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {a b \left (A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a b \left (A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(477\) |
1/d*(-A*a^2*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+a^2*C*tan(d*x+c)+2*A*a*b*(1 /2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+2*C*a*b*ln(sec(d*x +c)+tan(d*x+c))+A*b^2*tan(d*x+c)+b^2*C*(d*x+c))
Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {6 \, C b^{2} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (A + 2 \, C\right )} a b \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (A + 2 \, C\right )} a b \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, A a b \cos \left (d x + c\right ) + A a^{2} + {\left ({\left (2 \, A + 3 \, C\right )} a^{2} + 3 \, A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \]
1/6*(6*C*b^2*d*x*cos(d*x + c)^3 + 3*(A + 2*C)*a*b*cos(d*x + c)^3*log(sin(d *x + c) + 1) - 3*(A + 2*C)*a*b*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*( 3*A*a*b*cos(d*x + c) + A*a^2 + ((2*A + 3*C)*a^2 + 3*A*b^2)*cos(d*x + c)^2) *sin(d*x + c))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} + 6 \, {\left (d x + c\right )} C b^{2} - 3 \, A a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{2} \tan \left (d x + c\right ) + 6 \, A b^{2} \tan \left (d x + c\right )}{6 \, d} \]
1/6*(2*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2 + 6*(d*x + c)*C*b^2 - 3*A*a *b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin( d*x + c) - 1)) + 6*C*a*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 6*C*a^2*tan(d*x + c) + 6*A*b^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (106) = 212\).
Time = 0.32 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.34 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 \, {\left (d x + c\right )} C b^{2} + 3 \, {\left (A a b + 2 \, C a b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (A a b + 2 \, C a b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]
1/3*(3*(d*x + c)*C*b^2 + 3*(A*a*b + 2*C*a*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(A*a*b + 2*C*a*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 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 - 3*A*a*b*tan(1/ 2*d*x + 1/2*c)^5 + 3*A*b^2*tan(1/2*d*x + 1/2*c)^5 - 2*A*a^2*tan(1/2*d*x + 1/2*c)^3 - 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 + 3*A*a^2*tan(1/2*d*x + 1/2*c) + 3*C*a^2*tan(1/2*d*x + 1/2*c) + 3*A*a*b*t an(1/2*d*x + 1/2*c) + 3*A*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^ 2 - 1)^3)/d
Time = 1.68 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.87 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {2\,C\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a^2\,\sin \left (c+d\,x\right )}{3\,d\,\cos \left (c+d\,x\right )}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {A\,b^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a\,b\,\sin \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2}-\frac {A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,4{}\mathrm {i}}{d} \]
(2*C*b^2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*A*a^2*sin(c + d*x))/(3*d*cos(c + d*x)) + (A*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^3) + (A *b^2*sin(c + d*x))/(d*cos(c + d*x)) + (C*a^2*sin(c + d*x))/(d*cos(c + d*x) ) - (A*a*b*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*2i)/d - (C*a*b *atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*4i)/d + (A*a*b*sin(c + d *x))/(d*cos(c + d*x)^2)