Integrand size = 29, antiderivative size = 180 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {\left (8 a c^3+12 b c^2 d+12 a c d^2+3 b d^3\right ) \text {arctanh}(\sin (e+f x))}{8 f}+\frac {\left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \tan (e+f x)}{6 f}+\frac {d \left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f} \]
1/8*(8*a*c^3+12*a*c*d^2+12*b*c^2*d+3*b*d^3)*arctanh(sin(f*x+e))/f+1/6*(4*a *d*(4*c^2+d^2)+3*b*(c^3+4*c*d^2))*tan(f*x+e)/f+1/24*d*(20*a*c*d+6*b*c^2+9* b*d^2)*sec(f*x+e)*tan(f*x+e)/f+1/12*(4*a*d+3*b*c)*(c+d*sec(f*x+e))^2*tan(f *x+e)/f+1/4*b*(c+d*sec(f*x+e))^3*tan(f*x+e)/f
Time = 0.78 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {3 \left (3 b d \left (4 c^2+d^2\right )+4 a \left (2 c^3+3 c d^2\right )\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (9 d \left (4 a c d+b \left (4 c^2+d^2\right )\right ) \sec (e+f x)+6 b d^3 \sec ^3(e+f x)+8 \left (3 a d \left (3 c^2+d^2\right )+3 b \left (c^3+3 c d^2\right )+d^2 (3 b c+a d) \tan ^2(e+f x)\right )\right )}{24 f} \]
(3*(3*b*d*(4*c^2 + d^2) + 4*a*(2*c^3 + 3*c*d^2))*ArcTanh[Sin[e + f*x]] + T an[e + f*x]*(9*d*(4*a*c*d + b*(4*c^2 + d^2))*Sec[e + f*x] + 6*b*d^3*Sec[e + f*x]^3 + 8*(3*a*d*(3*c^2 + d^2) + 3*b*(c^3 + 3*c*d^2) + d^2*(3*b*c + a*d )*Tan[e + f*x]^2)))/(24*f)
Time = 1.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {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.
| \(\displaystyle \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {1}{4} \int \sec (e+f x) (c+d \sec (e+f x))^2 (4 a c+3 b d+(3 b c+4 a d) \sec (e+f x))dx+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2 \left (4 a c+3 b d+(3 b c+4 a d) \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \sec (e+f x) (c+d \sec (e+f x)) \left (12 a c^2+15 b d c+8 a d^2+\left (6 b c^2+20 a d c+9 b d^2\right ) \sec (e+f x)\right )dx+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \left (12 a c^2+15 b d c+8 a d^2+\left (6 b c^2+20 a d c+9 b d^2\right ) \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (e+f x) \left (3 \left (3 b d \left (4 c^2+d^2\right )+4 a \left (2 c^3+3 d^2 c\right )\right )+4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 d^2 c\right )\right ) \sec (e+f x)\right )dx+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (3 \left (3 b d \left (4 c^2+d^2\right )+4 a \left (2 c^3+3 d^2 c\right )\right )+4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 d^2 c\right )\right ) \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \int \sec ^2(e+f x)dx+3 \left (8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right ) \int \sec (e+f x)dx\right )+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx+3 \left (8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right ) \int \csc \left (e+f x+\frac {\pi }{2}\right )dx\right )+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right ) \int \csc \left (e+f x+\frac {\pi }{2}\right )dx-\frac {4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \int 1d(-\tan (e+f x))}{f}\right )+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right ) \int \csc \left (e+f x+\frac {\pi }{2}\right )dx+\frac {4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \tan (e+f x)}{f}\right )+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {3 \left (8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right ) \text {arctanh}(\sin (e+f x))}{f}+\frac {4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \tan (e+f x)}{f}\right )+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{2 f}\right )+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f}\right )+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f}\) |
(b*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(4*f) + (((3*b*c + 4*a*d)*(c + d*S ec[e + f*x])^2*Tan[e + f*x])/(3*f) + ((d*(6*b*c^2 + 20*a*c*d + 9*b*d^2)*Se c[e + f*x]*Tan[e + f*x])/(2*f) + ((3*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*ArcTanh[Sin[e + f*x]])/f + (4*(4*a*d*(4*c^2 + d^2) + 3*b*(c^3 + 4 *c*d^2))*Tan[e + f*x])/f)/2)/3)/4
3.3.45.3.1 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]
Time = 3.81 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03
| method | result | size |
| parts | \(-\frac {\left (a \,d^{3}+3 b c \,d^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {\left (3 a \,c^{2} d +b \,c^{3}\right ) \tan \left (f x +e \right )}{f}+\frac {a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {b \,d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(185\) |
| derivativedivides | \(\frac {a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a \,c^{2} d \tan \left (f x +e \right )+3 a c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a \,d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,c^{3} \tan \left (f x +e \right )+3 b \,c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 b c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(223\) |
| default | \(\frac {a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a \,c^{2} d \tan \left (f x +e \right )+3 a c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a \,d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,c^{3} \tan \left (f x +e \right )+3 b \,c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 b c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(223\) |
| parallelrisch | \(\frac {-96 \left (\frac {3}{4}+\frac {\cos \left (4 f x +4 e \right )}{4}+\cos \left (2 f x +2 e \right )\right ) \left (a \,c^{3}+\frac {3}{2} a c \,d^{2}+\frac {3}{2} b \,c^{2} d +\frac {3}{8} b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+96 \left (\frac {3}{4}+\frac {\cos \left (4 f x +4 e \right )}{4}+\cos \left (2 f x +2 e \right )\right ) \left (a \,c^{3}+\frac {3}{2} a c \,d^{2}+\frac {3}{2} b \,c^{2} d +\frac {3}{8} b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (144 a \,c^{2} d +64 a \,d^{3}+48 b \,c^{3}+192 b c \,d^{2}\right ) \sin \left (2 f x +2 e \right )+\left (72 a \,c^{2} d +16 a \,d^{3}+24 b \,c^{3}+48 b c \,d^{2}\right ) \sin \left (4 f x +4 e \right )+72 d \left (\left (a c d +b \,c^{2}+\frac {1}{4} b \,d^{2}\right ) \sin \left (3 f x +3 e \right )+\sin \left (f x +e \right ) \left (a c d +b \,c^{2}+\frac {11}{12} b \,d^{2}\right )\right )}{24 f \left (3+\cos \left (4 f x +4 e \right )+4 \cos \left (2 f x +2 e \right )\right )}\) | \(282\) |
| norman | \(\frac {-\frac {\left (24 a \,c^{2} d -12 a c \,d^{2}+8 a \,d^{3}+8 b \,c^{3}-12 b \,c^{2} d +24 b c \,d^{2}-5 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {\left (24 a \,c^{2} d +12 a c \,d^{2}+8 a \,d^{3}+8 b \,c^{3}+12 b \,c^{2} d +24 b c \,d^{2}+5 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {\left (216 a \,c^{2} d -36 a c \,d^{2}+40 a \,d^{3}+72 b \,c^{3}-36 b \,c^{2} d +120 b c \,d^{2}+9 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{12 f}-\frac {\left (216 a \,c^{2} d +36 a c \,d^{2}+40 a \,d^{3}+72 b \,c^{3}+36 b \,c^{2} d +120 b c \,d^{2}-9 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{12 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}-\frac {\left (8 a \,c^{3}+12 a c \,d^{2}+12 b \,c^{2} d +3 b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {\left (8 a \,c^{3}+12 a c \,d^{2}+12 b \,c^{2} d +3 b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(357\) |
| risch | \(-\frac {i \left (-48 b c \,d^{2}-72 a \,c^{2} d -16 a \,d^{3}-24 b \,c^{3}-33 b \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-9 d^{3} b \,{\mathrm e}^{i \left (f x +e \right )}-72 b \,c^{3} {\mathrm e}^{4 i \left (f x +e \right )}+33 b \,d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-48 a \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-72 b \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-64 a \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 b \,d^{3} {\mathrm e}^{7 i \left (f x +e \right )}-24 b \,c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-36 a c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-192 b c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-144 b c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+36 a c \,d^{2} {\mathrm e}^{7 i \left (f x +e \right )}-36 a c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-36 b \,c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}-216 a \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}-216 a \,c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+36 b \,c^{2} d \,{\mathrm e}^{5 i \left (f x +e \right )}-36 b \,c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+36 b \,c^{2} d \,{\mathrm e}^{7 i \left (f x +e \right )}-72 a \,c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}+36 a c \,d^{2} {\mathrm e}^{5 i \left (f x +e \right )}\right )}{12 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{4}}-\frac {a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a c \,d^{2}}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b \,c^{2} d}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b \,d^{3}}{8 f}+\frac {a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a c \,d^{2}}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b \,c^{2} d}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b \,d^{3}}{8 f}\) | \(570\) |
-(a*d^3+3*b*c*d^2)/f*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e)+(3*a*c*d^2+3*b*c^2 *d)/f*(1/2*sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+tan(f*x+e)))+(3*a*c^2*d +b*c^3)/f*tan(f*x+e)+a*c^3/f*ln(sec(f*x+e)+tan(f*x+e))+b*d^3/f*(-(-1/4*sec (f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e)+3/8*ln(sec(f*x+e)+tan(f*x+e)))
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.17 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (6 \, b d^{3} + 8 \, {\left (3 \, b c^{3} + 9 \, a c^{2} d + 6 \, b c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + 9 \, {\left (4 \, b c^{2} d + 4 \, a c d^{2} + b d^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (3 \, b c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \]
1/48*(3*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*cos(f*x + e)^4*log(s in(f*x + e) + 1) - 3*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*cos(f*x + e)^4*log(-sin(f*x + e) + 1) + 2*(6*b*d^3 + 8*(3*b*c^3 + 9*a*c^2*d + 6*b *c*d^2 + 2*a*d^3)*cos(f*x + e)^3 + 9*(4*b*c^2*d + 4*a*c*d^2 + b*d^3)*cos(f *x + e)^2 + 8*(3*b*c*d^2 + a*d^3)*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4)
\[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right ) \left (c + d \sec {\left (e + f x \right )}\right )^{3} \sec {\left (e + f x \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.48 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {48 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} b c d^{2} + 16 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a d^{3} - 3 \, b d^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 36 \, b c^{2} d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 36 \, a c d^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 48 \, b c^{3} \tan \left (f x + e\right ) + 144 \, a c^{2} d \tan \left (f x + e\right )}{48 \, f} \]
1/48*(48*(tan(f*x + e)^3 + 3*tan(f*x + e))*b*c*d^2 + 16*(tan(f*x + e)^3 + 3*tan(f*x + e))*a*d^3 - 3*b*d^3*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(si n(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(sin (f*x + e) - 1)) - 36*b*c^2*d*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(si n(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 36*a*c*d^2*(2*sin(f*x + e)/(sin (f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 48*a*c ^3*log(sec(f*x + e) + tan(f*x + e)) + 48*b*c^3*tan(f*x + e) + 144*a*c^2*d* tan(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (170) = 340\).
Time = 0.36 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.26 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 72 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 72 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 24 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 15 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 72 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 216 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 40 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 9 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 72 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 216 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \]
1/24*(3*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 3*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*log(abs(t an(1/2*f*x + 1/2*e) - 1)) - 2*(24*b*c^3*tan(1/2*f*x + 1/2*e)^7 + 72*a*c^2* d*tan(1/2*f*x + 1/2*e)^7 - 36*b*c^2*d*tan(1/2*f*x + 1/2*e)^7 - 36*a*c*d^2* tan(1/2*f*x + 1/2*e)^7 + 72*b*c*d^2*tan(1/2*f*x + 1/2*e)^7 + 24*a*d^3*tan( 1/2*f*x + 1/2*e)^7 - 15*b*d^3*tan(1/2*f*x + 1/2*e)^7 - 72*b*c^3*tan(1/2*f* x + 1/2*e)^5 - 216*a*c^2*d*tan(1/2*f*x + 1/2*e)^5 + 36*b*c^2*d*tan(1/2*f*x + 1/2*e)^5 + 36*a*c*d^2*tan(1/2*f*x + 1/2*e)^5 - 120*b*c*d^2*tan(1/2*f*x + 1/2*e)^5 - 40*a*d^3*tan(1/2*f*x + 1/2*e)^5 - 9*b*d^3*tan(1/2*f*x + 1/2*e )^5 + 72*b*c^3*tan(1/2*f*x + 1/2*e)^3 + 216*a*c^2*d*tan(1/2*f*x + 1/2*e)^3 + 36*b*c^2*d*tan(1/2*f*x + 1/2*e)^3 + 36*a*c*d^2*tan(1/2*f*x + 1/2*e)^3 + 120*b*c*d^2*tan(1/2*f*x + 1/2*e)^3 + 40*a*d^3*tan(1/2*f*x + 1/2*e)^3 - 9* b*d^3*tan(1/2*f*x + 1/2*e)^3 - 24*b*c^3*tan(1/2*f*x + 1/2*e) - 72*a*c^2*d* tan(1/2*f*x + 1/2*e) - 36*b*c^2*d*tan(1/2*f*x + 1/2*e) - 36*a*c*d^2*tan(1/ 2*f*x + 1/2*e) - 72*b*c*d^2*tan(1/2*f*x + 1/2*e) - 24*a*d^3*tan(1/2*f*x + 1/2*e) - 15*b*d^3*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^4)/f
Time = 17.20 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.19 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c^3+\frac {3\,b\,c^2\,d}{2}+\frac {3\,a\,c\,d^2}{2}+\frac {3\,b\,d^3}{8}\right )}{4\,a\,c^3+6\,b\,c^2\,d+6\,a\,c\,d^2+\frac {3\,b\,d^3}{2}}\right )\,\left (2\,a\,c^3+3\,b\,c^2\,d+3\,a\,c\,d^2+\frac {3\,b\,d^3}{4}\right )}{f}-\frac {\left (2\,a\,d^3+2\,b\,c^3-\frac {5\,b\,d^3}{4}-3\,a\,c\,d^2+6\,a\,c^2\,d+6\,b\,c\,d^2-3\,b\,c^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (3\,a\,c\,d^2-6\,b\,c^3-\frac {3\,b\,d^3}{4}-\frac {10\,a\,d^3}{3}-18\,a\,c^2\,d-10\,b\,c\,d^2+3\,b\,c^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (\frac {10\,a\,d^3}{3}+6\,b\,c^3-\frac {3\,b\,d^3}{4}+3\,a\,c\,d^2+18\,a\,c^2\,d+10\,b\,c\,d^2+3\,b\,c^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (-2\,a\,d^3-2\,b\,c^3-\frac {5\,b\,d^3}{4}-3\,a\,c\,d^2-6\,a\,c^2\,d-6\,b\,c\,d^2-3\,b\,c^2\,d\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
(atanh((4*tan(e/2 + (f*x)/2)*(a*c^3 + (3*b*d^3)/8 + (3*a*c*d^2)/2 + (3*b*c ^2*d)/2))/(4*a*c^3 + (3*b*d^3)/2 + 6*a*c*d^2 + 6*b*c^2*d))*(2*a*c^3 + (3*b *d^3)/4 + 3*a*c*d^2 + 3*b*c^2*d))/f - (tan(e/2 + (f*x)/2)^7*(2*a*d^3 + 2*b *c^3 - (5*b*d^3)/4 - 3*a*c*d^2 + 6*a*c^2*d + 6*b*c*d^2 - 3*b*c^2*d) + tan( e/2 + (f*x)/2)^3*((10*a*d^3)/3 + 6*b*c^3 - (3*b*d^3)/4 + 3*a*c*d^2 + 18*a* c^2*d + 10*b*c*d^2 + 3*b*c^2*d) - tan(e/2 + (f*x)/2)^5*((10*a*d^3)/3 + 6*b *c^3 + (3*b*d^3)/4 - 3*a*c*d^2 + 18*a*c^2*d + 10*b*c*d^2 - 3*b*c^2*d) - ta n(e/2 + (f*x)/2)*(2*a*d^3 + 2*b*c^3 + (5*b*d^3)/4 + 3*a*c*d^2 + 6*a*c^2*d + 6*b*c*d^2 + 3*b*c^2*d))/(f*(6*tan(e/2 + (f*x)/2)^4 - 4*tan(e/2 + (f*x)/2 )^2 - 4*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1))