Integrand size = 26, antiderivative size = 188 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=a^3 c^5 x-\frac {5 a^3 c^5 \text {arctanh}(\sin (e+f x))}{8 f}-\frac {a^3 c^5 \tan (e+f x)}{f}+\frac {5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f} \]
a^3*c^5*x-5/8*a^3*c^5*arctanh(sin(f*x+e))/f-a^3*c^5*tan(f*x+e)/f+5/8*a^3*c ^5*sec(f*x+e)*tan(f*x+e)/f+1/3*a^3*c^5*tan(f*x+e)^3/f-5/12*a^3*c^5*sec(f*x +e)*tan(f*x+e)^3/f-1/5*a^3*c^5*tan(f*x+e)^5/f+1/3*a^3*c^5*sec(f*x+e)*tan(f *x+e)^5/f-1/7*a^3*c^5*tan(f*x+e)^7/f
Time = 1.98 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {a^3 c^5 \sec ^7(e+f x) \left (14700 (e+f x) \cos (e+f x)-16800 \text {arctanh}(\sin (e+f x)) \cos ^7(e+f x)+8820 e \cos (3 (e+f x))+8820 f x \cos (3 (e+f x))+2940 e \cos (5 (e+f x))+2940 f x \cos (5 (e+f x))+420 e \cos (7 (e+f x))+420 f x \cos (7 (e+f x))-4200 \sin (e+f x)+2975 \sin (2 (e+f x))-2184 \sin (3 (e+f x))+980 \sin (4 (e+f x))-2408 \sin (5 (e+f x))+1155 \sin (6 (e+f x))-584 \sin (7 (e+f x))\right )}{26880 f} \]
(a^3*c^5*Sec[e + f*x]^7*(14700*(e + f*x)*Cos[e + f*x] - 16800*ArcTanh[Sin[ e + f*x]]*Cos[e + f*x]^7 + 8820*e*Cos[3*(e + f*x)] + 8820*f*x*Cos[3*(e + f *x)] + 2940*e*Cos[5*(e + f*x)] + 2940*f*x*Cos[5*(e + f*x)] + 420*e*Cos[7*( e + f*x)] + 420*f*x*Cos[7*(e + f*x)] - 4200*Sin[e + f*x] + 2975*Sin[2*(e + f*x)] - 2184*Sin[3*(e + f*x)] + 980*Sin[4*(e + f*x)] - 2408*Sin[5*(e + f* x)] + 1155*Sin[6*(e + f*x)] - 584*Sin[7*(e + f*x)]))/(26880*f)
Time = 0.50 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3042, 4392, 3042, 4374, 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.
| \(\displaystyle \int (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle -a^3 c^3 \int (c-c \sec (e+f x))^2 \tan ^6(e+f x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^3 c^3 \int \cot \left (e+f x+\frac {\pi }{2}\right )^6 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle -a^3 c^3 \int \left (c^2 \tan ^6(e+f x)+c^2 \sec ^2(e+f x) \tan ^6(e+f x)-2 c^2 \sec (e+f x) \tan ^6(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a^3 c^3 \left (\frac {5 c^2 \text {arctanh}(\sin (e+f x))}{8 f}+\frac {c^2 \tan ^7(e+f x)}{7 f}+\frac {c^2 \tan ^5(e+f x)}{5 f}-\frac {c^2 \tan ^3(e+f x)}{3 f}+\frac {c^2 \tan (e+f x)}{f}-\frac {c^2 \tan ^5(e+f x) \sec (e+f x)}{3 f}+\frac {5 c^2 \tan ^3(e+f x) \sec (e+f x)}{12 f}-\frac {5 c^2 \tan (e+f x) \sec (e+f x)}{8 f}-c^2 x\right )\) |
-(a^3*c^3*(-(c^2*x) + (5*c^2*ArcTanh[Sin[e + f*x]])/(8*f) + (c^2*Tan[e + f *x])/f - (5*c^2*Sec[e + f*x]*Tan[e + f*x])/(8*f) - (c^2*Tan[e + f*x]^3)/(3 *f) + (5*c^2*Sec[e + f*x]*Tan[e + f*x]^3)/(12*f) + (c^2*Tan[e + f*x]^5)/(5 *f) - (c^2*Sec[e + f*x]*Tan[e + f*x]^5)/(3*f) + (c^2*Tan[e + f*x]^7)/(7*f) ))
3.1.11.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[Cot[e + f*x]^(2*m)*( c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !( IntegerQ[n] && GtQ[m - n, 0])
Result contains complex when optimal does not.
Time = 4.59 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(a^{3} c^{5} x -\frac {i c^{5} a^{3} \left (1155 \,{\mathrm e}^{13 i \left (f x +e \right )}+1680 \,{\mathrm e}^{12 i \left (f x +e \right )}+980 \,{\mathrm e}^{11 i \left (f x +e \right )}+10080 \,{\mathrm e}^{10 i \left (f x +e \right )}+2975 \,{\mathrm e}^{9 i \left (f x +e \right )}+16240 \,{\mathrm e}^{8 i \left (f x +e \right )}+24640 \,{\mathrm e}^{6 i \left (f x +e \right )}-2975 \,{\mathrm e}^{5 i \left (f x +e \right )}+14448 \,{\mathrm e}^{4 i \left (f x +e \right )}-980 \,{\mathrm e}^{3 i \left (f x +e \right )}+6496 \,{\mathrm e}^{2 i \left (f x +e \right )}-1155 \,{\mathrm e}^{i \left (f x +e \right )}+1168\right )}{420 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{7}}+\frac {5 c^{5} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{8 f}-\frac {5 c^{5} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{8 f}\) | \(217\) |
| parts | \(a^{3} c^{5} x +\frac {c^{5} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {2 a^{3} c^{5} \tan \left (f x +e \right )}{f}+\frac {3 a^{3} c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}-\frac {6 c^{5} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}-\frac {2 c^{5} a^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}+\frac {2 c^{5} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}+\frac {c^{5} a^{3} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )}{f}\) | \(281\) |
| parallelrisch | \(-\frac {10 a^{3} c^{5} \left (\frac {\left (-\cos \left (7 f x +7 e \right )-7 \cos \left (5 f x +5 e \right )-21 \cos \left (3 f x +3 e \right )-35 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16}+\frac {\left (\cos \left (7 f x +7 e \right )+7 \cos \left (5 f x +5 e \right )+21 \cos \left (3 f x +3 e \right )+35 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16}-\frac {7 f x \cos \left (f x +e \right )}{2}-\frac {21 f x \cos \left (3 f x +3 e \right )}{10}-\frac {7 f x \cos \left (5 f x +5 e \right )}{10}-\frac {f x \cos \left (7 f x +7 e \right )}{10}+\sin \left (f x +e \right )-\frac {17 \sin \left (2 f x +2 e \right )}{24}+\frac {13 \sin \left (3 f x +3 e \right )}{25}-\frac {7 \sin \left (4 f x +4 e \right )}{30}+\frac {43 \sin \left (5 f x +5 e \right )}{75}-\frac {11 \sin \left (6 f x +6 e \right )}{40}+\frac {73 \sin \left (7 f x +7 e \right )}{525}\right )}{f \left (\cos \left (7 f x +7 e \right )+7 \cos \left (5 f x +5 e \right )+21 \cos \left (3 f x +3 e \right )+35 \cos \left (f x +e \right )\right )}\) | \(286\) |
| derivativedivides | \(\frac {c^{5} a^{3} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )+2 c^{5} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-2 c^{5} a^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-6 c^{5} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+6 c^{5} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{5} a^{3} \tan \left (f x +e \right )-2 c^{5} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{3} \left (f x +e \right )}{f}\) | \(288\) |
| default | \(\frac {c^{5} a^{3} \left (-\frac {16}{35}-\frac {\sec \left (f x +e \right )^{6}}{7}-\frac {6 \sec \left (f x +e \right )^{4}}{35}-\frac {8 \sec \left (f x +e \right )^{2}}{35}\right ) \tan \left (f x +e \right )+2 c^{5} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-2 c^{5} a^{3} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-6 c^{5} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+6 c^{5} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{5} a^{3} \tan \left (f x +e \right )-2 c^{5} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{3} \left (f x +e \right )}{f}\) | \(288\) |
| norman | \(\frac {a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}-a^{3} c^{5} x +7 a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-21 a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+35 a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-35 a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+21 a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-7 a^{3} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}+\frac {3 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {19 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}+\frac {1409 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{60 f}-\frac {1768 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{35 f}+\frac {1413 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{20 f}-\frac {23 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f}+\frac {13 c^{5} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{4 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{7}}+\frac {5 c^{5} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}-\frac {5 c^{5} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(365\) |
a^3*c^5*x-1/420*I*c^5*a^3*(1155*exp(13*I*(f*x+e))+1680*exp(12*I*(f*x+e))+9 80*exp(11*I*(f*x+e))+10080*exp(10*I*(f*x+e))+2975*exp(9*I*(f*x+e))+16240*e xp(8*I*(f*x+e))+24640*exp(6*I*(f*x+e))-2975*exp(5*I*(f*x+e))+14448*exp(4*I *(f*x+e))-980*exp(3*I*(f*x+e))+6496*exp(2*I*(f*x+e))-1155*exp(I*(f*x+e))+1 168)/f/(1+exp(2*I*(f*x+e)))^7+5/8*c^5*a^3/f*ln(exp(I*(f*x+e))-I)-5/8*c^5*a ^3/f*ln(exp(I*(f*x+e))+I)
Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.04 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {1680 \, a^{3} c^{5} f x \cos \left (f x + e\right )^{7} - 525 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} \log \left (\sin \left (f x + e\right ) + 1\right ) + 525 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (1168 \, a^{3} c^{5} \cos \left (f x + e\right )^{6} - 1155 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 256 \, a^{3} c^{5} \cos \left (f x + e\right )^{4} + 910 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 192 \, a^{3} c^{5} \cos \left (f x + e\right )^{2} - 280 \, a^{3} c^{5} \cos \left (f x + e\right ) + 120 \, a^{3} c^{5}\right )} \sin \left (f x + e\right )}{1680 \, f \cos \left (f x + e\right )^{7}} \]
1/1680*(1680*a^3*c^5*f*x*cos(f*x + e)^7 - 525*a^3*c^5*cos(f*x + e)^7*log(s in(f*x + e) + 1) + 525*a^3*c^5*cos(f*x + e)^7*log(-sin(f*x + e) + 1) - 2*( 1168*a^3*c^5*cos(f*x + e)^6 - 1155*a^3*c^5*cos(f*x + e)^5 - 256*a^3*c^5*co s(f*x + e)^4 + 910*a^3*c^5*cos(f*x + e)^3 - 192*a^3*c^5*cos(f*x + e)^2 - 2 80*a^3*c^5*cos(f*x + e) + 120*a^3*c^5)*sin(f*x + e))/(f*cos(f*x + e)^7)
\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=- a^{3} c^{5} \left (\int \left (-1\right )\, dx + \int 2 \sec {\left (e + f x \right )}\, dx + \int 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 6 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 6 \sec ^{5}{\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{7}{\left (e + f x \right )}\right )\, dx + \int \sec ^{8}{\left (e + f x \right )}\, dx\right ) \]
-a**3*c**5*(Integral(-1, x) + Integral(2*sec(e + f*x), x) + Integral(2*sec (e + f*x)**2, x) + Integral(-6*sec(e + f*x)**3, x) + Integral(6*sec(e + f* x)**5, x) + Integral(-2*sec(e + f*x)**6, x) + Integral(-2*sec(e + f*x)**7, x) + Integral(sec(e + f*x)**8, x))
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (174) = 348\).
Time = 0.20 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.89 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=-\frac {48 \, {\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 224 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 1680 \, {\left (f x + e\right )} a^{3} c^{5} + 35 \, a^{3} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 630 \, a^{3} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 2520 \, a^{3} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 3360 \, a^{3} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 3360 \, a^{3} c^{5} \tan \left (f x + e\right )}{1680 \, f} \]
-1/1680*(48*(5*tan(f*x + e)^7 + 21*tan(f*x + e)^5 + 35*tan(f*x + e)^3 + 35 *tan(f*x + e))*a^3*c^5 - 224*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*ta n(f*x + e))*a^3*c^5 - 1680*(f*x + e)*a^3*c^5 + 35*a^3*c^5*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e))/(sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) - 1)) - 630*a^3*c^5*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(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)) + 2520*a^3*c^5*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 3360*a^3*c^5*log(sec(f*x + e) + tan( f*x + e)) + 3360*a^3*c^5*tan(f*x + e))/f
Time = 0.42 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.12 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {840 \, {\left (f x + e\right )} a^{3} c^{5} - 525 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 525 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (1365 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 9660 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 29673 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 21216 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 9863 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2660 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 315 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{7}}}{840 \, f} \]
1/840*(840*(f*x + e)*a^3*c^5 - 525*a^3*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1)) + 525*a^3*c^5*log(abs(tan(1/2*f*x + 1/2*e) - 1)) + 2*(1365*a^3*c^5*tan (1/2*f*x + 1/2*e)^13 - 9660*a^3*c^5*tan(1/2*f*x + 1/2*e)^11 + 29673*a^3*c^ 5*tan(1/2*f*x + 1/2*e)^9 - 21216*a^3*c^5*tan(1/2*f*x + 1/2*e)^7 + 9863*a^3 *c^5*tan(1/2*f*x + 1/2*e)^5 - 2660*a^3*c^5*tan(1/2*f*x + 1/2*e)^3 + 315*a^ 3*c^5*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^7)/f
Time = 15.85 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.38 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx=\frac {\frac {13\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{4}-23\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+\frac {1413\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{20}-\frac {1768\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{35}+\frac {1409\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{60}-\frac {19\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+\frac {3\,a^3\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+a^3\,c^5\,x-\frac {5\,a^3\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{4\,f} \]
((1409*a^3*c^5*tan(e/2 + (f*x)/2)^5)/60 - (19*a^3*c^5*tan(e/2 + (f*x)/2)^3 )/3 - (1768*a^3*c^5*tan(e/2 + (f*x)/2)^7)/35 + (1413*a^3*c^5*tan(e/2 + (f* x)/2)^9)/20 - 23*a^3*c^5*tan(e/2 + (f*x)/2)^11 + (13*a^3*c^5*tan(e/2 + (f* x)/2)^13)/4 + (3*a^3*c^5*tan(e/2 + (f*x)/2))/4)/(f*(7*tan(e/2 + (f*x)/2)^2 - 21*tan(e/2 + (f*x)/2)^4 + 35*tan(e/2 + (f*x)/2)^6 - 35*tan(e/2 + (f*x)/ 2)^8 + 21*tan(e/2 + (f*x)/2)^10 - 7*tan(e/2 + (f*x)/2)^12 + tan(e/2 + (f*x )/2)^14 - 1)) + a^3*c^5*x - (5*a^3*c^5*atanh(tan(e/2 + (f*x)/2)))/(4*f)