Integrand size = 32, antiderivative size = 162 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac {3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}-\frac {2 (a+a \sec (e+f x))^3 \tan (e+f x)}{429 c^2 f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^3 \tan (e+f x)}{3003 c^3 f (c-c \sec (e+f x))^4} \]
-1/13*(a+a*sec(f*x+e))^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^7-3/143*(a+a*sec(f* x+e))^3*tan(f*x+e)/c/f/(c-c*sec(f*x+e))^6-2/429*(a+a*sec(f*x+e))^3*tan(f*x +e)/c^2/f/(c-c*sec(f*x+e))^5-2/3003*(a+a*sec(f*x+e))^3*tan(f*x+e)/c^3/f/(c -c*sec(f*x+e))^4
Time = 5.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.43 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=-\frac {a^3 (1+\sec (e+f x))^3 \left (-310+97 \sec (e+f x)-20 \sec ^2(e+f x)+2 \sec ^3(e+f x)\right ) \tan (e+f x)}{3003 c^7 f (-1+\sec (e+f x))^7} \]
-1/3003*(a^3*(1 + Sec[e + f*x])^3*(-310 + 97*Sec[e + f*x] - 20*Sec[e + f*x ]^2 + 2*Sec[e + f*x]^3)*Tan[e + f*x])/(c^7*f*(-1 + Sec[e + f*x])^7)
Time = 0.85 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4439, 3042, 4439, 3042, 4439, 3042, 4438}
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.
| \(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^3}{(c-c \sec (e+f x))^7} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^7}dx\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {3 \int \frac {\sec (e+f x) (\sec (e+f x) a+a)^3}{(c-c \sec (e+f x))^6}dx}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^6}dx}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\sec (e+f x) (\sec (e+f x) a+a)^3}{(c-c \sec (e+f x))^5}dx}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\right )}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5}dx}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\right )}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
| \(\Big \downarrow \) 4439 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\sec (e+f x) (\sec (e+f x) a+a)^3}{(c-c \sec (e+f x))^4}dx}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\right )}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^3}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx}{9 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\right )}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
| \(\Big \downarrow \) 4438 |
| \(\displaystyle \frac {3 \left (\frac {2 \left (-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{63 c f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5}\right )}{11 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6}\right )}{13 c}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7}\) |
-1/13*((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^7) + ( 3*(-1/11*((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^6) + (2*(-1/9*((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^5 ) - ((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(63*c*f*(c - c*Sec[e + f*x])^4)) )/(11*c)))/(13*c)
3.1.33.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] /; Fre eQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] & & EqQ[m + n + 1, 0] && NeQ[2*m + 1, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[b*Cot[e + f*x] *(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp [(m + n + 1)/(a*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* (c + d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[m + n + 1, 0] && NeQ[2*m + 1, 0 ] && !LtQ[n, 0] && !(IGtQ[n + 1/2, 0] && LtQ[n + 1/2, -(m + n)])
Time = 1.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.40
| method | result | size |
| parallelrisch | \(\frac {a^{3} \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} \left (231 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-819 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+1001 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-429\right )}{24024 c^{7} f}\) | \(64\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {3}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {1}{13 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\right )}{8 f \,c^{7}}\) | \(65\) |
| default | \(\frac {a^{3} \left (\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {3}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {1}{13 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\right )}{8 f \,c^{7}}\) | \(65\) |
| risch | \(\frac {2 i a^{3} \left (3003 \,{\mathrm e}^{12 i \left (f x +e \right )}-9009 \,{\mathrm e}^{11 i \left (f x +e \right )}+51051 \,{\mathrm e}^{10 i \left (f x +e \right )}-99099 \,{\mathrm e}^{9 i \left (f x +e \right )}+216216 \,{\mathrm e}^{8 i \left (f x +e \right )}-246246 \,{\mathrm e}^{7 i \left (f x +e \right )}+285714 \,{\mathrm e}^{6 i \left (f x +e \right )}-182754 \,{\mathrm e}^{5 i \left (f x +e \right )}+122551 \,{\mathrm e}^{4 i \left (f x +e \right )}-37609 \,{\mathrm e}^{3 i \left (f x +e \right )}+15171 \,{\mathrm e}^{2 i \left (f x +e \right )}-1027 \,{\mathrm e}^{i \left (f x +e \right )}+310\right )}{3003 f \,c^{7} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{13}}\) | \(160\) |
1/24024*a^3*cot(1/2*f*x+1/2*e)^7*(231*cot(1/2*f*x+1/2*e)^6-819*cot(1/2*f*x +1/2*e)^4+1001*cot(1/2*f*x+1/2*e)^2-429)/c^7/f
Time = 0.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.20 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=\frac {310 \, a^{3} \cos \left (f x + e\right )^{7} + 1143 \, a^{3} \cos \left (f x + e\right )^{6} + 1492 \, a^{3} \cos \left (f x + e\right )^{5} + 736 \, a^{3} \cos \left (f x + e\right )^{4} + 34 \, a^{3} \cos \left (f x + e\right )^{3} - 29 \, a^{3} \cos \left (f x + e\right )^{2} + 12 \, a^{3} \cos \left (f x + e\right ) - 2 \, a^{3}}{3003 \, {\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} + 15 \, c^{7} f \cos \left (f x + e\right )^{4} - 20 \, c^{7} f \cos \left (f x + e\right )^{3} + 15 \, c^{7} f \cos \left (f x + e\right )^{2} - 6 \, c^{7} f \cos \left (f x + e\right ) + c^{7} f\right )} \sin \left (f x + e\right )} \]
1/3003*(310*a^3*cos(f*x + e)^7 + 1143*a^3*cos(f*x + e)^6 + 1492*a^3*cos(f* x + e)^5 + 736*a^3*cos(f*x + e)^4 + 34*a^3*cos(f*x + e)^3 - 29*a^3*cos(f*x + e)^2 + 12*a^3*cos(f*x + e) - 2*a^3)/((c^7*f*cos(f*x + e)^6 - 6*c^7*f*co s(f*x + e)^5 + 15*c^7*f*cos(f*x + e)^4 - 20*c^7*f*cos(f*x + e)^3 + 15*c^7* f*cos(f*x + e)^2 - 6*c^7*f*cos(f*x + e) + c^7*f)*sin(f*x + e))
\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=- \frac {a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{7}} \]
-a**3*(Integral(sec(e + f*x)/(sec(e + f*x)**7 - 7*sec(e + f*x)**6 + 21*sec (e + f*x)**5 - 35*sec(e + f*x)**4 + 35*sec(e + f*x)**3 - 21*sec(e + f*x)** 2 + 7*sec(e + f*x) - 1), x) + Integral(3*sec(e + f*x)**2/(sec(e + f*x)**7 - 7*sec(e + f*x)**6 + 21*sec(e + f*x)**5 - 35*sec(e + f*x)**4 + 35*sec(e + f*x)**3 - 21*sec(e + f*x)**2 + 7*sec(e + f*x) - 1), x) + Integral(3*sec(e + f*x)**3/(sec(e + f*x)**7 - 7*sec(e + f*x)**6 + 21*sec(e + f*x)**5 - 35* sec(e + f*x)**4 + 35*sec(e + f*x)**3 - 21*sec(e + f*x)**2 + 7*sec(e + f*x) - 1), x) + Integral(sec(e + f*x)**4/(sec(e + f*x)**7 - 7*sec(e + f*x)**6 + 21*sec(e + f*x)**5 - 35*sec(e + f*x)**4 + 35*sec(e + f*x)**3 - 21*sec(e + f*x)**2 + 7*sec(e + f*x) - 1), x))/c**7
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (158) = 316\).
Time = 0.24 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.19 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=-\frac {\frac {a^{3} {\left (\frac {8190 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5005 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {25740 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {9009 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {30030 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {45045 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} - 3465\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{13}}{c^{7} \sin \left (f x + e\right )^{13}} + \frac {5 \, a^{3} {\left (\frac {1638 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5005 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {8580 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {9009 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {6006 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {3003 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} - 231\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{13}}{c^{7} \sin \left (f x + e\right )^{13}} + \frac {35 \, a^{3} {\left (\frac {468 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {715 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {1287 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {1716 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {1287 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} - 99\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{13}}{c^{7} \sin \left (f x + e\right )^{13}} + \frac {77 \, a^{3} {\left (\frac {65 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {117 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {195 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{13}}{c^{7} \sin \left (f x + e\right )^{13}}}{960960 \, f} \]
-1/960960*(a^3*(8190*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5005*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 25740*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 90 09*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 30030*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 45045*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 3465)*(cos(f*x + e) + 1)^13/(c^7*sin(f*x + e)^13) + 5*a^3*(1638*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5005*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 8580*sin(f*x + e)^ 6/(cos(f*x + e) + 1)^6 - 9009*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 6006*s in(f*x + e)^10/(cos(f*x + e) + 1)^10 - 3003*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 231)*(cos(f*x + e) + 1)^13/(c^7*sin(f*x + e)^13) + 35*a^3*(468*s in(f*x + e)^2/(cos(f*x + e) + 1)^2 - 715*sin(f*x + e)^4/(cos(f*x + e) + 1) ^4 + 1287*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 1716*sin(f*x + e)^10/(cos( f*x + e) + 1)^10 + 1287*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 99)*(cos(f *x + e) + 1)^13/(c^7*sin(f*x + e)^13) + 77*a^3*(65*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 117*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^ 12/(cos(f*x + e) + 1)^12 - 15)*(cos(f*x + e) + 1)^13/(c^7*sin(f*x + e)^13) )/f
Time = 0.50 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.45 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=-\frac {429 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1001 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 819 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 231 \, a^{3}}{24024 \, c^{7} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13}} \]
-1/24024*(429*a^3*tan(1/2*f*x + 1/2*e)^6 - 1001*a^3*tan(1/2*f*x + 1/2*e)^4 + 819*a^3*tan(1/2*f*x + 1/2*e)^2 - 231*a^3)/(c^7*f*tan(1/2*f*x + 1/2*e)^1 3)
Time = 13.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^7} \, dx=\frac {a^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (231\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-819\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1001\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-429\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{24024\,c^7\,f\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}} \]