Integrand size = 39, antiderivative size = 286 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (15 a^3 b B+60 a b^3 B-3 a^4 C+4 b^4 (5 A+4 C)+4 a^2 b^2 (20 A+13 C)\right ) \tan (c+d x)}{30 b d}+\frac {\left (30 a^2 b B+45 b^3 B-6 a^3 C+a b^2 (100 A+71 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (4 b^2 (5 A+4 C)+3 a (5 b B-a C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 b d}+\frac {(5 b B-a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 b d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 b d} \]
1/8*(12*B*a^2*b+3*B*b^3+4*a^3*(2*A+C)+3*a*b^2*(4*A+3*C))*arctanh(sin(d*x+c ))/d+1/30*(15*B*a^3*b+60*B*a*b^3-3*a^4*C+4*b^4*(5*A+4*C)+4*a^2*b^2*(20*A+1 3*C))*tan(d*x+c)/b/d+1/120*(30*B*a^2*b+45*B*b^3-6*a^3*C+a*b^2*(100*A+71*C) )*sec(d*x+c)*tan(d*x+c)/d+1/60*(4*b^2*(5*A+4*C)+3*a*(5*B*b-C*a))*(a+b*sec( d*x+c))^2*tan(d*x+c)/b/d+1/20*(5*B*b-C*a)*(a+b*sec(d*x+c))^3*tan(d*x+c)/b/ d+1/5*C*(a+b*sec(d*x+c))^4*tan(d*x+c)/b/d
Time = 5.78 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \left (12 a^2 b B+3 b^3 B+4 a^3 (2 A+C)+3 a b^2 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (12 a^2 b B+3 b^3 B+4 a^3 C+3 a b^2 (4 A+3 C)\right ) \sec (c+d x)+30 b^2 (b B+3 a C) \sec ^3(c+d x)+8 \left (15 \left (a^3 B+3 a b^2 B+3 a^2 b (A+C)+b^3 (A+C)\right )+5 b \left (A b^2+3 a b B+3 a^2 C+2 b^2 C\right ) \tan ^2(c+d x)+3 b^3 C \tan ^4(c+d x)\right )\right )}{120 d} \]
(15*(12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*ArcTanh [Sin[c + d*x]] + Tan[c + d*x]*(15*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 3*a*b^ 2*(4*A + 3*C))*Sec[c + d*x] + 30*b^2*(b*B + 3*a*C)*Sec[c + d*x]^3 + 8*(15* (a^3*B + 3*a*b^2*B + 3*a^2*b*(A + C) + b^3*(A + C)) + 5*b*(A*b^2 + 3*a*b*B + 3*a^2*C + 2*b^2*C)*Tan[c + d*x]^2 + 3*b^3*C*Tan[c + d*x]^4)))/(120*d)
Time = 1.54 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.359, Rules used = {3042, 4570, 3042, 4490, 3042, 4490, 3042, 4485, 3042, 4274, 3042, 4254, 24, 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.
| \(\displaystyle \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 (b (5 A+4 C)+(5 b B-a C) \sec (c+d x))dx}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (5 A+4 C)+(5 b B-a C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {\frac {1}{4} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (b (20 a A+15 b B+13 a C)+\left (4 (5 A+4 C) b^2+3 a (5 b B-a C)\right ) \sec (c+d x)\right )dx+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (20 a A+15 b B+13 a C)+\left (4 (5 A+4 C) b^2+3 a (5 b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (b \left ((60 A+33 C) a^2+75 b B a+8 b^2 (5 A+4 C)\right )+\left (-6 C a^3+30 b B a^2+b^2 (100 A+71 C) a+45 b^3 B\right ) \sec (c+d x)\right )dx+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left ((60 A+33 C) a^2+75 b B a+8 b^2 (5 A+4 C)\right )+\left (-6 C a^3+30 b B a^2+b^2 (100 A+71 C) a+45 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (c+d x) \left (15 b \left (4 (2 A+C) a^3+12 b B a^2+3 b^2 (4 A+3 C) a+3 b^3 B\right )+4 \left (-3 C a^4+15 b B a^3+4 b^2 (20 A+13 C) a^2+60 b^3 B a+4 b^4 (5 A+4 C)\right ) \sec (c+d x)\right )dx+\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (15 b \left (4 (2 A+C) a^3+12 b B a^2+3 b^2 (4 A+3 C) a+3 b^3 B\right )+4 \left (-3 C a^4+15 b B a^3+4 b^2 (20 A+13 C) a^2+60 b^3 B a+4 b^4 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right ) \int \sec (c+d x)dx+4 \left (-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right ) \int \sec ^2(c+d x)dx\right )+\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+4 \left (-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {4 \left (-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right ) \int 1d(-\tan (c+d x))}{d}\right )+\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 \tan (c+d x) \left (-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )+\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {b \tan (c+d x) \sec (c+d x) \left (-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right )}{2 d}+\frac {1}{2} \left (\frac {15 b \left (4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {4 \tan (c+d x) \left (-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )\right )+\frac {\tan (c+d x) \left (3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}\) |
(C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d) + (((5*b*B - a*C)*(a + b*S ec[c + d*x])^3*Tan[c + d*x])/(4*d) + (((4*b^2*(5*A + 4*C) + 3*a*(5*b*B - a *C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((b*(30*a^2*b*B + 45*b^3 *B - 6*a^3*C + a*b^2*(100*A + 71*C))*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (( 15*b*(12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*ArcTan h[Sin[c + d*x]])/d + (4*(15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 4*b^4*(5*A + 4*C) + 4*a^2*b^2*(20*A + 13*C))*Tan[c + d*x])/d)/2)/3)/4)/(5*b)
3.9.79.3.1 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_)], 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[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[1/(m + 1) Int[Csc[e + f*x]* (a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1 ))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a* B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 1.40 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.85
| method | result | size |
| parts | \(\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {C \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{3}}{d}\) | \(243\) |
| derivativedivides | \(\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \tan \left (d x +c \right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \tan \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{2} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 C a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(369\) |
| default | \(\frac {a^{3} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \tan \left (d x +c \right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{2} b \tan \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 a^{2} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 C a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(369\) |
| parallelrisch | \(\frac {-600 \left (\left (A +\frac {C}{2}\right ) a^{3}+\frac {3 B \,a^{2} b}{2}+\frac {3 \left (A +\frac {3 C}{4}\right ) b^{2} a}{2}+\frac {3 B \,b^{3}}{8}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+600 \left (\left (A +\frac {C}{2}\right ) a^{3}+\frac {3 B \,a^{2} b}{2}+\frac {3 \left (A +\frac {3 C}{4}\right ) b^{2} a}{2}+\frac {3 B \,b^{3}}{8}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (360 B \,a^{3}+1080 \left (A +\frac {10 C}{9}\right ) b \,a^{2}+1200 B a \,b^{2}+400 b^{3} \left (A +\frac {4 C}{5}\right )\right ) \sin \left (3 d x +3 c \right )+\left (120 B \,a^{3}+360 b \left (A +\frac {2 C}{3}\right ) a^{2}+240 B a \,b^{2}+80 b^{3} \left (A +\frac {4 C}{5}\right )\right ) \sin \left (5 d x +5 c \right )+\left (240 a^{3} C +720 B \,a^{2} b +720 b^{2} \left (A +\frac {7 C}{4}\right ) a +420 B \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (120 a^{3} C +360 B \,a^{2} b +360 \left (A +\frac {3 C}{4}\right ) b^{2} a +90 B \,b^{3}\right ) \sin \left (4 d x +4 c \right )+720 \left (\frac {B \,a^{3}}{3}+b \left (A +\frac {4 C}{3}\right ) a^{2}+\frac {4 B a \,b^{2}}{3}+\frac {4 b^{3} \left (A +2 C \right )}{9}\right ) \sin \left (d x +c \right )}{600 d \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right )}\) | \(406\) |
| norman | \(\frac {-\frac {4 \left (135 A \,a^{2} b +25 A \,b^{3}+45 B \,a^{3}+75 B a \,b^{2}+75 a^{2} b C +29 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {\left (24 A \,a^{2} b -12 a A \,b^{2}+8 A \,b^{3}+8 B \,a^{3}-12 B \,a^{2} b +24 B a \,b^{2}-5 B \,b^{3}-4 a^{3} C +24 a^{2} b C -15 C a \,b^{2}+8 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (24 A \,a^{2} b +12 a A \,b^{2}+8 A \,b^{3}+8 B \,a^{3}+12 B \,a^{2} b +24 B a \,b^{2}+5 B \,b^{3}+4 a^{3} C +24 a^{2} b C +15 C a \,b^{2}+8 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (144 A \,a^{2} b -36 a A \,b^{2}+32 A \,b^{3}+48 B \,a^{3}-36 B \,a^{2} b +96 B a \,b^{2}-3 B \,b^{3}-12 a^{3} C +96 a^{2} b C -9 C a \,b^{2}+16 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (144 A \,a^{2} b +36 a A \,b^{2}+32 A \,b^{3}+48 B \,a^{3}+36 B \,a^{2} b +96 B a \,b^{2}+3 B \,b^{3}+12 a^{3} C +96 a^{2} b C +9 C a \,b^{2}+16 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {\left (8 a^{3} A +12 a A \,b^{2}+12 B \,a^{2} b +3 B \,b^{3}+4 a^{3} C +9 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (8 a^{3} A +12 a A \,b^{2}+12 B \,a^{2} b +3 B \,b^{3}+4 a^{3} C +9 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(543\) |
| risch | \(\text {Expression too large to display}\) | \(937\) |
(3*A*a^2*b+B*a^3)/d*tan(d*x+c)+(B*b^3+3*C*a*b^2)/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)))-(A*b^3+3*B*a*b^2+3 *C*a^2*b)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(3*A*a*b^2+3*B*a^2*b+C*a^3) /d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-C*b^3/d*(-8/1 5-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+1/d*A*ln(sec(d*x+c)+tan(d *x+c))*a^3
Time = 0.29 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.02 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 12 \, B a^{2} b + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, {\left (2 \, A + C\right )} a^{3} + 12 \, B a^{2} b + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (15 \, B a^{3} + 15 \, {\left (3 \, A + 2 \, C\right )} a^{2} b + 30 \, B a b^{2} + 2 \, {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 24 \, C b^{3} + 15 \, {\left (4 \, C a^{3} + 12 \, B a^{2} b + 3 \, {\left (4 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (15 \, C a^{2} b + 15 \, B a b^{2} + {\left (5 \, A + 4 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
integrate(sec(d*x+c)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
1/240*(15*(4*(2*A + C)*a^3 + 12*B*a^2*b + 3*(4*A + 3*C)*a*b^2 + 3*B*b^3)*c os(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(4*(2*A + C)*a^3 + 12*B*a^2*b + 3 *(4*A + 3*C)*a*b^2 + 3*B*b^3)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(8 *(15*B*a^3 + 15*(3*A + 2*C)*a^2*b + 30*B*a*b^2 + 2*(5*A + 4*C)*b^3)*cos(d* x + c)^4 + 24*C*b^3 + 15*(4*C*a^3 + 12*B*a^2*b + 3*(4*A + 3*C)*a*b^2 + 3*B *b^3)*cos(d*x + c)^3 + 8*(15*C*a^2*b + 15*B*a*b^2 + (5*A + 4*C)*b^3)*cos(d *x + c)^2 + 30*(3*C*a*b^2 + B*b^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)
\[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Time = 0.24 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.56 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b^{2} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{3} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C b^{3} - 45 \, C a b^{2} {\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 )} - 15 \, B b^{3} {\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 )} - 60 \, C a^{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 )} - 180 \, B a^{2} 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 )} - 180 \, A 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 )} + 240 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, B a^{3} \tan \left (d x + c\right ) + 720 \, A a^{2} b \tan \left (d x + c\right )}{240 \, d} \]
integrate(sec(d*x+c)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
1/240*(240*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^2*b + 240*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a*b^2 + 80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*b^3 + 16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*b^3 - 45*C*a *b^2*(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)) - 15*B*b^3* (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)) - 60*C*a^3*(2*si n(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 180*B*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 180*A*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)) + 240*A*a^3*log (sec(d*x + c) + tan(d*x + c)) + 240*B*a^3*tan(d*x + c) + 720*A*a^2*b*tan(d *x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (273) = 546\).
Time = 0.37 (sec) , antiderivative size = 989, normalized size of antiderivative = 3.46 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
1/120*(15*(8*A*a^3 + 4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 9*C*a*b^2 + 3*B*b ^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(8*A*a^3 + 4*C*a^3 + 12*B*a^2* b + 12*A*a*b^2 + 9*C*a*b^2 + 3*B*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*B*a^3*tan(1/2*d*x + 1/2*c)^9 - 60*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 3 60*A*a^2*b*tan(1/2*d*x + 1/2*c)^9 - 180*B*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 3 60*C*a^2*b*tan(1/2*d*x + 1/2*c)^9 - 180*A*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 3 60*B*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 225*C*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 1 20*A*b^3*tan(1/2*d*x + 1/2*c)^9 - 75*B*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*C* b^3*tan(1/2*d*x + 1/2*c)^9 - 480*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 120*C*a^3* tan(1/2*d*x + 1/2*c)^7 - 1440*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 360*B*a^2*b *tan(1/2*d*x + 1/2*c)^7 - 960*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 360*A*a*b^2 *tan(1/2*d*x + 1/2*c)^7 - 960*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 90*C*a*b^2* tan(1/2*d*x + 1/2*c)^7 - 320*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 30*B*b^3*tan(1 /2*d*x + 1/2*c)^7 - 160*C*b^3*tan(1/2*d*x + 1/2*c)^7 + 720*B*a^3*tan(1/2*d *x + 1/2*c)^5 + 2160*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 1200*C*a^2*b*tan(1/2 *d*x + 1/2*c)^5 + 1200*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 400*A*b^3*tan(1/2* d*x + 1/2*c)^5 + 464*C*b^3*tan(1/2*d*x + 1/2*c)^5 - 480*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 120*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1440*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 360*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 960*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 360*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 960*B*a*b^2*tan(1/2*d*...
Time = 19.04 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.10 \[ \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a^3+\frac {3\,B\,b^3}{8}+\frac {C\,a^3}{2}+\frac {3\,A\,a\,b^2}{2}+\frac {3\,B\,a^2\,b}{2}+\frac {9\,C\,a\,b^2}{8}\right )}{4\,A\,a^3+\frac {3\,B\,b^3}{2}+2\,C\,a^3+6\,A\,a\,b^2+6\,B\,a^2\,b+\frac {9\,C\,a\,b^2}{2}}\right )\,\left (2\,A\,a^3+\frac {3\,B\,b^3}{4}+C\,a^3+3\,A\,a\,b^2+3\,B\,a^2\,b+\frac {9\,C\,a\,b^2}{4}\right )}{d}-\frac {\left (2\,A\,b^3+2\,B\,a^3-\frac {5\,B\,b^3}{4}-C\,a^3+2\,C\,b^3-3\,A\,a\,b^2+6\,A\,a^2\,b+6\,B\,a\,b^2-3\,B\,a^2\,b-\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {B\,b^3}{2}-8\,B\,a^3-\frac {16\,A\,b^3}{3}+2\,C\,a^3-\frac {8\,C\,b^3}{3}+6\,A\,a\,b^2-24\,A\,a^2\,b-16\,B\,a\,b^2+6\,B\,a^2\,b+\frac {3\,C\,a\,b^2}{2}-16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {20\,A\,b^3}{3}+12\,B\,a^3+\frac {116\,C\,b^3}{15}+36\,A\,a^2\,b+20\,B\,a\,b^2+20\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {16\,A\,b^3}{3}-8\,B\,a^3-\frac {B\,b^3}{2}-2\,C\,a^3-\frac {8\,C\,b^3}{3}-6\,A\,a\,b^2-24\,A\,a^2\,b-16\,B\,a\,b^2-6\,B\,a^2\,b-\frac {3\,C\,a\,b^2}{2}-16\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,b^3+2\,B\,a^3+\frac {5\,B\,b^3}{4}+C\,a^3+2\,C\,b^3+3\,A\,a\,b^2+6\,A\,a^2\,b+6\,B\,a\,b^2+3\,B\,a^2\,b+\frac {15\,C\,a\,b^2}{4}+6\,C\,a^2\,b\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 )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
(atanh((4*tan(c/2 + (d*x)/2)*(A*a^3 + (3*B*b^3)/8 + (C*a^3)/2 + (3*A*a*b^2 )/2 + (3*B*a^2*b)/2 + (9*C*a*b^2)/8))/(4*A*a^3 + (3*B*b^3)/2 + 2*C*a^3 + 6 *A*a*b^2 + 6*B*a^2*b + (9*C*a*b^2)/2))*(2*A*a^3 + (3*B*b^3)/4 + C*a^3 + 3* A*a*b^2 + 3*B*a^2*b + (9*C*a*b^2)/4))/d - (tan(c/2 + (d*x)/2)*(2*A*b^3 + 2 *B*a^3 + (5*B*b^3)/4 + C*a^3 + 2*C*b^3 + 3*A*a*b^2 + 6*A*a^2*b + 6*B*a*b^2 + 3*B*a^2*b + (15*C*a*b^2)/4 + 6*C*a^2*b) + tan(c/2 + (d*x)/2)^5*((20*A*b ^3)/3 + 12*B*a^3 + (116*C*b^3)/15 + 36*A*a^2*b + 20*B*a*b^2 + 20*C*a^2*b) + tan(c/2 + (d*x)/2)^9*(2*A*b^3 + 2*B*a^3 - (5*B*b^3)/4 - C*a^3 + 2*C*b^3 - 3*A*a*b^2 + 6*A*a^2*b + 6*B*a*b^2 - 3*B*a^2*b - (15*C*a*b^2)/4 + 6*C*a^2 *b) - tan(c/2 + (d*x)/2)^3*((16*A*b^3)/3 + 8*B*a^3 + (B*b^3)/2 + 2*C*a^3 + (8*C*b^3)/3 + 6*A*a*b^2 + 24*A*a^2*b + 16*B*a*b^2 + 6*B*a^2*b + (3*C*a*b^ 2)/2 + 16*C*a^2*b) - tan(c/2 + (d*x)/2)^7*((16*A*b^3)/3 + 8*B*a^3 - (B*b^3 )/2 - 2*C*a^3 + (8*C*b^3)/3 - 6*A*a*b^2 + 24*A*a^2*b + 16*B*a*b^2 - 6*B*a^ 2*b - (3*C*a*b^2)/2 + 16*C*a^2*b))/(d*(5*tan(c/2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 - 1))