Integrand size = 38, antiderivative size = 114 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(4 a B+3 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(4 a B+3 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {b C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d} \]
1/8*(4*B*a+3*C*b)*arctanh(sin(d*x+c))/d+(B*b+C*a)*tan(d*x+c)/d+1/8*(4*B*a+ 3*C*b)*sec(d*x+c)*tan(d*x+c)/d+1/4*b*C*sec(d*x+c)^3*tan(d*x+c)/d+1/3*(B*b+ C*a)*tan(d*x+c)^3/d
Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 (4 a B+3 b C) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (12 a B+9 b C+8 (b B+a C) (2+\cos (2 (c+d x))) \sec (c+d x)+6 b C \sec ^2(c+d x)\right ) \tan (c+d x)}{24 d} \]
(3*(4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(12*a*B + 9*b*C + 8*(b*B + a*C)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*b*C*Sec[c + d*x]^2)* Tan[c + d*x])/(24*d)
Time = 0.70 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 4560, 3042, 4485, 3042, 4274, 3042, 4254, 2009, 4255, 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.
| \(\displaystyle \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \sec ^3(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {1}{4} \int \sec ^3(c+d x) (4 a B+3 b C+4 (b B+a C) \sec (c+d x))dx+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (4 a B+3 b C+4 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{4} \left (4 (a C+b B) \int \sec ^4(c+d x)dx+(4 a B+3 b C) \int \sec ^3(c+d x)dx\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left ((4 a B+3 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+4 (a C+b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{4} \left ((4 a B+3 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 (a C+b B) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left ((4 a B+3 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{4} \left ((4 a B+3 b C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left ((4 a B+3 b C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left ((4 a B+3 b C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {b C \tan (c+d x) \sec ^3(c+d x)}{4 d}\) |
(b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((4*a*B + 3*b*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)) - (4*(b*B + a*C)*(-Tan[ c + d*x] - Tan[c + d*x]^3/3))/d)/4
3.8.66.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[ e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Simp[1/(n + 1) Int[(d*Csc [e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x ], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[ n, -1]
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 = 0.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(-\frac {\left (B b +C a \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(119\) |
| derivativedivides | \(\frac {a B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(131\) |
| default | \(\frac {a B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-C a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-B b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+C b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(131\) |
| parallelrisch | \(\frac {-24 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a B +\frac {3 C b}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+24 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a B +\frac {3 C b}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+32 \left (B b +C a \right ) \sin \left (2 d x +2 c \right )+3 \left (4 a B +3 C b \right ) \sin \left (3 d x +3 c \right )+8 \left (B b +C a \right ) \sin \left (4 d x +4 c \right )+3 \sin \left (d x +c \right ) \left (4 a B +11 C b \right )}{12 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(192\) |
| norman | \(\frac {\frac {\left (4 a B -8 B b -8 C a +5 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {\left (4 a B +8 B b +8 C a +5 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (12 a B -40 B b -40 C a -9 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {\left (12 a B +40 B b +40 C a -9 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {\left (4 a B +3 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (4 a B +3 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(201\) |
| risch | \(-\frac {i \left (12 a B \,{\mathrm e}^{7 i \left (d x +c \right )}+9 C b \,{\mathrm e}^{7 i \left (d x +c \right )}+12 B a \,{\mathrm e}^{5 i \left (d x +c \right )}+33 C b \,{\mathrm e}^{5 i \left (d x +c \right )}-48 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-48 C a \,{\mathrm e}^{4 i \left (d x +c \right )}-12 a B \,{\mathrm e}^{3 i \left (d x +c \right )}-33 C b \,{\mathrm e}^{3 i \left (d x +c \right )}-64 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-64 C a \,{\mathrm e}^{2 i \left (d x +c \right )}-12 a B \,{\mathrm e}^{i \left (d x +c \right )}-9 C b \,{\mathrm e}^{i \left (d x +c \right )}-16 B b -16 C a \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a B}{2 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a B}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(266\) |
-(B*b+C*a)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+a*B/d*(1/2*sec(d*x+c)*tan( d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+C*b/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d *x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )^{2} + 6 \, C b + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
1/48*(3*(4*B*a + 3*C*b)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(4*B*a + 3*C*b)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(16*(C*a + B*b)*cos(d*x + c)^3 + 3*(4*B*a + 3*C*b)*cos(d*x + c)^2 + 6*C*b + 8*(C*a + B*b)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4)
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 3 \, C b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, B a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a + 16*(tan(d*x + c)^3 + 3*ta n(d*x + c))*B*b - 3*C*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c ) - 1)) - 12*B*a*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)))/d
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (106) = 212\).
Time = 0.32 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.67 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (4 \, B a + 3 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (4 \, B a + 3 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
1/24*(3*(4*B*a + 3*C*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(4*B*a + 3* C*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(12*B*a*tan(1/2*d*x + 1/2*c)^7 - 24*C*a*tan(1/2*d*x + 1/2*c)^7 - 24*B*b*tan(1/2*d*x + 1/2*c)^7 + 15*C*b* tan(1/2*d*x + 1/2*c)^7 - 12*B*a*tan(1/2*d*x + 1/2*c)^5 + 40*C*a*tan(1/2*d* x + 1/2*c)^5 + 40*B*b*tan(1/2*d*x + 1/2*c)^5 + 9*C*b*tan(1/2*d*x + 1/2*c)^ 5 - 12*B*a*tan(1/2*d*x + 1/2*c)^3 - 40*C*a*tan(1/2*d*x + 1/2*c)^3 - 40*B*b *tan(1/2*d*x + 1/2*c)^3 + 9*C*b*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*tan(1/2*d* x + 1/2*c) + 24*C*a*tan(1/2*d*x + 1/2*c) + 24*B*b*tan(1/2*d*x + 1/2*c) + 1 5*C*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d
Time = 21.01 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.70 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (B\,a-2\,B\,b-2\,C\,a+\frac {5\,C\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {10\,B\,b}{3}-B\,a+\frac {10\,C\,a}{3}+\frac {3\,C\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,C\,b}{4}-\frac {10\,B\,b}{3}-\frac {10\,C\,a}{3}-B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (B\,a+2\,B\,b+2\,C\,a+\frac {5\,C\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B\,a+\frac {3\,C\,b}{4}\right )}{d} \]
(tan(c/2 + (d*x)/2)*(B*a + 2*B*b + 2*C*a + (5*C*b)/4) + tan(c/2 + (d*x)/2) ^7*(B*a - 2*B*b - 2*C*a + (5*C*b)/4) - tan(c/2 + (d*x)/2)^3*(B*a + (10*B*b )/3 + (10*C*a)/3 - (3*C*b)/4) + tan(c/2 + (d*x)/2)^5*((10*B*b)/3 - B*a + ( 10*C*a)/3 + (3*C*b)/4))/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^ 2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) + (atanh(tan(c/2 + (d*x)/2))*(B*a + (3*C*b)/4))/d