Integrand size = 30, antiderivative size = 244 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^5 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
-4/5*a^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^(11/2)/(a+a*sec(f*x+e))^(1/2)-1/3*a ^3*tan(f*x+e)/c^2/f/(c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(1/2)-1/2*a^3* tan(f*x+e)/c^3/f/(c-c*sec(f*x+e))^(5/2)/(a+a*sec(f*x+e))^(1/2)-a^3*tan(f*x +e)/c^4/f/(c-c*sec(f*x+e))^(3/2)/(a+a*sec(f*x+e))^(1/2)+a^3*ln(1-cos(f*x+e ))*tan(f*x+e)/c^5/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(1/2)
Time = 5.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.49 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=-\frac {a^3 \left (-30 \log (\cos (e+f x))-30 \log (1-\sec (e+f x))+\frac {-24-10 (-1+\sec (e+f x))^2+15 (-1+\sec (e+f x))^3-30 (-1+\sec (e+f x))^4}{(-1+\sec (e+f x))^5}\right ) \tan (e+f x)}{30 c^5 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
-1/30*(a^3*(-30*Log[Cos[e + f*x]] - 30*Log[1 - Sec[e + f*x]] + (-24 - 10*( -1 + Sec[e + f*x])^2 + 15*(-1 + Sec[e + f*x])^3 - 30*(-1 + Sec[e + f*x])^4 )/(-1 + Sec[e + f*x])^5)*Tan[e + f*x])/(c^5*f*Sqrt[a*(1 + Sec[e + f*x])]*S qrt[c - c*Sec[e + f*x]])
Time = 1.21 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {3042, 4398, 3042, 4395, 3042, 4395, 3042, 4395, 3042, 4399, 16}
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.
\(\displaystyle \int \frac {(a \sec (e+f x)+a)^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{11/2}}dx\) |
\(\Big \downarrow \) 4398 |
\(\displaystyle \frac {a^2 \int \frac {\sqrt {\sec (e+f x) a+a}}{(c-c \sec (e+f x))^{7/2}}dx}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 4395 |
\(\displaystyle \frac {a^2 \left (\frac {\int \frac {\sqrt {\sec (e+f x) a+a}}{(c-c \sec (e+f x))^{5/2}}dx}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \left (\frac {\int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 4395 |
\(\displaystyle \frac {a^2 \left (\frac {\frac {\int \frac {\sqrt {\sec (e+f x) a+a}}{(c-c \sec (e+f x))^{3/2}}dx}{c}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \left (\frac {\frac {\int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 4395 |
\(\displaystyle \frac {a^2 \left (\frac {\frac {\frac {\int \frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {c-c \sec (e+f x)}}dx}{c}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}}{c}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a^2 \left (\frac {\frac {\frac {\int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{c}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}}{c}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 4399 |
\(\displaystyle \frac {a^2 \left (\frac {\frac {\frac {a \tan (e+f x) \int \frac {1}{c \cos (e+f x)-c}d\cos (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}}{c}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {a^2 \left (\frac {\frac {\frac {a \tan (e+f x) \log (1-\cos (e+f x))}{c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}}{c}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}}{c}-\frac {a \tan (e+f x)}{3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}\right )}{c^2}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}}\) |
(-4*a^3*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^( 11/2)) + (a^2*(-1/3*(a*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Se c[e + f*x])^(7/2)) + (-1/2*(a*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) + (-((a*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x] ]*(c - c*Sec[e + f*x])^(3/2))) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c *f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]))/c)/c))/c^2
3.2.9.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_ .) + (c_))^(n_), x_Symbol] :> Simp[-2*a*Cot[e + f*x]*((c + d*Csc[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[1/c Int[Sqrt[a + b*C sc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(5/2)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[-8*a^3*Cot[e + f*x]*((c + d*Csc[e + f *x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[a^2/c^2 Int[Sqr t[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1) ]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_))^(n_), x_Symbol] :> Simp[(-a)*c*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[ e + f*x]]*Sqrt[c + d*Csc[e + f*x]])) Subst[Int[(b + a*x)^(m - 1/2)*((d + c*x)^(n - 1/2)/x^(m + n)), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e , f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && EqQ[m + n, 0]
Time = 2.43 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\sqrt {2}\, a^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right ) \left (240 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{10} \csc \left (f x +e \right )^{10}-120 \ln \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \left (1-\cos \left (f x +e \right )\right )^{10} \csc \left (f x +e \right )^{10}+120 \left (1-\cos \left (f x +e \right )\right )^{8} \csc \left (f x +e \right )^{8}-60 \left (1-\cos \left (f x +e \right )\right )^{6} \csc \left (f x +e \right )^{6}+35 \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}-15 \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+3\right ) \csc \left (f x +e \right )}{240 f \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{5} \left (\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )^{\frac {11}{2}}}\) | \(306\) |
risch | \(\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}}-\frac {2 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}+\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (105 \,{\mathrm e}^{9 i \left (f x +e \right )}-555 \,{\mathrm e}^{8 i \left (f x +e \right )}+1730 \,{\mathrm e}^{7 i \left (f x +e \right )}-3125 \,{\mathrm e}^{6 i \left (f x +e \right )}+3882 \,{\mathrm e}^{5 i \left (f x +e \right )}-3125 \,{\mathrm e}^{4 i \left (f x +e \right )}+1730 \,{\mathrm e}^{3 i \left (f x +e \right )}-555 \,{\mathrm e}^{2 i \left (f x +e \right )}+105 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(500\) |
1/240/f*2^(1/2)*a^2*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)/((1-cos (f*x+e))^2*csc(f*x+e)^2-1)^5/(c*(1-cos(f*x+e))^2/((1-cos(f*x+e))^2*csc(f*x +e)^2-1)*csc(f*x+e)^2)^(11/2)*(1-cos(f*x+e))*(240*ln(-cot(f*x+e)+csc(f*x+e ))*(1-cos(f*x+e))^10*csc(f*x+e)^10-120*ln((1-cos(f*x+e))^2*csc(f*x+e)^2+1) *(1-cos(f*x+e))^10*csc(f*x+e)^10+120*(1-cos(f*x+e))^8*csc(f*x+e)^8-60*(1-c os(f*x+e))^6*csc(f*x+e)^6+35*(1-cos(f*x+e))^4*csc(f*x+e)^4-15*(1-cos(f*x+e ))^2*csc(f*x+e)^2+3)*csc(f*x+e)
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
integral((a^2*sec(f*x + e)^2 + 2*a^2*sec(f*x + e) + a^2)*sqrt(a*sec(f*x + e) + a)*sqrt(-c*sec(f*x + e) + c)/(c^6*sec(f*x + e)^6 - 6*c^6*sec(f*x + e) ^5 + 15*c^6*sec(f*x + e)^4 - 20*c^6*sec(f*x + e)^3 + 15*c^6*sec(f*x + e)^2 - 6*c^6*sec(f*x + e) + c^6), x)
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 9150 vs. \(2 (218) = 436\).
Time = 70.10 (sec) , antiderivative size = 9150, normalized size of antiderivative = 37.50 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\text {Too large to display} \]
-1/15*(15*(f*x + e)*a^2*cos(10*f*x + 10*e)^2 + 30375*(f*x + e)*a^2*cos(8*f *x + 8*e)^2 + 661500*(f*x + e)*a^2*cos(6*f*x + 6*e)^2 + 661500*(f*x + e)*a ^2*cos(4*f*x + 4*e)^2 + 30375*(f*x + e)*a^2*cos(2*f*x + 2*e)^2 + 1500*(f*x + e)*a^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 216000* (f*x + e)*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 952 560*(f*x + e)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 216000*(f*x + e)*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) ^2 + 1500*(f*x + e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) ))^2 + 15*(f*x + e)*a^2*sin(10*f*x + 10*e)^2 + 30375*(f*x + e)*a^2*sin(8*f *x + 8*e)^2 + 661500*(f*x + e)*a^2*sin(6*f*x + 6*e)^2 + 661500*(f*x + e)*a ^2*sin(4*f*x + 4*e)^2 + 30375*(f*x + e)*a^2*sin(2*f*x + 2*e)^2 + 1500*(f*x + e)*a^2*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 216000* (f*x + e)*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 952 560*(f*x + e)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 216000*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) ^2 + 1500*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) ))^2 + 1350*(f*x + e)*a^2*cos(2*f*x + 2*e) + 15*(f*x + e)*a^2 - 1110*a^2*s in(2*f*x + 2*e) - 30*(a^2*cos(10*f*x + 10*e)^2 + 2025*a^2*cos(8*f*x + 8*e) ^2 + 44100*a^2*cos(6*f*x + 6*e)^2 + 44100*a^2*cos(4*f*x + 4*e)^2 + 2025*a^ 2*cos(2*f*x + 2*e)^2 + 100*a^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*...
\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{11/2}} \,d x \]