Integrand size = 32, antiderivative size = 227 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=\frac {55 a^3 c^6 \text {arctanh}(\sin (e+f x))}{128 f}-\frac {25 a^3 c^6 \sec (e+f x) \tan (e+f x)}{128 f}-\frac {15 a^3 c^6 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac {5 a^3 c^6 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac {5 a^3 c^6 \sec ^3(e+f x) \tan ^3(e+f x)}{16 f}-\frac {a^3 c^6 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac {3 a^3 c^6 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac {4 a^3 c^6 \tan ^7(e+f x)}{7 f}+\frac {a^3 c^6 \tan ^9(e+f x)}{9 f} \]
55/128*a^3*c^6*arctanh(sin(f*x+e))/f-25/128*a^3*c^6*sec(f*x+e)*tan(f*x+e)/ f-15/64*a^3*c^6*sec(f*x+e)^3*tan(f*x+e)/f+5/24*a^3*c^6*sec(f*x+e)*tan(f*x+ e)^3/f+5/16*a^3*c^6*sec(f*x+e)^3*tan(f*x+e)^3/f-1/6*a^3*c^6*sec(f*x+e)*tan (f*x+e)^5/f-3/8*a^3*c^6*sec(f*x+e)^3*tan(f*x+e)^5/f+4/7*a^3*c^6*tan(f*x+e) ^7/f+1/9*a^3*c^6*tan(f*x+e)^9/f
Time = 3.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.54 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=\frac {a^3 c^6 \left (443520 \text {arctanh}(\sin (e+f x))-\sec ^9(e+f x) (-88704 \sin (e+f x)+88074 \sin (2 (e+f x))+37632 \sin (3 (e+f x))-2142 \sin (4 (e+f x))+2304 \sin (5 (e+f x))+39858 \sin (6 (e+f x))-7488 \sin (7 (e+f x))+4599 \sin (8 (e+f x))+1856 \sin (9 (e+f x)))\right )}{1032192 f} \]
(a^3*c^6*(443520*ArcTanh[Sin[e + f*x]] - Sec[e + f*x]^9*(-88704*Sin[e + f* x] + 88074*Sin[2*(e + f*x)] + 37632*Sin[3*(e + f*x)] - 2142*Sin[4*(e + f*x )] + 2304*Sin[5*(e + f*x)] + 39858*Sin[6*(e + f*x)] - 7488*Sin[7*(e + f*x) ] + 4599*Sin[8*(e + f*x)] + 1856*Sin[9*(e + f*x)])))/(1032192*f)
Time = 0.54 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3042, 4446, 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.
| \(\displaystyle \int \sec (e+f x) (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^6 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^6dx\) |
| \(\Big \downarrow \) 4446 |
| \(\displaystyle -a^3 c^3 \int \left (-c^3 \sec ^4(e+f x) \tan ^6(e+f x)+3 c^3 \sec ^3(e+f x) \tan ^6(e+f x)-3 c^3 \sec ^2(e+f x) \tan ^6(e+f x)+c^3 \sec (e+f x) \tan ^6(e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -a^3 c^3 \left (-\frac {55 c^3 \text {arctanh}(\sin (e+f x))}{128 f}-\frac {c^3 \tan ^9(e+f x)}{9 f}-\frac {4 c^3 \tan ^7(e+f x)}{7 f}+\frac {3 c^3 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}-\frac {5 c^3 \tan ^3(e+f x) \sec ^3(e+f x)}{16 f}+\frac {15 c^3 \tan (e+f x) \sec ^3(e+f x)}{64 f}+\frac {c^3 \tan ^5(e+f x) \sec (e+f x)}{6 f}-\frac {5 c^3 \tan ^3(e+f x) \sec (e+f x)}{24 f}+\frac {25 c^3 \tan (e+f x) \sec (e+f x)}{128 f}\right )\) |
-(a^3*c^3*((-55*c^3*ArcTanh[Sin[e + f*x]])/(128*f) + (25*c^3*Sec[e + f*x]* Tan[e + f*x])/(128*f) + (15*c^3*Sec[e + f*x]^3*Tan[e + f*x])/(64*f) - (5*c ^3*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) - (5*c^3*Sec[e + f*x]^3*Tan[e + f*x ]^3)/(16*f) + (c^3*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) + (3*c^3*Sec[e + f*x ]^3*Tan[e + f*x]^5)/(8*f) - (4*c^3*Tan[e + f*x]^7)/(7*f) - (c^3*Tan[e + f* x]^9)/(9*f)))
3.1.21.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n - m ), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && Eq Q[a^2 - b^2, 0] && IntegersQ[m, n] && GeQ[n - m, 0] && GtQ[m*n, 0]
Result contains complex when optimal does not.
Time = 9.32 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {i a^{3} c^{6} \left (4599 \,{\mathrm e}^{17 i \left (f x +e \right )}-24192 \,{\mathrm e}^{16 i \left (f x +e \right )}+39858 \,{\mathrm e}^{15 i \left (f x +e \right )}-64512 \,{\mathrm e}^{14 i \left (f x +e \right )}-2142 \,{\mathrm e}^{13 i \left (f x +e \right )}-118272 \,{\mathrm e}^{12 i \left (f x +e \right )}+88074 \,{\mathrm e}^{11 i \left (f x +e \right )}-322560 \,{\mathrm e}^{10 i \left (f x +e \right )}-145152 \,{\mathrm e}^{8 i \left (f x +e \right )}-88074 \,{\mathrm e}^{7 i \left (f x +e \right )}-193536 \,{\mathrm e}^{6 i \left (f x +e \right )}+2142 \,{\mathrm e}^{5 i \left (f x +e \right )}-69120 \,{\mathrm e}^{4 i \left (f x +e \right )}-39858 \,{\mathrm e}^{3 i \left (f x +e \right )}-9216 \,{\mathrm e}^{2 i \left (f x +e \right )}-4599 \,{\mathrm e}^{i \left (f x +e \right )}-3712\right )}{4032 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{9}}-\frac {55 a^{3} c^{6} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{128 f}+\frac {55 a^{3} c^{6} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{128 f}\) | \(253\) |
| parallelrisch | \(-\frac {55 a^{3} c^{6} \left (\left (\cos \left (9 f x +9 e \right )+9 \cos \left (7 f x +7 e \right )+36 \cos \left (5 f x +5 e \right )+84 \cos \left (3 f x +3 e \right )+126 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+\left (-\cos \left (9 f x +9 e \right )-9 \cos \left (7 f x +7 e \right )-36 \cos \left (5 f x +5 e \right )-84 \cos \left (3 f x +3 e \right )-126 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {256 \sin \left (f x +e \right )}{5}+\frac {2796 \sin \left (2 f x +2 e \right )}{55}+\frac {3584 \sin \left (3 f x +3 e \right )}{165}-\frac {68 \sin \left (4 f x +4 e \right )}{55}+\frac {512 \sin \left (5 f x +5 e \right )}{385}+\frac {3796 \sin \left (6 f x +6 e \right )}{165}-\frac {1664 \sin \left (7 f x +7 e \right )}{385}+\frac {146 \sin \left (8 f x +8 e \right )}{55}+\frac {3712 \sin \left (9 f x +9 e \right )}{3465}\right )}{128 f \left (\cos \left (9 f x +9 e \right )+9 \cos \left (7 f x +7 e \right )+36 \cos \left (5 f x +5 e \right )+84 \cos \left (3 f x +3 e \right )+126 \cos \left (f x +e \right )\right )}\) | \(292\) |
| derivativedivides | \(\frac {-a^{3} c^{6} \left (-\frac {128}{315}-\frac {\sec \left (f x +e \right )^{8}}{9}-\frac {8 \sec \left (f x +e \right )^{6}}{63}-\frac {16 \sec \left (f x +e \right )^{4}}{105}-\frac {64 \sec \left (f x +e \right )^{2}}{315}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \tan \left (f x +e \right )+a^{3} c^{6} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+8 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+6 a^{3} c^{6} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-6 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-8 a^{3} c^{6} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(345\) |
| default | \(\frac {-a^{3} c^{6} \left (-\frac {128}{315}-\frac {\sec \left (f x +e \right )^{8}}{9}-\frac {8 \sec \left (f x +e \right )^{6}}{63}-\frac {16 \sec \left (f x +e \right )^{4}}{105}-\frac {64 \sec \left (f x +e \right )^{2}}{315}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \tan \left (f x +e \right )+a^{3} c^{6} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+8 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+6 a^{3} c^{6} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-6 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-8 a^{3} c^{6} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-3 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(345\) |
| parts | \(\frac {a^{3} c^{6} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {a^{3} c^{6} \left (-\frac {128}{315}-\frac {\sec \left (f x +e \right )^{8}}{9}-\frac {8 \sec \left (f x +e \right )^{6}}{63}-\frac {16 \sec \left (f x +e \right )^{4}}{105}-\frac {64 \sec \left (f x +e \right )^{2}}{315}\right ) \tan \left (f x +e \right )}{f}-\frac {3 a^{3} c^{6} \tan \left (f x +e \right )}{f}-\frac {8 a^{3} c^{6} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}-\frac {6 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}+\frac {6 a^{3} c^{6} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}+\frac {8 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}-\frac {3 a^{3} c^{6} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(365\) |
1/4032*I*a^3*c^6*(4599*exp(17*I*(f*x+e))-24192*exp(16*I*(f*x+e))+39858*exp (15*I*(f*x+e))-64512*exp(14*I*(f*x+e))-2142*exp(13*I*(f*x+e))-118272*exp(1 2*I*(f*x+e))+88074*exp(11*I*(f*x+e))-322560*exp(10*I*(f*x+e))-145152*exp(8 *I*(f*x+e))-88074*exp(7*I*(f*x+e))-193536*exp(6*I*(f*x+e))+2142*exp(5*I*(f *x+e))-69120*exp(4*I*(f*x+e))-39858*exp(3*I*(f*x+e))-9216*exp(2*I*(f*x+e)) -4599*exp(I*(f*x+e))-3712)/f/(1+exp(2*I*(f*x+e)))^9-55/128*a^3*c^6/f*ln(ex p(I*(f*x+e))-I)+55/128*a^3*c^6/f*ln(exp(I*(f*x+e))+I)
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.92 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=\frac {3465 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3465 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (3712 \, a^{3} c^{6} \cos \left (f x + e\right )^{8} + 4599 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 10240 \, a^{3} c^{6} \cos \left (f x + e\right )^{6} + 3066 \, a^{3} c^{6} \cos \left (f x + e\right )^{5} + 8448 \, a^{3} c^{6} \cos \left (f x + e\right )^{4} - 7224 \, a^{3} c^{6} \cos \left (f x + e\right )^{3} - 1024 \, a^{3} c^{6} \cos \left (f x + e\right )^{2} + 3024 \, a^{3} c^{6} \cos \left (f x + e\right ) - 896 \, a^{3} c^{6}\right )} \sin \left (f x + e\right )}{16128 \, f \cos \left (f x + e\right )^{9}} \]
1/16128*(3465*a^3*c^6*cos(f*x + e)^9*log(sin(f*x + e) + 1) - 3465*a^3*c^6* cos(f*x + e)^9*log(-sin(f*x + e) + 1) - 2*(3712*a^3*c^6*cos(f*x + e)^8 + 4 599*a^3*c^6*cos(f*x + e)^7 - 10240*a^3*c^6*cos(f*x + e)^6 + 3066*a^3*c^6*c os(f*x + e)^5 + 8448*a^3*c^6*cos(f*x + e)^4 - 7224*a^3*c^6*cos(f*x + e)^3 - 1024*a^3*c^6*cos(f*x + e)^2 + 3024*a^3*c^6*cos(f*x + e) - 896*a^3*c^6)*s in(f*x + e))/(f*cos(f*x + e)^9)
\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=a^{3} c^{6} \left (\int \sec {\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int 8 \sec ^{4}{\left (e + f x \right )}\, dx + \int \left (- 6 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- 6 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int 8 \sec ^{7}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{9}{\left (e + f x \right )}\right )\, dx + \int \sec ^{10}{\left (e + f x \right )}\, dx\right ) \]
a**3*c**6*(Integral(sec(e + f*x), x) + Integral(-3*sec(e + f*x)**2, x) + I ntegral(8*sec(e + f*x)**4, x) + Integral(-6*sec(e + f*x)**5, x) + Integral (-6*sec(e + f*x)**6, x) + Integral(8*sec(e + f*x)**7, x) + Integral(-3*sec (e + f*x)**9, x) + Integral(sec(e + f*x)**10, x))
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (209) = 418\).
Time = 0.23 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.95 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=\frac {256 \, {\left (35 \, \tan \left (f x + e\right )^{9} + 180 \, \tan \left (f x + e\right )^{7} + 378 \, \tan \left (f x + e\right )^{5} + 420 \, \tan \left (f x + e\right )^{3} + 315 \, \tan \left (f x + e\right )\right )} a^{3} c^{6} - 32256 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{6} + 215040 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{6} + 315 \, a^{3} c^{6} {\left (\frac {2 \, {\left (105 \, \sin \left (f x + e\right )^{7} - 385 \, \sin \left (f x + e\right )^{5} + 511 \, \sin \left (f x + e\right )^{3} - 279 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{8} - 4 \, \sin \left (f x + e\right )^{6} + 6 \, \sin \left (f x + e\right )^{4} - 4 \, \sin \left (f x + e\right )^{2} + 1} - 105 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 105 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 6720 \, a^{3} c^{6} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 30240 \, a^{3} c^{6} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 80640 \, a^{3} c^{6} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 241920 \, a^{3} c^{6} \tan \left (f x + e\right )}{80640 \, f} \]
1/80640*(256*(35*tan(f*x + e)^9 + 180*tan(f*x + e)^7 + 378*tan(f*x + e)^5 + 420*tan(f*x + e)^3 + 315*tan(f*x + e))*a^3*c^6 - 32256*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*c^6 + 215040*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c^6 + 315*a^3*c^6*(2*(105*sin(f*x + e)^7 - 385*sin(f* x + e)^5 + 511*sin(f*x + e)^3 - 279*sin(f*x + e))/(sin(f*x + e)^8 - 4*sin( f*x + e)^6 + 6*sin(f*x + e)^4 - 4*sin(f*x + e)^2 + 1) - 105*log(sin(f*x + e) + 1) + 105*log(sin(f*x + e) - 1)) - 6720*a^3*c^6*(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)) + 30240*a^3*c^6*(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)) + 80640*a^3*c^6*log(sec(f*x + e) + tan(f*x + e)) - 241920*a^3*c^6*tan (f*x + e))/f
Time = 0.42 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=\frac {3465 \, a^{3} c^{6} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3465 \, a^{3} c^{6} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3465 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{17} - 30030 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{15} + 115038 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 334602 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 360448 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 255222 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 115038 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30030 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3465 \, a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{9}}}{8064 \, f} \]
1/8064*(3465*a^3*c^6*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 3465*a^3*c^6*log (abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*(3465*a^3*c^6*tan(1/2*f*x + 1/2*e)^17 - 30030*a^3*c^6*tan(1/2*f*x + 1/2*e)^15 + 115038*a^3*c^6*tan(1/2*f*x + 1/2 *e)^13 + 334602*a^3*c^6*tan(1/2*f*x + 1/2*e)^11 - 360448*a^3*c^6*tan(1/2*f *x + 1/2*e)^9 + 255222*a^3*c^6*tan(1/2*f*x + 1/2*e)^7 - 115038*a^3*c^6*tan (1/2*f*x + 1/2*e)^5 + 30030*a^3*c^6*tan(1/2*f*x + 1/2*e)^3 - 3465*a^3*c^6* tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^9)/f
Time = 17.26 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.39 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6 \, dx=\frac {55\,a^3\,c^6\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{64\,f}-\frac {\frac {55\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{17}}{64}-\frac {715\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}}{96}+\frac {913\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{32}+\frac {18589\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{224}-\frac {5632\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{63}+\frac {14179\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{224}-\frac {913\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{32}+\frac {715\,a^3\,c^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{96}-\frac {55\,a^3\,c^6\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{64}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{18}-9\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-84\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-36\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
(55*a^3*c^6*atanh(tan(e/2 + (f*x)/2)))/(64*f) - ((715*a^3*c^6*tan(e/2 + (f *x)/2)^3)/96 - (913*a^3*c^6*tan(e/2 + (f*x)/2)^5)/32 + (14179*a^3*c^6*tan( e/2 + (f*x)/2)^7)/224 - (5632*a^3*c^6*tan(e/2 + (f*x)/2)^9)/63 + (18589*a^ 3*c^6*tan(e/2 + (f*x)/2)^11)/224 + (913*a^3*c^6*tan(e/2 + (f*x)/2)^13)/32 - (715*a^3*c^6*tan(e/2 + (f*x)/2)^15)/96 + (55*a^3*c^6*tan(e/2 + (f*x)/2)^ 17)/64 - (55*a^3*c^6*tan(e/2 + (f*x)/2))/64)/(f*(9*tan(e/2 + (f*x)/2)^2 - 36*tan(e/2 + (f*x)/2)^4 + 84*tan(e/2 + (f*x)/2)^6 - 126*tan(e/2 + (f*x)/2) ^8 + 126*tan(e/2 + (f*x)/2)^10 - 84*tan(e/2 + (f*x)/2)^12 + 36*tan(e/2 + ( f*x)/2)^14 - 9*tan(e/2 + (f*x)/2)^16 + tan(e/2 + (f*x)/2)^18 - 1))