Integrand size = 31, antiderivative size = 167 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=3 a^2 A b x+\frac {a \left (6 A b^2+2 a^2 C+3 b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {A (a+b \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \tan (c+d x)}{3 d}-\frac {a b^2 (6 A-5 C) \sec (c+d x) \tan (c+d x)}{6 d}-\frac {b (3 A-C) (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d} \]
3*a^2*A*b*x+1/2*a*(6*A*b^2+2*C*a^2+3*C*b^2)*arctanh(sin(d*x+c))/d+A*(a+b*s ec(d*x+c))^3*sin(d*x+c)/d-1/3*b*(a^2*(6*A-8*C)-b^2*(3*A+2*C))*tan(d*x+c)/d -1/6*a*b^2*(6*A-5*C)*sec(d*x+c)*tan(d*x+c)/d-1/3*b*(3*A-C)*(a+b*sec(d*x+c) )^2*tan(d*x+c)/d
Time = 4.77 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.95 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\sec ^3(c+d x) \left (9 a \cos (c+d x) \left (6 a A b (c+d x)-\left (6 A b^2+2 a^2 C+3 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+\left (6 A b^2+2 a^2 C+3 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 a \cos (3 (c+d x)) \left (6 a A b (c+d x)-\left (6 A b^2+2 a^2 C+3 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+\left (6 A b^2+2 a^2 C+3 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+2 \left (6 A b^3+18 a^2 b C+8 b^3 C+9 a \left (a^2 A+2 b^2 C\right ) \cos (c+d x)+2 \left (3 A b^3+9 a^2 b C+2 b^3 C\right ) \cos (2 (c+d x))+3 a^3 A \cos (3 (c+d x))\right ) \sin (c+d x)\right )}{24 d} \]
(Sec[c + d*x]^3*(9*a*Cos[c + d*x]*(6*a*A*b*(c + d*x) - (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*a*Cos[3*(c + d*x)] *(6*a*A*b*(c + d*x) - (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(6*A*b^3 + 18*a^2*b*C + 8*b^3*C + 9*a*(a^2*A + 2*b^ 2*C)*Cos[c + d*x] + 2*(3*A*b^3 + 9*a^2*b*C + 2*b^3*C)*Cos[2*(c + d*x)] + 3 *a^3*A*Cos[3*(c + d*x)])*Sin[c + d*x]))/(24*d)
Time = 0.70 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3042, 4583, 3042, 4544, 3042, 4536, 2009}
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.
| \(\displaystyle \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4583 |
| \(\displaystyle \int (a+b \sec (c+d x))^2 \left (-b (3 A-C) \sec ^2(c+d x)+a C \sec (c+d x)+3 A b\right )dx+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b (3 A-C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a C \csc \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )dx+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d}\) |
| \(\Big \downarrow \) 4544 |
| \(\displaystyle \frac {1}{3} \int (a+b \sec (c+d x)) \left (-a b (6 A-5 C) \sec ^2(c+d x)+\left (3 C a^2+3 A b^2+2 b^2 C\right ) \sec (c+d x)+9 a A b\right )dx-\frac {b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-a b (6 A-5 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 C a^2+3 A b^2+2 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+9 a A b\right )dx-\frac {b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \left (18 A b a^2+3 \left (2 C a^2+6 A b^2+3 b^2 C\right ) \sec (c+d x) a-2 b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right )dx-\frac {a b^2 (6 A-5 C) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {3 a \left (2 a^2 C+6 A b^2+3 b^2 C\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {2 b \left (a^2 (6 A-8 C)-b^2 (3 A+2 C)\right ) \tan (c+d x)}{d}+18 a^2 A b x\right )-\frac {a b^2 (6 A-5 C) \tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^3}{d}\) |
(A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d - (b*(3*A - C)*(a + b*Sec[c + d* x])^2*Tan[c + d*x])/(3*d) + (-1/2*(a*b^2*(6*A - 5*C)*Sec[c + d*x]*Tan[c + d*x])/d + (18*a^2*A*b*x + (3*a*(6*A*b^2 + 2*a^2*C + 3*b^2*C)*ArcTanh[Sin[c + d*x]])/d - (2*b*(a^2*(6*A - 8*C) - b^2*(3*A + 2*C))*Tan[c + d*x])/d)/2) /3
3.7.57.3.1 Defintions of rubi rules used
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Simp[1/2 Int[Simp[2*A*a + (2*B*a + b*(2* A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, e, f, A, B, C}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot [e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[1/(m + 1) Int[( a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m )*Csc[e + f*x] + (b*B*(m + 1) + a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/( d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^ 2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && Gt Q[m, 0] && LeQ[n, -1]
Time = 0.90 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(\frac {a^{3} A \sin \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \,a^{2} b \left (d x +c \right )+3 C \tan \left (d x +c \right ) a^{2} b +3 a A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \tan \left (d x +c \right ) b^{3}-C \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(155\) |
| default | \(\frac {a^{3} A \sin \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 A \,a^{2} b \left (d x +c \right )+3 C \tan \left (d x +c \right ) a^{2} b +3 a A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+A \tan \left (d x +c \right ) b^{3}-C \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(155\) |
| parallelrisch | \(\frac {-18 a \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (\frac {C \,a^{2}}{3}+\left (A +\frac {C}{2}\right ) b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+18 a \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (\frac {C \,a^{2}}{3}+\left (A +\frac {C}{2}\right ) b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 a^{2} A b x d \cos \left (3 d x +3 c \right )+2 b \left (3 C \,a^{2}+b^{2} \left (A +\frac {2 C}{3}\right )\right ) \sin \left (3 d x +3 c \right )+2 \left (a^{3} A +3 C a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+a^{3} A \sin \left (4 d x +4 c \right )+2 b \left (9 a^{2} A x d \cos \left (d x +c \right )+\left (3 C \,a^{2}+b^{2} \left (A +2 C \right )\right ) \sin \left (d x +c \right )\right )}{2 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(250\) |
| risch | \(3 a^{2} A b x -\frac {i a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i b \left (9 C a b \,{\mathrm e}^{5 i \left (d x +c \right )}-6 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 C \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 C b a \,{\mathrm e}^{i \left (d x +c \right )}-6 A \,b^{2}-18 C \,a^{2}-4 C \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{2}}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{2}}{2 d}\) | \(325\) |
| norman | \(\frac {\frac {\left (2 a^{3} A -2 A \,b^{3}-6 a^{2} b C +3 C a \,b^{2}-2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {\left (2 a^{3} A +2 A \,b^{3}+6 a^{2} b C +3 C a \,b^{2}+2 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 \left (6 a^{3} A -3 A \,b^{3}-9 a^{2} b C -C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {4 \left (6 a^{3} A +3 A \,b^{3}+9 a^{2} b C +C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {6 a \left (2 a^{2} A -C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+3 a^{2} A b x -9 a^{2} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6 a^{2} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+6 a^{2} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-9 a^{2} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+3 a^{2} A b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {a \left (6 A \,b^{2}+2 C \,a^{2}+3 C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a \left (6 A \,b^{2}+2 C \,a^{2}+3 C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(418\) |
1/d*(a^3*A*sin(d*x+c)+a^3*C*ln(sec(d*x+c)+tan(d*x+c))+3*A*a^2*b*(d*x+c)+3* C*tan(d*x+c)*a^2*b+3*a*A*b^2*ln(sec(d*x+c)+tan(d*x+c))+3*C*a*b^2*(1/2*sec( d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+A*tan(d*x+c)*b^3-C*b^3*(- 2/3-1/3*sec(d*x+c)^2)*tan(d*x+c))
Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {36 \, A a^{2} b d x \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, C a^{3} + 3 \, {\left (2 \, A + C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, C a^{3} + 3 \, {\left (2 \, A + C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 9 \, C a b^{2} \cos \left (d x + c\right ) + 2 \, C b^{3} + 2 \, {\left (9 \, C a^{2} b + {\left (3 \, A + 2 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
1/12*(36*A*a^2*b*d*x*cos(d*x + c)^3 + 3*(2*C*a^3 + 3*(2*A + C)*a*b^2)*cos( d*x + c)^3*log(sin(d*x + c) + 1) - 3*(2*C*a^3 + 3*(2*A + C)*a*b^2)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(6*A*a^3*cos(d*x + c)^3 + 9*C*a*b^2*cos (d*x + c) + 2*C*b^3 + 2*(9*C*a^2*b + (3*A + 2*C)*b^3)*cos(d*x + c)^2)*sin( d*x + c))/(d*cos(d*x + c)^3)
\[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \cos {\left (c + d x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {36 \, {\left (d x + c\right )} A a^{2} b + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{3} - 9 \, C a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, A a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 36 \, C a^{2} b \tan \left (d x + c\right ) + 12 \, A b^{3} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(36*(d*x + c)*A*a^2*b + 4*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*b^3 - 9 *C*a*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + lo g(sin(d*x + c) - 1)) + 6*C*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*A*a*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 12*A*a ^3*sin(d*x + c) + 36*C*a^2*b*tan(d*x + c) + 12*A*b^3*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (160) = 320\).
Time = 0.36 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.93 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {18 \, {\left (d x + c\right )} A a^{2} b + \frac {12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (2 \, C a^{3} + 6 \, A a b^{2} + 3 \, C a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, C a^{3} + 6 \, A a b^{2} + 3 \, C a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(18*(d*x + c)*A*a^2*b + 12*A*a^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1 /2*c)^2 + 1) + 3*(2*C*a^3 + 6*A*a*b^2 + 3*C*a*b^2)*log(abs(tan(1/2*d*x + 1 /2*c) + 1)) - 3*(2*C*a^3 + 6*A*a*b^2 + 3*C*a*b^2)*log(abs(tan(1/2*d*x + 1/ 2*c) - 1)) - 2*(18*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 9*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*b^3*tan(1/2*d*x + 1/2*c) ^5 - 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^3*tan(1/2*d*x + 1/2*c)^3 - 4*C*b^3*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^2*b*tan(1/2*d*x + 1/2*c) + 9*C*a* b^2*tan(1/2*d*x + 1/2*c) + 6*A*b^3*tan(1/2*d*x + 1/2*c) + 6*C*b^3*tan(1/2* d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d
Time = 17.72 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.78 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {A\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {A\,b^3\,\sin \left (c+d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (c+d\,x\right )}{2}+\frac {3\,C\,a^2\,b\,\sin \left (c+d\,x\right )}{4}-\frac {C\,a^3\,\cos \left (c+d\,x\right )\,\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 )\,3{}\mathrm {i}}{2}+\frac {3\,C\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,C\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4}-\frac {C\,a^3\,\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 )\,\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{2}+\frac {3\,A\,a^2\,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 )\,\cos \left (3\,c+3\,d\,x\right )}{2}-\frac {A\,a\,b^2\,\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 )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{2}-\frac {C\,a\,b^2\,\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 )\,\cos \left (3\,c+3\,d\,x\right )\,3{}\mathrm {i}}{4}+\frac {9\,A\,a^2\,b\,\cos \left (c+d\,x\right )\,\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 )}{2}-\frac {A\,a\,b^2\,\cos \left (c+d\,x\right )\,\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 )\,9{}\mathrm {i}}{2}-\frac {C\,a\,b^2\,\cos \left (c+d\,x\right )\,\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 )\,9{}\mathrm {i}}{4}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \]
((A*a^3*sin(2*c + 2*d*x))/4 + (A*a^3*sin(4*c + 4*d*x))/8 + (A*b^3*sin(3*c + 3*d*x))/4 + (C*b^3*sin(3*c + 3*d*x))/6 + (A*b^3*sin(c + d*x))/4 + (C*b^3 *sin(c + d*x))/2 + (3*C*a^2*b*sin(c + d*x))/4 - (C*a^3*cos(c + d*x)*atan(( sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*3i)/2 + (3*C*a*b^2*sin(2*c + 2* d*x))/4 + (3*C*a^2*b*sin(3*c + 3*d*x))/4 - (C*a^3*atan((sin(c/2 + (d*x)/2) *1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*1i)/2 + (3*A*a^2*b*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/2 - (A*a*b^2*atan((sin(c/ 2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/2 - (C*a*b^2*ata n((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/4 + (9* A*a^2*b*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 - (A*a *b^2*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*9i)/2 - (C*a*b^2*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*9i )/4)/(d*((3*cos(c + d*x))/4 + cos(3*c + 3*d*x)/4))