Integrand size = 33, antiderivative size = 290 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^4 A x+\frac {\left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \tan (c+d x)}{30 d}+\frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d} \]
a^4*A*x+1/8*(8*B*a^4+24*B*a^2*b^2+3*B*b^4+16*a^3*b*(2*A+C)+4*a*b^3*(4*A+3* C))*arctanh(sin(d*x+c))/d+1/30*(95*B*a^3*b+80*B*a*b^3+12*a^4*C+4*b^4*(5*A+ 4*C)+2*a^2*b^2*(85*A+56*C))*tan(d*x+c)/d+1/120*b*(130*B*a^2*b+45*B*b^3+24* a^3*C+4*a*b^2*(40*A+29*C))*sec(d*x+c)*tan(d*x+c)/d+1/60*(20*A*b^2+35*B*a*b +12*C*a^2+16*C*b^2)*(a+b*sec(d*x+c))^2*tan(d*x+c)/d+1/20*(5*B*b+4*C*a)*(a+ b*sec(d*x+c))^3*tan(d*x+c)/d+1/5*C*(a+b*sec(d*x+c))^4*tan(d*x+c)/d
Time = 6.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.80 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {120 a^4 A d x+15 \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))+15 \left (8 \left (4 a^3 b B+4 a b^3 B+a^4 C+6 a^2 b^2 (A+C)+b^4 (A+C)\right )+b \left (24 a^2 b B+3 b^3 B+16 a^3 C+4 a b^2 (4 A+3 C)\right ) \sec (c+d x)+2 b^3 (b B+4 a C) \sec ^3(c+d x)\right ) \tan (c+d x)+40 b^2 \left (A b^2+4 a b B+6 a^2 C+2 b^2 C\right ) \tan ^3(c+d x)+24 b^4 C \tan ^5(c+d x)}{120 d} \]
(120*a^4*A*d*x + 15*(8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]] + 15*(8*(4*a^3*b*B + 4*a*b^3 *B + a^4*C + 6*a^2*b^2*(A + C) + b^4*(A + C)) + b*(24*a^2*b*B + 3*b^3*B + 16*a^3*C + 4*a*b^2*(4*A + 3*C))*Sec[c + d*x] + 2*b^3*(b*B + 4*a*C)*Sec[c + d*x]^3)*Tan[c + d*x] + 40*b^2*(A*b^2 + 4*a*b*B + 6*a^2*C + 2*b^2*C)*Tan[c + d*x]^3 + 24*b^4*C*Tan[c + d*x]^5)/(120*d)
Time = 1.12 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3042, 4544, 3042, 4544, 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.
| \(\displaystyle \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4544 |
| \(\displaystyle \frac {1}{5} \int (a+b \sec (c+d x))^3 \left ((5 b B+4 a C) \sec ^2(c+d x)+(5 A b+4 C b+5 a B) \sec (c+d x)+5 a A\right )dx+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((5 b B+4 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(5 A b+4 C b+5 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+5 a A\right )dx+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4544 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (20 A a^2+\left (12 C a^2+35 b B a+20 A b^2+16 b^2 C\right ) \sec ^2(c+d x)+\left (20 B a^2+40 A b a+28 b C a+15 b^2 B\right ) \sec (c+d x)\right )dx+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (20 A a^2+\left (12 C a^2+35 b B a+20 A b^2+16 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (20 B a^2+40 A b a+28 b C a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4544 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int (a+b \sec (c+d x)) \left (60 A a^3+\left (24 C a^3+130 b B a^2+4 b^2 (40 A+29 C) a+45 b^3 B\right ) \sec ^2(c+d x)+\left (60 B a^3+36 b (5 A+3 C) a^2+115 b^2 B a+8 b^3 (5 A+4 C)\right ) \sec (c+d x)\right )dx+\frac {\tan (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (60 A a^3+\left (24 C a^3+130 b B a^2+4 b^2 (40 A+29 C) a+45 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (60 B a^3+36 b (5 A+3 C) a^2+115 b^2 B a+8 b^3 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\tan (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (120 A a^4+4 \left (12 C a^4+95 b B a^3+2 b^2 (85 A+56 C) a^2+80 b^3 B a+4 b^4 (5 A+4 C)\right ) \sec ^2(c+d x)+15 \left (8 B a^4+16 b (2 A+C) a^3+24 b^2 B a^2+4 b^3 (4 A+3 C) a+3 b^4 B\right ) \sec (c+d x)\right )dx+\frac {b \tan (c+d x) \sec (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\tan (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {\tan (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \sec (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {b \tan (c+d x) \sec (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}+\frac {1}{2} \left (120 a^4 A x+\frac {15 \left (8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {4 \tan (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )\right )\right )+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
(C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + (((5*b*B + 4*a*C)*(a + b*S ec[c + d*x])^3*Tan[c + d*x])/(4*d) + (((20*A*b^2 + 35*a*b*B + 12*a^2*C + 1 6*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((b*(130*a^2*b*B + 4 5*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Sec[c + d*x]*Tan[c + d*x])/(2* d) + (120*a^4*A*x + (15*(8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/d + (4*(95*a^3*b*B + 80 *a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Tan[c + d*x])/d)/2)/3)/4)/5
3.9.89.3.1 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]
Time = 1.70 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94
| method | result | size |
| parts | \(a^{4} A x +\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \tan \left (d x +c \right )}{d}-\frac {C \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(273\) |
| derivativedivides | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \tan \left (d x +c \right ) a^{3} b +4 a^{3} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \tan \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(421\) |
| default | \(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \tan \left (d x +c \right ) a^{3} b +4 a^{3} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \tan \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(421\) |
| parallelrisch | \(\frac {-2400 \left (\frac {3 B \,b^{4}}{32}+\frac {a \left (A +\frac {3 C}{4}\right ) b^{3}}{2}+\frac {3 B \,a^{2} b^{2}}{4}+a^{3} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{4}}{4}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2400 \left (\frac {3 B \,b^{4}}{32}+\frac {a \left (A +\frac {3 C}{4}\right ) b^{3}}{2}+\frac {3 B \,a^{2} b^{2}}{4}+a^{3} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{4}}{4}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+600 a^{4} A x d \cos \left (3 d x +3 c \right )+120 a^{4} A x d \cos \left (5 d x +5 c \right )+\left (\left (400 A +320 C \right ) b^{4}+1600 B a \,b^{3}+2160 \left (A +\frac {10 C}{9}\right ) a^{2} b^{2}+1440 B \,a^{3} b +360 a^{4} C \right ) \sin \left (3 d x +3 c \right )+\left (\left (80 A +64 C \right ) b^{4}+320 B a \,b^{3}+720 a^{2} \left (A +\frac {2 C}{3}\right ) b^{2}+480 B \,a^{3} b +120 a^{4} C \right ) \sin \left (5 d x +5 c \right )+960 b \left (\frac {7 B \,b^{3}}{16}+b^{2} \left (A +\frac {7 C}{4}\right ) a +\frac {3 B \,a^{2} b}{2}+a^{3} C \right ) \sin \left (2 d x +2 c \right )+480 \left (\frac {3 B \,b^{3}}{16}+\left (A +\frac {3 C}{4}\right ) b^{2} a +\frac {3 B \,a^{2} b}{2}+a^{3} C \right ) b \sin \left (4 d x +4 c \right )+1200 a^{4} A x d \cos \left (d x +c \right )+1440 \left (\left (\frac {2 A}{9}+\frac {4 C}{9}\right ) b^{4}+\frac {8 B a \,b^{3}}{9}+b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}+\frac {2 B \,a^{3} b}{3}+\frac {a^{4} C}{6}\right ) \sin \left (d x +c \right )}{600 d \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right )}\) | \(502\) |
| norman | \(\frac {a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-a^{4} A x -\frac {4 \left (270 A \,a^{2} b^{2}+25 A \,b^{4}+180 B \,a^{3} b +100 B a \,b^{3}+45 a^{4} C +150 C \,a^{2} b^{2}+29 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {\left (48 A \,a^{2} b^{2}-16 a A \,b^{3}+8 A \,b^{4}+32 B \,a^{3} b -24 B \,a^{2} b^{2}+32 B a \,b^{3}-5 B \,b^{4}+8 a^{4} C -16 a^{3} b C +48 C \,a^{2} b^{2}-20 C a \,b^{3}+8 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (48 A \,a^{2} b^{2}+16 a A \,b^{3}+8 A \,b^{4}+32 B \,a^{3} b +24 B \,a^{2} b^{2}+32 B a \,b^{3}+5 B \,b^{4}+8 a^{4} C +16 a^{3} b C +48 C \,a^{2} b^{2}+20 C a \,b^{3}+8 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (288 A \,a^{2} b^{2}-48 a A \,b^{3}+32 A \,b^{4}+192 B \,a^{3} b -72 B \,a^{2} b^{2}+128 B a \,b^{3}-3 B \,b^{4}+48 a^{4} C -48 a^{3} b C +192 C \,a^{2} b^{2}-12 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (288 A \,a^{2} b^{2}+48 a A \,b^{3}+32 A \,b^{4}+192 B \,a^{3} b +72 B \,a^{2} b^{2}+128 B a \,b^{3}+3 B \,b^{4}+48 a^{4} C +48 a^{3} b C +192 C \,a^{2} b^{2}+12 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}+5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {\left (32 A \,a^{3} b +16 a A \,b^{3}+8 B \,a^{4}+24 B \,a^{2} b^{2}+3 B \,b^{4}+16 a^{3} b C +12 C a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (32 A \,a^{3} b +16 a A \,b^{3}+8 B \,a^{4}+24 B \,a^{2} b^{2}+3 B \,b^{4}+16 a^{3} b C +12 C a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(726\) |
| risch | \(\text {Expression too large to display}\) | \(1096\) |
a^4*A*x+(4*A*a^3*b+B*a^4)/d*ln(sec(d*x+c)+tan(d*x+c))+(B*b^4+4*C*a*b^3)/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^4+4*B*a*b^3+6*C*a^2*b^2)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(4 *A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d* x+c)+tan(d*x+c)))+(6*A*a^2*b^2+4*B*a^3*b+C*a^4)/d*tan(d*x+c)-C*b^4/d*(-8/1 5-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)
Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, A a^{4} d x \cos \left (d x + c\right )^{5} + 15 \, {\left (8 \, B a^{4} + 16 \, {\left (2 \, A + C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{4} + 16 \, {\left (2 \, A + C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} + 8 \, {\left (15 \, C a^{4} + 60 \, B a^{3} b + 30 \, {\left (3 \, A + 2 \, C\right )} a^{2} b^{2} + 40 \, B a b^{3} + 2 \, {\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (16 \, C a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (30 \, C a^{2} b^{2} + 20 \, B a b^{3} + {\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
1/240*(240*A*a^4*d*x*cos(d*x + c)^5 + 15*(8*B*a^4 + 16*(2*A + C)*a^3*b + 2 4*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^3 + 3*B*b^4)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(8*B*a^4 + 16*(2*A + C)*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)* a*b^3 + 3*B*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(24*C*b^4 + 8*( 15*C*a^4 + 60*B*a^3*b + 30*(3*A + 2*C)*a^2*b^2 + 40*B*a*b^3 + 2*(5*A + 4*C )*b^4)*cos(d*x + c)^4 + 15*(16*C*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^ 3 + 3*B*b^4)*cos(d*x + c)^3 + 8*(30*C*a^2*b^2 + 20*B*a*b^3 + (5*A + 4*C)*b ^4)*cos(d*x + c)^2 + 30*(4*C*a*b^3 + B*b^4)*cos(d*x + c))*sin(d*x + c))/(d *cos(d*x + c)^5)
\[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{4} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
Time = 0.24 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.71 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, {\left (d x + c\right )} A a^{4} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b^{3} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} + 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^{4} - 60 \, C a 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 )} - 15 \, B b^{4} {\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 )} - 240 \, C a^{3} 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 )} - 360 \, B a^{2} 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 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 )} + 240 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 960 \, A a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, C a^{4} \tan \left (d x + c\right ) + 960 \, B a^{3} b \tan \left (d x + c\right ) + 1440 \, A a^{2} b^{2} \tan \left (d x + c\right )}{240 \, d} \]
1/240*(240*(d*x + c)*A*a^4 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^2*b ^2 + 320*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a*b^3 + 80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*b^4 + 16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan( d*x + c))*C*b^4 - 60*C*a*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)) - 15*B*b^4*(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)) - 240*C*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 360*B*a^2*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*b^ 3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d *x + c) - 1)) + 240*B*a^4*log(sec(d*x + c) + tan(d*x + c)) + 960*A*a^3*b*l og(sec(d*x + c) + tan(d*x + c)) + 240*C*a^4*tan(d*x + c) + 960*B*a^3*b*tan (d*x + c) + 1440*A*a^2*b^2*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1140 vs. \(2 (278) = 556\).
Time = 0.39 (sec) , antiderivative size = 1140, normalized size of antiderivative = 3.93 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
1/120*(120*(d*x + c)*A*a^4 + 15*(8*B*a^4 + 32*A*a^3*b + 16*C*a^3*b + 24*B* a^2*b^2 + 16*A*a*b^3 + 12*C*a*b^3 + 3*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(8*B*a^4 + 32*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 12*C*a*b^3 + 3*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*C*a^4* tan(1/2*d*x + 1/2*c)^9 + 480*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b* tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 360*B*a^2* b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 240*A* a*b^3*tan(1/2*d*x + 1/2*c)^9 + 480*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 300*C* a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 75*B*b^4 *tan(1/2*d*x + 1/2*c)^9 + 120*C*b^4*tan(1/2*d*x + 1/2*c)^9 - 480*C*a^4*tan (1/2*d*x + 1/2*c)^7 - 1920*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 480*C*a^3*b*ta n(1/2*d*x + 1/2*c)^7 - 2880*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 720*B*a^2*b ^2*tan(1/2*d*x + 1/2*c)^7 - 1920*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 480*A* a*b^3*tan(1/2*d*x + 1/2*c)^7 - 1280*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 120*C *a*b^3*tan(1/2*d*x + 1/2*c)^7 - 320*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 30*B*b^ 4*tan(1/2*d*x + 1/2*c)^7 - 160*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 720*C*a^4*ta n(1/2*d*x + 1/2*c)^5 + 2880*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 4320*A*a^2*b^ 2*tan(1/2*d*x + 1/2*c)^5 + 2400*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 1600*B* a*b^3*tan(1/2*d*x + 1/2*c)^5 + 400*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 464*C*b^ 4*tan(1/2*d*x + 1/2*c)^5 - 480*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 1920*B*a^...
Time = 20.10 (sec) , antiderivative size = 4068, normalized size of antiderivative = 14.03 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
(atan(((tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128* A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2 *a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^ 6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480* A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A* C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + (B*a^4 + (3*B*b^4)/8 + 3* B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^ 3 + 64*C*a^3*b))*(B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3* b + (3*C*a*b^3)/2 + 2*C*a^3*b)*1i + (tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B ^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b ^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C *a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b ^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3 ) - (B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^ 3)/2 + 2*C*a^3*b)*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a* b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b))*(B*a^4 + (3*B*b^4)/8 + 3*B*a ^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*1i)/((tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 +...