Integrand size = 38, antiderivative size = 137 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^3 B x+\frac {\left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b \left (9 a b B+8 a^2 C+2 b^2 C\right ) \tan (c+d x)}{3 d}+\frac {b^2 (3 b B+5 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {b C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d} \]
a^3*B*x+1/2*(6*B*a^2*b+B*b^3+2*C*a^3+3*C*a*b^2)*arctanh(sin(d*x+c))/d+1/3* b*(9*B*a*b+8*C*a^2+2*C*b^2)*tan(d*x+c)/d+1/6*b^2*(3*B*b+5*C*a)*sec(d*x+c)* tan(d*x+c)/d+1/3*b*C*(a+b*sec(d*x+c))^2*tan(d*x+c)/d
Time = 0.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 a^3 B d x+3 \left (6 a^2 b B+b^3 B+2 a^3 C+3 a b^2 C\right ) \text {arctanh}(\sin (c+d x))+3 b \left (6 a b B+6 a^2 C+2 b^2 C+b (b B+3 a C) \sec (c+d x)\right ) \tan (c+d x)+2 b^3 C \tan ^3(c+d x)}{6 d} \]
(6*a^3*B*d*x + 3*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*ArcTanh[Sin[c + d*x]] + 3*b*(6*a*b*B + 6*a^2*C + 2*b^2*C + b*(b*B + 3*a*C)*Sec[c + d*x])* Tan[c + d*x] + 2*b^3*C*Tan[c + d*x]^3)/(6*d)
Time = 0.61 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3042, 4560, 3042, 4406, 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.
| \(\displaystyle \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int (a+b \sec (c+d x))^3 (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4406 |
| \(\displaystyle \frac {1}{3} \int (a+b \sec (c+d x)) \left (3 B a^2+b (3 b B+5 a C) \sec ^2(c+d x)+\left (3 C a^2+6 b B a+2 b^2 C\right ) \sec (c+d x)\right )dx+\frac {b C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 B a^2+b (3 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 C a^2+6 b B a+2 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \left (6 B a^3+2 b \left (8 C a^2+9 b B a+2 b^2 C\right ) \sec ^2(c+d x)+3 \left (2 C a^3+6 b B a^2+3 b^2 C a+b^3 B\right ) \sec (c+d x)\right )dx+\frac {b^2 (5 a C+3 b B) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {b C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (6 a^3 B x+\frac {2 b \left (8 a^2 C+9 a b B+2 b^2 C\right ) \tan (c+d x)}{d}+\frac {3 \left (2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right ) \text {arctanh}(\sin (c+d x))}{d}\right )+\frac {b^2 (5 a C+3 b B) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {b C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\) |
(b*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((b^2*(3*b*B + 5*a*C)*Se c[c + d*x]*Tan[c + d*x])/(2*d) + (6*a^3*B*x + (3*(6*a^2*b*B + b^3*B + 2*a^ 3*C + 3*a*b^2*C)*ArcTanh[Sin[c + d*x]])/d + (2*b*(9*a*b*B + 8*a^2*C + 2*b^ 2*C)*Tan[c + d*x])/d)/2)/3
3.8.87.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-b)*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[1/m Int[(a + b*Csc[e + f*x])^(m - 2)*Simp[a^2*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b* c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
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_.))^(m_.)*((A_.) + csc[(e_.) + (f_. )*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.) *(x_)]*(d_.))^(n_.), x_Symbol] :> Simp[1/b^2 Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Time = 0.92 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {B \,a^{3} \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C \tan \left (d x +c \right ) a^{2} b +3 B \tan \left (d x +c \right ) a \,b^{2}+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(180\) |
| default | \(\frac {B \,a^{3} \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 B \,a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 C \tan \left (d x +c \right ) a^{2} b +3 B \tan \left (d x +c \right ) a \,b^{2}+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(180\) |
| parallelrisch | \(\frac {-27 \left (B \,a^{2} b +\frac {1}{6} B \,b^{3}+\frac {1}{3} a^{3} C +\frac {1}{2} C a \,b^{2}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+27 \left (B \,a^{2} b +\frac {1}{6} B \,b^{3}+\frac {1}{3} a^{3} C +\frac {1}{2} C a \,b^{2}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 a^{3} B x d \cos \left (3 d x +3 c \right )+\left (9 B a \,b^{2}+9 a^{2} b C +2 C \,b^{3}\right ) \sin \left (3 d x +3 c \right )+3 \left (B \,b^{3}+3 C a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+9 a^{3} B x d \cos \left (d x +c \right )+9 \left (B a b +C \,a^{2}+\frac {2}{3} C \,b^{2}\right ) b \sin \left (d x +c \right )}{3 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(254\) |
| risch | \(a^{3} B x -\frac {i b \left (3 B \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+9 C a b \,{\mathrm e}^{5 i \left (d x +c \right )}-18 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}-18 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-36 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-36 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 C \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 B \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-9 C b a \,{\mathrm e}^{i \left (d x +c \right )}-18 B a b -18 C \,a^{2}-4 C \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{2} b}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{3}}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{2}}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{2} b}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{3}}{2 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{2}}{2 d}\) | \(356\) |
| norman | \(\frac {a^{3} B x +a^{3} B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {b \left (6 B a b +B \,b^{2}+6 C \,a^{2}+3 C a b +2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-3 a^{3} B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a^{3} B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 a^{3} B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-3 a^{3} B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {4 b \left (9 B a b +9 C \,a^{2}+C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {4 b \left (9 B a b +9 C \,a^{2}+C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {b \left (6 B a b -B \,b^{2}+6 C \,a^{2}-3 C a b +2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {2 b^{2} \left (B b +3 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {\left (6 B \,a^{2} b +B \,b^{3}+2 a^{3} C +3 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (6 B \,a^{2} b +B \,b^{3}+2 a^{3} C +3 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(398\) |
1/d*(B*a^3*(d*x+c)+a^3*C*ln(sec(d*x+c)+tan(d*x+c))+3*B*a^2*b*ln(sec(d*x+c) +tan(d*x+c))+3*C*tan(d*x+c)*a^2*b+3*B*tan(d*x+c)*a*b^2+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)))+B*b^3*(1/2*sec(d*x+c)*tan (d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-C*b^3*(-2/3-1/3*sec(d*x+c)^2)*tan(d *x+c))
Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.38 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, B a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{3} + 2 \, {\left (9 \, C a^{2} b + 9 \, B a b^{2} + 2 \, C b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
1/12*(12*B*a^3*d*x*cos(d*x + c)^3 + 3*(2*C*a^3 + 6*B*a^2*b + 3*C*a*b^2 + B *b^3)*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(2*C*a^3 + 6*B*a^2*b + 3*C* a*b^2 + B*b^3)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(2*C*b^3 + 2*(9*C *a^2*b + 9*B*a*b^2 + 2*C*b^3)*cos(d*x + c)^2 + 3*(3*C*a*b^2 + B*b^3)*cos(d *x + c))*sin(d*x + c))/(d*cos(d*x + c)^3)
\[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.58 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \, {\left (d x + c\right )} B a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{3} - 9 \, C a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, B b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a^{2} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, C a^{2} b \tan \left (d x + c\right ) + 36 \, B a b^{2} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(12*(d*x + c)*B*a^3 + 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) + log( sin(d*x + c) - 1)) - 3*B*b^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(si n(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 6*C*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*B*a^2*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*C*a^2*b*tan(d*x + c) + 36*B*a*b^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (129) = 258\).
Time = 0.34 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.45 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 \, {\left (d x + c\right )} B a^{3} + 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, C a^{3} + 6 \, B a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
1/6*(6*(d*x + c)*B*a^3 + 3*(2*C*a^3 + 6*B*a^2*b + 3*C*a*b^2 + B*b^3)*log(a bs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(2*C*a^3 + 6*B*a^2*b + 3*C*a*b^2 + B*b^3 )*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 + 18*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 9*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 3*B*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 - 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 4*C*b^3*t an(1/2*d*x + 1/2*c)^3 + 18*C*a^2*b*tan(1/2*d*x + 1/2*c) + 18*B*a*b^2*tan(1 /2*d*x + 1/2*c) + 9*C*a*b^2*tan(1/2*d*x + 1/2*c) + 3*B*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 = 18.68 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.84 \[ \int \cos (c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {B\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {C\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{6}+\frac {C\,b^3\,\sin \left (c+d\,x\right )}{2}+\frac {3\,B\,a\,b^2\,\sin \left (c+d\,x\right )}{4}+\frac {3\,C\,a^2\,b\,\sin \left (c+d\,x\right )}{4}+\frac {3\,B\,a^3\,\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 {B\,b^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}}{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\,B\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4}+\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 {B\,a^3\,\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 {B\,b^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}}{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 {B\,a^2\,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 )\,\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 {B\,a^2\,b\,\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 )} \]
((B*b^3*sin(2*c + 2*d*x))/4 + (C*b^3*sin(3*c + 3*d*x))/6 + (C*b^3*sin(c + d*x))/2 + (3*B*a*b^2*sin(c + d*x))/4 + (3*C*a^2*b*sin(c + d*x))/4 + (3*B*a ^3*cos(c + d*x)*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 - (B*b^3*co s(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*3i)/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* B*a*b^2*sin(3*c + 3*d*x))/4 + (3*C*a*b^2*sin(2*c + 2*d*x))/4 + (3*C*a^2*b* sin(3*c + 3*d*x))/4 + (B*a^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*c os(3*c + 3*d*x))/2 - (B*b^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2 ))*cos(3*c + 3*d*x)*1i)/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 - (B*a^2*b*atan((sin(c/2 + (d*x)/2)*1i)/c os(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/2 - (C*a*b^2*atan((sin(c/2 + (d*x) /2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*3i)/4 - (B*a^2*b*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))