\(\int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx\) [122]

Optimal result

Integrand size = 36, antiderivative size = 92 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=-\frac {a \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{5 f (c-c \sec (e+f x))^{11/2}}+\frac {a^2 \tan (e+f x)}{20 c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}} \] Output:

-1/5*a*(a+a*sec(f*x+e))^(1/2)*tan(f*x+e)/f/(c-c*sec(f*x+e))^(11/2)+1/20*a^ 
2*tan(f*x+e)/c/f/(a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(9/2)
 

Mathematica [A] (verified)

Time = 5.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {a^2 (3+5 \sec (e+f x)) \tan (e+f x)}{20 c^5 f (-1+\sec (e+f x))^5 \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \] Input:

Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^( 
11/2),x]
 

Output:

(a^2*(3 + 5*Sec[e + f*x])*Tan[e + f*x])/(20*c^5*f*(-1 + Sec[e + f*x])^5*Sq 
rt[a*(1 + Sec[e + f*x])]*Sqrt[c - c*Sec[e + f*x]])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3042, 4442, 3042, 4441}

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.

\(\displaystyle \int \frac {\sec (e+f x) (a \sec (e+f x)+a)^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{11/2}}dx\)

\(\Big \downarrow \) 4442

\(\displaystyle -\frac {a \int \frac {\sec (e+f x) \sqrt {\sec (e+f x) a+a}}{(c-c \sec (e+f x))^{9/2}}dx}{5 c}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{5 f (c-c \sec (e+f x))^{11/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{9/2}}dx}{5 c}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{5 f (c-c \sec (e+f x))^{11/2}}\)

\(\Big \downarrow \) 4441

\(\displaystyle \frac {a^2 \tan (e+f x)}{20 c f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}}-\frac {a \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{5 f (c-c \sec (e+f x))^{11/2}}\)

Input:

Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(11/2), 
x]
 

Output:

-1/5*(a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(11 
/2)) + (a^2*Tan[e + f*x])/(20*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + 
f*x])^(9/2))
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4441

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sq 
rt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[2*a*c*Cot[e + f 
*x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]])), x] / 
; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] 
 && NeQ[m, -2^(-1)]
 

rule 4442

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs 
c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Simp[2*a*c*Cot[e + 
f*x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(b*f*(2*m + 1))), 
 x] - Simp[d*((2*n - 1)/(b*(2*m + 1)))   Int[Csc[e + f*x]*(a + b*Csc[e + f* 
x])^(m + 1)*(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] && IGtQ[n - 1/2, 0] && LtQ[ 
m, -2^(-1)]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(161\) vs. \(2(80)=160\).

Time = 2.00 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.76

method	result	size
default	\(\frac {\left (4341 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-1225 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-12910 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+18030 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-9655 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1931\right ) a \sqrt {\frac {a \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, \sec \left (\frac {f x}{2}+\frac {e}{2}\right ) \csc \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{40960 f \sqrt {-\frac {c \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}}\, c^{5}}\)	\(162\)
risch	\(\frac {2 i a \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (5 \,{\mathrm e}^{9 i \left (f x +e \right )}-15 \,{\mathrm e}^{8 i \left (f x +e \right )}+50 \,{\mathrm e}^{7 i \left (f x +e \right )}-75 \,{\mathrm e}^{6 i \left (f x +e \right )}+102 \,{\mathrm e}^{5 i \left (f x +e \right )}-75 \,{\mathrm e}^{4 i \left (f x +e \right )}+50 \,{\mathrm e}^{3 i \left (f x +e \right )}-15 \,{\mathrm e}^{2 i \left (f x +e \right )}+5 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{5 c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(197\)

Input:

int(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(11/2),x,method=_RE 
TURNVERBOSE)
 

Output:

1/40960/f*(4341*cos(1/2*f*x+1/2*e)^10-1225*cos(1/2*f*x+1/2*e)^8-12910*cos( 
1/2*f*x+1/2*e)^6+18030*cos(1/2*f*x+1/2*e)^4-9655*cos(1/2*f*x+1/2*e)^2+1931 
)*a*(a/(2*cos(1/2*f*x+1/2*e)^2-1)*cos(1/2*f*x+1/2*e)^2)^(1/2)/(-c/(2*cos(1 
/2*f*x+1/2*e)^2-1)*sin(1/2*f*x+1/2*e)^2)^(1/2)/c^5*sec(1/2*f*x+1/2*e)*csc( 
1/2*f*x+1/2*e)^9
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (80) = 160\).

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.00 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {{\left (20 \, a \cos \left (f x + e\right )^{5} - 30 \, a \cos \left (f x + e\right )^{4} + 30 \, a \cos \left (f x + e\right )^{3} - 15 \, a \cos \left (f x + e\right )^{2} + 3 \, a \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{20 \, {\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - c^{6} f\right )} \sin \left (f x + e\right )} \] Input:

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(11/2),x, alg 
orithm="fricas")
 

Output:

1/20*(20*a*cos(f*x + e)^5 - 30*a*cos(f*x + e)^4 + 30*a*cos(f*x + e)^3 - 15 
*a*cos(f*x + e)^2 + 3*a*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + 
e))*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/((c^6*f*cos(f*x + e)^5 - 5*c^6 
*f*cos(f*x + e)^4 + 10*c^6*f*cos(f*x + e)^3 - 10*c^6*f*cos(f*x + e)^2 + 5* 
c^6*f*cos(f*x + e) - c^6*f)*sin(f*x + e))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\text {Timed out} \] Input:

integrate(sec(f*x+e)*(a+a*sec(f*x+e))**(3/2)/(c-c*sec(f*x+e))**(11/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3906 vs. \(2 (80) = 160\).

Time = 16.31 (sec) , antiderivative size = 3906, normalized size of antiderivative = 42.46 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\text {Too large to display} \] Input:

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(11/2),x, alg 
orithm="maxima")
 

Output:

-2/5*(225*a*cos(6*f*x + 6*e)*sin(2*f*x + 2*e) + 225*a*cos(4*f*x + 4*e)*sin 
(2*f*x + 2*e) - 15*(a*sin(8*f*x + 8*e) + 5*a*sin(6*f*x + 6*e) + 5*a*sin(4* 
f*x + 4*e) + a*sin(2*f*x + 2*e))*cos(10*f*x + 10*e) - 225*(a*sin(6*f*x + 6 
*e) + a*sin(4*f*x + 4*e))*cos(8*f*x + 8*e) - 5*(a*sin(10*f*x + 10*e) + 15* 
a*sin(8*f*x + 8*e) + 60*a*sin(6*f*x + 6*e) + 60*a*sin(4*f*x + 4*e) + 15*a* 
sin(2*f*x + 2*e) - 20*a*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) 
)) - 48*a*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*a*sin( 
3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(9/2*arctan2(sin(2*f* 
x + 2*e), cos(2*f*x + 2*e))) - 10*(5*a*sin(10*f*x + 10*e) + 45*a*sin(8*f*x 
 + 8*e) + 150*a*sin(6*f*x + 6*e) + 150*a*sin(4*f*x + 4*e) + 45*a*sin(2*f*x 
 + 2*e) - 36*a*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 10*a 
*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(7/2*arctan2(sin 
(2*f*x + 2*e), cos(2*f*x + 2*e))) - 6*(17*a*sin(10*f*x + 10*e) + 135*a*sin 
(8*f*x + 8*e) + 420*a*sin(6*f*x + 6*e) + 420*a*sin(4*f*x + 4*e) + 135*a*si 
n(2*f*x + 2*e) + 60*a*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) 
 + 40*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(5/2*arct 
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 50*(a*sin(10*f*x + 10*e) + 9*a* 
sin(8*f*x + 8*e) + 30*a*sin(6*f*x + 6*e) + 30*a*sin(4*f*x + 4*e) + 9*a*sin 
(2*f*x + 2*e) + 2*a*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))* 
cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 5*(a*sin(10*f*x ...
 

Giac [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.49 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {{\left (a - \frac {10 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} a^{3} + 10 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} a^{4} + 5 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} a^{5} + a^{6}}{a^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}}\right )} a^{2}}{320 \, \sqrt {-a c} c^{5} f {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \] Input:

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(11/2),x, alg 
orithm="giac")
 

Output:

1/320*(a - (10*(a*tan(1/2*f*x + 1/2*e)^2 - a)^3*a^3 + 10*(a*tan(1/2*f*x + 
1/2*e)^2 - a)^2*a^4 + 5*(a*tan(1/2*f*x + 1/2*e)^2 - a)*a^5 + a^6)/(a^5*tan 
(1/2*f*x + 1/2*e)^10))*a^2/(sqrt(-a*c)*c^5*f*abs(a)*sgn(tan(1/2*f*x + 1/2* 
e)^3 + tan(1/2*f*x + 1/2*e)))
 

Mupad [B] (verification not implemented)

Time = 16.74 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.42 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}\,\left (\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,60{}\mathrm {i}}{c^6\,f}-\frac {a\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,608{}\mathrm {i}}{5\,c^6\,f}+\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,72{}\mathrm {i}}{c^6\,f}-\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,44{}\mathrm {i}}{c^6\,f}+\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,12{}\mathrm {i}}{c^6\,f}-\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,4{}\mathrm {i}}{c^6\,f}\right )}{{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,264{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,330{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,220{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,88{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,20{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )\,2{}\mathrm {i}} \] Input:

int((a + a/cos(e + f*x))^(3/2)/(cos(e + f*x)*(c - c/cos(e + f*x))^(11/2)), 
x)
 

Output:

((c - c/cos(e + f*x))^(1/2)*((a*exp(e*6i + f*x*6i)*(a + a/cos(e + f*x))^(1 
/2)*60i)/(c^6*f) - (a*cos(e + f*x)*exp(e*6i + f*x*6i)*(a + a/cos(e + f*x)) 
^(1/2)*608i)/(5*c^6*f) + (a*exp(e*6i + f*x*6i)*cos(2*e + 2*f*x)*(a + a/cos 
(e + f*x))^(1/2)*72i)/(c^6*f) - (a*exp(e*6i + f*x*6i)*cos(3*e + 3*f*x)*(a 
+ a/cos(e + f*x))^(1/2)*44i)/(c^6*f) + (a*exp(e*6i + f*x*6i)*cos(4*e + 4*f 
*x)*(a + a/cos(e + f*x))^(1/2)*12i)/(c^6*f) - (a*exp(e*6i + f*x*6i)*cos(5* 
e + 5*f*x)*(a + a/cos(e + f*x))^(1/2)*4i)/(c^6*f)))/(exp(e*6i + f*x*6i)*si 
n(e + f*x)*264i - exp(e*6i + f*x*6i)*sin(2*e + 2*f*x)*330i + exp(e*6i + f* 
x*6i)*sin(3*e + 3*f*x)*220i - exp(e*6i + f*x*6i)*sin(4*e + 4*f*x)*88i + ex 
p(e*6i + f*x*6i)*sin(5*e + 5*f*x)*20i - exp(e*6i + f*x*6i)*sin(6*e + 6*f*x 
)*2i)
 

Reduce [F]

\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{11/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a \left (\int \frac {\sqrt {\sec \left (f x +e \right )+1}\, \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )^{2}}{\sec \left (f x +e \right )^{6}-6 \sec \left (f x +e \right )^{5}+15 \sec \left (f x +e \right )^{4}-20 \sec \left (f x +e \right )^{3}+15 \sec \left (f x +e \right )^{2}-6 \sec \left (f x +e \right )+1}d x +\int \frac {\sqrt {\sec \left (f x +e \right )+1}\, \sqrt {-\sec \left (f x +e \right )+1}\, \sec \left (f x +e \right )}{\sec \left (f x +e \right )^{6}-6 \sec \left (f x +e \right )^{5}+15 \sec \left (f x +e \right )^{4}-20 \sec \left (f x +e \right )^{3}+15 \sec \left (f x +e \right )^{2}-6 \sec \left (f x +e \right )+1}d x \right )}{c^{6}} \] Input:

int(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(11/2),x)
 

Output:

(sqrt(c)*sqrt(a)*a*(int((sqrt(sec(e + f*x) + 1)*sqrt( - sec(e + f*x) + 1)* 
sec(e + f*x)**2)/(sec(e + f*x)**6 - 6*sec(e + f*x)**5 + 15*sec(e + f*x)**4 
 - 20*sec(e + f*x)**3 + 15*sec(e + f*x)**2 - 6*sec(e + f*x) + 1),x) + int( 
(sqrt(sec(e + f*x) + 1)*sqrt( - sec(e + f*x) + 1)*sec(e + f*x))/(sec(e + f 
*x)**6 - 6*sec(e + f*x)**5 + 15*sec(e + f*x)**4 - 20*sec(e + f*x)**3 + 15* 
sec(e + f*x)**2 - 6*sec(e + f*x) + 1),x)))/c**6