Integrand size = 34, antiderivative size = 246 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=-\frac {63 \arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{128 \sqrt {2} a^3 c^{5/2} f}-\frac {21 \tan (e+f x)}{32 a^3 f (c-c \sec (e+f x))^{5/2}}+\frac {\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}}+\frac {3 \tan (e+f x)}{10 a f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac {21 \tan (e+f x)}{20 f \left (a^3+a^3 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac {63 \tan (e+f x)}{128 a^3 c f (c-c \sec (e+f x))^{3/2}} \]
-63/256*arctan(1/2*c^(1/2)*tan(f*x+e)*2^(1/2)/(c-c*sec(f*x+e))^(1/2))/a^3/ c^(5/2)/f*2^(1/2)-21/32*tan(f*x+e)/a^3/f/(c-c*sec(f*x+e))^(5/2)+1/5*tan(f* x+e)/f/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(5/2)+3/10*tan(f*x+e)/a/f/(a+a* sec(f*x+e))^2/(c-c*sec(f*x+e))^(5/2)+21/20*tan(f*x+e)/f/(a^3+a^3*sec(f*x+e ))/(c-c*sec(f*x+e))^(5/2)-63/128*tan(f*x+e)/a^3/c/f/(c-c*sec(f*x+e))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},3,-\frac {3}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \tan (e+f x)}{20 a^3 c^2 f (1+\sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}} \]
(Hypergeometric2F1[-5/2, 3, -3/2, (1 + Sec[e + f*x])/2]*Tan[e + f*x])/(20* a^3*c^2*f*(1 + Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])
Time = 1.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {3042, 4448, 3042, 4448, 3042, 4448, 3042, 4283, 3042, 4283, 3042, 4282, 216}
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.
| \(\displaystyle \int \frac {\sec (e+f x)}{(a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4448 |
| \(\displaystyle \frac {9 \int \frac {\sec (e+f x)}{(\sec (e+f x) a+a)^2 (c-c \sec (e+f x))^{5/2}}dx}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4448 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {\sec (e+f x)}{(\sec (e+f x) a+a) (c-c \sec (e+f x))^{5/2}}dx}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right ) \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4448 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{5/2}}dx}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4283 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}}dx}{8 c}-\frac {\tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\right )}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{8 c}-\frac {\tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\right )}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4283 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\sec (e+f x)}{\sqrt {c-c \sec (e+f x)}}dx}{4 c}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{8 c}-\frac {\tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\right )}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {c-c \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{4 c}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{8 c}-\frac {\tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\right )}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {\int \frac {1}{\frac {c^2 \tan ^2(e+f x)}{c-c \sec (e+f x)}+2 c}d\frac {c \tan (e+f x)}{\sqrt {c-c \sec (e+f x)}}}{2 c f}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{8 c}-\frac {\tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\right )}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {\arctan \left (\frac {\sqrt {c} \tan (e+f x)}{\sqrt {2} \sqrt {c-c \sec (e+f x)}}\right )}{2 \sqrt {2} c^{3/2} f}-\frac {\tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\right )}{8 c}-\frac {\tan (e+f x)}{4 f (c-c \sec (e+f x))^{5/2}}\right )}{2 a}+\frac {\tan (e+f x)}{f (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}\right )}{6 a}+\frac {\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}}\right )}{10 a}+\frac {\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^{5/2}}\) |
Tan[e + f*x]/(5*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)) + (9* (Tan[e + f*x]/(3*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)) + (7 *(Tan[e + f*x]/(f*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)) + (5*(- 1/4*Tan[e + f*x]/(f*(c - c*Sec[e + f*x])^(5/2)) + (3*(-1/2*ArcTan[(Sqrt[c] *Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])]/(Sqrt[2]*c^(3/2)*f) - T an[e + f*x]/(2*f*(c - c*Sec[e + f*x])^(3/2))))/(8*c)))/(2*a)))/(6*a)))/(10 *a)
3.2.6.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_ Symbol] :> Simp[b*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[(m + 1)/(a*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1) ] && IntegerQ[2*m]
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, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ((ILtQ[m, 0] && ILtQ[n - 1/2, 0]) || (IL tQ[m - 1/2, 0] && ILtQ[n - 1/2, 0] && LtQ[m, n]))
Time = 3.54 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (257 \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )^{4}-354 \cos \left (f x +e \right )^{3} \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+315 \sin \left (f x +e \right )^{4} \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )-588 \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )^{2}+210 \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )+315 \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\right ) \tan \left (f x +e \right ) \sec \left (f x +e \right )}{1280 a^{3} f \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right )^{2} c^{2} \left (\cos \left (f x +e \right )+1\right )^{3}}\) | \(272\) |
1/1280/a^3/f*2^(1/2)*(257*2^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f *x+e)^4-354*cos(f*x+e)^3*2^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+315*si n(f*x+e)^4*arctan(1/2*2^(1/2)/(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2))-588*2^(1 /2)*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2+210*2^(1/2)*(-cos(f*x+ e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)+315*2^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1 ))^(1/2))/(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)/(-c*(sec(f*x+e)-1))^(1/2)/(se c(f*x+e)-1)^2/c^2/(cos(f*x+e)+1)^3*tan(f*x+e)*sec(f*x+e)
Time = 0.37 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.87 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=\left [-\frac {315 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-c} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {-c} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} + {\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 4 \, {\left (257 \, \cos \left (f x + e\right )^{5} - 354 \, \cos \left (f x + e\right )^{4} - 588 \, \cos \left (f x + e\right )^{3} + 210 \, \cos \left (f x + e\right )^{2} + 315 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{2560 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )}, \frac {315 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left (257 \, \cos \left (f x + e\right )^{5} - 354 \, \cos \left (f x + e\right )^{4} - 588 \, \cos \left (f x + e\right )^{3} + 210 \, \cos \left (f x + e\right )^{2} + 315 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{1280 \, {\left (a^{3} c^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{3} f \cos \left (f x + e\right )^{2} + a^{3} c^{3} f\right )} \sin \left (f x + e\right )}\right ] \]
[-1/2560*(315*sqrt(2)*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-c)*log ((2*sqrt(2)*(cos(f*x + e)^2 + cos(f*x + e))*sqrt(-c)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)) + (3*c*cos(f*x + e) + c)*sin(f*x + e))/((cos(f*x + e) - 1)*sin(f*x + e)))*sin(f*x + e) + 4*(257*cos(f*x + e)^5 - 354*cos(f*x + e) ^4 - 588*cos(f*x + e)^3 + 210*cos(f*x + e)^2 + 315*cos(f*x + e))*sqrt((c*c os(f*x + e) - c)/cos(f*x + e)))/((a^3*c^3*f*cos(f*x + e)^4 - 2*a^3*c^3*f*c os(f*x + e)^2 + a^3*c^3*f)*sin(f*x + e)), 1/1280*(315*sqrt(2)*(cos(f*x + e )^4 - 2*cos(f*x + e)^2 + 1)*sqrt(c)*arctan(sqrt(2)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/(sqrt(c)*sin(f*x + e)))*sin(f*x + e) - 2*(25 7*cos(f*x + e)^5 - 354*cos(f*x + e)^4 - 588*cos(f*x + e)^3 + 210*cos(f*x + e)^2 + 315*cos(f*x + e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e)))/((a^3*c ^3*f*cos(f*x + e)^4 - 2*a^3*c^3*f*cos(f*x + e)^2 + a^3*c^3*f)*sin(f*x + e) )]
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=\frac {\int \frac {\sec {\left (e + f x \right )}}{c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{5}{\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{4}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} - 2 c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c} \sec {\left (e + f x \right )} + c^{2} \sqrt {- c \sec {\left (e + f x \right )} + c}}\, dx}{a^{3}} \]
Integral(sec(e + f*x)/(c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**5 + c* *2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**4 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3 - 2*c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**2 + c**2*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x) + c**2*sqrt(-c*sec(e + f*x) + c)), x)/a**3
\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sec \left (f x + e\right )}{{\left (a \sec \left (f x + e\right ) + a\right )}^{3} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.71 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.75 \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (315 \, \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) - \frac {5 \, {\left (17 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}\right )}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} - \frac {8 \, {\left ({\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} c^{8} - 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{9} + 30 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{10}\right )}}{c^{10}}\right )}}{1280 \, a^{3} c^{3} f} \]
1/1280*sqrt(2)*(315*sqrt(c)*arctan(sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)/sqrt (c)) - 5*(17*(c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2)*c + 15*sqrt(c*tan(1/2*f* x + 1/2*e)^2 - c)*c^2)/(c^2*tan(1/2*f*x + 1/2*e)^4) - 8*((c*tan(1/2*f*x + 1/2*e)^2 - c)^(5/2)*c^8 - 5*(c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2)*c^9 + 30* sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)*c^10)/c^10)/(a^3*c^3*f)
Timed out. \[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]