1,1,75,0,1.084573," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e)^7,x, algorithm=""fricas"")","\frac{5 \, a \cos\left(f x + e\right)^{8} - 7 \, {\left(3 \, a - b\right)} \cos\left(f x + e\right)^{6} + 35 \, {\left(a - b\right)} \cos\left(f x + e\right)^{4} - 35 \, {\left(a - 3 \, b\right)} \cos\left(f x + e\right)^{2} + 35 \, b}{35 \, f \cos\left(f x + e\right)}"," ",0,"1/35*(5*a*cos(f*x + e)^8 - 7*(3*a - b)*cos(f*x + e)^6 + 35*(a - b)*cos(f*x + e)^4 - 35*(a - 3*b)*cos(f*x + e)^2 + 35*b)/(f*cos(f*x + e))","A",0
2,1,60,0,0.816255," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e)^5,x, algorithm=""fricas"")","-\frac{3 \, a \cos\left(f x + e\right)^{6} - 5 \, {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{4} + 15 \, {\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} - 15 \, b}{15 \, f \cos\left(f x + e\right)}"," ",0,"-1/15*(3*a*cos(f*x + e)^6 - 5*(2*a - b)*cos(f*x + e)^4 + 15*(a - 2*b)*cos(f*x + e)^2 - 15*b)/(f*cos(f*x + e))","A",0
3,1,42,0,1.174169," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e)^3,x, algorithm=""fricas"")","\frac{a \cos\left(f x + e\right)^{4} - 3 \, {\left(a - b\right)} \cos\left(f x + e\right)^{2} + 3 \, b}{3 \, f \cos\left(f x + e\right)}"," ",0,"1/3*(a*cos(f*x + e)^4 - 3*(a - b)*cos(f*x + e)^2 + 3*b)/(f*cos(f*x + e))","A",0
4,1,27,0,1.303994," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e),x, algorithm=""fricas"")","-\frac{a \cos\left(f x + e\right)^{2} - b}{f \cos\left(f x + e\right)}"," ",0,"-(a*cos(f*x + e)^2 - b)/(f*cos(f*x + e))","A",0
5,1,60,0,0.739606," ","integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a + b\right)} \cos\left(f x + e\right) \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - {\left(a + b\right)} \cos\left(f x + e\right) \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 2 \, b}{2 \, f \cos\left(f x + e\right)}"," ",0,"-1/2*((a + b)*cos(f*x + e)*log(1/2*cos(f*x + e) + 1/2) - (a + b)*cos(f*x + e)*log(-1/2*cos(f*x + e) + 1/2) - 2*b)/(f*cos(f*x + e))","B",0
6,1,124,0,0.814555," ","integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{2 \, {\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 4 \, b}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}"," ",0,"1/4*(2*(a + 3*b)*cos(f*x + e)^2 - ((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*log(1/2*cos(f*x + e) + 1/2) + ((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*log(-1/2*cos(f*x + e) + 1/2) - 4*b)/(f*cos(f*x + e)^3 - f*cos(f*x + e))","B",0
7,1,178,0,0.570372," ","integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{6 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{4} - 10 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left({\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{3} + {\left(a + 5 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{3} + {\left(a + 5 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 16 \, b}{16 \, {\left(f \cos\left(f x + e\right)^{5} - 2 \, f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)\right)}}"," ",0,"1/16*(6*(a + 5*b)*cos(f*x + e)^4 - 10*(a + 5*b)*cos(f*x + e)^2 - 3*((a + 5*b)*cos(f*x + e)^5 - 2*(a + 5*b)*cos(f*x + e)^3 + (a + 5*b)*cos(f*x + e))*log(1/2*cos(f*x + e) + 1/2) + 3*((a + 5*b)*cos(f*x + e)^5 - 2*(a + 5*b)*cos(f*x + e)^3 + (a + 5*b)*cos(f*x + e))*log(-1/2*cos(f*x + e) + 1/2) + 16*b)/(f*cos(f*x + e)^5 - 2*f*cos(f*x + e)^3 + f*cos(f*x + e))","B",0
8,1,86,0,1.105422," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e)^6,x, algorithm=""fricas"")","\frac{15 \, {\left(a - 6 \, b\right)} f x \cos\left(f x + e\right) - {\left(8 \, a \cos\left(f x + e\right)^{6} - 2 \, {\left(13 \, a - 6 \, b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(11 \, a - 18 \, b\right)} \cos\left(f x + e\right)^{2} - 48 \, b\right)} \sin\left(f x + e\right)}{48 \, f \cos\left(f x + e\right)}"," ",0,"1/48*(15*(a - 6*b)*f*x*cos(f*x + e) - (8*a*cos(f*x + e)^6 - 2*(13*a - 6*b)*cos(f*x + e)^4 + 3*(11*a - 18*b)*cos(f*x + e)^2 - 48*b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
9,1,68,0,0.685256," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e)^4,x, algorithm=""fricas"")","\frac{3 \, {\left(a - 4 \, b\right)} f x \cos\left(f x + e\right) + {\left(2 \, a \cos\left(f x + e\right)^{4} - {\left(5 \, a - 4 \, b\right)} \cos\left(f x + e\right)^{2} + 8 \, b\right)} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}"," ",0,"1/8*(3*(a - 4*b)*f*x*cos(f*x + e) + (2*a*cos(f*x + e)^4 - (5*a - 4*b)*cos(f*x + e)^2 + 8*b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
10,1,50,0,0.679805," ","integrate((a+b*sec(f*x+e)^2)*sin(f*x+e)^2,x, algorithm=""fricas"")","\frac{{\left(a - 2 \, b\right)} f x \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} - 2 \, b\right)} \sin\left(f x + e\right)}{2 \, f \cos\left(f x + e\right)}"," ",0,"1/2*((a - 2*b)*f*x*cos(f*x + e) - (a*cos(f*x + e)^2 - 2*b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
11,1,31,0,0.761011," ","integrate(a+b*sec(f*x+e)^2,x, algorithm=""fricas"")","\frac{a f x \cos\left(f x + e\right) + b \sin\left(f x + e\right)}{f \cos\left(f x + e\right)}"," ",0,"(a*f*x*cos(f*x + e) + b*sin(f*x + e))/(f*cos(f*x + e))","B",0
12,1,39,0,0.933288," ","integrate(csc(f*x+e)^2*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{f \cos\left(f x + e\right) \sin\left(f x + e\right)}"," ",0,"-((a + 2*b)*cos(f*x + e)^2 - b)/(f*cos(f*x + e)*sin(f*x + e))","A",0
13,1,66,0,0.724103," ","integrate(csc(f*x+e)^4*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b}{3 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a + 4*b)*cos(f*x + e)^4 - 3*(a + 4*b)*cos(f*x + e)^2 + 3*b)/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e))","A",0
14,1,91,0,0.552936," ","integrate(csc(f*x+e)^6*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{8 \, {\left(a + 6 \, b\right)} \cos\left(f x + e\right)^{6} - 20 \, {\left(a + 6 \, b\right)} \cos\left(f x + e\right)^{4} + 15 \, {\left(a + 6 \, b\right)} \cos\left(f x + e\right)^{2} - 15 \, b}{15 \, {\left(f \cos\left(f x + e\right)^{5} - 2 \, f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a + 6*b)*cos(f*x + e)^6 - 20*(a + 6*b)*cos(f*x + e)^4 + 15*(a + 6*b)*cos(f*x + e)^2 - 15*b)/((f*cos(f*x + e)^5 - 2*f*cos(f*x + e)^3 + f*cos(f*x + e))*sin(f*x + e))","A",0
15,1,90,0,0.504012," ","integrate((a+b*sec(f*x+e)^2)^2*sin(f*x+e)^5,x, algorithm=""fricas"")","-\frac{3 \, a^{2} \cos\left(f x + e\right)^{8} - 10 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{6} + 15 \, {\left(a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 30 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 5 \, b^{2}}{15 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/15*(3*a^2*cos(f*x + e)^8 - 10*(a^2 - a*b)*cos(f*x + e)^6 + 15*(a^2 - 4*a*b + b^2)*cos(f*x + e)^4 - 30*(a*b - b^2)*cos(f*x + e)^2 - 5*b^2)/(f*cos(f*x + e)^3)","A",0
16,1,67,0,0.461918," ","integrate((a+b*sec(f*x+e)^2)^2*sin(f*x+e)^3,x, algorithm=""fricas"")","\frac{a^{2} \cos\left(f x + e\right)^{6} - 3 \, {\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*(a^2*cos(f*x + e)^6 - 3*(a^2 - 2*a*b)*cos(f*x + e)^4 + 3*(2*a*b - b^2)*cos(f*x + e)^2 + b^2)/(f*cos(f*x + e)^3)","A",0
17,1,44,0,0.577157," ","integrate((a+b*sec(f*x+e)^2)^2*sin(f*x+e),x, algorithm=""fricas"")","-\frac{3 \, a^{2} \cos\left(f x + e\right)^{4} - 6 \, a b \cos\left(f x + e\right)^{2} - b^{2}}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/3*(3*a^2*cos(f*x + e)^4 - 6*a*b*cos(f*x + e)^2 - b^2)/(f*cos(f*x + e)^3)","A",0
18,1,101,0,0.661538," ","integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 6 \, {\left(2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, b^{2}}{6 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/6*(3*(a^2 + 2*a*b + b^2)*cos(f*x + e)^3*log(1/2*cos(f*x + e) + 1/2) - 3*(a^2 + 2*a*b + b^2)*cos(f*x + e)^3*log(-1/2*cos(f*x + e) + 1/2) - 6*(2*a*b + b^2)*cos(f*x + e)^2 - 2*b^2)/(f*cos(f*x + e)^3)","B",0
19,1,193,0,0.613763," ","integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{6 \, {\left(a^{2} + 6 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 4 \, {\left(6 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, b^{2} - 3 \, {\left({\left(a^{2} + 6 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(a^{2} + 6 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{2} + 6 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(a^{2} + 6 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{12 \, {\left(f \cos\left(f x + e\right)^{5} - f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/12*(6*(a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^4 - 4*(6*a*b + 5*b^2)*cos(f*x + e)^2 - 4*b^2 - 3*((a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^5 - (a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^5 - (a^2 + 6*a*b + 5*b^2)*cos(f*x + e)^3)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^5 - f*cos(f*x + e)^3)","B",0
20,1,286,0,0.634950," ","integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{6 \, {\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - 10 \, {\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 16 \, {\left(6 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 16 \, b^{2} - 3 \, {\left({\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(3 \, a^{2} + 30 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{48 \, {\left(f \cos\left(f x + e\right)^{7} - 2 \, f \cos\left(f x + e\right)^{5} + f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/48*(6*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^6 - 10*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^4 + 16*(6*a*b + 7*b^2)*cos(f*x + e)^2 + 16*b^2 - 3*((3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^5 + (3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^5 + (3*a^2 + 30*a*b + 35*b^2)*cos(f*x + e)^3)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^7 - 2*f*cos(f*x + e)^5 + f*cos(f*x + e)^3)","B",0
21,1,131,0,0.564259," ","integrate((a+b*sec(f*x+e)^2)^2*sin(f*x+e)^6,x, algorithm=""fricas"")","\frac{15 \, {\left(a^{2} - 12 \, a b + 8 \, b^{2}\right)} f x \cos\left(f x + e\right)^{3} - {\left(8 \, a^{2} \cos\left(f x + e\right)^{8} - 2 \, {\left(13 \, a^{2} - 12 \, a b\right)} \cos\left(f x + e\right)^{6} + 3 \, {\left(11 \, a^{2} - 36 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 16 \, {\left(6 \, a b - 7 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 16 \, b^{2}\right)} \sin\left(f x + e\right)}{48 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/48*(15*(a^2 - 12*a*b + 8*b^2)*f*x*cos(f*x + e)^3 - (8*a^2*cos(f*x + e)^8 - 2*(13*a^2 - 12*a*b)*cos(f*x + e)^6 + 3*(11*a^2 - 36*a*b + 8*b^2)*cos(f*x + e)^4 - 16*(6*a*b - 7*b^2)*cos(f*x + e)^2 - 16*b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
22,1,107,0,0.647440," ","integrate((a+b*sec(f*x+e)^2)^2*sin(f*x+e)^4,x, algorithm=""fricas"")","\frac{3 \, {\left(3 \, a^{2} - 24 \, a b + 8 \, b^{2}\right)} f x \cos\left(f x + e\right)^{3} + {\left(6 \, a^{2} \cos\left(f x + e\right)^{6} - 3 \, {\left(5 \, a^{2} - 8 \, a b\right)} \cos\left(f x + e\right)^{4} + 16 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, b^{2}\right)} \sin\left(f x + e\right)}{24 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/24*(3*(3*a^2 - 24*a*b + 8*b^2)*f*x*cos(f*x + e)^3 + (6*a^2*cos(f*x + e)^6 - 3*(5*a^2 - 8*a*b)*cos(f*x + e)^4 + 16*(3*a*b - 2*b^2)*cos(f*x + e)^2 + 8*b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
23,1,81,0,0.576239," ","integrate((a+b*sec(f*x+e)^2)^2*sin(f*x+e)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(a^{2} - 4 \, a b\right)} f x \cos\left(f x + e\right)^{3} - {\left(3 \, a^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, b^{2}\right)} \sin\left(f x + e\right)}{6 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/6*(3*(a^2 - 4*a*b)*f*x*cos(f*x + e)^3 - (3*a^2*cos(f*x + e)^4 - 2*(6*a*b - b^2)*cos(f*x + e)^2 - 2*b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
24,1,58,0,1.656115," ","integrate((a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} f x \cos\left(f x + e\right)^{3} + {\left(2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*(3*a^2*f*x*cos(f*x + e)^3 + (2*(3*a*b + b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
25,1,71,0,0.620341," ","integrate(csc(f*x+e)^2*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - b^{2}}{3 \, f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right)}"," ",0,"-1/3*((3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 2*b^2)*cos(f*x + e)^2 - b^2)/(f*cos(f*x + e)^3*sin(f*x + e))","A",0
26,1,101,0,0.707109," ","integrate(csc(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - 3 \, {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 6 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}{3 \, {\left(f \cos\left(f x + e\right)^{5} - f \cos\left(f x + e\right)^{3}\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^6 - 3*(a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 + 6*(a*b + b^2)*cos(f*x + e)^2 + b^2)/((f*cos(f*x + e)^5 - f*cos(f*x + e)^3)*sin(f*x + e))","A",0
27,1,139,0,0.618887," ","integrate(csc(f*x+e)^6*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{8 \, {\left(a^{2} + 12 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{8} - 20 \, {\left(a^{2} + 12 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + 15 \, {\left(a^{2} + 12 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 10 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 5 \, b^{2}}{15 \, {\left(f \cos\left(f x + e\right)^{7} - 2 \, f \cos\left(f x + e\right)^{5} + f \cos\left(f x + e\right)^{3}\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a^2 + 12*a*b + 16*b^2)*cos(f*x + e)^8 - 20*(a^2 + 12*a*b + 16*b^2)*cos(f*x + e)^6 + 15*(a^2 + 12*a*b + 16*b^2)*cos(f*x + e)^4 - 10*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 5*b^2)/((f*cos(f*x + e)^7 - 2*f*cos(f*x + e)^5 + f*cos(f*x + e)^3)*sin(f*x + e))","A",0
28,1,229,0,0.717001," ","integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{6 \, a^{2} \cos\left(f x + e\right)^{5} - 10 \, {\left(2 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 30 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)}{30 \, a^{3} f}, -\frac{3 \, a^{2} \cos\left(f x + e\right)^{5} - 5 \, {\left(2 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + 15 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)}{15 \, a^{3} f}\right]"," ",0,"[-1/30*(6*a^2*cos(f*x + e)^5 - 10*(2*a^2 + a*b)*cos(f*x + e)^3 - 15*(a^2 + 2*a*b + b^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 30*(a^2 + 2*a*b + b^2)*cos(f*x + e))/(a^3*f), -1/15*(3*a^2*cos(f*x + e)^5 - 5*(2*a^2 + a*b)*cos(f*x + e)^3 - 15*(a^2 + 2*a*b + b^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + 15*(a^2 + 2*a*b + b^2)*cos(f*x + e))/(a^3*f)]","A",0
29,1,154,0,0.873986," ","integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, a \cos\left(f x + e\right)^{3} + 3 \, {\left(a + b\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 6 \, {\left(a + b\right)} \cos\left(f x + e\right)}{6 \, a^{2} f}, \frac{a \cos\left(f x + e\right)^{3} + 3 \, {\left(a + b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - 3 \, {\left(a + b\right)} \cos\left(f x + e\right)}{3 \, a^{2} f}\right]"," ",0,"[1/6*(2*a*cos(f*x + e)^3 + 3*(a + b)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 6*(a + b)*cos(f*x + e))/(a^2*f), 1/3*(a*cos(f*x + e)^3 + 3*(a + b)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - 3*(a + b)*cos(f*x + e))/(a^2*f)]","A",0
30,1,118,0,0.561367," ","integrate(sin(f*x+e)/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 2 \, \cos\left(f x + e\right)}{2 \, a f}, \frac{\sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - \cos\left(f x + e\right)}{a f}\right]"," ",0,"[1/2*(sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 2*cos(f*x + e))/(a*f), (sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - cos(f*x + e))/(a*f)]","A",0
31,1,156,0,0.762865," ","integrate(csc(f*x+e)/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, {\left(a + b\right)} f}, \frac{2 \, \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, {\left(a + b\right)} f}\right]"," ",0,"[1/2*(sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - log(1/2*cos(f*x + e) + 1/2) + log(-1/2*cos(f*x + e) + 1/2))/((a + b)*f), 1/2*(2*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - log(1/2*cos(f*x + e) + 1/2) + log(-1/2*cos(f*x + e) + 1/2))/((a + b)*f)]","A",0
32,1,327,0,1.190760," ","integrate(csc(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{-a b} {\left(\cos\left(f x + e\right)^{2} - 1\right)} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-a b} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(a + b\right)} \cos\left(f x + e\right) - {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, \frac{4 \, \sqrt{a b} {\left(\cos\left(f x + e\right)^{2} - 1\right)} \arctan\left(\frac{\sqrt{a b} \cos\left(f x + e\right)}{b}\right) + 2 \, {\left(a + b\right)} \cos\left(f x + e\right) - {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}\right]"," ",0,"[1/4*(2*sqrt(-a*b)*(cos(f*x + e)^2 - 1)*log(-(a*cos(f*x + e)^2 + 2*sqrt(-a*b)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 2*(a + b)*cos(f*x + e) - ((a - b)*cos(f*x + e)^2 - a + b)*log(1/2*cos(f*x + e) + 1/2) + ((a - b)*cos(f*x + e)^2 - a + b)*log(-1/2*cos(f*x + e) + 1/2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f), 1/4*(4*sqrt(a*b)*(cos(f*x + e)^2 - 1)*arctan(sqrt(a*b)*cos(f*x + e)/b) + 2*(a + b)*cos(f*x + e) - ((a - b)*cos(f*x + e)^2 - a + b)*log(1/2*cos(f*x + e) + 1/2) + ((a - b)*cos(f*x + e)^2 - a + b)*log(-1/2*cos(f*x + e) + 1/2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)]","A",0
33,1,693,0,0.844579," ","integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} + 8 \, {\left(a \cos\left(f x + e\right)^{4} - 2 \, a \cos\left(f x + e\right)^{2} + a\right)} \sqrt{-a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-a b} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 2 \, {\left(5 \, a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right) - {\left({\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 6 \, a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 6 \, a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)}}, \frac{2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} + 16 \, {\left(a \cos\left(f x + e\right)^{4} - 2 \, a \cos\left(f x + e\right)^{2} + a\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \cos\left(f x + e\right)}{b}\right) - 2 \, {\left(5 \, a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right) - {\left({\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 6 \, a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 6 \, a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)}}\right]"," ",0,"[1/16*(2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^3 + 8*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 + a)*sqrt(-a*b)*log(-(a*cos(f*x + e)^2 + 2*sqrt(-a*b)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 2*(5*a^2 + 6*a*b + b^2)*cos(f*x + e) - ((3*a^2 - 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 - 6*a*b - b^2)*log(1/2*cos(f*x + e) + 1/2) + ((3*a^2 - 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 - 6*a*b - b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f), 1/16*(2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^3 + 16*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 + a)*sqrt(a*b)*arctan(sqrt(a*b)*cos(f*x + e)/b) - 2*(5*a^2 + 6*a*b + b^2)*cos(f*x + e) - ((3*a^2 - 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 - 6*a*b - b^2)*log(1/2*cos(f*x + e) + 1/2) + ((3*a^2 - 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 - 6*a*b - b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f)]","B",0
34,1,428,0,0.748507," ","integrate(sin(f*x+e)^6/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{3} + 30 \, a^{2} b + 40 \, a b^{2} + 16 \, b^{3}\right)} f x + 12 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} + 6 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(11 \, a^{3} + 18 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, a^{4} f}, \frac{3 \, {\left(5 \, a^{3} + 30 \, a^{2} b + 40 \, a b^{2} + 16 \, b^{3}\right)} f x + 24 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} + 6 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(11 \, a^{3} + 18 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, a^{4} f}\right]"," ",0,"[1/48*(3*(5*a^3 + 30*a^2*b + 40*a*b^2 + 16*b^3)*f*x + 12*(a^2 + 2*a*b + b^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - (8*a^3*cos(f*x + e)^5 - 2*(13*a^3 + 6*a^2*b)*cos(f*x + e)^3 + 3*(11*a^3 + 18*a^2*b + 8*a*b^2)*cos(f*x + e))*sin(f*x + e))/(a^4*f), 1/48*(3*(5*a^3 + 30*a^2*b + 40*a*b^2 + 16*b^3)*f*x + 24*(a^2 + 2*a*b + b^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - (8*a^3*cos(f*x + e)^5 - 2*(13*a^3 + 6*a^2*b)*cos(f*x + e)^3 + 3*(11*a^3 + 18*a^2*b + 8*a*b^2)*cos(f*x + e))*sin(f*x + e))/(a^4*f)]","A",0
35,1,332,0,0.566769," ","integrate(sin(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} f x + 2 \, \sqrt{-a b - b^{2}} {\left(a + b\right)} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, a^{3} f}, \frac{{\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} f x + 4 \, \sqrt{a b + b^{2}} {\left(a + b\right)} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, a^{3} f}\right]"," ",0,"[1/8*((3*a^2 + 12*a*b + 8*b^2)*f*x + 2*sqrt(-a*b - b^2)*(a + b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + (2*a^2*cos(f*x + e)^3 - (5*a^2 + 4*a*b)*cos(f*x + e))*sin(f*x + e))/(a^3*f), 1/8*((3*a^2 + 12*a*b + 8*b^2)*f*x + 4*sqrt(a*b + b^2)*(a + b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) + (2*a^2*cos(f*x + e)^3 - (5*a^2 + 4*a*b)*cos(f*x + e))*sin(f*x + e))/(a^3*f)]","A",0
36,1,257,0,0.531545," ","integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a + 2 \, b\right)} f x - 2 \, a \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, a^{2} f}, \frac{{\left(a + 2 \, b\right)} f x - a \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, a^{2} f}\right]"," ",0,"[1/4*(2*(a + 2*b)*f*x - 2*a*cos(f*x + e)*sin(f*x + e) + sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a^2*f), 1/2*((a + 2*b)*f*x - a*cos(f*x + e)*sin(f*x + e) + sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))))/(a^2*f)]","A",0
37,1,231,0,0.503645," ","integrate(1/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, f x + \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, a f}, \frac{2 \, f x + \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, a f}\right]"," ",0,"[1/4*(4*f*x + sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a*f), 1/2*(2*f*x + sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/(a*f)]","A",0
38,1,271,0,0.594072," ","integrate(csc(f*x+e)^2/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 4 \, \cos\left(f x + e\right)}{4 \, {\left(a + b\right)} f \sin\left(f x + e\right)}, \frac{\sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)}{2 \, {\left(a + b\right)} f \sin\left(f x + e\right)}\right]"," ",0,"[1/4*(sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 4*cos(f*x + e))/((a + b)*f*sin(f*x + e)), 1/2*(sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 2*cos(f*x + e))/((a + b)*f*sin(f*x + e))]","A",0
39,1,397,0,0.614424," ","integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, a \cos\left(f x + e\right)}{12 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, a \cos\left(f x + e\right)}{6 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/12*(4*(2*a - b)*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 - a)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*a*cos(f*x + e))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)*sin(f*x + e)), -1/6*(2*(2*a - b)*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 - a)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*a*cos(f*x + e))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)*sin(f*x + e))]","B",0
40,1,587,0,0.532031," ","integrate(csc(f*x+e)^6/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{2} - 9 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 20 \, {\left(4 \, a^{2} - 3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, a^{2} \cos\left(f x + e\right)}{60 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(8 \, a^{2} - 9 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(4 \, a^{2} - 3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, a^{2} \cos\left(f x + e\right)}{30 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/60*(4*(8*a^2 - 9*a*b - 2*b^2)*cos(f*x + e)^5 - 20*(4*a^2 - 3*a*b - b^2)*cos(f*x + e)^3 - 15*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*a^2*cos(f*x + e))/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f)*sin(f*x + e)), -1/30*(2*(8*a^2 - 9*a*b - 2*b^2)*cos(f*x + e)^5 - 10*(4*a^2 - 3*a*b - b^2)*cos(f*x + e)^3 - 15*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*a^2*cos(f*x + e))/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f)*sin(f*x + e))]","B",0
41,1,405,0,0.696454," ","integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{12 \, a^{3} \cos\left(f x + e\right)^{7} - 4 \, {\left(10 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(3 \, a^{3} + 10 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(3 \, a^{2} b + 10 \, a b^{2} + 7 \, b^{3} + {\left(3 \, a^{3} + 10 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 30 \, {\left(3 \, a^{2} b + 10 \, a b^{2} + 7 \, b^{3}\right)} \cos\left(f x + e\right)}{60 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}, -\frac{6 \, a^{3} \cos\left(f x + e\right)^{7} - 2 \, {\left(10 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(3 \, a^{3} + 10 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(3 \, a^{2} b + 10 \, a b^{2} + 7 \, b^{3} + {\left(3 \, a^{3} + 10 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + 15 \, {\left(3 \, a^{2} b + 10 \, a b^{2} + 7 \, b^{3}\right)} \cos\left(f x + e\right)}{30 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}\right]"," ",0,"[-1/60*(12*a^3*cos(f*x + e)^7 - 4*(10*a^3 + 7*a^2*b)*cos(f*x + e)^5 + 20*(3*a^3 + 10*a^2*b + 7*a*b^2)*cos(f*x + e)^3 - 15*(3*a^2*b + 10*a*b^2 + 7*b^3 + (3*a^3 + 10*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 30*(3*a^2*b + 10*a*b^2 + 7*b^3)*cos(f*x + e))/(a^5*f*cos(f*x + e)^2 + a^4*b*f), -1/30*(6*a^3*cos(f*x + e)^7 - 2*(10*a^3 + 7*a^2*b)*cos(f*x + e)^5 + 10*(3*a^3 + 10*a^2*b + 7*a*b^2)*cos(f*x + e)^3 - 15*(3*a^2*b + 10*a*b^2 + 7*b^3 + (3*a^3 + 10*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + 15*(3*a^2*b + 10*a*b^2 + 7*b^3)*cos(f*x + e))/(a^5*f*cos(f*x + e)^2 + a^4*b*f)]","A",0
42,1,297,0,0.576796," ","integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, a^{2} \cos\left(f x + e\right)^{5} - 4 \, {\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 5 \, b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 6 \, {\left(3 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)}{12 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}, \frac{2 \, a^{2} \cos\left(f x + e\right)^{5} - 2 \, {\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 5 \, b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - 3 \, {\left(3 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)}{6 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}\right]"," ",0,"[1/12*(4*a^2*cos(f*x + e)^5 - 4*(3*a^2 + 5*a*b)*cos(f*x + e)^3 + 3*((3*a^2 + 5*a*b)*cos(f*x + e)^2 + 3*a*b + 5*b^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 6*(3*a*b + 5*b^2)*cos(f*x + e))/(a^4*f*cos(f*x + e)^2 + a^3*b*f), 1/6*(2*a^2*cos(f*x + e)^5 - 2*(3*a^2 + 5*a*b)*cos(f*x + e)^3 + 3*((3*a^2 + 5*a*b)*cos(f*x + e)^2 + 3*a*b + 5*b^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - 3*(3*a*b + 5*b^2)*cos(f*x + e))/(a^4*f*cos(f*x + e)^2 + a^3*b*f)]","A",0
43,1,201,0,0.474660," ","integrate(sin(f*x+e)/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, a \cos\left(f x + e\right)^{3} - 3 \, {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 6 \, b \cos\left(f x + e\right)}{4 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}, -\frac{2 \, a \cos\left(f x + e\right)^{3} - 3 \, {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + 3 \, b \cos\left(f x + e\right)}{2 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}\right]"," ",0,"[-1/4*(4*a*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 + b)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 6*b*cos(f*x + e))/(a^3*f*cos(f*x + e)^2 + a^2*b*f), -1/2*(2*a*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 + b)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + 3*b*cos(f*x + e))/(a^3*f*cos(f*x + e)^2 + a^2*b*f)]","A",0
44,1,390,0,1.040559," ","integrate(csc(f*x+e)/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right) - 2 \, {\left(a^{2} \cos\left(f x + e\right)^{2} + a b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 2 \, {\left(a^{2} \cos\left(f x + e\right)^{2} + a b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, \frac{{\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - {\left(a b + b^{2}\right)} \cos\left(f x + e\right) - {\left(a^{2} \cos\left(f x + e\right)^{2} + a b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left(a^{2} \cos\left(f x + e\right)^{2} + a b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[1/4*(((3*a^2 + a*b)*cos(f*x + e)^2 + 3*a*b + b^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 2*(a*b + b^2)*cos(f*x + e) - 2*(a^2*cos(f*x + e)^2 + a*b)*log(1/2*cos(f*x + e) + 1/2) + 2*(a^2*cos(f*x + e)^2 + a*b)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b + 2*a^2*b^2 + a*b^3)*f), 1/2*(((3*a^2 + a*b)*cos(f*x + e)^2 + 3*a*b + b^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - (a*b + b^2)*cos(f*x + e) - (a^2*cos(f*x + e)^2 + a*b)*log(1/2*cos(f*x + e) + 1/2) + (a^2*cos(f*x + e)^2 + a*b)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b + 2*a^2*b^2 + a*b^3)*f)]","B",0
45,1,698,0,0.591586," ","integrate(csc(f*x+e)^3/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(3 \, a^{2} - a b\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{a \cos\left(f x + e\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 4 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right) - {\left({\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 3 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 3 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)}}, \frac{2 \, {\left(a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left({\left(3 \, a^{2} - a b\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + 4 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right) - {\left({\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 3 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 3 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(a^2 - b^2)*cos(f*x + e)^3 - ((3*a^2 - a*b)*cos(f*x + e)^4 - (3*a^2 - 4*a*b + b^2)*cos(f*x + e)^2 - 3*a*b + b^2)*sqrt(-b/a)*log((a*cos(f*x + e)^2 - 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 4*(a*b + b^2)*cos(f*x + e) - ((a^2 - 3*a*b)*cos(f*x + e)^4 - (a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^2 - a*b + 3*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((a^2 - 3*a*b)*cos(f*x + e)^4 - (a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^2 - a*b + 3*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f), 1/4*(2*(a^2 - b^2)*cos(f*x + e)^3 + 2*((3*a^2 - a*b)*cos(f*x + e)^4 - (3*a^2 - 4*a*b + b^2)*cos(f*x + e)^2 - 3*a*b + b^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + 4*(a*b + b^2)*cos(f*x + e) - ((a^2 - 3*a*b)*cos(f*x + e)^4 - (a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^2 - a*b + 3*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((a^2 - 3*a*b)*cos(f*x + e)^4 - (a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^2 - a*b + 3*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f)]","B",0
46,1,1202,0,0.661948," ","integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{3} - 2 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(5 \, a^{3} - 9 \, a^{2} b - 9 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 12 \, {\left({\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \sqrt{-a b} \log\left(\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a b} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 6 \, {\left(3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{3} - 6 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 13 \, a^{2} b + 8 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 6 \, a b^{2} + b^{3} + {\left(a^{3} - 8 \, a^{2} b + 13 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{3} - 6 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 13 \, a^{2} b + 8 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 6 \, a b^{2} + b^{3} + {\left(a^{3} - 8 \, a^{2} b + 13 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)}}, \frac{6 \, {\left(a^{3} - 2 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(5 \, a^{3} - 9 \, a^{2} b - 9 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 24 \, {\left({\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \cos\left(f x + e\right)}{b}\right) - 6 \, {\left(3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{3} - 6 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 13 \, a^{2} b + 8 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 6 \, a b^{2} + b^{3} + {\left(a^{3} - 8 \, a^{2} b + 13 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{3} - 6 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 13 \, a^{2} b + 8 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 6 \, a b^{2} + b^{3} + {\left(a^{3} - 8 \, a^{2} b + 13 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[1/16*(6*(a^3 - 2*a^2*b - 3*a*b^2)*cos(f*x + e)^5 - 2*(5*a^3 - 9*a^2*b - 9*a*b^2 + 5*b^3)*cos(f*x + e)^3 - 12*((a^2 - a*b)*cos(f*x + e)^6 - (2*a^2 - 3*a*b + b^2)*cos(f*x + e)^4 + (a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 + a*b - b^2)*sqrt(-a*b)*log((a*cos(f*x + e)^2 - 2*sqrt(-a*b)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 6*(3*a^2*b + 2*a*b^2 - b^3)*cos(f*x + e) - 3*((a^3 - 6*a^2*b + a*b^2)*cos(f*x + e)^6 - (2*a^3 - 13*a^2*b + 8*a*b^2 - b^3)*cos(f*x + e)^4 + a^2*b - 6*a*b^2 + b^3 + (a^3 - 8*a^2*b + 13*a*b^2 - 2*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 - 6*a^2*b + a*b^2)*cos(f*x + e)^6 - (2*a^3 - 13*a^2*b + 8*a*b^2 - b^3)*cos(f*x + e)^4 + a^2*b - 6*a*b^2 + b^3 + (a^3 - 8*a^2*b + 13*a*b^2 - 2*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f), 1/16*(6*(a^3 - 2*a^2*b - 3*a*b^2)*cos(f*x + e)^5 - 2*(5*a^3 - 9*a^2*b - 9*a*b^2 + 5*b^3)*cos(f*x + e)^3 + 24*((a^2 - a*b)*cos(f*x + e)^6 - (2*a^2 - 3*a*b + b^2)*cos(f*x + e)^4 + (a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 + a*b - b^2)*sqrt(a*b)*arctan(sqrt(a*b)*cos(f*x + e)/b) - 6*(3*a^2*b + 2*a*b^2 - b^3)*cos(f*x + e) - 3*((a^3 - 6*a^2*b + a*b^2)*cos(f*x + e)^6 - (2*a^3 - 13*a^2*b + 8*a*b^2 - b^3)*cos(f*x + e)^4 + a^2*b - 6*a*b^2 + b^3 + (a^3 - 8*a^2*b + 13*a*b^2 - 2*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 - 6*a^2*b + a*b^2)*cos(f*x + e)^6 - (2*a^3 - 13*a^2*b + 8*a*b^2 - b^3)*cos(f*x + e)^4 + a^2*b - 6*a*b^2 + b^3 + (a^3 - 8*a^2*b + 13*a*b^2 - 2*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)]","B",0
47,1,674,0,0.675953," ","integrate(sin(f*x+e)^6/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{4} + 60 \, a^{3} b + 120 \, a^{2} b^{2} + 64 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(5 \, a^{3} b + 60 \, a^{2} b^{2} + 120 \, a b^{3} + 64 \, b^{4}\right)} f x + 6 \, {\left(3 \, a^{2} b + 11 \, a b^{2} + 8 \, b^{3} + {\left(3 \, a^{3} + 11 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - {\left(8 \, a^{4} \cos\left(f x + e\right)^{7} - 2 \, {\left(13 \, a^{4} + 8 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(33 \, a^{4} + 82 \, a^{3} b + 48 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(19 \, a^{3} b + 52 \, a^{2} b^{2} + 32 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left(a^{6} f \cos\left(f x + e\right)^{2} + a^{5} b f\right)}}, \frac{3 \, {\left(5 \, a^{4} + 60 \, a^{3} b + 120 \, a^{2} b^{2} + 64 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(5 \, a^{3} b + 60 \, a^{2} b^{2} + 120 \, a b^{3} + 64 \, b^{4}\right)} f x + 12 \, {\left(3 \, a^{2} b + 11 \, a b^{2} + 8 \, b^{3} + {\left(3 \, a^{3} + 11 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - {\left(8 \, a^{4} \cos\left(f x + e\right)^{7} - 2 \, {\left(13 \, a^{4} + 8 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(33 \, a^{4} + 82 \, a^{3} b + 48 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(19 \, a^{3} b + 52 \, a^{2} b^{2} + 32 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left(a^{6} f \cos\left(f x + e\right)^{2} + a^{5} b f\right)}}\right]"," ",0,"[1/48*(3*(5*a^4 + 60*a^3*b + 120*a^2*b^2 + 64*a*b^3)*f*x*cos(f*x + e)^2 + 3*(5*a^3*b + 60*a^2*b^2 + 120*a*b^3 + 64*b^4)*f*x + 6*(3*a^2*b + 11*a*b^2 + 8*b^3 + (3*a^3 + 11*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - (8*a^4*cos(f*x + e)^7 - 2*(13*a^4 + 8*a^3*b)*cos(f*x + e)^5 + (33*a^4 + 82*a^3*b + 48*a^2*b^2)*cos(f*x + e)^3 + 3*(19*a^3*b + 52*a^2*b^2 + 32*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^6*f*cos(f*x + e)^2 + a^5*b*f), 1/48*(3*(5*a^4 + 60*a^3*b + 120*a^2*b^2 + 64*a*b^3)*f*x*cos(f*x + e)^2 + 3*(5*a^3*b + 60*a^2*b^2 + 120*a*b^3 + 64*b^4)*f*x + 12*(3*a^2*b + 11*a*b^2 + 8*b^3 + (3*a^3 + 11*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - (8*a^4*cos(f*x + e)^7 - 2*(13*a^4 + 8*a^3*b)*cos(f*x + e)^5 + (33*a^4 + 82*a^3*b + 48*a^2*b^2)*cos(f*x + e)^3 + 3*(19*a^3*b + 52*a^2*b^2 + 32*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^6*f*cos(f*x + e)^2 + a^5*b*f)]","A",0
48,1,522,0,0.623116," ","integrate(sin(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} + 8 \, a^{2} b + 8 \, a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right)} f x + 3 \, {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(2 \, a^{3} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} + 6 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}, \frac{3 \, {\left(a^{3} + 8 \, a^{2} b + 8 \, a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right)} f x + 6 \, {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(2 \, a^{3} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} + 6 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}\right]"," ",0,"[1/8*(3*(a^3 + 8*a^2*b + 8*a*b^2)*f*x*cos(f*x + e)^2 + 3*(a^2*b + 8*a*b^2 + 8*b^3)*f*x + 3*((a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + (2*a^3*cos(f*x + e)^5 - (5*a^3 + 6*a^2*b)*cos(f*x + e)^3 - 3*(3*a^2*b + 4*a*b^2)*cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^2 + a^4*b*f), 1/8*(3*(a^3 + 8*a^2*b + 8*a*b^2)*f*x*cos(f*x + e)^2 + 3*(a^2*b + 8*a*b^2 + 8*b^3)*f*x + 6*((a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) + (2*a^3*cos(f*x + e)^5 - (5*a^3 + 6*a^2*b)*cos(f*x + e)^3 - 3*(3*a^2*b + 4*a*b^2)*cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^2 + a^4*b*f)]","A",0
49,1,441,0,0.657040," ","integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} + 4 \, a b\right)} f x \cos\left(f x + e\right)^{2} + 4 \, {\left(a b + 4 \, b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 4 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(a^{2} \cos\left(f x + e\right)^{3} + 2 \, a b \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}, \frac{2 \, {\left(a^{2} + 4 \, a b\right)} f x \cos\left(f x + e\right)^{2} + 2 \, {\left(a b + 4 \, b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 4 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(a^{2} \cos\left(f x + e\right)^{3} + 2 \, a b \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}\right]"," ",0,"[1/8*(4*(a^2 + 4*a*b)*f*x*cos(f*x + e)^2 + 4*(a*b + 4*b^2)*f*x + ((3*a^2 + 4*a*b)*cos(f*x + e)^2 + 3*a*b + 4*b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(a^2*cos(f*x + e)^3 + 2*a*b*cos(f*x + e))*sin(f*x + e))/(a^4*f*cos(f*x + e)^2 + a^3*b*f), 1/4*(2*(a^2 + 4*a*b)*f*x*cos(f*x + e)^2 + 2*(a*b + 4*b^2)*f*x + ((3*a^2 + 4*a*b)*cos(f*x + e)^2 + 3*a*b + 4*b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 2*(a^2*cos(f*x + e)^3 + 2*a*b*cos(f*x + e))*sin(f*x + e))/(a^4*f*cos(f*x + e)^2 + a^3*b*f)]","A",0
50,1,435,0,0.789867," ","integrate(1/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} + a b\right)} f x \cos\left(f x + e\right)^{2} - 4 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 8 \, {\left(a b + b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{4 \, {\left(a^{2} + a b\right)} f x \cos\left(f x + e\right)^{2} - 2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 4 \, {\left(a b + b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(8*(a^2 + a*b)*f*x*cos(f*x + e)^2 - 4*a*b*cos(f*x + e)*sin(f*x + e) + 8*(a*b + b^2)*f*x + ((3*a^2 + 2*a*b)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), 1/4*(4*(a^2 + a*b)*f*x*cos(f*x + e)^2 - 2*a*b*cos(f*x + e)*sin(f*x + e) + 4*(a*b + b^2)*f*x + ((3*a^2 + 2*a*b)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]","B",0
51,1,407,0,0.641925," ","integrate(csc(f*x+e)^2/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 12 \, b \cos\left(f x + e\right)}{8 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 6 \, b \cos\left(f x + e\right)}{4 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/8*(4*(2*a - b)*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 + b)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 12*b*cos(f*x + e))/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3)*f)*sin(f*x + e)), -1/4*(2*(2*a - b)*cos(f*x + e)^3 - 3*(a*cos(f*x + e)^2 + b)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 6*b*cos(f*x + e))/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3)*f)*sin(f*x + e))]","B",0
52,1,663,0,0.627894," ","integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(4 \, a^{2} - 11 \, a b\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(3 \, a^{2} - 8 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(3 \, a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + 2 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)}{24 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(4 \, a^{2} - 11 \, a b\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(3 \, a^{2} - 8 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(3 \, a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + 2 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)}{12 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/24*(4*(4*a^2 - 11*a*b)*cos(f*x + e)^5 - 8*(3*a^2 - 8*a*b + 4*b^2)*cos(f*x + e)^3 + 3*((3*a^2 - 2*a*b)*cos(f*x + e)^4 - (3*a^2 - 5*a*b + 2*b^2)*cos(f*x + e)^2 - 3*a*b + 2*b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(3*a*b - 2*b^2)*cos(f*x + e))/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f)*sin(f*x + e)), -1/12*(2*(4*a^2 - 11*a*b)*cos(f*x + e)^5 - 4*(3*a^2 - 8*a*b + 4*b^2)*cos(f*x + e)^3 - 3*((3*a^2 - 2*a*b)*cos(f*x + e)^4 - (3*a^2 - 5*a*b + 2*b^2)*cos(f*x + e)^2 - 3*a*b + 2*b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*(3*a*b - 2*b^2)*cos(f*x + e))/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f)*sin(f*x + e))]","B",0
53,1,987,0,0.745569," ","integrate(csc(f*x+e)^6/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(16 \, a^{3} - 83 \, a^{2} b + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(40 \, a^{3} - 201 \, a^{2} b + 68 \, a b^{2} - 6 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(6 \, a^{3} - 29 \, a^{2} b + 28 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(3 \, a^{3} - 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{6} - {\left(6 \, a^{3} - 11 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b - 4 \, a b^{2} + {\left(3 \, a^{3} - 10 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)}{120 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(16 \, a^{3} - 83 \, a^{2} b + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(40 \, a^{3} - 201 \, a^{2} b + 68 \, a b^{2} - 6 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(6 \, a^{3} - 29 \, a^{2} b + 28 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(3 \, a^{3} - 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{6} - {\left(6 \, a^{3} - 11 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b - 4 \, a b^{2} + {\left(3 \, a^{3} - 10 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)}{60 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/120*(4*(16*a^3 - 83*a^2*b + 6*a*b^2)*cos(f*x + e)^7 - 4*(40*a^3 - 201*a^2*b + 68*a*b^2 - 6*b^3)*cos(f*x + e)^5 + 20*(6*a^3 - 29*a^2*b + 28*a*b^2)*cos(f*x + e)^3 + 15*((3*a^3 - 4*a^2*b)*cos(f*x + e)^6 - (6*a^3 - 11*a^2*b + 4*a*b^2)*cos(f*x + e)^4 + 3*a^2*b - 4*a*b^2 + (3*a^3 - 10*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(3*a^2*b - 4*a*b^2)*cos(f*x + e))/(((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)*sin(f*x + e)), -1/60*(2*(16*a^3 - 83*a^2*b + 6*a*b^2)*cos(f*x + e)^7 - 2*(40*a^3 - 201*a^2*b + 68*a*b^2 - 6*b^3)*cos(f*x + e)^5 + 10*(6*a^3 - 29*a^2*b + 28*a*b^2)*cos(f*x + e)^3 - 15*((3*a^3 - 4*a^2*b)*cos(f*x + e)^6 - (6*a^3 - 11*a^2*b + 4*a*b^2)*cos(f*x + e)^4 + 3*a^2*b - 4*a*b^2 + (3*a^3 - 10*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(3*a^2*b - 4*a*b^2)*cos(f*x + e))/(((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)*sin(f*x + e))]","B",0
54,1,579,0,0.676949," ","integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{48 \, a^{4} \cos\left(f x + e\right)^{9} - 16 \, {\left(10 \, a^{4} + 9 \, a^{3} b\right)} \cos\left(f x + e\right)^{7} + 16 \, {\left(15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{5} + 50 \, {\left(15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 30 \, {\left(15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4}\right)} \cos\left(f x + e\right)}{240 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}, -\frac{24 \, a^{4} \cos\left(f x + e\right)^{9} - 8 \, {\left(10 \, a^{4} + 9 \, a^{3} b\right)} \cos\left(f x + e\right)^{7} + 8 \, {\left(15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{5} + 25 \, {\left(15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(15 \, a^{4} + 70 \, a^{3} b + 63 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 70 \, a^{2} b^{2} + 63 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + 15 \, {\left(15 \, a^{2} b^{2} + 70 \, a b^{3} + 63 \, b^{4}\right)} \cos\left(f x + e\right)}{120 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}\right]"," ",0,"[-1/240*(48*a^4*cos(f*x + e)^9 - 16*(10*a^4 + 9*a^3*b)*cos(f*x + e)^7 + 16*(15*a^4 + 70*a^3*b + 63*a^2*b^2)*cos(f*x + e)^5 + 50*(15*a^3*b + 70*a^2*b^2 + 63*a*b^3)*cos(f*x + e)^3 - 15*((15*a^4 + 70*a^3*b + 63*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 70*a*b^3 + 63*b^4 + 2*(15*a^3*b + 70*a^2*b^2 + 63*a*b^3)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 30*(15*a^2*b^2 + 70*a*b^3 + 63*b^4)*cos(f*x + e))/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f), -1/120*(24*a^4*cos(f*x + e)^9 - 8*(10*a^4 + 9*a^3*b)*cos(f*x + e)^7 + 8*(15*a^4 + 70*a^3*b + 63*a^2*b^2)*cos(f*x + e)^5 + 25*(15*a^3*b + 70*a^2*b^2 + 63*a*b^3)*cos(f*x + e)^3 - 15*((15*a^4 + 70*a^3*b + 63*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 70*a*b^3 + 63*b^4 + 2*(15*a^3*b + 70*a^2*b^2 + 63*a*b^3)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + 15*(15*a^2*b^2 + 70*a*b^3 + 63*b^4)*cos(f*x + e))/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f)]","A",0
55,1,439,0,0.684218," ","integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{16 \, a^{3} \cos\left(f x + e\right)^{7} - 16 \, {\left(3 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{5} - 50 \, {\left(3 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(3 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} + 3 \, a b^{2} + 7 \, b^{3} + 2 \, {\left(3 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 30 \, {\left(3 \, a b^{2} + 7 \, b^{3}\right)} \cos\left(f x + e\right)}{48 \, {\left(a^{6} f \cos\left(f x + e\right)^{4} + 2 \, a^{5} b f \cos\left(f x + e\right)^{2} + a^{4} b^{2} f\right)}}, \frac{8 \, a^{3} \cos\left(f x + e\right)^{7} - 8 \, {\left(3 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{5} - 25 \, {\left(3 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(3 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} + 3 \, a b^{2} + 7 \, b^{3} + 2 \, {\left(3 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - 15 \, {\left(3 \, a b^{2} + 7 \, b^{3}\right)} \cos\left(f x + e\right)}{24 \, {\left(a^{6} f \cos\left(f x + e\right)^{4} + 2 \, a^{5} b f \cos\left(f x + e\right)^{2} + a^{4} b^{2} f\right)}}\right]"," ",0,"[1/48*(16*a^3*cos(f*x + e)^7 - 16*(3*a^3 + 7*a^2*b)*cos(f*x + e)^5 - 50*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^3 + 15*((3*a^3 + 7*a^2*b)*cos(f*x + e)^4 + 3*a*b^2 + 7*b^3 + 2*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 30*(3*a*b^2 + 7*b^3)*cos(f*x + e))/(a^6*f*cos(f*x + e)^4 + 2*a^5*b*f*cos(f*x + e)^2 + a^4*b^2*f), 1/24*(8*a^3*cos(f*x + e)^7 - 8*(3*a^3 + 7*a^2*b)*cos(f*x + e)^5 - 25*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^3 + 15*((3*a^3 + 7*a^2*b)*cos(f*x + e)^4 + 3*a*b^2 + 7*b^3 + 2*(3*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - 15*(3*a*b^2 + 7*b^3)*cos(f*x + e))/(a^6*f*cos(f*x + e)^4 + 2*a^5*b*f*cos(f*x + e)^2 + a^4*b^2*f)]","A",0
56,1,299,0,0.649600," ","integrate(sin(f*x+e)/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{16 \, a^{2} \cos\left(f x + e\right)^{5} + 50 \, a b \cos\left(f x + e\right)^{3} + 30 \, b^{2} \cos\left(f x + e\right) - 15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right)}{16 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}, -\frac{8 \, a^{2} \cos\left(f x + e\right)^{5} + 25 \, a b \cos\left(f x + e\right)^{3} + 15 \, b^{2} \cos\left(f x + e\right) - 15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right)}{8 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}\right]"," ",0,"[-1/16*(16*a^2*cos(f*x + e)^5 + 50*a*b*cos(f*x + e)^3 + 30*b^2*cos(f*x + e) - 15*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f), -1/8*(8*a^2*cos(f*x + e)^5 + 25*a*b*cos(f*x + e)^3 + 15*b^2*cos(f*x + e) - 15*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)]","A",0
57,1,779,0,0.805110," ","integrate(csc(f*x+e)/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(9 \, a^{3} b + 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(15 \, a^{4} + 10 \, a^{3} b + 3 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(7 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right) + 8 \, {\left(a^{4} \cos\left(f x + e\right)^{4} + 2 \, a^{3} b \cos\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 8 \, {\left(a^{4} \cos\left(f x + e\right)^{4} + 2 \, a^{3} b \cos\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, -\frac{{\left(9 \, a^{3} b + 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(15 \, a^{4} + 10 \, a^{3} b + 3 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + {\left(7 \, a^{2} b^{2} + 10 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right) + 4 \, {\left(a^{4} \cos\left(f x + e\right)^{4} + 2 \, a^{3} b \cos\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 4 \, {\left(a^{4} \cos\left(f x + e\right)^{4} + 2 \, a^{3} b \cos\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{8 \, {\left({\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[-1/16*(2*(9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^3 - ((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 + 3*b^4 + 2*(15*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 2*(7*a^2*b^2 + 10*a*b^3 + 3*b^4)*cos(f*x + e) + 8*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*log(1/2*cos(f*x + e) + 1/2) - 8*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f), -1/8*((9*a^3*b + 14*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^3 - ((15*a^4 + 10*a^3*b + 3*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 + 3*b^4 + 2*(15*a^3*b + 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + (7*a^2*b^2 + 10*a*b^3 + 3*b^4)*cos(f*x + e) + 4*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*log(1/2*cos(f*x + e) + 1/2) - 4*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f)]","B",0
58,1,1332,0,0.878978," ","integrate(csc(f*x+e)^3/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(4 \, a^{4} - 5 \, a^{3} b - 10 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(17 \, a^{3} b + 11 \, a^{2} b^{2} - 5 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(15 \, a^{4} - 10 \, a^{3} b - a^{2} b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(15 \, a^{4} - 40 \, a^{3} b + 19 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} - 15 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4} - {\left(30 \, a^{3} b - 35 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{a \cos\left(f x + e\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(11 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right) - 4 \, {\left({\left(a^{4} - 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 7 \, a^{3} b + 10 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 5 \, a b^{3} - {\left(2 \, a^{3} b - 11 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 4 \, {\left({\left(a^{4} - 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 7 \, a^{3} b + 10 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 5 \, a b^{3} - {\left(2 \, a^{3} b - 11 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b + 7 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 2 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6}\right)} f\right)}}, \frac{{\left(4 \, a^{4} - 5 \, a^{3} b - 10 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(17 \, a^{3} b + 11 \, a^{2} b^{2} - 5 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left({\left(15 \, a^{4} - 10 \, a^{3} b - a^{2} b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(15 \, a^{4} - 40 \, a^{3} b + 19 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} - 15 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4} - {\left(30 \, a^{3} b - 35 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) + {\left(11 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right) - 2 \, {\left({\left(a^{4} - 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 7 \, a^{3} b + 10 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 5 \, a b^{3} - {\left(2 \, a^{3} b - 11 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 2 \, {\left({\left(a^{4} - 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 7 \, a^{3} b + 10 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 5 \, a b^{3} - {\left(2 \, a^{3} b - 11 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{8 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b + 7 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 2 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6}\right)} f\right)}}\right]"," ",0,"[1/16*(2*(4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cos(f*x + e)^5 + 2*(17*a^3*b + 11*a^2*b^2 - 5*a*b^3 + b^4)*cos(f*x + e)^3 - ((15*a^4 - 10*a^3*b - a^2*b^2)*cos(f*x + e)^6 - (15*a^4 - 40*a^3*b + 19*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 15*a^2*b^2 + 10*a*b^3 + b^4 - (30*a^3*b - 35*a^2*b^2 + 8*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-b/a)*log((a*cos(f*x + e)^2 - 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) + 2*(11*a^2*b^2 + 10*a*b^3 - b^4)*cos(f*x + e) - 4*((a^4 - 5*a^3*b)*cos(f*x + e)^6 - (a^4 - 7*a^3*b + 10*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 5*a*b^3 - (2*a^3*b - 11*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 4*((a^4 - 5*a^3*b)*cos(f*x + e)^6 - (a^4 - 7*a^3*b + 10*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 5*a*b^3 - (2*a^3*b - 11*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^4 - (2*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 - (a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4 + 4*a^2*b^5 + a*b^6)*f), 1/8*((4*a^4 - 5*a^3*b - 10*a^2*b^2 - a*b^3)*cos(f*x + e)^5 + (17*a^3*b + 11*a^2*b^2 - 5*a*b^3 + b^4)*cos(f*x + e)^3 + ((15*a^4 - 10*a^3*b - a^2*b^2)*cos(f*x + e)^6 - (15*a^4 - 40*a^3*b + 19*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 15*a^2*b^2 + 10*a*b^3 + b^4 - (30*a^3*b - 35*a^2*b^2 + 8*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) + (11*a^2*b^2 + 10*a*b^3 - b^4)*cos(f*x + e) - 2*((a^4 - 5*a^3*b)*cos(f*x + e)^6 - (a^4 - 7*a^3*b + 10*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 5*a*b^3 - (2*a^3*b - 11*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 2*((a^4 - 5*a^3*b)*cos(f*x + e)^6 - (a^4 - 7*a^3*b + 10*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 5*a*b^3 - (2*a^3*b - 11*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^4 - (2*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 - (a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4 + 4*a^2*b^5 + a*b^6)*f)]","B",0
59,1,1833,0,0.999821," ","integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{4} - 5 \, a^{3} b - 5 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(5 \, a^{4} - 26 \, a^{3} b + 26 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(19 \, a^{3} b - 15 \, a^{2} b^{2} - 15 \, a b^{3} + 19 \, b^{4}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(5 \, a^{4} - 10 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(5 \, a^{4} - 15 \, a^{3} b + 11 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(5 \, a^{4} - 30 \, a^{3} b + 46 \, a^{2} b^{2} - 14 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4} + 2 \, {\left(5 \, a^{3} b - 15 \, a^{2} b^{2} + 11 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, a \sqrt{-\frac{b}{a}} \cos\left(f x + e\right) - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 24 \, {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{4} - 10 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 11 \, a^{3} b + 15 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 14 \, a^{3} b + 46 \, a^{2} b^{2} - 30 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 10 \, a b^{3} + 5 \, b^{4} + 2 \, {\left(a^{3} b - 11 \, a^{2} b^{2} + 15 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{4} - 10 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 11 \, a^{3} b + 15 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 14 \, a^{3} b + 46 \, a^{2} b^{2} - 30 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 10 \, a b^{3} + 5 \, b^{4} + 2 \, {\left(a^{3} b - 11 \, a^{2} b^{2} + 15 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)}}, \frac{6 \, {\left(a^{4} - 5 \, a^{3} b - 5 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(5 \, a^{4} - 26 \, a^{3} b + 26 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(19 \, a^{3} b - 15 \, a^{2} b^{2} - 15 \, a b^{3} + 19 \, b^{4}\right)} \cos\left(f x + e\right)^{3} + 6 \, {\left({\left(5 \, a^{4} - 10 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(5 \, a^{4} - 15 \, a^{3} b + 11 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(5 \, a^{4} - 30 \, a^{3} b + 46 \, a^{2} b^{2} - 14 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4} + 2 \, {\left(5 \, a^{3} b - 15 \, a^{2} b^{2} + 11 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{a \sqrt{\frac{b}{a}} \cos\left(f x + e\right)}{b}\right) - 24 \, {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{4} - 10 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 11 \, a^{3} b + 15 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 14 \, a^{3} b + 46 \, a^{2} b^{2} - 30 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 10 \, a b^{3} + 5 \, b^{4} + 2 \, {\left(a^{3} b - 11 \, a^{2} b^{2} + 15 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{4} - 10 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 11 \, a^{3} b + 15 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 14 \, a^{3} b + 46 \, a^{2} b^{2} - 30 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 10 \, a b^{3} + 5 \, b^{4} + 2 \, {\left(a^{3} b - 11 \, a^{2} b^{2} + 15 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)}}\right]"," ",0,"[1/16*(6*(a^4 - 5*a^3*b - 5*a^2*b^2 + a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 26*a^3*b + 26*a*b^3 - 5*b^4)*cos(f*x + e)^5 - 2*(19*a^3*b - 15*a^2*b^2 - 15*a*b^3 + 19*b^4)*cos(f*x + e)^3 + 3*((5*a^4 - 10*a^3*b + a^2*b^2)*cos(f*x + e)^8 - 2*(5*a^4 - 15*a^3*b + 11*a^2*b^2 - a*b^3)*cos(f*x + e)^6 + (5*a^4 - 30*a^3*b + 46*a^2*b^2 - 14*a*b^3 + b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 10*a*b^3 + b^4 + 2*(5*a^3*b - 15*a^2*b^2 + 11*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-b/a)*log(-(a*cos(f*x + e)^2 + 2*a*sqrt(-b/a)*cos(f*x + e) - b)/(a*cos(f*x + e)^2 + b)) - 24*(a^2*b^2 - b^4)*cos(f*x + e) - 3*((a^4 - 10*a^3*b + 5*a^2*b^2)*cos(f*x + e)^8 - 2*(a^4 - 11*a^3*b + 15*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^6 + (a^4 - 14*a^3*b + 46*a^2*b^2 - 30*a*b^3 + 5*b^4)*cos(f*x + e)^4 + a^2*b^2 - 10*a*b^3 + 5*b^4 + 2*(a^3*b - 11*a^2*b^2 + 15*a*b^3 - 5*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 10*a^3*b + 5*a^2*b^2)*cos(f*x + e)^8 - 2*(a^4 - 11*a^3*b + 15*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^6 + (a^4 - 14*a^3*b + 46*a^2*b^2 - 30*a*b^3 + 5*b^4)*cos(f*x + e)^4 + a^2*b^2 - 10*a*b^3 + 5*b^4 + 2*(a^3*b - 11*a^2*b^2 + 15*a*b^3 - 5*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f), 1/16*(6*(a^4 - 5*a^3*b - 5*a^2*b^2 + a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 26*a^3*b + 26*a*b^3 - 5*b^4)*cos(f*x + e)^5 - 2*(19*a^3*b - 15*a^2*b^2 - 15*a*b^3 + 19*b^4)*cos(f*x + e)^3 + 6*((5*a^4 - 10*a^3*b + a^2*b^2)*cos(f*x + e)^8 - 2*(5*a^4 - 15*a^3*b + 11*a^2*b^2 - a*b^3)*cos(f*x + e)^6 + (5*a^4 - 30*a^3*b + 46*a^2*b^2 - 14*a*b^3 + b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 10*a*b^3 + b^4 + 2*(5*a^3*b - 15*a^2*b^2 + 11*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(b/a)*arctan(a*sqrt(b/a)*cos(f*x + e)/b) - 24*(a^2*b^2 - b^4)*cos(f*x + e) - 3*((a^4 - 10*a^3*b + 5*a^2*b^2)*cos(f*x + e)^8 - 2*(a^4 - 11*a^3*b + 15*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^6 + (a^4 - 14*a^3*b + 46*a^2*b^2 - 30*a*b^3 + 5*b^4)*cos(f*x + e)^4 + a^2*b^2 - 10*a*b^3 + 5*b^4 + 2*(a^3*b - 11*a^2*b^2 + 15*a*b^3 - 5*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 10*a^3*b + 5*a^2*b^2)*cos(f*x + e)^8 - 2*(a^4 - 11*a^3*b + 15*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^6 + (a^4 - 14*a^3*b + 46*a^2*b^2 - 30*a*b^3 + 5*b^4)*cos(f*x + e)^4 + a^2*b^2 - 10*a*b^3 + 5*b^4 + 2*(a^3*b - 11*a^2*b^2 + 15*a*b^3 - 5*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f)]","B",0
60,1,930,0,0.634280," ","integrate(sin(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{30 \, {\left(a^{5} + 18 \, a^{4} b + 48 \, a^{3} b^{2} + 32 \, a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 60 \, {\left(a^{4} b + 18 \, a^{3} b^{2} + 48 \, a^{2} b^{3} + 32 \, a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 30 \, {\left(a^{3} b^{2} + 18 \, a^{2} b^{3} + 48 \, a b^{4} + 32 \, b^{5}\right)} f x + 15 \, {\left({\left(3 \, a^{4} + 16 \, a^{3} b + 16 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 16 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 16 \, a^{2} b^{2} + 16 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 2 \, {\left(8 \, a^{5} \cos\left(f x + e\right)^{9} - 2 \, {\left(13 \, a^{5} + 10 \, a^{4} b\right)} \cos\left(f x + e\right)^{7} + {\left(33 \, a^{5} + 110 \, a^{4} b + 80 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(6 \, a^{4} b + 23 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left(5 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 16 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{96 \, {\left(a^{8} f \cos\left(f x + e\right)^{4} + 2 \, a^{7} b f \cos\left(f x + e\right)^{2} + a^{6} b^{2} f\right)}}, \frac{15 \, {\left(a^{5} + 18 \, a^{4} b + 48 \, a^{3} b^{2} + 32 \, a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 30 \, {\left(a^{4} b + 18 \, a^{3} b^{2} + 48 \, a^{2} b^{3} + 32 \, a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 15 \, {\left(a^{3} b^{2} + 18 \, a^{2} b^{3} + 48 \, a b^{4} + 32 \, b^{5}\right)} f x + 15 \, {\left({\left(3 \, a^{4} + 16 \, a^{3} b + 16 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 16 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 16 \, a^{2} b^{2} + 16 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - {\left(8 \, a^{5} \cos\left(f x + e\right)^{9} - 2 \, {\left(13 \, a^{5} + 10 \, a^{4} b\right)} \cos\left(f x + e\right)^{7} + {\left(33 \, a^{5} + 110 \, a^{4} b + 80 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(6 \, a^{4} b + 23 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left(5 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 16 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left(a^{8} f \cos\left(f x + e\right)^{4} + 2 \, a^{7} b f \cos\left(f x + e\right)^{2} + a^{6} b^{2} f\right)}}\right]"," ",0,"[1/96*(30*(a^5 + 18*a^4*b + 48*a^3*b^2 + 32*a^2*b^3)*f*x*cos(f*x + e)^4 + 60*(a^4*b + 18*a^3*b^2 + 48*a^2*b^3 + 32*a*b^4)*f*x*cos(f*x + e)^2 + 30*(a^3*b^2 + 18*a^2*b^3 + 48*a*b^4 + 32*b^5)*f*x + 15*((3*a^4 + 16*a^3*b + 16*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 16*a*b^3 + 16*b^4 + 2*(3*a^3*b + 16*a^2*b^2 + 16*a*b^3)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 2*(8*a^5*cos(f*x + e)^9 - 2*(13*a^5 + 10*a^4*b)*cos(f*x + e)^7 + (33*a^5 + 110*a^4*b + 80*a^3*b^2)*cos(f*x + e)^5 + 20*(6*a^4*b + 23*a^3*b^2 + 18*a^2*b^3)*cos(f*x + e)^3 + 15*(5*a^3*b^2 + 20*a^2*b^3 + 16*a*b^4)*cos(f*x + e))*sin(f*x + e))/(a^8*f*cos(f*x + e)^4 + 2*a^7*b*f*cos(f*x + e)^2 + a^6*b^2*f), 1/48*(15*(a^5 + 18*a^4*b + 48*a^3*b^2 + 32*a^2*b^3)*f*x*cos(f*x + e)^4 + 30*(a^4*b + 18*a^3*b^2 + 48*a^2*b^3 + 32*a*b^4)*f*x*cos(f*x + e)^2 + 15*(a^3*b^2 + 18*a^2*b^3 + 48*a*b^4 + 32*b^5)*f*x + 15*((3*a^4 + 16*a^3*b + 16*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 16*a*b^3 + 16*b^4 + 2*(3*a^3*b + 16*a^2*b^2 + 16*a*b^3)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - (8*a^5*cos(f*x + e)^9 - 2*(13*a^5 + 10*a^4*b)*cos(f*x + e)^7 + (33*a^5 + 110*a^4*b + 80*a^3*b^2)*cos(f*x + e)^5 + 20*(6*a^4*b + 23*a^3*b^2 + 18*a^2*b^3)*cos(f*x + e)^3 + 15*(5*a^3*b^2 + 20*a^2*b^3 + 16*a*b^4)*cos(f*x + e))*sin(f*x + e))/(a^8*f*cos(f*x + e)^4 + 2*a^7*b*f*cos(f*x + e)^2 + a^6*b^2*f)]","A",0
61,1,803,0,0.596002," ","integrate(sin(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{4} + 12 \, a^{3} b + 16 \, a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 24 \, {\left(a^{3} b + 12 \, a^{2} b^{2} + 16 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 12 \, {\left(a^{2} b^{2} + 12 \, a b^{3} + 16 \, b^{4}\right)} f x + 3 \, {\left({\left(5 \, a^{4} + 20 \, a^{3} b + 16 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(5 \, a^{3} b + 20 \, a^{2} b^{2} + 16 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left(2 \, a^{4} \cos\left(f x + e\right)^{7} - {\left(5 \, a^{4} + 8 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} - {\left(19 \, a^{3} b + 36 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, {\left(a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}, \frac{6 \, {\left(a^{4} + 12 \, a^{3} b + 16 \, a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 12 \, {\left(a^{3} b + 12 \, a^{2} b^{2} + 16 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 6 \, {\left(a^{2} b^{2} + 12 \, a b^{3} + 16 \, b^{4}\right)} f x + 3 \, {\left({\left(5 \, a^{4} + 20 \, a^{3} b + 16 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(5 \, a^{3} b + 20 \, a^{2} b^{2} + 16 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(2 \, a^{4} \cos\left(f x + e\right)^{7} - {\left(5 \, a^{4} + 8 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} - {\left(19 \, a^{3} b + 36 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, {\left(a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}\right]"," ",0,"[1/32*(12*(a^4 + 12*a^3*b + 16*a^2*b^2)*f*x*cos(f*x + e)^4 + 24*(a^3*b + 12*a^2*b^2 + 16*a*b^3)*f*x*cos(f*x + e)^2 + 12*(a^2*b^2 + 12*a*b^3 + 16*b^4)*f*x + 3*((5*a^4 + 20*a^3*b + 16*a^2*b^2)*cos(f*x + e)^4 + 5*a^2*b^2 + 20*a*b^3 + 16*b^4 + 2*(5*a^3*b + 20*a^2*b^2 + 16*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*(2*a^4*cos(f*x + e)^7 - (5*a^4 + 8*a^3*b)*cos(f*x + e)^5 - (19*a^3*b + 36*a^2*b^2)*cos(f*x + e)^3 - 12*(a^2*b^2 + 2*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f), 1/16*(6*(a^4 + 12*a^3*b + 16*a^2*b^2)*f*x*cos(f*x + e)^4 + 12*(a^3*b + 12*a^2*b^2 + 16*a*b^3)*f*x*cos(f*x + e)^2 + 6*(a^2*b^2 + 12*a*b^3 + 16*b^4)*f*x + 3*((5*a^4 + 20*a^3*b + 16*a^2*b^2)*cos(f*x + e)^4 + 5*a^2*b^2 + 20*a*b^3 + 16*b^4 + 2*(5*a^3*b + 20*a^2*b^2 + 16*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + 2*(2*a^4*cos(f*x + e)^7 - (5*a^4 + 8*a^3*b)*cos(f*x + e)^5 - (19*a^3*b + 36*a^2*b^2)*cos(f*x + e)^3 - 12*(a^2*b^2 + 2*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f)]","A",0
62,1,815,0,0.702603," ","integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{16 \, {\left(a^{4} + 7 \, a^{3} b + 6 \, a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{2} b^{2} + 7 \, a b^{3} + 6 \, b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 40 \, a^{3} b + 24 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 40 \, a b^{3} + 24 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 40 \, a^{2} b^{2} + 24 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(4 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(17 \, a^{3} b + 18 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(11 \, a^{2} b^{2} + 12 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + a^{4} b^{3}\right)} f\right)}}, \frac{8 \, {\left(a^{4} + 7 \, a^{3} b + 6 \, a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 16 \, {\left(a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 8 \, {\left(a^{2} b^{2} + 7 \, a b^{3} + 6 \, b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 40 \, a^{3} b + 24 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 40 \, a b^{3} + 24 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 40 \, a^{2} b^{2} + 24 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(4 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(17 \, a^{3} b + 18 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(11 \, a^{2} b^{2} + 12 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[1/32*(16*(a^4 + 7*a^3*b + 6*a^2*b^2)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 7*a^2*b^2 + 6*a*b^3)*f*x*cos(f*x + e)^2 + 16*(a^2*b^2 + 7*a*b^3 + 6*b^4)*f*x + ((15*a^4 + 40*a^3*b + 24*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 40*a*b^3 + 24*b^4 + 2*(15*a^3*b + 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(4*(a^4 + a^3*b)*cos(f*x + e)^5 + (17*a^3*b + 18*a^2*b^2)*cos(f*x + e)^3 + (11*a^2*b^2 + 12*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^4 + 2*(a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^5*b^2 + a^4*b^3)*f), 1/16*(8*(a^4 + 7*a^3*b + 6*a^2*b^2)*f*x*cos(f*x + e)^4 + 16*(a^3*b + 7*a^2*b^2 + 6*a*b^3)*f*x*cos(f*x + e)^2 + 8*(a^2*b^2 + 7*a*b^3 + 6*b^4)*f*x + ((15*a^4 + 40*a^3*b + 24*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 40*a*b^3 + 24*b^4 + 2*(15*a^3*b + 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 2*(4*(a^4 + a^3*b)*cos(f*x + e)^5 + (17*a^3*b + 18*a^2*b^2)*cos(f*x + e)^3 + (11*a^2*b^2 + 12*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^4 + 2*(a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^5*b^2 + a^4*b^3)*f)]","B",0
63,1,819,0,0.593474," ","integrate(1/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 64 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 32 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 20 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(7 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, \frac{16 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 20 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(7 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[1/32*(32*(a^4 + 2*a^3*b + a^2*b^2)*f*x*cos(f*x + e)^4 + 64*(a^3*b + 2*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^2 + 32*(a^2*b^2 + 2*a*b^3 + b^4)*f*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 20*a*b^3 + 8*b^4 + 2*(15*a^3*b + 20*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(3*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (7*a^2*b^2 + 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f), 1/16*(16*(a^4 + 2*a^3*b + a^2*b^2)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 2*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^2 + 16*(a^2*b^2 + 2*a*b^3 + b^4)*f*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 20*a*b^3 + 8*b^4 + 2*(15*a^3*b + 20*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 2*(3*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (7*a^2*b^2 + 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f)]","B",0
64,1,615,0,0.610348," ","integrate(csc(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{2} - 9 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(5 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, b^{2} \cos\left(f x + e\right)}{32 \, {\left({\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(8 \, a^{2} - 9 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(5 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, b^{2} \cos\left(f x + e\right)}{16 \, {\left({\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/32*(4*(8*a^2 - 9*a*b - 2*b^2)*cos(f*x + e)^5 + 20*(5*a*b - b^2)*cos(f*x + e)^3 - 15*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*b^2*cos(f*x + e))/(((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*f*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2 + (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f)*sin(f*x + e)), -1/16*(2*(8*a^2 - 9*a*b - 2*b^2)*cos(f*x + e)^5 + 10*(5*a*b - b^2)*cos(f*x + e)^3 - 15*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*b^2*cos(f*x + e))/(((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*f*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2 + (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f)*sin(f*x + e))]","B",0
65,1,1009,0,0.649759," ","integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(16 \, a^{3} - 83 \, a^{2} b + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(24 \, a^{3} - 134 \, a^{2} b + 145 \, a b^{2} - 12 \, b^{3}\right)} \cos\left(f x + e\right)^{5} - 20 \, {\left(15 \, a^{2} b - 32 \, a b^{2} + 16 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(3 \, a^{3} - 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{6} - {\left(3 \, a^{3} - 10 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - 3 \, a b^{2} + 4 \, b^{3} - {\left(6 \, a^{2} b - 11 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 60 \, {\left(3 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)}{96 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(16 \, a^{3} - 83 \, a^{2} b + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(24 \, a^{3} - 134 \, a^{2} b + 145 \, a b^{2} - 12 \, b^{3}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(15 \, a^{2} b - 32 \, a b^{2} + 16 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(3 \, a^{3} - 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{6} - {\left(3 \, a^{3} - 10 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - 3 \, a b^{2} + 4 \, b^{3} - {\left(6 \, a^{2} b - 11 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 30 \, {\left(3 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)}{48 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/96*(4*(16*a^3 - 83*a^2*b + 6*a*b^2)*cos(f*x + e)^7 - 4*(24*a^3 - 134*a^2*b + 145*a*b^2 - 12*b^3)*cos(f*x + e)^5 - 20*(15*a^2*b - 32*a*b^2 + 16*b^3)*cos(f*x + e)^3 + 15*((3*a^3 - 4*a^2*b)*cos(f*x + e)^6 - (3*a^3 - 10*a^2*b + 8*a*b^2)*cos(f*x + e)^4 - 3*a*b^2 + 4*b^3 - (6*a^2*b - 11*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 60*(3*a*b^2 - 4*b^3)*cos(f*x + e))/(((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f)*sin(f*x + e)), -1/48*(2*(16*a^3 - 83*a^2*b + 6*a*b^2)*cos(f*x + e)^7 - 2*(24*a^3 - 134*a^2*b + 145*a*b^2 - 12*b^3)*cos(f*x + e)^5 - 10*(15*a^2*b - 32*a*b^2 + 16*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 4*a^2*b)*cos(f*x + e)^6 - (3*a^3 - 10*a^2*b + 8*a*b^2)*cos(f*x + e)^4 - 3*a*b^2 + 4*b^3 - (6*a^2*b - 11*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 30*(3*a*b^2 - 4*b^3)*cos(f*x + e))/(((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f)*sin(f*x + e))]","B",0
66,1,1423,0,0.668152," ","integrate(csc(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(64 \, a^{4} - 607 \, a^{3} b + 274 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{9} - 4 \, {\left(160 \, a^{4} - 1533 \, a^{3} b + 1599 \, a^{2} b^{2} - 488 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} + 4 \, {\left(120 \, a^{4} - 1205 \, a^{3} b + 2769 \, a^{2} b^{2} - 1392 \, a b^{3} + 184 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(75 \, a^{3} b - 305 \, a^{2} b^{2} + 320 \, a b^{3} - 56 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(15 \, a^{4} - 40 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(15 \, a^{4} - 55 \, a^{3} b + 48 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(15 \, a^{4} - 100 \, a^{3} b + 183 \, a^{2} b^{2} - 72 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} - 40 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b - 55 \, a^{2} b^{2} + 48 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(15 \, a^{2} b^{2} - 40 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)}{480 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(64 \, a^{4} - 607 \, a^{3} b + 274 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{9} - 2 \, {\left(160 \, a^{4} - 1533 \, a^{3} b + 1599 \, a^{2} b^{2} - 488 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(120 \, a^{4} - 1205 \, a^{3} b + 2769 \, a^{2} b^{2} - 1392 \, a b^{3} + 184 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(75 \, a^{3} b - 305 \, a^{2} b^{2} + 320 \, a b^{3} - 56 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(15 \, a^{4} - 40 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(15 \, a^{4} - 55 \, a^{3} b + 48 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(15 \, a^{4} - 100 \, a^{3} b + 183 \, a^{2} b^{2} - 72 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} - 40 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b - 55 \, a^{2} b^{2} + 48 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(15 \, a^{2} b^{2} - 40 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)}{240 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/480*(4*(64*a^4 - 607*a^3*b + 274*a^2*b^2)*cos(f*x + e)^9 - 4*(160*a^4 - 1533*a^3*b + 1599*a^2*b^2 - 488*a*b^3)*cos(f*x + e)^7 + 4*(120*a^4 - 1205*a^3*b + 2769*a^2*b^2 - 1392*a*b^3 + 184*b^4)*cos(f*x + e)^5 + 20*(75*a^3*b - 305*a^2*b^2 + 320*a*b^3 - 56*b^4)*cos(f*x + e)^3 - 15*((15*a^4 - 40*a^3*b + 8*a^2*b^2)*cos(f*x + e)^8 - 2*(15*a^4 - 55*a^3*b + 48*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 + (15*a^4 - 100*a^3*b + 183*a^2*b^2 - 72*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 - 40*a*b^3 + 8*b^4 + 2*(15*a^3*b - 55*a^2*b^2 + 48*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(15*a^2*b^2 - 40*a*b^3 + 8*b^4)*cos(f*x + e))/(((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f)*sin(f*x + e)), -1/240*(2*(64*a^4 - 607*a^3*b + 274*a^2*b^2)*cos(f*x + e)^9 - 2*(160*a^4 - 1533*a^3*b + 1599*a^2*b^2 - 488*a*b^3)*cos(f*x + e)^7 + 2*(120*a^4 - 1205*a^3*b + 2769*a^2*b^2 - 1392*a*b^3 + 184*b^4)*cos(f*x + e)^5 + 10*(75*a^3*b - 305*a^2*b^2 + 320*a*b^3 - 56*b^4)*cos(f*x + e)^3 - 15*((15*a^4 - 40*a^3*b + 8*a^2*b^2)*cos(f*x + e)^8 - 2*(15*a^4 - 55*a^3*b + 48*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 + (15*a^4 - 100*a^3*b + 183*a^2*b^2 - 72*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 - 40*a*b^3 + 8*b^4 + 2*(15*a^3*b - 55*a^2*b^2 + 48*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(15*a^2*b^2 - 40*a*b^3 + 8*b^4)*cos(f*x + e))/(((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f)*sin(f*x + e))]","B",0
67,1,295,0,0.876618," ","integrate(sin(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{15 \, a^{2} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(3 \, a^{2} \cos\left(f x + e\right)^{5} - {\left(10 \, a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{30 \, a^{2} f}, -\frac{15 \, a^{2} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left(3 \, a^{2} \cos\left(f x + e\right)^{5} - {\left(10 \, a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, a^{2} f}\right]"," ",0,"[1/30*(15*a^2*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - 2*(3*a^2*cos(f*x + e)^5 - (10*a^2 - a*b)*cos(f*x + e)^3 + (15*a^2 - 10*a*b - 2*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f), -1/15*(15*a^2*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + (3*a^2*cos(f*x + e)^5 - (10*a^2 - a*b)*cos(f*x + e)^3 + (15*a^2 - 10*a*b - 2*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f)]","A",0
68,1,232,0,0.846117," ","integrate(sin(f*x+e)^3*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(a \cos\left(f x + e\right)^{3} - {\left(3 \, a - b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, a f}, -\frac{3 \, a \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - {\left(a \cos\left(f x + e\right)^{3} - {\left(3 \, a - b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, a f}\right]"," ",0,"[1/6*(3*a*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(a*cos(f*x + e)^3 - (3*a - b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f), -1/3*(3*a*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - (a*cos(f*x + e)^3 - (3*a - b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f)]","A",0
69,1,182,0,0.836319," ","integrate(sin(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, f}, -\frac{\sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{f}\right]"," ",0,"[-1/2*(2*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/f, -(sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/f]","A",0
70,1,496,0,0.784964," ","integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, f}, \frac{2 \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, f}, -\frac{2 \, \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, f}, \frac{\sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right)}{f}\right]"," ",0,"[1/2*(sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/f, 1/2*(2*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/f, -1/2*(2*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)))/f, (sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b))/f]","A",0
71,1,867,0,0.719687," ","integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - a - 2 \, b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{4 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}, \frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - a - 2 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}, -\frac{4 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - 2 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - a - 2 \, b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{4 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}, \frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - a - 2 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - 2 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{2 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}\right]"," ",0,"[1/4*(2*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + ((a + 2*b)*cos(f*x + e)^2 - a - 2*b)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + 2*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f), 1/2*(((a + 2*b)*cos(f*x + e)^2 - a - 2*b)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f), -1/4*(4*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - 2*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - ((a + 2*b)*cos(f*x + e)^2 - a - 2*b)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f), 1/2*(((a + 2*b)*cos(f*x + e)^2 - a - 2*b)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - 2*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + (a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f)]","A",0
72,1,1476,0,1.296780," ","integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 8 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(3 \, a^{2} + 7 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, \frac{{\left({\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + 4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + {\left({\left(3 \, a^{2} + 7 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, -\frac{16 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - {\left({\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(3 \, a^{2} + 7 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, \frac{{\left({\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - 8 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left({\left(3 \, a^{2} + 7 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}\right]"," ",0,"[1/16*(((3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 + 12*a*b + 8*b^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + 8*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((3*a^2 + 7*a*b + 4*b^2)*cos(f*x + e)^3 - (5*a^2 + 11*a*b + 6*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f), 1/8*(((3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 + 12*a*b + 8*b^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + 4*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + ((3*a^2 + 7*a*b + 4*b^2)*cos(f*x + e)^3 - (5*a^2 + 11*a*b + 6*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f), -1/16*(16*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - ((3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 + 12*a*b + 8*b^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*((3*a^2 + 7*a*b + 4*b^2)*cos(f*x + e)^3 - (5*a^2 + 11*a*b + 6*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f), 1/8*(((3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 + 12*a*b + 8*b^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - 8*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + ((3*a^2 + 7*a*b + 4*b^2)*cos(f*x + e)^3 - (5*a^2 + 11*a*b + 6*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f)]","B",0
73,1,1715,0,4.909025," ","integrate(sin(f*x+e)^6*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{96 \, a^{3} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 3 \, {\left(5 \, a^{3} - 15 \, a^{2} b - 5 \, a b^{2} - b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(33 \, a^{3} - 14 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{3} f}, \frac{192 \, a^{3} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + 3 \, {\left(5 \, a^{3} - 15 \, a^{2} b - 5 \, a b^{2} - b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(33 \, a^{3} - 14 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{3} f}, \frac{48 \, a^{3} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 3 \, {\left(5 \, a^{3} - 15 \, a^{2} b - 5 \, a b^{2} - b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(33 \, a^{3} - 14 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{3} f}, \frac{96 \, a^{3} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - 3 \, {\left(5 \, a^{3} - 15 \, a^{2} b - 5 \, a b^{2} - b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(33 \, a^{3} - 14 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{3} f}\right]"," ",0,"[1/384*(96*a^3*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 3*(5*a^3 - 15*a^2*b - 5*a*b^2 - b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(8*a^3*cos(f*x + e)^5 - 2*(13*a^3 - a^2*b)*cos(f*x + e)^3 + (33*a^3 - 14*a^2*b - 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f), 1/384*(192*a^3*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + 3*(5*a^3 - 15*a^2*b - 5*a*b^2 - b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(8*a^3*cos(f*x + e)^5 - 2*(13*a^3 - a^2*b)*cos(f*x + e)^3 + (33*a^3 - 14*a^2*b - 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f), 1/192*(48*a^3*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 3*(5*a^3 - 15*a^2*b - 5*a*b^2 - b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*a^3*cos(f*x + e)^5 - 2*(13*a^3 - a^2*b)*cos(f*x + e)^3 + (33*a^3 - 14*a^2*b - 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f), 1/192*(96*a^3*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - 3*(5*a^3 - 15*a^2*b - 5*a*b^2 - b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*a^3*cos(f*x + e)^5 - 2*(13*a^3 - a^2*b)*cos(f*x + e)^3 + (33*a^3 - 14*a^2*b - 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f)]","A",0
74,1,1565,0,1.833533," ","integrate(sin(f*x+e)^4*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{16 \, a^{2} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a^{2} f}, \frac{32 \, a^{2} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a^{2} f}, \frac{8 \, a^{2} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a^{2} f}, \frac{16 \, a^{2} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a^{2} f}\right]"," ",0,"[1/64*(16*a^2*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + (3*a^2 - 6*a*b - b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*a^2*cos(f*x + e)^3 - (5*a^2 - a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f), 1/64*(32*a^2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + (3*a^2 - 6*a*b - b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*a^2*cos(f*x + e)^3 - (5*a^2 - a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f), 1/32*(8*a^2*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - (3*a^2 - 6*a*b - b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*a^2*cos(f*x + e)^3 - (5*a^2 - a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f), 1/32*(16*a^2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - (3*a^2 - 6*a*b - b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*a^2*cos(f*x + e)^3 - (5*a^2 - a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f)]","A",0
75,1,1417,0,1.086786," ","integrate(sin(f*x+e)^2*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - \sqrt{-a} {\left(a - b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 4 \, a \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{16 \, a f}, -\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - 8 \, a \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{-a} {\left(a - b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, a f}, -\frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a - b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, a \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, a f}, -\frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a - b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, a \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, a f}\right]"," ",0,"[-1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - sqrt(-a)*(a - b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 4*a*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/(a*f), -1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - 8*a*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - sqrt(-a)*(a - b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a*f), -1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a - b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 2*a*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/(a*f), -1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a - b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*a*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/(a*f)]","B",0
76,1,1227,0,0.907388," ","integrate((a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 2 \, \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, f}, \frac{4 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, f}\right]"," ",0,"[1/8*(sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 2*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, 1/8*(4*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/f]","B",0
77,1,306,0,0.669495," ","integrate(csc(f*x+e)^2*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{4 \, f \sin\left(f x + e\right)}, \frac{\sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{2 \, f \sin\left(f x + e\right)}\right]"," ",0,"[1/4*(sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e)), 1/2*(sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) - 2*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e))]","B",0
78,1,436,0,1.039538," ","integrate(csc(f*x+e)^4*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(2 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)} \sin\left(f x + e\right)}, \frac{3 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, {\left({\left(2 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/12*(3*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((2*a + 3*b)*cos(f*x + e)^3 - (3*a + 4*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^2 - (a + b)*f)*sin(f*x + e)), 1/6*(3*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) - 2*((2*a + 3*b)*cos(f*x + e)^3 - (3*a + 4*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^2 - (a + b)*f)*sin(f*x + e))]","B",0
79,1,656,0,2.469581," ","integrate(csc(f*x+e)^6*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(8 \, a^{2} + 25 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(20 \, a^{2} + 59 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, {\left({\left(8 \, a^{2} + 25 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(20 \, a^{2} + 59 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 40 \, a b + 23 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{30 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/60*(15*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((8*a^2 + 25*a*b + 15*b^2)*cos(f*x + e)^5 - (20*a^2 + 59*a*b + 35*b^2)*cos(f*x + e)^3 + (15*a^2 + 40*a*b + 23*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f)*sin(f*x + e)), 1/30*(15*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) - 2*((8*a^2 + 25*a*b + 15*b^2)*cos(f*x + e)^5 - (20*a^2 + 59*a*b + 35*b^2)*cos(f*x + e)^3 + (15*a^2 + 40*a*b + 23*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f)*sin(f*x + e))]","B",0
80,1,351,0,1.282733," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*sin(f*x+e)^5,x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(3 \, a^{2} - 4 \, a b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(6 \, a^{2} \cos\left(f x + e\right)^{6} - 4 \, {\left(5 \, a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(15 \, a^{2} - 40 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, a f \cos\left(f x + e\right)}, -\frac{15 \, {\left(3 \, a^{2} - 4 \, a b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) + {\left(6 \, a^{2} \cos\left(f x + e\right)^{6} - 4 \, {\left(5 \, a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(15 \, a^{2} - 40 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{30 \, a f \cos\left(f x + e\right)}\right]"," ",0,"[-1/60*(15*(3*a^2 - 4*a*b)*sqrt(b)*cos(f*x + e)*log((a*cos(f*x + e)^2 - 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(6*a^2*cos(f*x + e)^6 - 4*(5*a^2 - 3*a*b)*cos(f*x + e)^4 + 2*(15*a^2 - 40*a*b + 3*b^2)*cos(f*x + e)^2 - 15*a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)), -1/30*(15*(3*a^2 - 4*a*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) + (6*a^2*cos(f*x + e)^6 - 4*(5*a^2 - 3*a*b)*cos(f*x + e)^4 + 2*(15*a^2 - 40*a*b + 3*b^2)*cos(f*x + e)^2 - 15*a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e))]","A",0
81,1,278,0,0.850494," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*sin(f*x+e)^3,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(3 \, a - 2 \, b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(2 \, a \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a - 4 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, f \cos\left(f x + e\right)}, -\frac{3 \, {\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) - {\left(2 \, a \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a - 4 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, f \cos\left(f x + e\right)}\right]"," ",0,"[-1/12*(3*(3*a - 2*b)*sqrt(b)*cos(f*x + e)*log((a*cos(f*x + e)^2 - 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - 2*(2*a*cos(f*x + e)^4 - 2*(3*a - 4*b)*cos(f*x + e)^2 + 3*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), -1/6*(3*(3*a - 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) - (2*a*cos(f*x + e)^4 - 2*(3*a - 4*b)*cos(f*x + e)^2 + 3*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e))]","A",0
82,1,231,0,0.873944," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*sin(f*x+e),x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(2 \, a \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right)}, -\frac{3 \, a \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) + {\left(2 \, a \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, f \cos\left(f x + e\right)}\right]"," ",0,"[1/4*(3*a*sqrt(b)*cos(f*x + e)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - 2*(2*a*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), -1/2*(3*a*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) + (2*a*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e))]","A",0
83,1,715,0,0.859933," ","integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a + b\right)}^{\frac{3}{2}} \cos\left(f x + e\right) \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + {\left(3 \, a + 2 \, b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right)}, \frac{4 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) \cos\left(f x + e\right) + {\left(3 \, a + 2 \, b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right)}, -\frac{{\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) - {\left(a + b\right)}^{\frac{3}{2}} \cos\left(f x + e\right) \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, f \cos\left(f x + e\right)}, \frac{2 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) \cos\left(f x + e\right) - {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) + b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, f \cos\left(f x + e\right)}\right]"," ",0,"[1/4*(2*(a + b)^(3/2)*cos(f*x + e)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + (3*a + 2*b)*sqrt(b)*cos(f*x + e)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), 1/4*(4*(a + b)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b))*cos(f*x + e) + (3*a + 2*b)*sqrt(b)*cos(f*x + e)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), -1/2*((3*a + 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) - (a + b)^(3/2)*cos(f*x + e)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), 1/2*(2*(a + b)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b))*cos(f*x + e) - (3*a + 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) + b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e))]","A",0
84,1,984,0,1.002797," ","integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + {\left({\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}, \frac{2 \, {\left({\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left({\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}, -\frac{2 \, {\left({\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - {\left({\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}, \frac{{\left({\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - {\left({\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*(((a + 4*b)*cos(f*x + e)^3 - (a + 4*b)*cos(f*x + e))*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + ((3*a + 4*b)*cos(f*x + e)^3 - (3*a + 4*b)*cos(f*x + e))*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), 1/4*(2*((a + 4*b)*cos(f*x + e)^3 - (a + 4*b)*cos(f*x + e))*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + ((3*a + 4*b)*cos(f*x + e)^3 - (3*a + 4*b)*cos(f*x + e))*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), -1/4*(2*((3*a + 4*b)*cos(f*x + e)^3 - (3*a + 4*b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - ((a + 4*b)*cos(f*x + e)^3 - (a + 4*b)*cos(f*x + e))*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), 1/2*(((a + 4*b)*cos(f*x + e)^3 - (a + 4*b)*cos(f*x + e))*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - ((3*a + 4*b)*cos(f*x + e)^3 - (3*a + 4*b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + ((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e))]","A",0
85,1,1511,0,0.913696," ","integrate(csc(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 12 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(3 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 23 \, a b + 18 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b + 4 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{5} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{3} + {\left(a + b\right)} f \cos\left(f x + e\right)\right)}}, \frac{3 \, {\left({\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + 6 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + {\left(3 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 23 \, a b + 18 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b + 4 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{5} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{3} + {\left(a + b\right)} f \cos\left(f x + e\right)\right)}}, -\frac{24 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - 3 \, {\left({\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 23 \, a b + 18 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b + 4 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{5} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{3} + {\left(a + b\right)} f \cos\left(f x + e\right)\right)}}, \frac{3 \, {\left({\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - 12 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left(3 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 23 \, a b + 18 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b + 4 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{5} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{3} + {\left(a + b\right)} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/16*(3*((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^5 - 2*(a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^3 + (a^2 + 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + 12*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^5 - 2*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 + 3*a*b + 2*b^2)*cos(f*x + e))*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(3*(a^2 + 5*a*b + 4*b^2)*cos(f*x + e)^4 - (5*a^2 + 23*a*b + 18*b^2)*cos(f*x + e)^2 + 4*a*b + 4*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^5 - 2*(a + b)*f*cos(f*x + e)^3 + (a + b)*f*cos(f*x + e)), 1/8*(3*((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^5 - 2*(a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^3 + (a^2 + 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + 6*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^5 - 2*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 + 3*a*b + 2*b^2)*cos(f*x + e))*sqrt(b)*log((a*cos(f*x + e)^2 + 2*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + (3*(a^2 + 5*a*b + 4*b^2)*cos(f*x + e)^4 - (5*a^2 + 23*a*b + 18*b^2)*cos(f*x + e)^2 + 4*a*b + 4*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^5 - 2*(a + b)*f*cos(f*x + e)^3 + (a + b)*f*cos(f*x + e)), -1/16*(24*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^5 - 2*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 + 3*a*b + 2*b^2)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - 3*((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^5 - 2*(a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^3 + (a^2 + 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*(3*(a^2 + 5*a*b + 4*b^2)*cos(f*x + e)^4 - (5*a^2 + 23*a*b + 18*b^2)*cos(f*x + e)^2 + 4*a*b + 4*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^5 - 2*(a + b)*f*cos(f*x + e)^3 + (a + b)*f*cos(f*x + e)), 1/8*(3*((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^5 - 2*(a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^3 + (a^2 + 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - 12*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^5 - 2*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 + 3*a*b + 2*b^2)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + (3*(a^2 + 5*a*b + 4*b^2)*cos(f*x + e)^4 - (5*a^2 + 23*a*b + 18*b^2)*cos(f*x + e)^2 + 4*a*b + 4*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^5 - 2*(a + b)*f*cos(f*x + e)^3 + (a + b)*f*cos(f*x + e))]","A",0
86,1,1855,0,17.852498," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*sin(f*x+e)^6,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(5 \, a^{3} - 45 \, a^{2} b + 15 \, a b^{2} + b^{3}\right)} \sqrt{-a} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 48 \, {\left(3 \, a^{3} - 5 \, a^{2} b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{6} - 2 \, {\left(13 \, a^{3} - 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} - 24 \, a^{2} b + {\left(33 \, a^{3} - 68 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{2} f \cos\left(f x + e\right)}, \frac{96 \, {\left(3 \, a^{3} - 5 \, a^{2} b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 3 \, {\left(5 \, a^{3} - 45 \, a^{2} b + 15 \, a b^{2} + b^{3}\right)} \sqrt{-a} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{6} - 2 \, {\left(13 \, a^{3} - 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} - 24 \, a^{2} b + {\left(33 \, a^{3} - 68 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{2} f \cos\left(f x + e\right)}, -\frac{3 \, {\left(5 \, a^{3} - 45 \, a^{2} b + 15 \, a b^{2} + b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 24 \, {\left(3 \, a^{3} - 5 \, a^{2} b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{6} - 2 \, {\left(13 \, a^{3} - 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} - 24 \, a^{2} b + {\left(33 \, a^{3} - 68 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{2} f \cos\left(f x + e\right)}, -\frac{3 \, {\left(5 \, a^{3} - 45 \, a^{2} b + 15 \, a b^{2} + b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 48 \, {\left(3 \, a^{3} - 5 \, a^{2} b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{6} - 2 \, {\left(13 \, a^{3} - 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} - 24 \, a^{2} b + {\left(33 \, a^{3} - 68 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{2} f \cos\left(f x + e\right)}\right]"," ",0,"[-1/384*(3*(5*a^3 - 45*a^2*b + 15*a*b^2 + b^3)*sqrt(-a)*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 48*(3*a^3 - 5*a^2*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 8*(8*a^3*cos(f*x + e)^6 - 2*(13*a^3 - 7*a^2*b)*cos(f*x + e)^4 - 24*a^2*b + (33*a^3 - 68*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f*cos(f*x + e)), 1/384*(96*(3*a^3 - 5*a^2*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 3*(5*a^3 - 45*a^2*b + 15*a*b^2 + b^3)*sqrt(-a)*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(8*a^3*cos(f*x + e)^6 - 2*(13*a^3 - 7*a^2*b)*cos(f*x + e)^4 - 24*a^2*b + (33*a^3 - 68*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f*cos(f*x + e)), -1/192*(3*(5*a^3 - 45*a^2*b + 15*a*b^2 + b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) + 24*(3*a^3 - 5*a^2*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*(8*a^3*cos(f*x + e)^6 - 2*(13*a^3 - 7*a^2*b)*cos(f*x + e)^4 - 24*a^2*b + (33*a^3 - 68*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f*cos(f*x + e)), -1/192*(3*(5*a^3 - 45*a^2*b + 15*a*b^2 + b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - 48*(3*a^3 - 5*a^2*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + 4*(8*a^3*cos(f*x + e)^6 - 2*(13*a^3 - 7*a^2*b)*cos(f*x + e)^4 - 24*a^2*b + (33*a^3 - 68*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f*cos(f*x + e))]","A",0
87,1,1667,0,6.160775," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*sin(f*x+e)^4,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} - 6 \, a b + b^{2}\right)} \sqrt{-a} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 24 \, {\left(a^{2} - a b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a f \cos\left(f x + e\right)}, \frac{48 \, {\left(a^{2} - a b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 3 \, {\left(a^{2} - 6 \, a b + b^{2}\right)} \sqrt{-a} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a f \cos\left(f x + e\right)}, -\frac{3 \, {\left(a^{2} - 6 \, a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 12 \, {\left(a^{2} - a b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a f \cos\left(f x + e\right)}, -\frac{3 \, {\left(a^{2} - 6 \, a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 24 \, {\left(a^{2} - a b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a f \cos\left(f x + e\right)}\right]"," ",0,"[-1/64*(3*(a^2 - 6*a*b + b^2)*sqrt(-a)*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 24*(a^2 - a*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 8*(2*a^2*cos(f*x + e)^4 - 5*(a^2 - a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f*cos(f*x + e)), 1/64*(48*(a^2 - a*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 3*(a^2 - 6*a*b + b^2)*sqrt(-a)*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*a^2*cos(f*x + e)^4 - 5*(a^2 - a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f*cos(f*x + e)), -1/32*(3*(a^2 - 6*a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) + 12*(a^2 - a*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*(2*a^2*cos(f*x + e)^4 - 5*(a^2 - a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f*cos(f*x + e)), -1/32*(3*(a^2 - 6*a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - 24*(a^2 - a*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 4*(2*a^2*cos(f*x + e)^4 - 5*(a^2 - a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f*cos(f*x + e))]","A",0
88,1,1535,0,2.069147," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*sin(f*x+e)^2,x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} {\left(a - 3 \, b\right)} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 2 \, {\left(3 \, a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 8 \, {\left(a \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, f \cos\left(f x + e\right)}, \frac{4 \, {\left(3 \, a - b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - \sqrt{-a} {\left(a - 3 \, b\right)} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(a \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, f \cos\left(f x + e\right)}, -\frac{{\left(a - 3 \, b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + {\left(3 \, a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(a \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{{\left(a - 3 \, b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, {\left(3 \, a - b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 4 \, {\left(a \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}\right]"," ",0,"[-1/16*(sqrt(-a)*(a - 3*b)*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 2*(3*a - b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 8*(a*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), 1/16*(4*(3*a - b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - sqrt(-a)*(a - 3*b)*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(a*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/8*((a - 3*b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) + (3*a - b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*(a*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/8*((a - 3*b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - 2*(3*a - b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + 4*(a*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e))]","B",0
89,1,1457,0,1.397398," ","integrate((a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + {\left(3 \, a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, \frac{2 \, {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + \sqrt{-a} a \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{2 \, a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, f \cos\left(f x + e\right)}\right]"," ",0,"[1/8*(sqrt(-a)*a*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + (3*a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), 1/8*(2*(3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + sqrt(-a)*a*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/8*(2*a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/4*(a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e))]","B",0
90,1,370,0,1.287451," ","integrate(csc(f*x+e)^2*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(2 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, f \cos\left(f x + e\right) \sin\left(f x + e\right)}, \frac{3 \, {\left(a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, {\left({\left(2 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right) \sin\left(f x + e\right)}\right]"," ",0,"[1/8*(3*(a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((2*a + 3*b)*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)*sin(f*x + e)), 1/4*(3*(a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)*sin(f*x + e) - 2*((2*a + 3*b)*cos(f*x + e)^2 - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)*sin(f*x + e))]","A",0
91,1,472,0,2.526416," ","integrate(csc(f*x+e)^4*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(3 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 5 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(4 \, a + 15 \, b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a + 10 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}, \frac{3 \, {\left({\left(3 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 5 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, {\left({\left(4 \, a + 15 \, b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a + 10 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/24*(3*((3*a + 5*b)*cos(f*x + e)^3 - (3*a + 5*b)*cos(f*x + e))*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((4*a + 15*b)*cos(f*x + e)^4 - 2*(3*a + 10*b)*cos(f*x + e)^2 + 3*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e)), 1/12*(3*((3*a + 5*b)*cos(f*x + e)^3 - (3*a + 5*b)*cos(f*x + e))*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) - 2*((4*a + 15*b)*cos(f*x + e)^4 - 2*(3*a + 10*b)*cos(f*x + e)^2 + 3*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e))]","A",0
92,1,682,0,10.591512," ","integrate(csc(f*x+e)^6*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left({\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(16 \, a^{2} + 115 \, a b + 105 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(40 \, a^{2} + 273 \, a b + 245 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(30 \, a^{2} + 185 \, a b + 161 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b - 15 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{5} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{3} + {\left(a + b\right)} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, {\left({\left(16 \, a^{2} + 115 \, a b + 105 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(40 \, a^{2} + 273 \, a b + 245 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(30 \, a^{2} + 185 \, a b + 161 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b - 15 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{5} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{3} + {\left(a + b\right)} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/120*(15*((3*a^2 + 10*a*b + 7*b^2)*cos(f*x + e)^5 - 2*(3*a^2 + 10*a*b + 7*b^2)*cos(f*x + e)^3 + (3*a^2 + 10*a*b + 7*b^2)*cos(f*x + e))*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((16*a^2 + 115*a*b + 105*b^2)*cos(f*x + e)^6 - (40*a^2 + 273*a*b + 245*b^2)*cos(f*x + e)^4 + (30*a^2 + 185*a*b + 161*b^2)*cos(f*x + e)^2 - 15*a*b - 15*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^5 - 2*(a + b)*f*cos(f*x + e)^3 + (a + b)*f*cos(f*x + e))*sin(f*x + e)), 1/60*(15*((3*a^2 + 10*a*b + 7*b^2)*cos(f*x + e)^5 - 2*(3*a^2 + 10*a*b + 7*b^2)*cos(f*x + e)^3 + (3*a^2 + 10*a*b + 7*b^2)*cos(f*x + e))*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) - 2*((16*a^2 + 115*a*b + 105*b^2)*cos(f*x + e)^6 - (40*a^2 + 273*a*b + 245*b^2)*cos(f*x + e)^4 + (30*a^2 + 185*a*b + 161*b^2)*cos(f*x + e)^2 - 15*a*b - 15*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^5 - 2*(a + b)*f*cos(f*x + e)^3 + (a + b)*f*cos(f*x + e))*sin(f*x + e))]","A",0
93,1,87,0,0.544400," ","integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} \cos\left(f x + e\right)^{5} - 2 \, {\left(5 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 20 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, a^{3} f}"," ",0,"-1/15*(3*a^2*cos(f*x + e)^5 - 2*(5*a^2 + 2*a*b)*cos(f*x + e)^3 + (15*a^2 + 20*a*b + 8*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^3*f)","A",0
94,1,57,0,0.502285," ","integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(a \cos\left(f x + e\right)^{3} - {\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, a^{2} f}"," ",0,"1/3*(a*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^2*f)","A",0
95,1,37,0,0.514360," ","integrate(sin(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a f}"," ",0,"-sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a*f)","A",0
96,1,140,0,0.627848," ","integrate(csc(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, \sqrt{a + b} f}, \frac{\sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right)}{{\left(a + b\right)} f}\right]"," ",0,"[1/2*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1))/(sqrt(a + b)*f), sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b))/((a + b)*f)]","A",0
97,1,305,0,0.619396," ","integrate(csc(f*x+e)^3/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, \frac{{\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + (a*cos(f*x + e)^2 - a)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f), 1/2*((a*cos(f*x + e)^2 - a)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)]","A",0
98,1,491,0,0.704575," ","integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(3 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 7 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)}}, \frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left(3 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 7 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)}}\right]"," ",0,"[1/16*(3*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + 2*(3*(a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + 7*a*b + 2*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f), 1/8*(3*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (3*(a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + 7*a*b + 2*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f)]","A",0
99,1,639,0,2.243872," ","integrate(sin(f*x+e)^6/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} + 5 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(33 \, a^{3} + 40 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{4} f}, -\frac{15 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} + 5 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(33 \, a^{3} + 40 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{4} f}\right]"," ",0,"[-1/384*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*a^3*cos(f*x + e)^5 - 2*(13*a^3 + 5*a^2*b)*cos(f*x + e)^3 + (33*a^3 + 40*a^2*b + 15*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^4*f), -1/192*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(8*a^3*cos(f*x + e)^5 - 2*(13*a^3 + 5*a^2*b)*cos(f*x + e)^3 + (33*a^3 + 40*a^2*b + 15*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^4*f)]","A",0
100,1,565,0,0.944871," ","integrate(sin(f*x+e)^4/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a^{3} f}, -\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a^{3} f}\right]"," ",0,"[-1/64*(3*(a^2 + 2*a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*a^2*cos(f*x + e)^3 - (5*a^2 + 3*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f), -1/32*(3*(a^2 + 2*a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*a^2*cos(f*x + e)^3 - (5*a^2 + 3*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f)]","A",0
101,1,497,0,0.713045," ","integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{-a} {\left(a + b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, a^{2} f}, -\frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a + b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, a^{2} f}\right]"," ",0,"[-1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + sqrt(-a)*(a + b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a^2*f), -1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a + b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/(a^2*f)]","B",0
102,1,408,0,0.597665," ","integrate(1/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, a f}, -\frac{\arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, \sqrt{a} f}\right]"," ",0,"[-1/8*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f), -1/4*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))/(sqrt(a)*f)]","B",0
103,1,47,0,0.666319," ","integrate(csc(f*x+e)^2/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{{\left(a + b\right)} f \sin\left(f x + e\right)}"," ",0,"-sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/((a + b)*f*sin(f*x + e))","A",0
104,1,96,0,0.551233," ","integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, a \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*a*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)*sin(f*x + e))","A",0
105,1,171,0,1.151907," ","integrate(csc(f*x+e)^6/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 4 \, {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*a^2*cos(f*x + e)^5 - 4*(5*a^2 + a*b)*cos(f*x + e)^3 + (15*a^2 + 10*a*b + 3*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f)*sin(f*x + e))","A",0
106,1,136,0,0.750346," ","integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{7} - 2 \, {\left(5 \, a^{3} + 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{3} + 40 \, a^{2} b + 24 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(15 \, a^{2} b + 40 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}"," ",0,"-1/15*(3*a^3*cos(f*x + e)^7 - 2*(5*a^3 + 3*a^2*b)*cos(f*x + e)^5 + (15*a^3 + 40*a^2*b + 24*a*b^2)*cos(f*x + e)^3 + 2*(15*a^2*b + 40*a*b^2 + 24*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^5*f*cos(f*x + e)^2 + a^4*b*f)","A",0
107,1,98,0,0.560516," ","integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(a^{2} \cos\left(f x + e\right)^{5} - {\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}"," ",0,"1/3*(a^2*cos(f*x + e)^5 - (3*a^2 + 4*a*b)*cos(f*x + e)^3 - 2*(3*a*b + 4*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^4*f*cos(f*x + e)^2 + a^3*b*f)","A",0
108,1,67,0,0.506532," ","integrate(sin(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(a \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f}"," ",0,"-(a*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^3*f*cos(f*x + e)^2 + a^2*b*f)","A",0
109,1,344,0,0.667499," ","integrate(csc(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a b + b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - {\left(a^{2} \cos\left(f x + e\right)^{2} + a b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, \frac{{\left(a^{2} \cos\left(f x + e\right)^{2} + a b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - {\left(a b + b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{{\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f}\right]"," ",0,"[-1/2*(2*(a*b + b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - (a^2*cos(f*x + e)^2 + a*b)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)))/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b + 2*a^2*b^2 + a*b^3)*f), ((a^2*cos(f*x + e)^2 + a*b)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - (a*b + b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b + 2*a^2*b^2 + a*b^3)*f)]","B",0
110,1,547,0,0.732290," ","integrate(csc(f*x+e)^3/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 2 \, b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)}}, \frac{{\left({\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 2 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left({\left(a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)}}\right]"," ",0,"[-1/4*(((a^2 - 2*a*b)*cos(f*x + e)^4 - (a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 - a*b + 2*b^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 + 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*((a^2 - a*b - 2*b^2)*cos(f*x + e)^3 + 3*(a*b + b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f), 1/2*(((a^2 - 2*a*b)*cos(f*x + e)^4 - (a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 - a*b + 2*b^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + ((a^2 - a*b - 2*b^2)*cos(f*x + e)^3 + 3*(a*b + b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f)]","B",0
111,1,873,0,1.039779," ","integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{3} - 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 9 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 4 \, a b^{2} + {\left(a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} - 16 \, a^{2} b - 17 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b + 11 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{3} - 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 9 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 4 \, a b^{2} + {\left(a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left(3 \, {\left(a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} - 16 \, a^{2} b - 17 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b + 11 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[-1/16*(3*((a^3 - 4*a^2*b)*cos(f*x + e)^6 - (2*a^3 - 9*a^2*b + 4*a*b^2)*cos(f*x + e)^4 + a^2*b - 4*a*b^2 + (a^3 - 6*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 + 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*(3*(a^3 - 3*a^2*b - 4*a*b^2)*cos(f*x + e)^5 - (5*a^3 - 16*a^2*b - 17*a*b^2 + 4*b^3)*cos(f*x + e)^3 - (13*a^2*b + 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f), 1/8*(3*((a^3 - 4*a^2*b)*cos(f*x + e)^6 - (2*a^3 - 9*a^2*b + 4*a*b^2)*cos(f*x + e)^4 + a^2*b - 4*a*b^2 + (a^3 - 6*a^2*b + 8*a*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (3*(a^3 - 3*a^2*b - 4*a*b^2)*cos(f*x + e)^5 - (5*a^3 - 16*a^2*b - 17*a*b^2 + 4*b^3)*cos(f*x + e)^3 - (13*a^2*b + 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)]","B",0
112,1,813,0,10.051246," ","integrate(sin(f*x+e)^6/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{3} b + 9 \, a^{2} b^{2} + 15 \, a b^{3} + 7 \, b^{4} + {\left(a^{4} + 9 \, a^{3} b + 15 \, a^{2} b^{2} + 7 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, a^{4} \cos\left(f x + e\right)^{7} - 2 \, {\left(13 \, a^{4} + 7 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(33 \, a^{4} + 68 \, a^{3} b + 35 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(81 \, a^{3} b + 190 \, a^{2} b^{2} + 105 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, {\left(a^{6} f \cos\left(f x + e\right)^{2} + a^{5} b f\right)}}, -\frac{15 \, {\left(a^{3} b + 9 \, a^{2} b^{2} + 15 \, a b^{3} + 7 \, b^{4} + {\left(a^{4} + 9 \, a^{3} b + 15 \, a^{2} b^{2} + 7 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(8 \, a^{4} \cos\left(f x + e\right)^{7} - 2 \, {\left(13 \, a^{4} + 7 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(33 \, a^{4} + 68 \, a^{3} b + 35 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(81 \, a^{3} b + 190 \, a^{2} b^{2} + 105 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, {\left(a^{6} f \cos\left(f x + e\right)^{2} + a^{5} b f\right)}}\right]"," ",0,"[-1/384*(15*(a^3*b + 9*a^2*b^2 + 15*a*b^3 + 7*b^4 + (a^4 + 9*a^3*b + 15*a^2*b^2 + 7*a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*a^4*cos(f*x + e)^7 - 2*(13*a^4 + 7*a^3*b)*cos(f*x + e)^5 + (33*a^4 + 68*a^3*b + 35*a^2*b^2)*cos(f*x + e)^3 + (81*a^3*b + 190*a^2*b^2 + 105*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^6*f*cos(f*x + e)^2 + a^5*b*f), -1/192*(15*(a^3*b + 9*a^2*b^2 + 15*a*b^3 + 7*b^4 + (a^4 + 9*a^3*b + 15*a^2*b^2 + 7*a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(8*a^4*cos(f*x + e)^7 - 2*(13*a^4 + 7*a^3*b)*cos(f*x + e)^5 + (33*a^4 + 68*a^3*b + 35*a^2*b^2)*cos(f*x + e)^3 + (81*a^3*b + 190*a^2*b^2 + 105*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^6*f*cos(f*x + e)^2 + a^5*b*f)]","A",0
113,1,703,0,3.210574," ","integrate(sin(f*x+e)^4/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} b + 6 \, a b^{2} + 5 \, b^{3} + {\left(a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{5} - 5 \, {\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}, -\frac{3 \, {\left(a^{2} b + 6 \, a b^{2} + 5 \, b^{3} + {\left(a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{5} - 5 \, {\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left(a^{5} f \cos\left(f x + e\right)^{2} + a^{4} b f\right)}}\right]"," ",0,"[-1/64*(3*(a^2*b + 6*a*b^2 + 5*b^3 + (a^3 + 6*a^2*b + 5*a*b^2)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*a^3*cos(f*x + e)^5 - 5*(a^3 + a^2*b)*cos(f*x + e)^3 - (13*a^2*b + 15*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*f*cos(f*x + e)^2 + a^4*b*f), -1/32*(3*(a^2*b + 6*a*b^2 + 5*b^3 + (a^3 + 6*a^2*b + 5*a*b^2)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*a^3*cos(f*x + e)^5 - 5*(a^3 + a^2*b)*cos(f*x + e)^3 - (13*a^2*b + 15*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*f*cos(f*x + e)^2 + a^4*b*f)]","A",0
114,1,607,0,1.088999," ","integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 3 \, b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(a^{2} \cos\left(f x + e\right)^{3} + 3 \, a b \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}, -\frac{{\left({\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 3 \, b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(a^{2} \cos\left(f x + e\right)^{3} + 3 \, a b \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left(a^{4} f \cos\left(f x + e\right)^{2} + a^{3} b f\right)}}\right]"," ",0,"[-1/16*(((a^2 + 3*a*b)*cos(f*x + e)^2 + a*b + 3*b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(a^2*cos(f*x + e)^3 + 3*a*b*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^4*f*cos(f*x + e)^2 + a^3*b*f), -1/8*(((a^2 + 3*a*b)*cos(f*x + e)^2 + a*b + 3*b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(a^2*cos(f*x + e)^3 + 3*a*b*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^4*f*cos(f*x + e)^2 + a^3*b*f)]","B",0
115,1,601,0,0.906652," ","integrate(1/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}, -\frac{4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), -1/4*(4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]","B",0
116,1,102,0,0.662990," ","integrate(csc(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{{\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3)*f)*sin(f*x + e))","A",0
117,1,189,0,1.463679," ","integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, {\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{5} - {\left(3 \, a^{2} - 10 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, a b - b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a^2 - 3*a*b)*cos(f*x + e)^5 - (3*a^2 - 10*a*b + 3*b^2)*cos(f*x + e)^3 - 2*(3*a*b - b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f)*sin(f*x + e))","A",0
118,1,314,0,4.672738," ","integrate(csc(f*x+e)^6/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, {\left(a^{3} - 5 \, a^{2} b\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(5 \, a^{3} - 26 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{3} - 85 \, a^{2} b + 49 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(15 \, a^{2} b - 10 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a^3 - 5*a^2*b)*cos(f*x + e)^7 - 4*(5*a^3 - 26*a^2*b + 5*a*b^2)*cos(f*x + e)^5 + (15*a^3 - 85*a^2*b + 49*a*b^2 + 5*b^3)*cos(f*x + e)^3 + 2*(15*a^2*b - 10*a*b^2 - b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f)*sin(f*x + e))","A",0
119,1,189,0,0.963524," ","integrate(sin(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{4} \cos\left(f x + e\right)^{9} - 2 \, {\left(5 \, a^{4} + 4 \, a^{3} b\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{4} + 20 \, a^{3} b + 16 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{5} + 12 \, {\left(5 \, a^{3} b + 20 \, a^{2} b^{2} + 16 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + 8 \, {\left(5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}"," ",0,"-1/15*(3*a^4*cos(f*x + e)^9 - 2*(5*a^4 + 4*a^3*b)*cos(f*x + e)^7 + 3*(5*a^4 + 20*a^3*b + 16*a^2*b^2)*cos(f*x + e)^5 + 12*(5*a^3*b + 20*a^2*b^2 + 16*a*b^3)*cos(f*x + e)^3 + 8*(5*a^2*b^2 + 20*a*b^3 + 16*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f)","A",0
120,1,138,0,0.709616," ","integrate(sin(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(a^{3} \cos\left(f x + e\right)^{7} - 3 \, {\left(a^{3} + 2 \, a^{2} b\right)} \cos\left(f x + e\right)^{5} - 12 \, {\left(a^{2} b + 2 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 8 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a^{6} f \cos\left(f x + e\right)^{4} + 2 \, a^{5} b f \cos\left(f x + e\right)^{2} + a^{4} b^{2} f\right)}}"," ",0,"1/3*(a^3*cos(f*x + e)^7 - 3*(a^3 + 2*a^2*b)*cos(f*x + e)^5 - 12*(a^2*b + 2*a*b^2)*cos(f*x + e)^3 - 8*(a*b^2 + 2*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^6*f*cos(f*x + e)^4 + 2*a^5*b*f*cos(f*x + e)^2 + a^4*b^2*f)","A",0
121,1,101,0,0.615631," ","integrate(sin(f*x+e)/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} \cos\left(f x + e\right)^{5} + 12 \, a b \cos\left(f x + e\right)^{3} + 8 \, b^{2} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}"," ",0,"-1/3*(3*a^2*cos(f*x + e)^5 + 12*a*b*cos(f*x + e)^3 + 8*b^2*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)","A",0
122,1,592,0,0.700165," ","integrate(csc(f*x+e)/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{4} \cos\left(f x + e\right)^{4} + 2 \, a^{3} b \cos\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left({\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, \frac{3 \, {\left(a^{4} \cos\left(f x + e\right)^{4} + 2 \, a^{3} b \cos\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) - {\left(3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[1/6*(3*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*(3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^3 + (5*a^2*b^2 + 7*a*b^3 + 2*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f), 1/3*(3*(a^4*cos(f*x + e)^4 + 2*a^3*b*cos(f*x + e)^2 + a^2*b^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) - (3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^3 + (5*a^2*b^2 + 7*a*b^3 + 2*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f)]","B",0
123,1,941,0,0.774486," ","integrate(csc(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} - 4 \, a^{3} b\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 6 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 4 \, a b^{3} - {\left(2 \, a^{3} b - 9 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(a^{4} - 3 \, a^{3} b - 4 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{3} b + 4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(13 \, a^{2} b^{2} + 11 \, a b^{3} - 2 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b + 7 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 2 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{4} - 4 \, a^{3} b\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 6 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 4 \, a b^{3} - {\left(2 \, a^{3} b - 9 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left(3 \, {\left(a^{4} - 3 \, a^{3} b - 4 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{3} b + 4 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(13 \, a^{2} b^{2} + 11 \, a b^{3} - 2 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b + 7 \, a^{5} b^{2} + 8 \, a^{4} b^{3} + 2 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6}\right)} f\right)}}\right]"," ",0,"[-1/12*(3*((a^4 - 4*a^3*b)*cos(f*x + e)^6 - (a^4 - 6*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 4*a*b^3 - (2*a^3*b - 9*a^2*b^2 + 4*a*b^3)*cos(f*x + e)^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 + 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) - 2*(3*(a^4 - 3*a^3*b - 4*a^2*b^2)*cos(f*x + e)^5 + 2*(9*a^3*b + 4*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^3 + (13*a^2*b^2 + 11*a*b^3 - 2*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^4 - (2*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 - (a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4 + 4*a^2*b^5 + a*b^6)*f), 1/6*(3*((a^4 - 4*a^3*b)*cos(f*x + e)^6 - (a^4 - 6*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 - a^2*b^2 + 4*a*b^3 - (2*a^3*b - 9*a^2*b^2 + 4*a*b^3)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (3*(a^4 - 3*a^3*b - 4*a^2*b^2)*cos(f*x + e)^5 + 2*(9*a^3*b + 4*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^3 + (13*a^2*b^2 + 11*a*b^3 - 2*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^4 - (2*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 - (a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4 + 4*a^2*b^5 + a*b^6)*f)]","B",0
124,1,1307,0,1.003633," ","integrate(csc(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(3 \, a^{4} - 24 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(3 \, a^{4} - 27 \, a^{3} b + 32 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(3 \, a^{4} - 36 \, a^{3} b + 107 \, a^{2} b^{2} - 56 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} - 24 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b - 27 \, a^{2} b^{2} + 32 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + 2 \, b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(3 \, {\left(3 \, a^{4} - 21 \, a^{3} b - 16 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} - {\left(15 \, a^{4} - 117 \, a^{3} b + 4 \, a^{2} b^{2} + 104 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{5} - {\left(78 \, a^{3} b - 71 \, a^{2} b^{2} - 61 \, a b^{3} + 88 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(11 \, a^{2} b^{2} + a b^{3} - 10 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{48 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)}}, \frac{3 \, {\left({\left(3 \, a^{4} - 24 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(3 \, a^{4} - 27 \, a^{3} b + 32 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(3 \, a^{4} - 36 \, a^{3} b + 107 \, a^{2} b^{2} - 56 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} - 24 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b - 27 \, a^{2} b^{2} + 32 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a + b}\right) + {\left(3 \, {\left(3 \, a^{4} - 21 \, a^{3} b - 16 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} - {\left(15 \, a^{4} - 117 \, a^{3} b + 4 \, a^{2} b^{2} + 104 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{5} - {\left(78 \, a^{3} b - 71 \, a^{2} b^{2} - 61 \, a b^{3} + 88 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(11 \, a^{2} b^{2} + a b^{3} - 10 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)}}\right]"," ",0,"[1/48*(3*((3*a^4 - 24*a^3*b + 8*a^2*b^2)*cos(f*x + e)^8 - 2*(3*a^4 - 27*a^3*b + 32*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 + (3*a^4 - 36*a^3*b + 107*a^2*b^2 - 56*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 24*a*b^3 + 8*b^4 + 2*(3*a^3*b - 27*a^2*b^2 + 32*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(a + b)*log(2*(a*cos(f*x + e)^2 - 2*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + 2*b)/(cos(f*x + e)^2 - 1)) + 2*(3*(3*a^4 - 21*a^3*b - 16*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 117*a^3*b + 4*a^2*b^2 + 104*a*b^3 - 32*b^4)*cos(f*x + e)^5 - (78*a^3*b - 71*a^2*b^2 - 61*a*b^3 + 88*b^4)*cos(f*x + e)^3 - 5*(11*a^2*b^2 + a*b^3 - 10*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f), 1/24*(3*((3*a^4 - 24*a^3*b + 8*a^2*b^2)*cos(f*x + e)^8 - 2*(3*a^4 - 27*a^3*b + 32*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^6 + (3*a^4 - 36*a^3*b + 107*a^2*b^2 - 56*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 24*a*b^3 + 8*b^4 + 2*(3*a^3*b - 27*a^2*b^2 + 32*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a + b)) + (3*(3*a^4 - 21*a^3*b - 16*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 117*a^3*b + 4*a^2*b^2 + 104*a*b^3 - 32*b^4)*cos(f*x + e)^5 - (78*a^3*b - 71*a^2*b^2 - 61*a*b^3 + 88*b^4)*cos(f*x + e)^3 - 5*(11*a^2*b^2 + a*b^3 - 10*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f)]","B",0
125,1,1003,0,39.594088," ","integrate(sin(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{3} b^{2} + 15 \, a^{2} b^{3} + 35 \, a b^{4} + 21 \, b^{5} + {\left(a^{5} + 15 \, a^{4} b + 35 \, a^{3} b^{2} + 21 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 15 \, a^{3} b^{2} + 35 \, a^{2} b^{3} + 21 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, a^{5} \cos\left(f x + e\right)^{9} - 2 \, {\left(13 \, a^{5} + 9 \, a^{4} b\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(11 \, a^{5} + 32 \, a^{4} b + 21 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(81 \, a^{4} b + 287 \, a^{3} b^{2} + 210 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(113 \, a^{3} b^{2} + 420 \, a^{2} b^{3} + 315 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, {\left(a^{8} f \cos\left(f x + e\right)^{4} + 2 \, a^{7} b f \cos\left(f x + e\right)^{2} + a^{6} b^{2} f\right)}}, -\frac{15 \, {\left(a^{3} b^{2} + 15 \, a^{2} b^{3} + 35 \, a b^{4} + 21 \, b^{5} + {\left(a^{5} + 15 \, a^{4} b + 35 \, a^{3} b^{2} + 21 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 15 \, a^{3} b^{2} + 35 \, a^{2} b^{3} + 21 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(8 \, a^{5} \cos\left(f x + e\right)^{9} - 2 \, {\left(13 \, a^{5} + 9 \, a^{4} b\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(11 \, a^{5} + 32 \, a^{4} b + 21 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(81 \, a^{4} b + 287 \, a^{3} b^{2} + 210 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(113 \, a^{3} b^{2} + 420 \, a^{2} b^{3} + 315 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, {\left(a^{8} f \cos\left(f x + e\right)^{4} + 2 \, a^{7} b f \cos\left(f x + e\right)^{2} + a^{6} b^{2} f\right)}}\right]"," ",0,"[-1/384*(15*(a^3*b^2 + 15*a^2*b^3 + 35*a*b^4 + 21*b^5 + (a^5 + 15*a^4*b + 35*a^3*b^2 + 21*a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 15*a^3*b^2 + 35*a^2*b^3 + 21*a*b^4)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*a^5*cos(f*x + e)^9 - 2*(13*a^5 + 9*a^4*b)*cos(f*x + e)^7 + 3*(11*a^5 + 32*a^4*b + 21*a^3*b^2)*cos(f*x + e)^5 + 2*(81*a^4*b + 287*a^3*b^2 + 210*a^2*b^3)*cos(f*x + e)^3 + (113*a^3*b^2 + 420*a^2*b^3 + 315*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^8*f*cos(f*x + e)^4 + 2*a^7*b*f*cos(f*x + e)^2 + a^6*b^2*f), -1/192*(15*(a^3*b^2 + 15*a^2*b^3 + 35*a*b^4 + 21*b^5 + (a^5 + 15*a^4*b + 35*a^3*b^2 + 21*a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 15*a^3*b^2 + 35*a^2*b^3 + 21*a*b^4)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(8*a^5*cos(f*x + e)^9 - 2*(13*a^5 + 9*a^4*b)*cos(f*x + e)^7 + 3*(11*a^5 + 32*a^4*b + 21*a^3*b^2)*cos(f*x + e)^5 + 2*(81*a^4*b + 287*a^3*b^2 + 210*a^2*b^3)*cos(f*x + e)^3 + (113*a^3*b^2 + 420*a^2*b^3 + 315*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^8*f*cos(f*x + e)^4 + 2*a^7*b*f*cos(f*x + e)^2 + a^6*b^2*f)]","A",0
126,1,873,0,12.091866," ","integrate(sin(f*x+e)^4/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(3 \, a^{4} + 30 \, a^{3} b + 35 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 30 \, a b^{3} + 35 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 30 \, a^{2} b^{2} + 35 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(6 \, a^{4} \cos\left(f x + e\right)^{7} - 3 \, {\left(5 \, a^{4} + 7 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(39 \, a^{3} b + 70 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(11 \, a^{2} b^{2} + 21 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}, -\frac{3 \, {\left({\left(3 \, a^{4} + 30 \, a^{3} b + 35 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 30 \, a b^{3} + 35 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 30 \, a^{2} b^{2} + 35 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(6 \, a^{4} \cos\left(f x + e\right)^{7} - 3 \, {\left(5 \, a^{4} + 7 \, a^{3} b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(39 \, a^{3} b + 70 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(11 \, a^{2} b^{2} + 21 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, {\left(a^{7} f \cos\left(f x + e\right)^{4} + 2 \, a^{6} b f \cos\left(f x + e\right)^{2} + a^{5} b^{2} f\right)}}\right]"," ",0,"[-1/192*(3*((3*a^4 + 30*a^3*b + 35*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 30*a*b^3 + 35*b^4 + 2*(3*a^3*b + 30*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(6*a^4*cos(f*x + e)^7 - 3*(5*a^4 + 7*a^3*b)*cos(f*x + e)^5 - 2*(39*a^3*b + 70*a^2*b^2)*cos(f*x + e)^3 - 5*(11*a^2*b^2 + 21*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f), -1/96*(3*((3*a^4 + 30*a^3*b + 35*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 30*a*b^3 + 35*b^4 + 2*(3*a^3*b + 30*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(6*a^4*cos(f*x + e)^7 - 3*(5*a^4 + 7*a^3*b)*cos(f*x + e)^5 - 2*(39*a^3*b + 70*a^2*b^2)*cos(f*x + e)^3 - 5*(11*a^2*b^2 + 21*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^7*f*cos(f*x + e)^4 + 2*a^6*b*f*cos(f*x + e)^2 + a^5*b^2*f)]","A",0
127,1,879,0,3.659777," ","integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} + 6 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 6 \, a b^{3} + 5 \, b^{4} + 2 \, {\left(a^{3} b + 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(3 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{3} b + 10 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(13 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + a^{4} b^{3}\right)} f\right)}}, -\frac{3 \, {\left({\left(a^{4} + 6 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 6 \, a b^{3} + 5 \, b^{4} + 2 \, {\left(a^{3} b + 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(3 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{3} b + 10 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(13 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/48*(3*((a^4 + 6*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 6*a*b^3 + 5*b^4 + 2*(a^3*b + 6*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(3*(a^4 + a^3*b)*cos(f*x + e)^5 + 2*(9*a^3*b + 10*a^2*b^2)*cos(f*x + e)^3 + (13*a^2*b^2 + 15*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^4 + 2*(a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^5*b^2 + a^4*b^3)*f), -1/24*(3*((a^4 + 6*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 6*a*b^3 + 5*b^4 + 2*(a^3*b + 6*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(3*(a^4 + a^3*b)*cos(f*x + e)^5 + 2*(9*a^3*b + 10*a^2*b^2)*cos(f*x + e)^3 + (13*a^2*b^2 + 15*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^4 + 2*(a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^5*b^2 + a^4*b^3)*f)]","B",0
128,1,881,0,1.230782," ","integrate(1/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, -\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[-1/24*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f), -1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f)]","B",0
129,1,192,0,1.461260," ","integrate(csc(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left({\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{5} + 4 \, {\left(3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} + 8 \, b^{2} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*((3*a^2 - 6*a*b - b^2)*cos(f*x + e)^5 + 4*(3*a*b - b^2)*cos(f*x + e)^3 + 8*b^2*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*f*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2 + (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f)*sin(f*x + e))","A",0
130,1,320,0,4.853825," ","integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, {\left(a^{3} - 6 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{7} - 3 \, {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - 12 \, {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - 8 \, {\left(a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a^3 - 6*a^2*b + a*b^2)*cos(f*x + e)^7 - 3*(a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e)^5 - 12*(a^2*b - 2*a*b^2 + b^3)*cos(f*x + e)^3 - 8*(a*b^2 - b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f)*sin(f*x + e))","B",0
131,1,460,0,15.770782," ","integrate(csc(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, {\left(a^{4} - 10 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{9} - 4 \, {\left(5 \, a^{4} - 53 \, a^{3} b + 55 \, a^{2} b^{2} - 15 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{4} - 60 \, a^{3} b + 126 \, a^{2} b^{2} - 60 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 4 \, {\left(15 \, a^{3} b - 55 \, a^{2} b^{2} + 53 \, a b^{3} - 5 \, b^{4}\right)} \cos\left(f x + e\right)^{3} + 8 \, {\left(5 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a^4 - 10*a^3*b + 5*a^2*b^2)*cos(f*x + e)^9 - 4*(5*a^4 - 53*a^3*b + 55*a^2*b^2 - 15*a*b^3)*cos(f*x + e)^7 + 3*(5*a^4 - 60*a^3*b + 126*a^2*b^2 - 60*a*b^3 + 5*b^4)*cos(f*x + e)^5 + 4*(15*a^3*b - 55*a^2*b^2 + 53*a*b^3 - 5*b^4)*cos(f*x + e)^3 + 8*(5*a^2*b^2 - 10*a*b^3 + b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f)*sin(f*x + e))","B",0
132,0,0,0,0.897595," ","integrate((a+b*sec(f*x+e)^2)^p*(d*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*(d*sin(f*x + e))^m, x)","F",0
133,0,0,0,0.562170," ","integrate((a+b*sec(f*x+e)^2)^p*sin(f*x+e)^5,x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} {\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(b*sec(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
134,0,0,0,0.521872," ","integrate((a+b*sec(f*x+e)^2)^p*sin(f*x+e)^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(b*sec(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
135,0,0,0,0.512189," ","integrate((a+b*sec(f*x+e)^2)^p*sin(f*x+e),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
136,0,0,0,0.574016," ","integrate(csc(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right), x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*csc(f*x + e), x)","F",0
137,0,0,0,0.587118," ","integrate(csc(f*x+e)^3*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*csc(f*x + e)^3, x)","F",0
138,0,0,0,0.608917," ","integrate((a+b*sec(f*x+e)^2)^p*sin(f*x+e)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} {\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(b*sec(f*x + e)^2 + a)^p, x)","F",0
139,0,0,0,0.719152," ","integrate((a+b*sec(f*x+e)^2)^p*sin(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(b*sec(f*x + e)^2 + a)^p, x)","F",0
140,0,0,0,0.623546," ","integrate((a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p, x)","F",0
141,0,0,0,0.627928," ","integrate(csc(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*csc(f*x + e)^2, x)","F",0
142,0,0,0,0.764775," ","integrate(csc(f*x+e)^4*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*csc(f*x + e)^4, x)","F",0
143,0,0,0,0.593081," ","integrate(csc(f*x+e)^6*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{6}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*csc(f*x + e)^6, x)","F",0
144,1,82,0,0.454326," ","integrate((a-a*sec(d*x+c)^2)^4,x, algorithm=""fricas"")","\frac{105 \, a^{4} d x \cos\left(d x + c\right)^{7} - {\left(176 \, a^{4} \cos\left(d x + c\right)^{6} - 122 \, a^{4} \cos\left(d x + c\right)^{4} + 66 \, a^{4} \cos\left(d x + c\right)^{2} - 15 \, a^{4}\right)} \sin\left(d x + c\right)}{105 \, d \cos\left(d x + c\right)^{7}}"," ",0,"1/105*(105*a^4*d*x*cos(d*x + c)^7 - (176*a^4*cos(d*x + c)^6 - 122*a^4*cos(d*x + c)^4 + 66*a^4*cos(d*x + c)^2 - 15*a^4)*sin(d*x + c))/(d*cos(d*x + c)^7)","A",0
145,1,69,0,0.441548," ","integrate((a-a*sec(d*x+c)^2)^3,x, algorithm=""fricas"")","\frac{15 \, a^{3} d x \cos\left(d x + c\right)^{5} - {\left(23 \, a^{3} \cos\left(d x + c\right)^{4} - 11 \, a^{3} \cos\left(d x + c\right)^{2} + 3 \, a^{3}\right)} \sin\left(d x + c\right)}{15 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/15*(15*a^3*d*x*cos(d*x + c)^5 - (23*a^3*cos(d*x + c)^4 - 11*a^3*cos(d*x + c)^2 + 3*a^3)*sin(d*x + c))/(d*cos(d*x + c)^5)","A",0
146,1,56,0,0.537603," ","integrate((a-a*sec(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} d x \cos\left(d x + c\right)^{3} - {\left(4 \, a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sin\left(d x + c\right)}{3 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/3*(3*a^2*d*x*cos(d*x + c)^3 - (4*a^2*cos(d*x + c)^2 - a^2)*sin(d*x + c))/(d*cos(d*x + c)^3)","A",0
147,1,32,0,0.498533," ","integrate(a-a*sec(d*x+c)^2,x, algorithm=""fricas"")","\frac{a d x \cos\left(d x + c\right) - a \sin\left(d x + c\right)}{d \cos\left(d x + c\right)}"," ",0,"(a*d*x*cos(d*x + c) - a*sin(d*x + c))/(d*cos(d*x + c))","A",0
148,1,31,0,0.413717," ","integrate(1/(a-a*sec(d*x+c)^2),x, algorithm=""fricas"")","\frac{d x \sin\left(d x + c\right) + \cos\left(d x + c\right)}{a d \sin\left(d x + c\right)}"," ",0,"(d*x*sin(d*x + c) + cos(d*x + c))/(a*d*sin(d*x + c))","A",0
149,1,75,0,0.451089," ","integrate(1/(a-a*sec(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{4 \, \cos\left(d x + c\right)^{3} + 3 \, {\left(d x \cos\left(d x + c\right)^{2} - d x\right)} \sin\left(d x + c\right) - 3 \, \cos\left(d x + c\right)}{3 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"1/3*(4*cos(d*x + c)^3 + 3*(d*x*cos(d*x + c)^2 - d*x)*sin(d*x + c) - 3*cos(d*x + c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))","B",0
150,1,109,0,0.462452," ","integrate(1/(a-a*sec(d*x+c)^2)^3,x, algorithm=""fricas"")","\frac{23 \, \cos\left(d x + c\right)^{5} - 35 \, \cos\left(d x + c\right)^{3} + 15 \, {\left(d x \cos\left(d x + c\right)^{4} - 2 \, d x \cos\left(d x + c\right)^{2} + d x\right)} \sin\left(d x + c\right) + 15 \, \cos\left(d x + c\right)}{15 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)} \sin\left(d x + c\right)}"," ",0,"1/15*(23*cos(d*x + c)^5 - 35*cos(d*x + c)^3 + 15*(d*x*cos(d*x + c)^4 - 2*d*x*cos(d*x + c)^2 + d*x)*sin(d*x + c) + 15*cos(d*x + c))/((a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))","B",0
151,1,147,0,0.456241," ","integrate(1/(a-a*sec(d*x+c)^2)^4,x, algorithm=""fricas"")","\frac{176 \, \cos\left(d x + c\right)^{7} - 406 \, \cos\left(d x + c\right)^{5} + 350 \, \cos\left(d x + c\right)^{3} + 105 \, {\left(d x \cos\left(d x + c\right)^{6} - 3 \, d x \cos\left(d x + c\right)^{4} + 3 \, d x \cos\left(d x + c\right)^{2} - d x\right)} \sin\left(d x + c\right) - 105 \, \cos\left(d x + c\right)}{105 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)}"," ",0,"1/105*(176*cos(d*x + c)^7 - 406*cos(d*x + c)^5 + 350*cos(d*x + c)^3 + 105*(d*x*cos(d*x + c)^6 - 3*d*x*cos(d*x + c)^4 + 3*d*x*cos(d*x + c)^2 - d*x)*sin(d*x + c) - 105*cos(d*x + c))/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))","B",0
152,1,114,0,0.488034," ","integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(6 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{6} \log\left(\sin\left(f x + e\right) + 1\right) - 3 \, {\left(6 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{6} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left(3 \, {\left(6 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(6 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{2} + 8 \, b\right)} \sin\left(f x + e\right)}{96 \, f \cos\left(f x + e\right)^{6}}"," ",0,"1/96*(3*(6*a + 5*b)*cos(f*x + e)^6*log(sin(f*x + e) + 1) - 3*(6*a + 5*b)*cos(f*x + e)^6*log(-sin(f*x + e) + 1) + 2*(3*(6*a + 5*b)*cos(f*x + e)^4 + 2*(6*a + 5*b)*cos(f*x + e)^2 + 8*b)*sin(f*x + e))/(f*cos(f*x + e)^6)","A",0
153,1,95,0,0.514584," ","integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(4 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{4} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(4 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{4} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left({\left(4 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{2} + 2 \, b\right)} \sin\left(f x + e\right)}{16 \, f \cos\left(f x + e\right)^{4}}"," ",0,"1/16*((4*a + 3*b)*cos(f*x + e)^4*log(sin(f*x + e) + 1) - (4*a + 3*b)*cos(f*x + e)^4*log(-sin(f*x + e) + 1) + 2*((4*a + 3*b)*cos(f*x + e)^2 + 2*b)*sin(f*x + e))/(f*cos(f*x + e)^4)","A",0
154,1,72,0,0.468819," ","integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, b \sin\left(f x + e\right)}{4 \, f \cos\left(f x + e\right)^{2}}"," ",0,"1/4*((2*a + b)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (2*a + b)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*b*sin(f*x + e))/(f*cos(f*x + e)^2)","A",0
155,1,40,0,0.504741," ","integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{b \log\left(\sin\left(f x + e\right) + 1\right) - b \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, a \sin\left(f x + e\right)}{2 \, f}"," ",0,"1/2*(b*log(sin(f*x + e) + 1) - b*log(-sin(f*x + e) + 1) + 2*a*sin(f*x + e))/f","A",0
156,1,28,0,0.493982," ","integrate(cos(f*x+e)^3*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(a \cos\left(f x + e\right)^{2} + 2 \, a + 3 \, b\right)} \sin\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(a*cos(f*x + e)^2 + 2*a + 3*b)*sin(f*x + e)/f","A",0
157,1,45,0,0.445345," ","integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(3 \, a \cos\left(f x + e\right)^{4} + {\left(4 \, a + 5 \, b\right)} \cos\left(f x + e\right)^{2} + 8 \, a + 10 \, b\right)} \sin\left(f x + e\right)}{15 \, f}"," ",0,"1/15*(3*a*cos(f*x + e)^4 + (4*a + 5*b)*cos(f*x + e)^2 + 8*a + 10*b)*sin(f*x + e)/f","A",0
158,1,74,0,0.444942," ","integrate(sec(f*x+e)^6*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(8 \, {\left(7 \, a + 6 \, b\right)} \cos\left(f x + e\right)^{6} + 4 \, {\left(7 \, a + 6 \, b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(7 \, a + 6 \, b\right)} \cos\left(f x + e\right)^{2} + 15 \, b\right)} \sin\left(f x + e\right)}{105 \, f \cos\left(f x + e\right)^{7}}"," ",0,"1/105*(8*(7*a + 6*b)*cos(f*x + e)^6 + 4*(7*a + 6*b)*cos(f*x + e)^4 + 3*(7*a + 6*b)*cos(f*x + e)^2 + 15*b)*sin(f*x + e)/(f*cos(f*x + e)^7)","A",0
159,1,56,0,0.443504," ","integrate(sec(f*x+e)^4*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(5 \, a + 4 \, b\right)} \cos\left(f x + e\right)^{4} + {\left(5 \, a + 4 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)^{5}}"," ",0,"1/15*(2*(5*a + 4*b)*cos(f*x + e)^4 + (5*a + 4*b)*cos(f*x + e)^2 + 3*b)*sin(f*x + e)/(f*cos(f*x + e)^5)","A",0
160,1,37,0,0.515085," ","integrate(sec(f*x+e)^2*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left({\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*((3*a + 2*b)*cos(f*x + e)^2 + b)*sin(f*x + e)/(f*cos(f*x + e)^3)","A",0
161,1,31,0,0.480569," ","integrate(a+b*sec(f*x+e)^2,x, algorithm=""fricas"")","\frac{a f x \cos\left(f x + e\right) + b \sin\left(f x + e\right)}{f \cos\left(f x + e\right)}"," ",0,"(a*f*x*cos(f*x + e) + b*sin(f*x + e))/(f*cos(f*x + e))","B",0
162,1,28,0,0.495875," ","integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(a + 2 \, b\right)} f x + a \cos\left(f x + e\right) \sin\left(f x + e\right)}{2 \, f}"," ",0,"1/2*((a + 2*b)*f*x + a*cos(f*x + e)*sin(f*x + e))/f","A",0
163,1,49,0,0.721381," ","integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(3 \, a + 4 \, b\right)} f x + {\left(2 \, a \cos\left(f x + e\right)^{3} + {\left(3 \, a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, f}"," ",0,"1/8*((3*a + 4*b)*f*x + (2*a*cos(f*x + e)^3 + (3*a + 4*b)*cos(f*x + e))*sin(f*x + e))/f","A",0
164,1,68,0,0.571750," ","integrate(cos(f*x+e)^6*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(5 \, a + 6 \, b\right)} f x + {\left(8 \, a \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a + 6 \, b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a + 6 \, b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, f}"," ",0,"1/48*(3*(5*a + 6*b)*f*x + (8*a*cos(f*x + e)^5 + 2*(5*a + 6*b)*cos(f*x + e)^3 + 3*(5*a + 6*b)*cos(f*x + e))*sin(f*x + e))/f","A",0
165,1,168,0,0.475337," ","integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{8} \log\left(\sin\left(f x + e\right) + 1\right) - 3 \, {\left(48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{8} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left(3 \, {\left(48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + 2 \, {\left(48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(16 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 48 \, b^{2}\right)} \sin\left(f x + e\right)}{768 \, f \cos\left(f x + e\right)^{8}}"," ",0,"1/768*(3*(48*a^2 + 80*a*b + 35*b^2)*cos(f*x + e)^8*log(sin(f*x + e) + 1) - 3*(48*a^2 + 80*a*b + 35*b^2)*cos(f*x + e)^8*log(-sin(f*x + e) + 1) + 2*(3*(48*a^2 + 80*a*b + 35*b^2)*cos(f*x + e)^6 + 2*(48*a^2 + 80*a*b + 35*b^2)*cos(f*x + e)^4 + 8*(16*a*b + 7*b^2)*cos(f*x + e)^2 + 48*b^2)*sin(f*x + e))/(f*cos(f*x + e)^8)","A",0
166,1,143,0,0.488997," ","integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{6} \log\left(\sin\left(f x + e\right) + 1\right) - 3 \, {\left(8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{6} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left(3 \, {\left(8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(12 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, b^{2}\right)} \sin\left(f x + e\right)}{96 \, f \cos\left(f x + e\right)^{6}}"," ",0,"1/96*(3*(8*a^2 + 12*a*b + 5*b^2)*cos(f*x + e)^6*log(sin(f*x + e) + 1) - 3*(8*a^2 + 12*a*b + 5*b^2)*cos(f*x + e)^6*log(-sin(f*x + e) + 1) + 2*(3*(8*a^2 + 12*a*b + 5*b^2)*cos(f*x + e)^4 + 2*(12*a*b + 5*b^2)*cos(f*x + e)^2 + 8*b^2)*sin(f*x + e))/(f*cos(f*x + e)^6)","A",0
167,1,116,0,0.620176," ","integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left({\left(8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sin\left(f x + e\right)}{16 \, f \cos\left(f x + e\right)^{4}}"," ",0,"1/16*((8*a^2 + 8*a*b + 3*b^2)*cos(f*x + e)^4*log(sin(f*x + e) + 1) - (8*a^2 + 8*a*b + 3*b^2)*cos(f*x + e)^4*log(-sin(f*x + e) + 1) + 2*((8*a*b + 3*b^2)*cos(f*x + e)^2 + 2*b^2)*sin(f*x + e))/(f*cos(f*x + e)^4)","A",0
168,1,94,0,0.665919," ","integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}{4 \, f \cos\left(f x + e\right)^{2}}"," ",0,"1/4*((4*a*b + b^2)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (4*a*b + b^2)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*(2*a^2*cos(f*x + e)^2 + b^2)*sin(f*x + e))/(f*cos(f*x + e)^2)","A",0
169,1,66,0,0.820977," ","integrate(cos(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} \log\left(\sin\left(f x + e\right) + 1\right) - 3 \, b^{2} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left(a^{2} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 6 \, a b\right)} \sin\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(3*b^2*log(sin(f*x + e) + 1) - 3*b^2*log(-sin(f*x + e) + 1) + 2*(a^2*cos(f*x + e)^2 + 2*a^2 + 6*a*b)*sin(f*x + e))/f","A",0
170,1,59,0,0.533293," ","integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(3 \, a^{2} \cos\left(f x + e\right)^{4} + 2 \, {\left(2 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 20 \, a b + 15 \, b^{2}\right)} \sin\left(f x + e\right)}{15 \, f}"," ",0,"1/15*(3*a^2*cos(f*x + e)^4 + 2*(2*a^2 + 5*a*b)*cos(f*x + e)^2 + 8*a^2 + 20*a*b + 15*b^2)*sin(f*x + e)/f","A",0
171,1,120,0,0.762846," ","integrate(sec(f*x+e)^6*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, {\left(21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{8} + 4 \, {\left(21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + 3 \, {\left(21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 10 \, {\left(9 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 35 \, b^{2}\right)} \sin\left(f x + e\right)}{315 \, f \cos\left(f x + e\right)^{9}}"," ",0,"1/315*(8*(21*a^2 + 36*a*b + 16*b^2)*cos(f*x + e)^8 + 4*(21*a^2 + 36*a*b + 16*b^2)*cos(f*x + e)^6 + 3*(21*a^2 + 36*a*b + 16*b^2)*cos(f*x + e)^4 + 10*(9*a*b + 4*b^2)*cos(f*x + e)^2 + 35*b^2)*sin(f*x + e)/(f*cos(f*x + e)^9)","A",0
172,1,94,0,0.594082," ","integrate(sec(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(35 \, a^{2} + 56 \, a b + 24 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + {\left(35 \, a^{2} + 56 \, a b + 24 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 6 \, {\left(7 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 15 \, b^{2}\right)} \sin\left(f x + e\right)}{105 \, f \cos\left(f x + e\right)^{7}}"," ",0,"1/105*(2*(35*a^2 + 56*a*b + 24*b^2)*cos(f*x + e)^6 + (35*a^2 + 56*a*b + 24*b^2)*cos(f*x + e)^4 + 6*(7*a*b + 3*b^2)*cos(f*x + e)^2 + 15*b^2)*sin(f*x + e)/(f*cos(f*x + e)^7)","A",0
173,1,69,0,0.902978," ","integrate(sec(f*x+e)^2*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left({\left(15 \, a^{2} + 20 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, b^{2}\right)} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)^{5}}"," ",0,"1/15*((15*a^2 + 20*a*b + 8*b^2)*cos(f*x + e)^4 + 2*(5*a*b + 2*b^2)*cos(f*x + e)^2 + 3*b^2)*sin(f*x + e)/(f*cos(f*x + e)^5)","A",0
174,1,58,0,1.114177," ","integrate((a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} f x \cos\left(f x + e\right)^{3} + {\left(2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*(3*a^2*f*x*cos(f*x + e)^3 + (2*(3*a*b + b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
175,1,56,0,1.208735," ","integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(a^{2} + 4 \, a b\right)} f x \cos\left(f x + e\right) + {\left(a^{2} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sin\left(f x + e\right)}{2 \, f \cos\left(f x + e\right)}"," ",0,"1/2*((a^2 + 4*a*b)*f*x*cos(f*x + e) + (a^2*cos(f*x + e)^2 + 2*b^2)*sin(f*x + e))/(f*cos(f*x + e))","A",0
176,1,62,0,1.389753," ","integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right)} f x + {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 8 \, a b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, f}"," ",0,"1/8*((3*a^2 + 8*a*b + 8*b^2)*f*x + (2*a^2*cos(f*x + e)^3 + (3*a^2 + 8*a*b)*cos(f*x + e))*sin(f*x + e))/f","A",0
177,1,89,0,1.734450," ","integrate(cos(f*x+e)^6*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} f x + {\left(8 \, a^{2} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{2} + 12 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, f}"," ",0,"1/48*(3*(5*a^2 + 12*a*b + 8*b^2)*f*x + (8*a^2*cos(f*x + e)^5 + 2*(5*a^2 + 12*a*b)*cos(f*x + e)^3 + 3*(5*a^2 + 12*a*b + 8*b^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
178,1,90,0,0.749245," ","integrate((a+b*sec(d*x+c)^2)^3,x, algorithm=""fricas"")","\frac{15 \, a^{3} d x \cos\left(d x + c\right)^{5} + {\left({\left(45 \, a^{2} b + 30 \, a b^{2} + 8 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + 3 \, b^{3} + {\left(15 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{15 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/15*(15*a^3*d*x*cos(d*x + c)^5 + ((45*a^2*b + 30*a*b^2 + 8*b^3)*cos(d*x + c)^4 + 3*b^3 + (15*a*b^2 + 4*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^5)","A",0
179,1,130,0,0.627728," ","integrate((a+b*sec(d*x+c)^2)^4,x, algorithm=""fricas"")","\frac{105 \, a^{4} d x \cos\left(d x + c\right)^{7} + {\left(4 \, {\left(105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right)} \cos\left(d x + c\right)^{6} + 2 \, {\left(105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 15 \, b^{4} + 6 \, {\left(14 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{105 \, d \cos\left(d x + c\right)^{7}}"," ",0,"1/105*(105*a^4*d*x*cos(d*x + c)^7 + (4*(105*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cos(d*x + c)^6 + 2*(105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cos(d*x + c)^4 + 15*b^4 + 6*(14*a*b^3 + 3*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^7)","A",0
180,1,272,0,1.990935," ","integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, a \sqrt{\frac{a}{a + b}} \cos\left(f x + e\right)^{2} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, {\left(a + b\right)} \sqrt{\frac{a}{a + b}} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) - {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} \log\left(\sin\left(f x + e\right) + 1\right) + {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, b \sin\left(f x + e\right)}{4 \, b^{2} f \cos\left(f x + e\right)^{2}}, -\frac{4 \, a \sqrt{-\frac{a}{a + b}} \arctan\left(\sqrt{-\frac{a}{a + b}} \sin\left(f x + e\right)\right) \cos\left(f x + e\right)^{2} + {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} \log\left(-\sin\left(f x + e\right) + 1\right) - 2 \, b \sin\left(f x + e\right)}{4 \, b^{2} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/4*(2*a*sqrt(a/(a + b))*cos(f*x + e)^2*log(-(a*cos(f*x + e)^2 - 2*(a + b)*sqrt(a/(a + b))*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) - (2*a - b)*cos(f*x + e)^2*log(sin(f*x + e) + 1) + (2*a - b)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*b*sin(f*x + e))/(b^2*f*cos(f*x + e)^2), -1/4*(4*a*sqrt(-a/(a + b))*arctan(sqrt(-a/(a + b))*sin(f*x + e))*cos(f*x + e)^2 + (2*a - b)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (2*a - b)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) - 2*b*sin(f*x + e))/(b^2*f*cos(f*x + e)^2)]","A",0
181,1,157,0,1.962658," ","integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{\frac{a}{a + b}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, {\left(a + b\right)} \sqrt{\frac{a}{a + b}} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + \log\left(\sin\left(f x + e\right) + 1\right) - \log\left(-\sin\left(f x + e\right) + 1\right)}{2 \, b f}, \frac{2 \, \sqrt{-\frac{a}{a + b}} \arctan\left(\sqrt{-\frac{a}{a + b}} \sin\left(f x + e\right)\right) + \log\left(\sin\left(f x + e\right) + 1\right) - \log\left(-\sin\left(f x + e\right) + 1\right)}{2 \, b f}\right]"," ",0,"[1/2*(sqrt(a/(a + b))*log(-(a*cos(f*x + e)^2 + 2*(a + b)*sqrt(a/(a + b))*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + log(sin(f*x + e) + 1) - log(-sin(f*x + e) + 1))/(b*f), 1/2*(2*sqrt(-a/(a + b))*arctan(sqrt(-a/(a + b))*sin(f*x + e)) + log(sin(f*x + e) + 1) - log(-sin(f*x + e) + 1))/(b*f)]","A",0
182,1,117,0,1.156134," ","integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right)}{2 \, \sqrt{a^{2} + a b} f}, -\frac{\sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right)}{{\left(a^{2} + a b\right)} f}\right]"," ",0,"[1/2*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b))/(sqrt(a^2 + a*b)*f), -sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b))/((a^2 + a*b)*f)]","A",0
183,1,164,0,0.460268," ","integrate(cos(f*x+e)/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a^{2} + a b} b \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(a^{2} + a b\right)} \sin\left(f x + e\right)}{2 \, {\left(a^{3} + a^{2} b\right)} f}, \frac{\sqrt{-a^{2} - a b} b \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(a^{2} + a b\right)} \sin\left(f x + e\right)}{{\left(a^{3} + a^{2} b\right)} f}\right]"," ",0,"[1/2*(sqrt(a^2 + a*b)*b*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(a^2 + a*b)*sin(f*x + e))/((a^3 + a^2*b)*f), (sqrt(-a^2 - a*b)*b*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (a^2 + a*b)*sin(f*x + e))/((a^3 + a^2*b)*f)]","A",0
184,1,230,0,0.697809," ","integrate(cos(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a^{2} + a b} b^{2} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(2 \, a^{3} - a^{2} b - 3 \, a b^{2} + {\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{6 \, {\left(a^{4} + a^{3} b\right)} f}, -\frac{3 \, \sqrt{-a^{2} - a b} b^{2} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) - {\left(2 \, a^{3} - a^{2} b - 3 \, a b^{2} + {\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{3 \, {\left(a^{4} + a^{3} b\right)} f}\right]"," ",0,"[1/6*(3*sqrt(a^2 + a*b)*b^2*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(2*a^3 - a^2*b - 3*a*b^2 + (a^3 + a^2*b)*cos(f*x + e)^2)*sin(f*x + e))/((a^4 + a^3*b)*f), -1/3*(3*sqrt(-a^2 - a*b)*b^2*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) - (2*a^3 - a^2*b - 3*a*b^2 + (a^3 + a^2*b)*cos(f*x + e)^2)*sin(f*x + e))/((a^4 + a^3*b)*f)]","A",0
185,1,305,0,0.474479," ","integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{a^{2} + a b} b^{3} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(3 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3} + {\left(4 \, a^{4} - a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{30 \, {\left(a^{5} + a^{4} b\right)} f}, \frac{15 \, \sqrt{-a^{2} - a b} b^{3} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(3 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3} + {\left(4 \, a^{4} - a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{15 \, {\left(a^{5} + a^{4} b\right)} f}\right]"," ",0,"[1/30*(15*sqrt(a^2 + a*b)*b^3*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(3*(a^4 + a^3*b)*cos(f*x + e)^4 + 8*a^4 - 2*a^3*b + 5*a^2*b^2 + 15*a*b^3 + (4*a^4 - a^3*b - 5*a^2*b^2)*cos(f*x + e)^2)*sin(f*x + e))/((a^5 + a^4*b)*f), 1/15*(15*sqrt(-a^2 - a*b)*b^3*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (3*(a^4 + a^3*b)*cos(f*x + e)^4 + 8*a^4 - 2*a^3*b + 5*a^2*b^2 + 15*a*b^3 + (4*a^4 - a^3*b - 5*a^2*b^2)*cos(f*x + e)^2)*sin(f*x + e))/((a^5 + a^4*b)*f)]","A",0
186,1,354,0,0.864665," ","integrate(sec(f*x+e)^6/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{3 \, \sqrt{-a b - b^{2}} a^{2} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(a b^{2} + b^{3} - {\left(3 \, a^{2} b + a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{12 \, {\left(a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{3}}, -\frac{3 \, \sqrt{a b + b^{2}} a^{2} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - 2 \, {\left(a b^{2} + b^{3} - {\left(3 \, a^{2} b + a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{6 \, {\left(a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[-1/12*(3*sqrt(-a*b - b^2)*a^2*cos(f*x + e)^3*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(a*b^2 + b^3 - (3*a^2*b + a*b^2 - 2*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a*b^3 + b^4)*f*cos(f*x + e)^3), -1/6*(3*sqrt(a*b + b^2)*a^2*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e)))*cos(f*x + e)^3 - 2*(a*b^2 + b^3 - (3*a^2*b + a*b^2 - 2*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a*b^3 + b^4)*f*cos(f*x + e)^3)]","B",0
187,1,286,0,0.538661," ","integrate(sec(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a b - b^{2}} a \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(a b + b^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left(a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)}, \frac{\sqrt{a b + b^{2}} a \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 2 \, {\left(a b + b^{2}\right)} \sin\left(f x + e\right)}{2 \, {\left(a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)}\right]"," ",0,"[-1/4*(sqrt(-a*b - b^2)*a*cos(f*x + e)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(a*b + b^2)*sin(f*x + e))/((a*b^2 + b^3)*f*cos(f*x + e)), 1/2*(sqrt(a*b + b^2)*a*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e)))*cos(f*x + e) + 2*(a*b + b^2)*sin(f*x + e))/((a*b^2 + b^3)*f*cos(f*x + e))]","B",0
188,1,209,0,0.564854," ","integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, {\left(a b + b^{2}\right)} f}, -\frac{\arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, \sqrt{a b + b^{2}} f}\right]"," ",0,"[-1/4*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))/((a*b + b^2)*f), -1/2*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e)))/(sqrt(a*b + b^2)*f)]","B",0
189,1,231,0,0.625447," ","integrate(1/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, f x + \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, a f}, \frac{2 \, f x + \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, a f}\right]"," ",0,"[1/4*(4*f*x + sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a*f), 1/2*(2*f*x + sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/(a*f)]","A",0
190,1,272,0,0.620808," ","integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a - 2 \, b\right)} f x + 2 \, a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, a^{2} f}, \frac{{\left(a - 2 \, b\right)} f x + a \cos\left(f x + e\right) \sin\left(f x + e\right) - b \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, a^{2} f}\right]"," ",0,"[1/4*(2*(a - 2*b)*f*x + 2*a*cos(f*x + e)*sin(f*x + e) + b*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a^2*f), 1/2*((a - 2*b)*f*x + a*cos(f*x + e)*sin(f*x + e) - b*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/(a^2*f)]","A",0
191,1,343,0,0.870210," ","integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, b^{2} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} f x + {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} - 4 \, a b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, a^{3} f}, \frac{4 \, b^{2} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} f x + {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} - 4 \, a b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, a^{3} f}\right]"," ",0,"[1/8*(2*b^2*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + (3*a^2 - 4*a*b + 8*b^2)*f*x + (2*a^2*cos(f*x + e)^3 + (3*a^2 - 4*a*b)*cos(f*x + e))*sin(f*x + e))/(a^3*f), 1/8*(4*b^2*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + (3*a^2 - 4*a*b + 8*b^2)*f*x + (2*a^2*cos(f*x + e)^3 + (3*a^2 - 4*a*b)*cos(f*x + e))*sin(f*x + e))/(a^3*f)]","A",0
192,1,424,0,0.782919," ","integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{12 \, b^{3} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 3 \, {\left(5 \, a^{3} - 6 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right)} f x + {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 6 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, a^{4} f}, -\frac{24 \, b^{3} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 3 \, {\left(5 \, a^{3} - 6 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right)} f x - {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 6 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{3} - 6 \, a^{2} b + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, a^{4} f}\right]"," ",0,"[1/48*(12*b^3*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 3*(5*a^3 - 6*a^2*b + 8*a*b^2 - 16*b^3)*f*x + (8*a^3*cos(f*x + e)^5 + 2*(5*a^3 - 6*a^2*b)*cos(f*x + e)^3 + 3*(5*a^3 - 6*a^2*b + 8*a*b^2)*cos(f*x + e))*sin(f*x + e))/(a^4*f), -1/48*(24*b^3*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 3*(5*a^3 - 6*a^2*b + 8*a*b^2 - 16*b^3)*f*x - (8*a^3*cos(f*x + e)^5 + 2*(5*a^3 - 6*a^2*b)*cos(f*x + e)^3 + 3*(5*a^3 - 6*a^2*b + 8*a*b^2)*cos(f*x + e))*sin(f*x + e))/(a^4*f)]","A",0
193,1,392,0,2.071357," ","integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{2 \, a b \sin\left(f x + e\right) - {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} + 2 \, a b + 3 \, b^{2}\right)} \sqrt{\frac{a}{a + b}} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, {\left(a + b\right)} \sqrt{\frac{a}{a + b}} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \log\left(\sin\left(f x + e\right) + 1\right) + 2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \log\left(-\sin\left(f x + e\right) + 1\right)}{4 \, {\left({\left(a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a b^{3} + b^{4}\right)} f\right)}}, -\frac{a b \sin\left(f x + e\right) - {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} + 2 \, a b + 3 \, b^{2}\right)} \sqrt{-\frac{a}{a + b}} \arctan\left(\sqrt{-\frac{a}{a + b}} \sin\left(f x + e\right)\right) - {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \log\left(\sin\left(f x + e\right) + 1\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \log\left(-\sin\left(f x + e\right) + 1\right)}{2 \, {\left({\left(a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a b^{3} + b^{4}\right)} f\right)}}\right]"," ",0,"[-1/4*(2*a*b*sin(f*x + e) - ((2*a^2 + 3*a*b)*cos(f*x + e)^2 + 2*a*b + 3*b^2)*sqrt(a/(a + b))*log(-(a*cos(f*x + e)^2 + 2*(a + b)*sqrt(a/(a + b))*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) - 2*((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*log(sin(f*x + e) + 1) + 2*((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*log(-sin(f*x + e) + 1))/((a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a*b^3 + b^4)*f), -1/2*(a*b*sin(f*x + e) - ((2*a^2 + 3*a*b)*cos(f*x + e)^2 + 2*a*b + 3*b^2)*sqrt(-a/(a + b))*arctan(sqrt(-a/(a + b))*sin(f*x + e)) - ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*log(sin(f*x + e) + 1) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*log(-sin(f*x + e) + 1))/((a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a*b^3 + b^4)*f)]","A",0
194,1,262,0,1.342778," ","integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(a^{2} + a b\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, -\frac{{\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) - {\left(a^{2} + a b\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[1/4*((a*cos(f*x + e)^2 + b)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(a^2 + a*b)*sin(f*x + e))/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b + 2*a^2*b^2 + a*b^3)*f), -1/2*((a*cos(f*x + e)^2 + b)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) - (a^2 + a*b)*sin(f*x + e))/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b + 2*a^2*b^2 + a*b^3)*f)]","A",0
195,1,301,0,1.079535," ","integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(2 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + 2 \, a b + b^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 2 \, {\left(a^{2} b + a b^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}, -\frac{{\left({\left(2 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(a^{2} b + a b^{2}\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}\right]"," ",0,"[1/4*(((2*a^2 + a*b)*cos(f*x + e)^2 + 2*a*b + b^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) - 2*(a^2*b + a*b^2)*sin(f*x + e))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f), -1/2*(((2*a^2 + a*b)*cos(f*x + e)^2 + 2*a*b + b^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (a^2*b + a*b^2)*sin(f*x + e))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)]","A",0
196,1,391,0,0.689642," ","integrate(cos(f*x+e)/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left(4 \, a b^{2} + 3 \, b^{3} + {\left(4 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(2 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, a b^{3} + 2 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f\right)}}, \frac{{\left(4 \, a b^{2} + 3 \, b^{3} + {\left(4 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(2 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, a b^{3} + 2 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f\right)}}\right]"," ",0,"[1/4*((4*a*b^2 + 3*b^3 + (4*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(2*a^3*b + 5*a^2*b^2 + 3*a*b^3 + 2*(a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^2)*sin(f*x + e))/((a^6 + 2*a^5*b + a^4*b^2)*f*cos(f*x + e)^2 + (a^5*b + 2*a^4*b^2 + a^3*b^3)*f), 1/2*((4*a*b^2 + 3*b^3 + (4*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (2*a^3*b + 5*a^2*b^2 + 3*a*b^3 + 2*(a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^2)*sin(f*x + e))/((a^6 + 2*a^5*b + a^4*b^2)*f*cos(f*x + e)^2 + (a^5*b + 2*a^4*b^2 + a^3*b^3)*f)]","A",0
197,1,490,0,0.741777," ","integrate(cos(f*x+e)^3/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(6 \, a b^{3} + 5 \, b^{4} + {\left(6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4} + 2 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(2 \, a^{5} - a^{4} b - 8 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f\right)}}, -\frac{3 \, {\left(6 \, a b^{3} + 5 \, b^{4} + {\left(6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) - {\left(4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4} + 2 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(2 \, a^{5} - a^{4} b - 8 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[1/12*(3*(6*a*b^3 + 5*b^4 + (6*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(4*a^4*b - 4*a^3*b^2 - 23*a^2*b^3 - 15*a*b^4 + 2*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^4 + 2*(2*a^5 - a^4*b - 8*a^3*b^2 - 5*a^2*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^6*b + 2*a^5*b^2 + a^4*b^3)*f), -1/6*(3*(6*a*b^3 + 5*b^4 + (6*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) - (4*a^4*b - 4*a^3*b^2 - 23*a^2*b^3 - 15*a*b^4 + 2*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^4 + 2*(2*a^5 - a^4*b - 8*a^3*b^2 - 5*a^2*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^6*b + 2*a^5*b^2 + a^4*b^3)*f)]","B",0
198,1,583,0,1.968053," ","integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{15 \, {\left(8 \, a b^{4} + 7 \, b^{5} + {\left(8 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(6 \, {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} \cos\left(f x + e\right)^{6} + 16 \, a^{5} b - 8 \, a^{4} b^{2} + 26 \, a^{3} b^{3} + 155 \, a^{2} b^{4} + 105 \, a b^{5} + 2 \, {\left(4 \, a^{6} + a^{5} b - 10 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{6} + 11 \, a^{4} b^{2} + 54 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{60 \, {\left({\left(a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f\right)}}, \frac{15 \, {\left(8 \, a b^{4} + 7 \, b^{5} + {\left(8 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(6 \, {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} \cos\left(f x + e\right)^{6} + 16 \, a^{5} b - 8 \, a^{4} b^{2} + 26 \, a^{3} b^{3} + 155 \, a^{2} b^{4} + 105 \, a b^{5} + 2 \, {\left(4 \, a^{6} + a^{5} b - 10 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{6} + 11 \, a^{4} b^{2} + 54 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{30 \, {\left({\left(a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f\right)}}\right]"," ",0,"[1/60*(15*(8*a*b^4 + 7*b^5 + (8*a^2*b^3 + 7*a*b^4)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(6*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^6 + 16*a^5*b - 8*a^4*b^2 + 26*a^3*b^3 + 155*a^2*b^4 + 105*a*b^5 + 2*(4*a^6 + a^5*b - 10*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^4 + 2*(8*a^6 + 11*a^4*b^2 + 54*a^3*b^3 + 35*a^2*b^4)*cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^2 + (a^7*b + 2*a^6*b^2 + a^5*b^3)*f), 1/30*(15*(8*a*b^4 + 7*b^5 + (8*a^2*b^3 + 7*a*b^4)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (6*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^6 + 16*a^5*b - 8*a^4*b^2 + 26*a^3*b^3 + 155*a^2*b^4 + 105*a*b^5 + 2*(4*a^6 + a^5*b - 10*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^4 + 2*(8*a^6 + 11*a^4*b^2 + 54*a^3*b^3 + 35*a^2*b^4)*cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^2 + (a^7*b + 2*a^6*b^2 + a^5*b^3)*f)]","A",0
199,1,516,0,0.699569," ","integrate(sec(f*x+e)^6/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(3 \, a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(2 \, a^{2} b^{2} + 4 \, a b^{3} + 2 \, b^{4} + {\left(3 \, a^{3} b + 5 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)\right)}}, \frac{{\left({\left(3 \, a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(2 \, a^{2} b^{2} + 4 \, a b^{3} + 2 \, b^{4} + {\left(3 \, a^{3} b + 5 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[-1/8*(((3*a^3 + 4*a^2*b)*cos(f*x + e)^3 + (3*a^2*b + 4*a*b^2)*cos(f*x + e))*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(2*a^2*b^2 + 4*a*b^3 + 2*b^4 + (3*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a^3*b^3 + 2*a^2*b^4 + a*b^5)*f*cos(f*x + e)^3 + (a^2*b^4 + 2*a*b^5 + b^6)*f*cos(f*x + e)), 1/4*(((3*a^3 + 4*a^2*b)*cos(f*x + e)^3 + (3*a^2*b + 4*a*b^2)*cos(f*x + e))*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) + 2*(2*a^2*b^2 + 4*a*b^3 + 2*b^4 + (3*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a^3*b^3 + 2*a^2*b^4 + a*b^5)*f*cos(f*x + e)^3 + (a^2*b^4 + 2*a*b^5 + b^6)*f*cos(f*x + e))]","B",0
200,1,406,0,0.768049," ","integrate(sec(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left({\left(a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} f\right)}}, -\frac{2 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[-1/8*(4*(a^2*b + a*b^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/((a^3*b^2 + 2*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2 + (a^2*b^3 + 2*a*b^4 + b^5)*f), -1/4*(2*(a^2*b + a*b^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))))/((a^3*b^2 + 2*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2 + (a^2*b^3 + 2*a*b^4 + b^5)*f)]","B",0
201,1,368,0,0.572753," ","integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f\right)}}, \frac{2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f\right)}}\right]"," ",0,"[1/8*(4*(a*b + b^2)*cos(f*x + e)*sin(f*x + e) - (a*cos(f*x + e)^2 + b)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/((a^3*b + 2*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^2*b^2 + 2*a*b^3 + b^4)*f), 1/4*(2*(a*b + b^2)*cos(f*x + e)*sin(f*x + e) - (a*cos(f*x + e)^2 + b)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))))/((a^3*b + 2*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^2*b^2 + 2*a*b^3 + b^4)*f)]","B",0
202,1,435,0,0.772669," ","integrate(1/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} + a b\right)} f x \cos\left(f x + e\right)^{2} - 4 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 8 \, {\left(a b + b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{4 \, {\left(a^{2} + a b\right)} f x \cos\left(f x + e\right)^{2} - 2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 4 \, {\left(a b + b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(8*(a^2 + a*b)*f*x*cos(f*x + e)^2 - 4*a*b*cos(f*x + e)*sin(f*x + e) + 8*(a*b + b^2)*f*x + ((3*a^2 + 2*a*b)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), 1/4*(4*(a^2 + a*b)*f*x*cos(f*x + e)^2 - 2*a*b*cos(f*x + e)*sin(f*x + e) + 4*(a*b + b^2)*f*x + ((3*a^2 + 2*a*b)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]","B",0
203,1,544,0,0.794633," ","integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 4 \, {\left(a^{2} b - 3 \, a b^{2} - 4 \, b^{3}\right)} f x + {\left(5 \, a b^{2} + 4 \, b^{3} + {\left(5 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left({\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{5} + a^{4} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + a^{3} b^{2}\right)} f\right)}}, \frac{2 \, {\left(a^{3} - 3 \, a^{2} b - 4 \, a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{2} b - 3 \, a b^{2} - 4 \, b^{3}\right)} f x - {\left(5 \, a b^{2} + 4 \, b^{3} + {\left(5 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left({\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{5} + a^{4} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + a^{3} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(4*(a^3 - 3*a^2*b - 4*a*b^2)*f*x*cos(f*x + e)^2 + 4*(a^2*b - 3*a*b^2 - 4*b^3)*f*x + (5*a*b^2 + 4*b^3 + (5*a^2*b + 4*a*b^2)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*((a^3 + a^2*b)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2)*cos(f*x + e))*sin(f*x + e))/((a^5 + a^4*b)*f*cos(f*x + e)^2 + (a^4*b + a^3*b^2)*f), 1/4*(2*(a^3 - 3*a^2*b - 4*a*b^2)*f*x*cos(f*x + e)^2 + 2*(a^2*b - 3*a*b^2 - 4*b^3)*f*x - (5*a*b^2 + 4*b^3 + (5*a^2*b + 4*a*b^2)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + 2*((a^3 + a^2*b)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2)*cos(f*x + e))*sin(f*x + e))/((a^5 + a^4*b)*f*cos(f*x + e)^2 + (a^4*b + a^3*b^2)*f)]","A",0
204,1,656,0,0.771413," ","integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{4} - 5 \, a^{3} b + 16 \, a^{2} b^{2} + 24 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + {\left(3 \, a^{3} b - 5 \, a^{2} b^{2} + 16 \, a b^{3} + 24 \, b^{4}\right)} f x + {\left(7 \, a b^{3} + 6 \, b^{4} + {\left(7 \, a^{2} b^{2} + 6 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(2 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + 3 \, {\left(a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 5 \, a^{2} b^{2} - 12 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{6} + a^{5} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b + a^{4} b^{2}\right)} f\right)}}, \frac{{\left(3 \, a^{4} - 5 \, a^{3} b + 16 \, a^{2} b^{2} + 24 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + {\left(3 \, a^{3} b - 5 \, a^{2} b^{2} + 16 \, a b^{3} + 24 \, b^{4}\right)} f x + 2 \, {\left(7 \, a b^{3} + 6 \, b^{4} + {\left(7 \, a^{2} b^{2} + 6 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(2 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + 3 \, {\left(a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 5 \, a^{2} b^{2} - 12 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{6} + a^{5} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b + a^{4} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*((3*a^4 - 5*a^3*b + 16*a^2*b^2 + 24*a*b^3)*f*x*cos(f*x + e)^2 + (3*a^3*b - 5*a^2*b^2 + 16*a*b^3 + 24*b^4)*f*x + (7*a*b^3 + 6*b^4 + (7*a^2*b^2 + 6*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + (2*(a^4 + a^3*b)*cos(f*x + e)^5 + 3*(a^4 - a^3*b - 2*a^2*b^2)*cos(f*x + e)^3 + (3*a^3*b - 5*a^2*b^2 - 12*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^6 + a^5*b)*f*cos(f*x + e)^2 + (a^5*b + a^4*b^2)*f), 1/8*((3*a^4 - 5*a^3*b + 16*a^2*b^2 + 24*a*b^3)*f*x*cos(f*x + e)^2 + (3*a^3*b - 5*a^2*b^2 + 16*a*b^3 + 24*b^4)*f*x + 2*(7*a*b^3 + 6*b^4 + (7*a^2*b^2 + 6*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + (2*(a^4 + a^3*b)*cos(f*x + e)^5 + 3*(a^4 - a^3*b - 2*a^2*b^2)*cos(f*x + e)^3 + (3*a^3*b - 5*a^2*b^2 - 12*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^6 + a^5*b)*f*cos(f*x + e)^2 + (a^5*b + a^4*b^2)*f)]","A",0
205,1,789,0,0.951573," ","integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{5} - 7 \, a^{4} b + 12 \, a^{3} b^{2} - 40 \, a^{2} b^{3} - 64 \, a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} - 40 \, a b^{4} - 64 \, b^{5}\right)} f x + 6 \, {\left(9 \, a b^{4} + 8 \, b^{5} + {\left(9 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(8 \, {\left(a^{5} + a^{4} b\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(5 \, a^{5} - 3 \, a^{4} b - 8 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{5} - 11 \, a^{4} b + 22 \, a^{3} b^{2} + 48 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} + 32 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + a^{5} b^{2}\right)} f\right)}}, \frac{3 \, {\left(5 \, a^{5} - 7 \, a^{4} b + 12 \, a^{3} b^{2} - 40 \, a^{2} b^{3} - 64 \, a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} - 40 \, a b^{4} - 64 \, b^{5}\right)} f x - 12 \, {\left(9 \, a b^{4} + 8 \, b^{5} + {\left(9 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(8 \, {\left(a^{5} + a^{4} b\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(5 \, a^{5} - 3 \, a^{4} b - 8 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{5} - 11 \, a^{4} b + 22 \, a^{3} b^{2} + 48 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{4} b - 7 \, a^{3} b^{2} + 12 \, a^{2} b^{3} + 32 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + a^{5} b^{2}\right)} f\right)}}\right]"," ",0,"[1/48*(3*(5*a^5 - 7*a^4*b + 12*a^3*b^2 - 40*a^2*b^3 - 64*a*b^4)*f*x*cos(f*x + e)^2 + 3*(5*a^4*b - 7*a^3*b^2 + 12*a^2*b^3 - 40*a*b^4 - 64*b^5)*f*x + 6*(9*a*b^4 + 8*b^5 + (9*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + (8*(a^5 + a^4*b)*cos(f*x + e)^7 + 2*(5*a^5 - 3*a^4*b - 8*a^3*b^2)*cos(f*x + e)^5 + (15*a^5 - 11*a^4*b + 22*a^3*b^2 + 48*a^2*b^3)*cos(f*x + e)^3 + 3*(5*a^4*b - 7*a^3*b^2 + 12*a^2*b^3 + 32*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^2 + (a^6*b + a^5*b^2)*f), 1/48*(3*(5*a^5 - 7*a^4*b + 12*a^3*b^2 - 40*a^2*b^3 - 64*a*b^4)*f*x*cos(f*x + e)^2 + 3*(5*a^4*b - 7*a^3*b^2 + 12*a^2*b^3 - 40*a*b^4 - 64*b^5)*f*x - 12*(9*a*b^4 + 8*b^5 + (9*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + (8*(a^5 + a^4*b)*cos(f*x + e)^7 + 2*(5*a^5 - 3*a^4*b - 8*a^3*b^2)*cos(f*x + e)^5 + (15*a^5 - 11*a^4*b + 22*a^3*b^2 + 48*a^2*b^3)*cos(f*x + e)^3 + 3*(5*a^4*b - 7*a^3*b^2 + 12*a^2*b^3 + 32*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^2 + (a^6*b + a^5*b^2)*f)]","A",0
206,1,472,0,1.676246," ","integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2} + 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} f\right)}}, -\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) - {\left(2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2} + 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} f\right)}}\right]"," ",0,"[1/16*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(2*a^3 + 7*a^2*b + 5*a*b^2 + 3*(a^3 + a^2*b)*cos(f*x + e)^2)*sin(f*x + e))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 + 2*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^2 + (a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*f), -1/8*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) - (2*a^3 + 7*a^2*b + 5*a*b^2 + 3*(a^3 + a^2*b)*cos(f*x + e)^2)*sin(f*x + e))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 + 2*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^2 + (a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*f)]","B",0
207,1,544,0,1.283918," ","integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{{\left({\left(4 \, a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{4} + 4 \, a b^{2} + b^{3} + 2 \, {\left(4 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(2 \, a^{3} b + a^{2} b^{2} - a b^{3} + {\left(4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, -\frac{{\left({\left(4 \, a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{4} + 4 \, a b^{2} + b^{3} + 2 \, {\left(4 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) - {\left(2 \, a^{3} b + a^{2} b^{2} - a b^{3} + {\left(4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[1/16*(((4*a^3 + a^2*b)*cos(f*x + e)^4 + 4*a*b^2 + b^3 + 2*(4*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(2*a^3*b + a^2*b^2 - a*b^3 + (4*a^4 + 5*a^3*b + a^2*b^2)*cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f), -1/8*(((4*a^3 + a^2*b)*cos(f*x + e)^4 + 4*a*b^2 + b^3 + 2*(4*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) - (2*a^3*b + a^2*b^2 - a*b^3 + (4*a^4 + 5*a^3*b + a^2*b^2)*cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f)]","B",0
208,1,613,0,1.198407," ","integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{{\left({\left(8 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4} + 2 \, {\left(8 \, a^{3} b + 8 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) - 2 \, {\left(6 \, a^{3} b^{2} + 9 \, a^{2} b^{3} + 3 \, a b^{4} + {\left(8 \, a^{4} b + 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}, -\frac{{\left({\left(8 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4} + 2 \, {\left(8 \, a^{3} b + 8 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(6 \, a^{3} b^{2} + 9 \, a^{2} b^{3} + 3 \, a b^{4} + {\left(8 \, a^{4} b + 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}\right]"," ",0,"[1/16*(((8*a^4 + 8*a^3*b + 3*a^2*b^2)*cos(f*x + e)^4 + 8*a^2*b^2 + 8*a*b^3 + 3*b^4 + 2*(8*a^3*b + 8*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) - 2*(6*a^3*b^2 + 9*a^2*b^3 + 3*a*b^4 + (8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f), -1/8*(((8*a^4 + 8*a^3*b + 3*a^2*b^2)*cos(f*x + e)^4 + 8*a^2*b^2 + 8*a*b^3 + 3*b^4 + 2*(8*a^3*b + 8*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (6*a^3*b^2 + 9*a^2*b^3 + 3*a*b^4 + (8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)]","B",0
209,1,727,0,1.640948," ","integrate(cos(f*x+e)/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(8 \, a^{2} b^{3} + 12 \, a b^{4} + 5 \, b^{5} + {\left(8 \, a^{4} b + 12 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{3} b^{2} + 12 \, a^{2} b^{3} + 5 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(8 \, a^{4} b^{2} + 34 \, a^{3} b^{3} + 41 \, a^{2} b^{4} + 15 \, a b^{5} + 8 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(16 \, a^{5} b + 60 \, a^{4} b^{2} + 69 \, a^{3} b^{3} + 25 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{9} + 3 \, a^{8} b + 3 \, a^{7} b^{2} + a^{6} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{8} b + 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b^{2} + 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} + a^{4} b^{5}\right)} f\right)}}, \frac{3 \, {\left(8 \, a^{2} b^{3} + 12 \, a b^{4} + 5 \, b^{5} + {\left(8 \, a^{4} b + 12 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{3} b^{2} + 12 \, a^{2} b^{3} + 5 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(8 \, a^{4} b^{2} + 34 \, a^{3} b^{3} + 41 \, a^{2} b^{4} + 15 \, a b^{5} + 8 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(16 \, a^{5} b + 60 \, a^{4} b^{2} + 69 \, a^{3} b^{3} + 25 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{9} + 3 \, a^{8} b + 3 \, a^{7} b^{2} + a^{6} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{8} b + 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b^{2} + 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} + a^{4} b^{5}\right)} f\right)}}\right]"," ",0,"[1/16*(3*(8*a^2*b^3 + 12*a*b^4 + 5*b^5 + (8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cos(f*x + e)^4 + 2*(8*a^3*b^2 + 12*a^2*b^3 + 5*a*b^4)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(8*a^4*b^2 + 34*a^3*b^3 + 41*a^2*b^4 + 15*a*b^5 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(f*x + e)^4 + (16*a^5*b + 60*a^4*b^2 + 69*a^3*b^3 + 25*a^2*b^4)*cos(f*x + e)^2)*sin(f*x + e))/((a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^4 + 2*(a^8*b + 3*a^7*b^2 + 3*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^2 + (a^7*b^2 + 3*a^6*b^3 + 3*a^5*b^4 + a^4*b^5)*f), 1/8*(3*(8*a^2*b^3 + 12*a*b^4 + 5*b^5 + (8*a^4*b + 12*a^3*b^2 + 5*a^2*b^3)*cos(f*x + e)^4 + 2*(8*a^3*b^2 + 12*a^2*b^3 + 5*a*b^4)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (8*a^4*b^2 + 34*a^3*b^3 + 41*a^2*b^4 + 15*a*b^5 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(f*x + e)^4 + (16*a^5*b + 60*a^4*b^2 + 69*a^3*b^3 + 25*a^2*b^4)*cos(f*x + e)^2)*sin(f*x + e))/((a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^4 + 2*(a^8*b + 3*a^7*b^2 + 3*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^2 + (a^7*b^2 + 3*a^6*b^3 + 3*a^5*b^4 + a^4*b^5)*f)]","B",0
210,1,856,0,0.903461," ","integrate(cos(f*x+e)^3/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(48 \, a^{2} b^{4} + 80 \, a b^{5} + 35 \, b^{6} + {\left(48 \, a^{4} b^{2} + 80 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(48 \, a^{3} b^{3} + 80 \, a^{2} b^{4} + 35 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(16 \, a^{5} b^{2} - 24 \, a^{4} b^{3} - 210 \, a^{3} b^{4} - 275 \, a^{2} b^{5} - 105 \, a b^{6} + 8 \, {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + 8 \, {\left(2 \, a^{7} - a^{6} b - 15 \, a^{5} b^{2} - 19 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + {\left(32 \, a^{6} b - 40 \, a^{5} b^{2} - 360 \, a^{4} b^{3} - 463 \, a^{3} b^{4} - 175 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + a^{6} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} + a^{5} b^{5}\right)} f\right)}}, -\frac{3 \, {\left(48 \, a^{2} b^{4} + 80 \, a b^{5} + 35 \, b^{6} + {\left(48 \, a^{4} b^{2} + 80 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(48 \, a^{3} b^{3} + 80 \, a^{2} b^{4} + 35 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) - {\left(16 \, a^{5} b^{2} - 24 \, a^{4} b^{3} - 210 \, a^{3} b^{4} - 275 \, a^{2} b^{5} - 105 \, a b^{6} + 8 \, {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + 8 \, {\left(2 \, a^{7} - a^{6} b - 15 \, a^{5} b^{2} - 19 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + {\left(32 \, a^{6} b - 40 \, a^{5} b^{2} - 360 \, a^{4} b^{3} - 463 \, a^{3} b^{4} - 175 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + a^{6} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} + a^{5} b^{5}\right)} f\right)}}\right]"," ",0,"[1/48*(3*(48*a^2*b^4 + 80*a*b^5 + 35*b^6 + (48*a^4*b^2 + 80*a^3*b^3 + 35*a^2*b^4)*cos(f*x + e)^4 + 2*(48*a^3*b^3 + 80*a^2*b^4 + 35*a*b^5)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(16*a^5*b^2 - 24*a^4*b^3 - 210*a^3*b^4 - 275*a^2*b^5 - 105*a*b^6 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*cos(f*x + e)^6 + 8*(2*a^7 - a^6*b - 15*a^5*b^2 - 19*a^4*b^3 - 7*a^3*b^4)*cos(f*x + e)^4 + (32*a^6*b - 40*a^5*b^2 - 360*a^4*b^3 - 463*a^3*b^4 - 175*a^2*b^5)*cos(f*x + e)^2)*sin(f*x + e))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^4 + 2*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*f*cos(f*x + e)^2 + (a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*f), -1/24*(3*(48*a^2*b^4 + 80*a*b^5 + 35*b^6 + (48*a^4*b^2 + 80*a^3*b^3 + 35*a^2*b^4)*cos(f*x + e)^4 + 2*(48*a^3*b^3 + 80*a^2*b^4 + 35*a*b^5)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) - (16*a^5*b^2 - 24*a^4*b^3 - 210*a^3*b^4 - 275*a^2*b^5 - 105*a*b^6 + 8*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*cos(f*x + e)^6 + 8*(2*a^7 - a^6*b - 15*a^5*b^2 - 19*a^4*b^3 - 7*a^3*b^4)*cos(f*x + e)^4 + (32*a^6*b - 40*a^5*b^2 - 360*a^4*b^3 - 463*a^3*b^4 - 175*a^2*b^5)*cos(f*x + e)^2)*sin(f*x + e))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^4 + 2*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*f*cos(f*x + e)^2 + (a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*f)]","B",0
211,1,995,0,0.925155," ","integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{15 \, {\left(80 \, a^{2} b^{5} + 140 \, a b^{6} + 63 \, b^{7} + {\left(80 \, a^{4} b^{3} + 140 \, a^{3} b^{4} + 63 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(80 \, a^{3} b^{4} + 140 \, a^{2} b^{5} + 63 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a^{2} + a b} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a^{2} + a b} \sin\left(f x + e\right) - 2 \, a - b}{a \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(24 \, {\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} \cos\left(f x + e\right)^{8} + 64 \, a^{6} b^{2} - 48 \, a^{5} b^{3} + 192 \, a^{4} b^{4} + 1774 \, a^{3} b^{5} + 2415 \, a^{2} b^{6} + 945 \, a b^{7} + 8 \, {\left(4 \, a^{8} + 3 \, a^{7} b - 15 \, a^{6} b^{2} - 23 \, a^{5} b^{3} - 9 \, a^{4} b^{4}\right)} \cos\left(f x + e\right)^{6} + 8 \, {\left(8 \, a^{8} + 2 \, a^{7} b + 21 \, a^{6} b^{2} + 131 \, a^{5} b^{3} + 167 \, a^{4} b^{4} + 63 \, a^{3} b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(128 \, a^{7} b - 64 \, a^{6} b^{2} + 360 \, a^{5} b^{3} + 3044 \, a^{4} b^{4} + 4067 \, a^{3} b^{5} + 1575 \, a^{2} b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{240 \, {\left({\left(a^{11} + 3 \, a^{10} b + 3 \, a^{9} b^{2} + a^{8} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{10} b + 3 \, a^{9} b^{2} + 3 \, a^{8} b^{3} + a^{7} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{9} b^{2} + 3 \, a^{8} b^{3} + 3 \, a^{7} b^{4} + a^{6} b^{5}\right)} f\right)}}, \frac{15 \, {\left(80 \, a^{2} b^{5} + 140 \, a b^{6} + 63 \, b^{7} + {\left(80 \, a^{4} b^{3} + 140 \, a^{3} b^{4} + 63 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(80 \, a^{3} b^{4} + 140 \, a^{2} b^{5} + 63 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a^{2} - a b} \arctan\left(\frac{\sqrt{-a^{2} - a b} \sin\left(f x + e\right)}{a + b}\right) + {\left(24 \, {\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} \cos\left(f x + e\right)^{8} + 64 \, a^{6} b^{2} - 48 \, a^{5} b^{3} + 192 \, a^{4} b^{4} + 1774 \, a^{3} b^{5} + 2415 \, a^{2} b^{6} + 945 \, a b^{7} + 8 \, {\left(4 \, a^{8} + 3 \, a^{7} b - 15 \, a^{6} b^{2} - 23 \, a^{5} b^{3} - 9 \, a^{4} b^{4}\right)} \cos\left(f x + e\right)^{6} + 8 \, {\left(8 \, a^{8} + 2 \, a^{7} b + 21 \, a^{6} b^{2} + 131 \, a^{5} b^{3} + 167 \, a^{4} b^{4} + 63 \, a^{3} b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(128 \, a^{7} b - 64 \, a^{6} b^{2} + 360 \, a^{5} b^{3} + 3044 \, a^{4} b^{4} + 4067 \, a^{3} b^{5} + 1575 \, a^{2} b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{120 \, {\left({\left(a^{11} + 3 \, a^{10} b + 3 \, a^{9} b^{2} + a^{8} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{10} b + 3 \, a^{9} b^{2} + 3 \, a^{8} b^{3} + a^{7} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{9} b^{2} + 3 \, a^{8} b^{3} + 3 \, a^{7} b^{4} + a^{6} b^{5}\right)} f\right)}}\right]"," ",0,"[1/240*(15*(80*a^2*b^5 + 140*a*b^6 + 63*b^7 + (80*a^4*b^3 + 140*a^3*b^4 + 63*a^2*b^5)*cos(f*x + e)^4 + 2*(80*a^3*b^4 + 140*a^2*b^5 + 63*a*b^6)*cos(f*x + e)^2)*sqrt(a^2 + a*b)*log(-(a*cos(f*x + e)^2 + 2*sqrt(a^2 + a*b)*sin(f*x + e) - 2*a - b)/(a*cos(f*x + e)^2 + b)) + 2*(24*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cos(f*x + e)^8 + 64*a^6*b^2 - 48*a^5*b^3 + 192*a^4*b^4 + 1774*a^3*b^5 + 2415*a^2*b^6 + 945*a*b^7 + 8*(4*a^8 + 3*a^7*b - 15*a^6*b^2 - 23*a^5*b^3 - 9*a^4*b^4)*cos(f*x + e)^6 + 8*(8*a^8 + 2*a^7*b + 21*a^6*b^2 + 131*a^5*b^3 + 167*a^4*b^4 + 63*a^3*b^5)*cos(f*x + e)^4 + (128*a^7*b - 64*a^6*b^2 + 360*a^5*b^3 + 3044*a^4*b^4 + 4067*a^3*b^5 + 1575*a^2*b^6)*cos(f*x + e)^2)*sin(f*x + e))/((a^11 + 3*a^10*b + 3*a^9*b^2 + a^8*b^3)*f*cos(f*x + e)^4 + 2*(a^10*b + 3*a^9*b^2 + 3*a^8*b^3 + a^7*b^4)*f*cos(f*x + e)^2 + (a^9*b^2 + 3*a^8*b^3 + 3*a^7*b^4 + a^6*b^5)*f), 1/120*(15*(80*a^2*b^5 + 140*a*b^6 + 63*b^7 + (80*a^4*b^3 + 140*a^3*b^4 + 63*a^2*b^5)*cos(f*x + e)^4 + 2*(80*a^3*b^4 + 140*a^2*b^5 + 63*a*b^6)*cos(f*x + e)^2)*sqrt(-a^2 - a*b)*arctan(sqrt(-a^2 - a*b)*sin(f*x + e)/(a + b)) + (24*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*cos(f*x + e)^8 + 64*a^6*b^2 - 48*a^5*b^3 + 192*a^4*b^4 + 1774*a^3*b^5 + 2415*a^2*b^6 + 945*a*b^7 + 8*(4*a^8 + 3*a^7*b - 15*a^6*b^2 - 23*a^5*b^3 - 9*a^4*b^4)*cos(f*x + e)^6 + 8*(8*a^8 + 2*a^7*b + 21*a^6*b^2 + 131*a^5*b^3 + 167*a^4*b^4 + 63*a^3*b^5)*cos(f*x + e)^4 + (128*a^7*b - 64*a^6*b^2 + 360*a^5*b^3 + 3044*a^4*b^4 + 4067*a^3*b^5 + 1575*a^2*b^6)*cos(f*x + e)^2)*sin(f*x + e))/((a^11 + 3*a^10*b + 3*a^9*b^2 + a^8*b^3)*f*cos(f*x + e)^4 + 2*(a^10*b + 3*a^9*b^2 + 3*a^8*b^3 + a^7*b^4)*f*cos(f*x + e)^2 + (a^9*b^2 + 3*a^8*b^3 + 3*a^7*b^4 + a^6*b^5)*f)]","B",0
212,1,722,0,1.447222," ","integrate(sec(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(3 \, a^{4} + 8 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 8 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left(3 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{3} b^{2} + 13 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{5} b^{3} + 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} + a^{2} b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{4} + 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} + a b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{5} + 3 \, a^{2} b^{6} + 3 \, a b^{7} + b^{8}\right)} f\right)}}, -\frac{{\left({\left(3 \, a^{4} + 8 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 8 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(3 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{3} b^{2} + 13 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{5} b^{3} + 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} + a^{2} b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{4} + 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} + a b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{5} + 3 \, a^{2} b^{6} + 3 \, a b^{7} + b^{8}\right)} f\right)}}\right]"," ",0,"[-1/32*(((3*a^4 + 8*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 8*a*b^3 + 8*b^4 + 2*(3*a^3*b + 8*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*(3*(a^4*b + 3*a^3*b^2 + 2*a^2*b^3)*cos(f*x + e)^3 + (5*a^3*b^2 + 13*a^2*b^3 + 8*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*f*cos(f*x + e)^4 + 2*(a^4*b^4 + 3*a^3*b^5 + 3*a^2*b^6 + a*b^7)*f*cos(f*x + e)^2 + (a^3*b^5 + 3*a^2*b^6 + 3*a*b^7 + b^8)*f), -1/16*(((3*a^4 + 8*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 8*a*b^3 + 8*b^4 + 2*(3*a^3*b + 8*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) + 2*(3*(a^4*b + 3*a^3*b^2 + 2*a^2*b^3)*cos(f*x + e)^3 + (5*a^3*b^2 + 13*a^2*b^3 + 8*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*f*cos(f*x + e)^4 + 2*(a^4*b^4 + 3*a^3*b^5 + 3*a^2*b^6 + a*b^7)*f*cos(f*x + e)^2 + (a^3*b^5 + 3*a^2*b^6 + 3*a*b^7 + b^8)*f)]","B",0
213,1,654,0,0.884633," ","integrate(sec(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} + a b^{2} + 4 \, b^{3} + 2 \, {\left(a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left({\left(a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{2} b^{2} + 5 \, a b^{3} + 4 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} + a b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} f\right)}}, -\frac{{\left({\left(a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{4} + a b^{2} + 4 \, b^{3} + 2 \, {\left(a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left({\left(a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{2} b^{2} + 5 \, a b^{3} + 4 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} + a b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} f\right)}}\right]"," ",0,"[-1/32*(((a^3 + 4*a^2*b)*cos(f*x + e)^4 + a*b^2 + 4*b^3 + 2*(a^2*b + 4*a*b^2)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*((a^3*b - a^2*b^2 - 2*a*b^3)*cos(f*x + e)^3 - (a^2*b^2 + 5*a*b^3 + 4*b^4)*cos(f*x + e))*sin(f*x + e))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^4 + 2*(a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 + a*b^6)*f*cos(f*x + e)^2 + (a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*f), -1/16*(((a^3 + 4*a^2*b)*cos(f*x + e)^4 + a*b^2 + 4*b^3 + 2*(a^2*b + 4*a*b^2)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) + 2*((a^3*b - a^2*b^2 - 2*a*b^3)*cos(f*x + e)^3 - (a^2*b^2 + 5*a*b^3 + 4*b^4)*cos(f*x + e))*sin(f*x + e))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^4 + 2*(a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 + a*b^6)*f*cos(f*x + e)^2 + (a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*f)]","B",0
214,1,580,0,0.956545," ","integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left({\left(5 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right)} f\right)}}, -\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left({\left(5 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right)} f\right)}}\right]"," ",0,"[-1/32*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*((5*a^2*b + 7*a*b^2 + 2*b^3)*cos(f*x + e)^3 + 3*(a*b^2 + b^3)*cos(f*x + e))*sin(f*x + e))/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^4 + 2*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*f*cos(f*x + e)^2 + (a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*f), -1/16*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - 2*((5*a^2*b + 7*a*b^2 + 2*b^3)*cos(f*x + e)^3 + 3*(a*b^2 + b^3)*cos(f*x + e))*sin(f*x + e))/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f*cos(f*x + e)^4 + 2*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*f*cos(f*x + e)^2 + (a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*f)]","B",0
215,1,819,0,0.779565," ","integrate(1/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 64 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 32 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 20 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(7 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, \frac{16 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 20 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(7 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[1/32*(32*(a^4 + 2*a^3*b + a^2*b^2)*f*x*cos(f*x + e)^4 + 64*(a^3*b + 2*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^2 + 32*(a^2*b^2 + 2*a*b^3 + b^4)*f*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 20*a*b^3 + 8*b^4 + 2*(15*a^3*b + 20*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(3*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (7*a^2*b^2 + 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f), 1/16*(16*(a^4 + 2*a^3*b + a^2*b^2)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 2*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^2 + 16*(a^2*b^2 + 2*a*b^3 + b^4)*f*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 20*a*b^3 + 8*b^4 + 2*(15*a^3*b + 20*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 2*(3*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (7*a^2*b^2 + 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f)]","B",0
216,1,970,0,0.600271," ","integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{16 \, {\left(a^{5} - 4 \, a^{4} b - 11 \, a^{3} b^{2} - 6 \, a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{4} b - 4 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 6 \, a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{3} b^{2} - 4 \, a^{2} b^{3} - 11 \, a b^{4} - 6 \, b^{5}\right)} f x + {\left(35 \, a^{2} b^{3} + 56 \, a b^{4} + 24 \, b^{5} + {\left(35 \, a^{4} b + 56 \, a^{3} b^{2} + 24 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(35 \, a^{3} b^{2} + 56 \, a^{2} b^{3} + 24 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left(4 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(8 \, a^{4} b + 29 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, a^{3} b^{2} + 19 \, a^{2} b^{3} + 12 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 2 \, a^{5} b^{3} + a^{4} b^{4}\right)} f\right)}}, \frac{8 \, {\left(a^{5} - 4 \, a^{4} b - 11 \, a^{3} b^{2} - 6 \, a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 16 \, {\left(a^{4} b - 4 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 6 \, a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 8 \, {\left(a^{3} b^{2} - 4 \, a^{2} b^{3} - 11 \, a b^{4} - 6 \, b^{5}\right)} f x - {\left(35 \, a^{2} b^{3} + 56 \, a b^{4} + 24 \, b^{5} + {\left(35 \, a^{4} b + 56 \, a^{3} b^{2} + 24 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(35 \, a^{3} b^{2} + 56 \, a^{2} b^{3} + 24 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(4 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(8 \, a^{4} b + 29 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, a^{3} b^{2} + 19 \, a^{2} b^{3} + 12 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 2 \, a^{5} b^{3} + a^{4} b^{4}\right)} f\right)}}\right]"," ",0,"[1/32*(16*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*f*x*cos(f*x + e)^4 + 32*(a^4*b - 4*a^3*b^2 - 11*a^2*b^3 - 6*a*b^4)*f*x*cos(f*x + e)^2 + 16*(a^3*b^2 - 4*a^2*b^3 - 11*a*b^4 - 6*b^5)*f*x + (35*a^2*b^3 + 56*a*b^4 + 24*b^5 + (35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cos(f*x + e)^4 + 2*(35*a^3*b^2 + 56*a^2*b^3 + 24*a*b^4)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*(4*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^5 + (8*a^4*b + 29*a^3*b^2 + 18*a^2*b^3)*cos(f*x + e)^3 + (4*a^3*b^2 + 19*a^2*b^3 + 12*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^4 + 2*(a^7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^2 + (a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f), 1/16*(8*(a^5 - 4*a^4*b - 11*a^3*b^2 - 6*a^2*b^3)*f*x*cos(f*x + e)^4 + 16*(a^4*b - 4*a^3*b^2 - 11*a^2*b^3 - 6*a*b^4)*f*x*cos(f*x + e)^2 + 8*(a^3*b^2 - 4*a^2*b^3 - 11*a*b^4 - 6*b^5)*f*x - (35*a^2*b^3 + 56*a*b^4 + 24*b^5 + (35*a^4*b + 56*a^3*b^2 + 24*a^2*b^3)*cos(f*x + e)^4 + 2*(35*a^3*b^2 + 56*a^2*b^3 + 24*a*b^4)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + 2*(4*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^5 + (8*a^4*b + 29*a^3*b^2 + 18*a^2*b^3)*cos(f*x + e)^3 + (4*a^3*b^2 + 19*a^2*b^3 + 12*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^4 + 2*(a^7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^2 + (a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f)]","B",0
217,1,1129,0,0.841112," ","integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{6} - 2 \, a^{5} b + 9 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} f x \cos\left(f x + e\right)^{4} + 24 \, {\left(a^{5} b - 2 \, a^{4} b^{2} + 9 \, a^{3} b^{3} + 28 \, a^{2} b^{4} + 16 \, a b^{5}\right)} f x \cos\left(f x + e\right)^{2} + 12 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + 9 \, a^{2} b^{4} + 28 \, a b^{5} + 16 \, b^{6}\right)} f x + 3 \, {\left(21 \, a^{2} b^{4} + 36 \, a b^{5} + 16 \, b^{6} + {\left(21 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(21 \, a^{3} b^{3} + 36 \, a^{2} b^{4} + 16 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left(2 \, {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} \cos\left(f x + e\right)^{7} + {\left(3 \, a^{6} - 2 \, a^{5} b - 13 \, a^{4} b^{2} - 8 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(6 \, a^{5} b - 10 \, a^{4} b^{2} - 55 \, a^{3} b^{3} - 36 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} - 12 \, a^{2} b^{4} - 8 \, a b^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{9} + 2 \, a^{8} b + a^{7} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{8} b + 2 \, a^{7} b^{2} + a^{6} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b^{2} + 2 \, a^{6} b^{3} + a^{5} b^{4}\right)} f\right)}}, \frac{6 \, {\left(a^{6} - 2 \, a^{5} b + 9 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} f x \cos\left(f x + e\right)^{4} + 12 \, {\left(a^{5} b - 2 \, a^{4} b^{2} + 9 \, a^{3} b^{3} + 28 \, a^{2} b^{4} + 16 \, a b^{5}\right)} f x \cos\left(f x + e\right)^{2} + 6 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + 9 \, a^{2} b^{4} + 28 \, a b^{5} + 16 \, b^{6}\right)} f x + 3 \, {\left(21 \, a^{2} b^{4} + 36 \, a b^{5} + 16 \, b^{6} + {\left(21 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(21 \, a^{3} b^{3} + 36 \, a^{2} b^{4} + 16 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(2 \, {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} \cos\left(f x + e\right)^{7} + {\left(3 \, a^{6} - 2 \, a^{5} b - 13 \, a^{4} b^{2} - 8 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(6 \, a^{5} b - 10 \, a^{4} b^{2} - 55 \, a^{3} b^{3} - 36 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} - 12 \, a^{2} b^{4} - 8 \, a b^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{9} + 2 \, a^{8} b + a^{7} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{8} b + 2 \, a^{7} b^{2} + a^{6} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b^{2} + 2 \, a^{6} b^{3} + a^{5} b^{4}\right)} f\right)}}\right]"," ",0,"[1/32*(12*(a^6 - 2*a^5*b + 9*a^4*b^2 + 28*a^3*b^3 + 16*a^2*b^4)*f*x*cos(f*x + e)^4 + 24*(a^5*b - 2*a^4*b^2 + 9*a^3*b^3 + 28*a^2*b^4 + 16*a*b^5)*f*x*cos(f*x + e)^2 + 12*(a^4*b^2 - 2*a^3*b^3 + 9*a^2*b^4 + 28*a*b^5 + 16*b^6)*f*x + 3*(21*a^2*b^4 + 36*a*b^5 + 16*b^6 + (21*a^4*b^2 + 36*a^3*b^3 + 16*a^2*b^4)*cos(f*x + e)^4 + 2*(21*a^3*b^3 + 36*a^2*b^4 + 16*a*b^5)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*(2*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^7 + (3*a^6 - 2*a^5*b - 13*a^4*b^2 - 8*a^3*b^3)*cos(f*x + e)^5 + (6*a^5*b - 10*a^4*b^2 - 55*a^3*b^3 - 36*a^2*b^4)*cos(f*x + e)^3 + 3*(a^4*b^2 - 2*a^3*b^3 - 12*a^2*b^4 - 8*a*b^5)*cos(f*x + e))*sin(f*x + e))/((a^9 + 2*a^8*b + a^7*b^2)*f*cos(f*x + e)^4 + 2*(a^8*b + 2*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^2 + (a^7*b^2 + 2*a^6*b^3 + a^5*b^4)*f), 1/16*(6*(a^6 - 2*a^5*b + 9*a^4*b^2 + 28*a^3*b^3 + 16*a^2*b^4)*f*x*cos(f*x + e)^4 + 12*(a^5*b - 2*a^4*b^2 + 9*a^3*b^3 + 28*a^2*b^4 + 16*a*b^5)*f*x*cos(f*x + e)^2 + 6*(a^4*b^2 - 2*a^3*b^3 + 9*a^2*b^4 + 28*a*b^5 + 16*b^6)*f*x + 3*(21*a^2*b^4 + 36*a*b^5 + 16*b^6 + (21*a^4*b^2 + 36*a^3*b^3 + 16*a^2*b^4)*cos(f*x + e)^4 + 2*(21*a^3*b^3 + 36*a^2*b^4 + 16*a*b^5)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + 2*(2*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^7 + (3*a^6 - 2*a^5*b - 13*a^4*b^2 - 8*a^3*b^3)*cos(f*x + e)^5 + (6*a^5*b - 10*a^4*b^2 - 55*a^3*b^3 - 36*a^2*b^4)*cos(f*x + e)^3 + 3*(a^4*b^2 - 2*a^3*b^3 - 12*a^2*b^4 - 8*a*b^5)*cos(f*x + e))*sin(f*x + e))/((a^9 + 2*a^8*b + a^7*b^2)*f*cos(f*x + e)^4 + 2*(a^8*b + 2*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^2 + (a^7*b^2 + 2*a^6*b^3 + a^5*b^4)*f)]","B",0
218,1,1296,0,0.705368," ","integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{6 \, {\left(5 \, a^{7} - 8 \, a^{6} b + 17 \, a^{5} b^{2} - 82 \, a^{4} b^{3} - 272 \, a^{3} b^{4} - 160 \, a^{2} b^{5}\right)} f x \cos\left(f x + e\right)^{4} + 12 \, {\left(5 \, a^{6} b - 8 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 82 \, a^{3} b^{4} - 272 \, a^{2} b^{5} - 160 \, a b^{6}\right)} f x \cos\left(f x + e\right)^{2} + 6 \, {\left(5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 82 \, a^{2} b^{5} - 272 \, a b^{6} - 160 \, b^{7}\right)} f x + 3 \, {\left(99 \, a^{2} b^{5} + 176 \, a b^{6} + 80 \, b^{7} + {\left(99 \, a^{4} b^{3} + 176 \, a^{3} b^{4} + 80 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(99 \, a^{3} b^{4} + 176 \, a^{2} b^{5} + 80 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 2 \, {\left(8 \, {\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} \cos\left(f x + e\right)^{9} + 10 \, {\left(a^{7} - 3 \, a^{5} b^{2} - 2 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{7} + {\left(15 \, a^{7} - 4 \, a^{6} b + 27 \, a^{5} b^{2} + 126 \, a^{4} b^{3} + 80 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(15 \, a^{6} b - 19 \, a^{5} b^{2} + 43 \, a^{4} b^{3} + 266 \, a^{3} b^{4} + 180 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} + 116 \, a^{2} b^{5} + 80 \, a b^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{96 \, {\left({\left(a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right)} f\right)}}, \frac{3 \, {\left(5 \, a^{7} - 8 \, a^{6} b + 17 \, a^{5} b^{2} - 82 \, a^{4} b^{3} - 272 \, a^{3} b^{4} - 160 \, a^{2} b^{5}\right)} f x \cos\left(f x + e\right)^{4} + 6 \, {\left(5 \, a^{6} b - 8 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 82 \, a^{3} b^{4} - 272 \, a^{2} b^{5} - 160 \, a b^{6}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 82 \, a^{2} b^{5} - 272 \, a b^{6} - 160 \, b^{7}\right)} f x - 3 \, {\left(99 \, a^{2} b^{5} + 176 \, a b^{6} + 80 \, b^{7} + {\left(99 \, a^{4} b^{3} + 176 \, a^{3} b^{4} + 80 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(99 \, a^{3} b^{4} + 176 \, a^{2} b^{5} + 80 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(8 \, {\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} \cos\left(f x + e\right)^{9} + 10 \, {\left(a^{7} - 3 \, a^{5} b^{2} - 2 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{7} + {\left(15 \, a^{7} - 4 \, a^{6} b + 27 \, a^{5} b^{2} + 126 \, a^{4} b^{3} + 80 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(15 \, a^{6} b - 19 \, a^{5} b^{2} + 43 \, a^{4} b^{3} + 266 \, a^{3} b^{4} + 180 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(5 \, a^{5} b^{2} - 8 \, a^{4} b^{3} + 17 \, a^{3} b^{4} + 116 \, a^{2} b^{5} + 80 \, a b^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right)} f\right)}}\right]"," ",0,"[1/96*(6*(5*a^7 - 8*a^6*b + 17*a^5*b^2 - 82*a^4*b^3 - 272*a^3*b^4 - 160*a^2*b^5)*f*x*cos(f*x + e)^4 + 12*(5*a^6*b - 8*a^5*b^2 + 17*a^4*b^3 - 82*a^3*b^4 - 272*a^2*b^5 - 160*a*b^6)*f*x*cos(f*x + e)^2 + 6*(5*a^5*b^2 - 8*a^4*b^3 + 17*a^3*b^4 - 82*a^2*b^5 - 272*a*b^6 - 160*b^7)*f*x + 3*(99*a^2*b^5 + 176*a*b^6 + 80*b^7 + (99*a^4*b^3 + 176*a^3*b^4 + 80*a^2*b^5)*cos(f*x + e)^4 + 2*(99*a^3*b^4 + 176*a^2*b^5 + 80*a*b^6)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 2*(8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 10*(a^7 - 3*a^5*b^2 - 2*a^4*b^3)*cos(f*x + e)^7 + (15*a^7 - 4*a^6*b + 27*a^5*b^2 + 126*a^4*b^3 + 80*a^3*b^4)*cos(f*x + e)^5 + 2*(15*a^6*b - 19*a^5*b^2 + 43*a^4*b^3 + 266*a^3*b^4 + 180*a^2*b^5)*cos(f*x + e)^3 + 3*(5*a^5*b^2 - 8*a^4*b^3 + 17*a^3*b^4 + 116*a^2*b^5 + 80*a*b^6)*cos(f*x + e))*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*f), 1/48*(3*(5*a^7 - 8*a^6*b + 17*a^5*b^2 - 82*a^4*b^3 - 272*a^3*b^4 - 160*a^2*b^5)*f*x*cos(f*x + e)^4 + 6*(5*a^6*b - 8*a^5*b^2 + 17*a^4*b^3 - 82*a^3*b^4 - 272*a^2*b^5 - 160*a*b^6)*f*x*cos(f*x + e)^2 + 3*(5*a^5*b^2 - 8*a^4*b^3 + 17*a^3*b^4 - 82*a^2*b^5 - 272*a*b^6 - 160*b^7)*f*x - 3*(99*a^2*b^5 + 176*a*b^6 + 80*b^7 + (99*a^4*b^3 + 176*a^3*b^4 + 80*a^2*b^5)*cos(f*x + e)^4 + 2*(99*a^3*b^4 + 176*a^2*b^5 + 80*a*b^6)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) + (8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 10*(a^7 - 3*a^5*b^2 - 2*a^4*b^3)*cos(f*x + e)^7 + (15*a^7 - 4*a^6*b + 27*a^5*b^2 + 126*a^4*b^3 + 80*a^3*b^4)*cos(f*x + e)^5 + 2*(15*a^6*b - 19*a^5*b^2 + 43*a^4*b^3 + 266*a^3*b^4 + 180*a^2*b^5)*cos(f*x + e)^3 + 3*(5*a^5*b^2 - 8*a^4*b^3 + 17*a^3*b^4 + 116*a^2*b^5 + 80*a*b^6)*cos(f*x + e))*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*f)]","A",0
219,1,1323,0,0.607735," ","integrate(1/(a+b*sec(d*x+c)^2)^4,x, algorithm=""fricas"")","\left[\frac{192 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} d x \cos\left(d x + c\right)^{6} + 576 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} d x \cos\left(d x + c\right)^{4} + 576 \, {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} d x \cos\left(d x + c\right)^{2} + 192 \, {\left(a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right)} d x + 3 \, {\left({\left(35 \, a^{6} + 70 \, a^{5} b + 56 \, a^{4} b^{2} + 16 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} + 35 \, a^{3} b^{3} + 70 \, a^{2} b^{4} + 56 \, a b^{5} + 16 \, b^{6} + 3 \, {\left(35 \, a^{5} b + 70 \, a^{4} b^{2} + 56 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(35 \, a^{4} b^{2} + 70 \, a^{3} b^{3} + 56 \, a^{2} b^{4} + 16 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(d x + c\right) + b^{2}}{a^{2} \cos\left(d x + c\right)^{4} + 2 \, a b \cos\left(d x + c\right)^{2} + b^{2}}\right) - 4 \, {\left({\left(87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(68 \, a^{4} b^{2} + 83 \, a^{3} b^{3} + 30 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(19 \, a^{3} b^{3} + 22 \, a^{2} b^{4} + 8 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{192 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} + 3 \, {\left(a^{9} b + 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} b^{3} + 3 \, a^{6} b^{4} + 3 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}, \frac{96 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} d x \cos\left(d x + c\right)^{6} + 288 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} d x \cos\left(d x + c\right)^{4} + 288 \, {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} d x \cos\left(d x + c\right)^{2} + 96 \, {\left(a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right)} d x + 3 \, {\left({\left(35 \, a^{6} + 70 \, a^{5} b + 56 \, a^{4} b^{2} + 16 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} + 35 \, a^{3} b^{3} + 70 \, a^{2} b^{4} + 56 \, a b^{5} + 16 \, b^{6} + 3 \, {\left(35 \, a^{5} b + 70 \, a^{4} b^{2} + 56 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(35 \, a^{4} b^{2} + 70 \, a^{3} b^{3} + 56 \, a^{2} b^{4} + 16 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left({\left(87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(68 \, a^{4} b^{2} + 83 \, a^{3} b^{3} + 30 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(19 \, a^{3} b^{3} + 22 \, a^{2} b^{4} + 8 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{96 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} + 3 \, {\left(a^{9} b + 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} b^{3} + 3 \, a^{6} b^{4} + 3 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}\right]"," ",0,"[1/192*(192*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*x*cos(d*x + c)^6 + 576*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 576*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 + 192*(a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*d*x + 3*((35*a^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x + c)^6 + 35*a^3*b^3 + 70*a^2*b^4 + 56*a*b^5 + 16*b^6 + 3*(35*a^5*b + 70*a^4*b^2 + 56*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*a^4*b^2 + 70*a^3*b^3 + 56*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(3*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + b^2)/(a^2*cos(d*x + c)^4 + 2*a*b*cos(d*x + c)^2 + b^2)) - 4*((87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)*cos(d*x + c)^5 + 2*(68*a^4*b^2 + 83*a^3*b^3 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(19*a^3*b^3 + 22*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 + 3*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6)*d), 1/96*(96*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*x*cos(d*x + c)^6 + 288*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 288*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 + 96*(a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*d*x + 3*((35*a^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x + c)^6 + 35*a^3*b^3 + 70*a^2*b^4 + 56*a*b^5 + 16*b^6 + 3*(35*a^5*b + 70*a^4*b^2 + 56*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*a^4*b^2 + 70*a^3*b^3 + 56*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - b)*sqrt(b/(a + b))/(b*cos(d*x + c)*sin(d*x + c))) - 2*((87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)*cos(d*x + c)^5 + 2*(68*a^4*b^2 + 83*a^3*b^3 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(19*a^3*b^3 + 22*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 + 3*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6)*d)]","B",0
220,1,100,0,0.451430," ","integrate((a-a*sec(d*x+c)^2)^(7/2),x, algorithm=""fricas"")","-\frac{{\left(12 \, a^{3} \cos\left(d x + c\right)^{6} \log\left(-\cos\left(d x + c\right)\right) + 18 \, a^{3} \cos\left(d x + c\right)^{4} - 9 \, a^{3} \cos\left(d x + c\right)^{2} + 2 \, a^{3}\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}}}{12 \, d \cos\left(d x + c\right)^{5} \sin\left(d x + c\right)}"," ",0,"-1/12*(12*a^3*cos(d*x + c)^6*log(-cos(d*x + c)) + 18*a^3*cos(d*x + c)^4 - 9*a^3*cos(d*x + c)^2 + 2*a^3)*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/(d*cos(d*x + c)^5*sin(d*x + c))","A",0
221,1,87,0,0.889668," ","integrate((a-a*sec(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(4 \, a^{2} \cos\left(d x + c\right)^{4} \log\left(-\cos\left(d x + c\right)\right) + 4 \, a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}}}{4 \, d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right)}"," ",0,"-1/4*(4*a^2*cos(d*x + c)^4*log(-cos(d*x + c)) + 4*a^2*cos(d*x + c)^2 - a^2)*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/(d*cos(d*x + c)^3*sin(d*x + c))","A",0
222,1,68,0,1.334340," ","integrate((a-a*sec(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, a \cos\left(d x + c\right)^{2} \log\left(-\cos\left(d x + c\right)\right) + a\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}}}{2 \, d \cos\left(d x + c\right) \sin\left(d x + c\right)}"," ",0,"-1/2*(2*a*cos(d*x + c)^2*log(-cos(d*x + c)) + a)*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/(d*cos(d*x + c)*sin(d*x + c))","A",0
223,1,53,0,0.625975," ","integrate((a-a*sec(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}} \cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right)}{d \sin\left(d x + c\right)}"," ",0,"-sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)*cos(d*x + c)*log(-cos(d*x + c))/(d*sin(d*x + c))","A",0
224,1,56,0,2.867533," ","integrate(1/(a-a*sec(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}} \cos\left(d x + c\right) \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right)}{a d \sin\left(d x + c\right)}"," ",0,"-sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)*cos(d*x + c)*log(1/2*sin(d*x + c))/(a*d*sin(d*x + c))","A",0
225,1,94,0,0.837386," ","integrate(1/(a-a*sec(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, {\left(\cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}}}{2 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/2*(2*(cos(d*x + c)^3 - cos(d*x + c))*log(1/2*sin(d*x + c)) - cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))","A",0
226,1,125,0,1.655192," ","integrate(1/(a-a*sec(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(4 \, \cos\left(d x + c\right)^{3} - 4 \, {\left(\cos\left(d x + c\right)^{5} - 2 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 3 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}}}{4 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)} \sin\left(d x + c\right)}"," ",0,"1/4*(4*cos(d*x + c)^3 - 4*(cos(d*x + c)^5 - 2*cos(d*x + c)^3 + cos(d*x + c))*log(1/2*sin(d*x + c)) - 3*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/((a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))","A",0
227,1,162,0,0.644215," ","integrate(1/(a-a*sec(d*x+c)^2)^(7/2),x, algorithm=""fricas"")","\frac{{\left(18 \, \cos\left(d x + c\right)^{5} - 27 \, \cos\left(d x + c\right)^{3} - 12 \, {\left(\cos\left(d x + c\right)^{7} - 3 \, \cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 11 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a}{\cos\left(d x + c\right)^{2}}}}{12 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)}"," ",0,"1/12*(18*cos(d*x + c)^5 - 27*cos(d*x + c)^3 - 12*(cos(d*x + c)^7 - 3*cos(d*x + c)^5 + 3*cos(d*x + c)^3 - cos(d*x + c))*log(1/2*sin(d*x + c)) + 11*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a)/cos(d*x + c)^2)/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))","A",0
228,0,0,0,0.635195," ","integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{5}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)^5, x)","F",0
229,0,0,0,1.198176," ","integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)^3, x)","F",0
230,0,0,0,0.830641," ","integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right), x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e), x)","F",0
231,0,0,0,0.697609," ","integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right), x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e), x)","F",0
232,0,0,0,1.432242," ","integrate(cos(f*x+e)^3*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^3, x)","F",0
233,0,0,0,0.806747," ","integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{5}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^5, x)","F",0
234,1,468,0,1.980188," ","integrate(sec(f*x+e)^6*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right)} \sqrt{b} \cos\left(f x + e\right)^{5} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{2} b - 4 \, a b^{2} - 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, b^{3} - 2 \, {\left(a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, b^{3} f \cos\left(f x + e\right)^{5}}, \frac{3 \, {\left(a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - 2 \, {\left({\left(3 \, a^{2} b - 4 \, a b^{2} - 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, b^{3} - 2 \, {\left(a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, b^{3} f \cos\left(f x + e\right)^{5}}\right]"," ",0,"[1/192*(3*(a^3 - a^2*b + 3*a*b^2 + 5*b^3)*sqrt(b)*cos(f*x + e)^5*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^2*b - 4*a*b^2 - 15*b^3)*cos(f*x + e)^4 - 8*b^3 - 2*(a*b^2 + 5*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^5), 1/96*(3*(a^3 - a^2*b + 3*a*b^2 + 5*b^3)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^5 - 2*((3*a^2*b - 4*a*b^2 - 15*b^3)*cos(f*x + e)^4 - 8*b^3 - 2*(a*b^2 + 5*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^5)]","A",0
235,1,390,0,0.846251," ","integrate(sec(f*x+e)^4*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b^{2} f \cos\left(f x + e\right)^{3}}, -\frac{{\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - 2 \, {\left({\left(a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b^{2} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[-1/32*((a^2 - 2*a*b - 3*b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((a*b + 3*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^3), -1/16*((a^2 - 2*a*b - 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 - 2*((a*b + 3*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^3)]","A",0
236,1,320,0,0.739601," ","integrate(sec(f*x+e)^2*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, b f \cos\left(f x + e\right)}, \frac{{\left(a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, b f \cos\left(f x + e\right)}\right]"," ",0,"[1/8*((a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)), 1/4*((a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e))]","B",0
237,1,1227,0,1.079838," ","integrate((a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 2 \, \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, f}, \frac{4 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, f}\right]"," ",0,"[1/8*(sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 2*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, 1/8*(4*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/f]","B",0
238,1,499,0,1.354291," ","integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - \sqrt{-a} {\left(a + b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, a f}, \frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a + b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, a f}\right]"," ",0,"[1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - sqrt(-a)*(a + b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a*f), 1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - (a + b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/(a*f)]","B",0
239,1,567,0,0.838339," ","integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a^{2} f}, -\frac{{\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a^{2} f}\right]"," ",0,"[1/64*((3*a^2 + 2*a*b - b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*a^2*cos(f*x + e)^3 + (3*a^2 + a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f), -1/32*((3*a^2 + 2*a*b - b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*a^2*cos(f*x + e)^3 + (3*a^2 + a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f)]","A",0
240,1,641,0,2.012474," ","integrate(cos(f*x+e)^6*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(5 \, a^{3} + 3 \, a^{2} b - a b^{2} + b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} + 4 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{3} f}, -\frac{3 \, {\left(5 \, a^{3} + 3 \, a^{2} b - a b^{2} + b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} + 4 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{3} f}\right]"," ",0,"[-1/384*(3*(5*a^3 + 3*a^2*b - a*b^2 + b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(8*a^3*cos(f*x + e)^5 + 2*(5*a^3 + a^2*b)*cos(f*x + e)^3 + (15*a^3 + 4*a^2*b - 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f), -1/192*(3*(5*a^3 + 3*a^2*b - a*b^2 + b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*a^3*cos(f*x + e)^5 + 2*(5*a^3 + a^2*b)*cos(f*x + e)^3 + (15*a^3 + 4*a^2*b - 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f)]","A",0
241,0,0,0,1.041310," ","integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{7} + a \sec\left(f x + e\right)^{5}\right)} \sqrt{b \sec\left(f x + e\right)^{2} + a}, x\right)"," ",0,"integral((b*sec(f*x + e)^7 + a*sec(f*x + e)^5)*sqrt(b*sec(f*x + e)^2 + a), x)","F",0
242,0,0,0,0.736275," ","integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{5} + a \sec\left(f x + e\right)^{3}\right)} \sqrt{b \sec\left(f x + e\right)^{2} + a}, x\right)"," ",0,"integral((b*sec(f*x + e)^5 + a*sec(f*x + e)^3)*sqrt(b*sec(f*x + e)^2 + a), x)","F",0
243,0,0,0,0.579390," ","integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{3} + a \sec\left(f x + e\right)\right)} \sqrt{b \sec\left(f x + e\right)^{2} + a}, x\right)"," ",0,"integral((b*sec(f*x + e)^3 + a*sec(f*x + e))*sqrt(b*sec(f*x + e)^2 + a), x)","F",0
244,0,0,0,0.577714," ","integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right) \sec\left(f x + e\right)^{2} + a \cos\left(f x + e\right)\right)} \sqrt{b \sec\left(f x + e\right)^{2} + a}, x\right)"," ",0,"integral((b*cos(f*x + e)*sec(f*x + e)^2 + a*cos(f*x + e))*sqrt(b*sec(f*x + e)^2 + a), x)","F",0
245,0,0,0,0.484992," ","integrate(cos(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{3} \sec\left(f x + e\right)^{2} + a \cos\left(f x + e\right)^{3}\right)} \sqrt{b \sec\left(f x + e\right)^{2} + a}, x\right)"," ",0,"integral((b*cos(f*x + e)^3*sec(f*x + e)^2 + a*cos(f*x + e)^3)*sqrt(b*sec(f*x + e)^2 + a), x)","F",0
246,0,0,0,0.684178," ","integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{5} \sec\left(f x + e\right)^{2} + a \cos\left(f x + e\right)^{5}\right)} \sqrt{b \sec\left(f x + e\right)^{2} + a}, x\right)"," ",0,"integral((b*cos(f*x + e)^5*sec(f*x + e)^2 + a*cos(f*x + e)^5)*sqrt(b*sec(f*x + e)^2 + a), x)","F",0
247,1,566,0,8.545011," ","integrate(sec(f*x+e)^6*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(3 \, a^{4} - 4 \, a^{3} b + 18 \, a^{2} b^{2} + 60 \, a b^{3} + 35 \, b^{4}\right)} \sqrt{b} \cos\left(f x + e\right)^{7} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(9 \, a^{3} b - 15 \, a^{2} b^{2} - 145 \, a b^{3} - 105 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(3 \, a^{2} b^{2} + 46 \, a b^{3} + 35 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 48 \, b^{4} - 8 \, {\left(9 \, a b^{3} + 7 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{1536 \, b^{3} f \cos\left(f x + e\right)^{7}}, \frac{3 \, {\left(3 \, a^{4} - 4 \, a^{3} b + 18 \, a^{2} b^{2} + 60 \, a b^{3} + 35 \, b^{4}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{7} - 2 \, {\left({\left(9 \, a^{3} b - 15 \, a^{2} b^{2} - 145 \, a b^{3} - 105 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(3 \, a^{2} b^{2} + 46 \, a b^{3} + 35 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 48 \, b^{4} - 8 \, {\left(9 \, a b^{3} + 7 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{768 \, b^{3} f \cos\left(f x + e\right)^{7}}\right]"," ",0,"[1/1536*(3*(3*a^4 - 4*a^3*b + 18*a^2*b^2 + 60*a*b^3 + 35*b^4)*sqrt(b)*cos(f*x + e)^7*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((9*a^3*b - 15*a^2*b^2 - 145*a*b^3 - 105*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 + 46*a*b^3 + 35*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 + 7*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7), 1/768*(3*(3*a^4 - 4*a^3*b + 18*a^2*b^2 + 60*a*b^3 + 35*b^4)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^7 - 2*((9*a^3*b - 15*a^2*b^2 - 145*a*b^3 - 105*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 + 46*a*b^3 + 35*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 + 7*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7)]","A",0
248,1,470,0,1.990473," ","integrate(sec(f*x+e)^4*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{b} \cos\left(f x + e\right)^{5} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{2} b + 22 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, b^{3} + 2 \, {\left(7 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, b^{2} f \cos\left(f x + e\right)^{5}}, -\frac{3 \, {\left(a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - 2 \, {\left({\left(3 \, a^{2} b + 22 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, b^{3} + 2 \, {\left(7 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, b^{2} f \cos\left(f x + e\right)^{5}}\right]"," ",0,"[-1/192*(3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*sqrt(b)*cos(f*x + e)^5*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^2*b + 22*a*b^2 + 15*b^3)*cos(f*x + e)^4 + 8*b^3 + 2*(7*a*b^2 + 5*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^5), -1/96*(3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^5 - 2*((3*a^2*b + 22*a*b^2 + 15*b^3)*cos(f*x + e)^4 + 8*b^3 + 2*(7*a*b^2 + 5*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^5)]","A",0
249,1,390,0,0.954369," ","integrate(sec(f*x+e)^2*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(5 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b f \cos\left(f x + e\right)^{3}}, \frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + 2 \, {\left({\left(5 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[1/32*(3*(a^2 + 2*a*b + b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*((5*a*b + 3*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)^3), 1/16*(3*(a^2 + 2*a*b + b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 2*((5*a*b + 3*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)^3)]","A",0
250,1,1457,0,1.143002," ","integrate((a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + {\left(3 \, a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, \frac{2 \, {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + \sqrt{-a} a \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{2 \, a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, f \cos\left(f x + e\right)}\right]"," ",0,"[1/8*(sqrt(-a)*a*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + (3*a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), 1/8*(2*(3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + sqrt(-a)*a*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/8*(2*a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/4*(a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e))]","B",0
251,1,1403,0,1.365688," ","integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{-a} {\left(a + 3 \, b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 4 \, b^{\frac{3}{2}} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{16 \, f}, \frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + 8 \, \sqrt{-b} b \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + \sqrt{-a} {\left(a + 3 \, b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, f}, \frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a + 3 \, b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 2 \, b^{\frac{3}{2}} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, f}, \frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a + 3 \, b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, \sqrt{-b} b \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, f}\right]"," ",0,"[1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + sqrt(-a)*(a + 3*b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 4*b^(3/2)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, 1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + 8*sqrt(-b)*b*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + sqrt(-a)*(a + 3*b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/f, 1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - (a + 3*b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 2*b^(3/2)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, 1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - (a + 3*b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*sqrt(-b)*b*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/f]","B",0
252,1,563,0,0.888255," ","integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a f}, -\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a f}\right]"," ",0,"[-1/64*(3*(a^2 + 2*a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*a^2*cos(f*x + e)^3 + (3*a^2 + 5*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f), -1/32*(3*(a^2 + 2*a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*a^2*cos(f*x + e)^3 + (3*a^2 + 5*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f)]","B",0
253,1,647,0,2.241884," ","integrate(cos(f*x+e)^6*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{2} f}, -\frac{3 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} + 7 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{2} f}\right]"," ",0,"[1/384*(3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*a^3*cos(f*x + e)^5 + 2*(5*a^3 + 7*a^2*b)*cos(f*x + e)^3 + (15*a^3 + 22*a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f), -1/192*(3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*a^3*cos(f*x + e)^5 + 2*(5*a^3 + 7*a^2*b)*cos(f*x + e)^3 + (15*a^3 + 22*a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^2*f)]","A",0
254,1,1611,0,2.250184," ","integrate((a+b*sec(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a} a^{2} \cos\left(d x + c\right)^{3} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(d x + c\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)\right) + {\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \sqrt{b} \cos\left(d x + c\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} + 2 \, b \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right) + 8 \, b^{2}}{\cos\left(d x + c\right)^{4}}\right) + 4 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)}{32 \, d \cos\left(d x + c\right)^{3}}, \frac{2 \, \sqrt{-a} a^{2} \cos\left(d x + c\right)^{3} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(d x + c\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)\right) + {\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} + 2 \, b \cos\left(d x + c\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}}}{2 \, {\left(a b \cos\left(d x + c\right)^{2} + b^{2}\right)} \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} + 2 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)}{16 \, d \cos\left(d x + c\right)^{3}}, -\frac{8 \, a^{\frac{5}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} - {\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \sqrt{b} \cos\left(d x + c\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} + 2 \, b \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right) + 8 \, b^{2}}{\cos\left(d x + c\right)^{4}}\right) - 4 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)}{32 \, d \cos\left(d x + c\right)^{3}}, -\frac{4 \, a^{\frac{5}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} - {\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} + 2 \, b \cos\left(d x + c\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}}}{2 \, {\left(a b \cos\left(d x + c\right)^{2} + b^{2}\right)} \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} - 2 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)}{16 \, d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[1/32*(4*sqrt(-a)*a^2*cos(d*x + c)^3*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 - a^3*b)*cos(d*x + c)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(d*x + c)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 - a^2*b)*cos(d*x + c)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(d*x + c)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c)) + (15*a^2 + 10*a*b + 3*b^2)*sqrt(b)*cos(d*x + c)^3*log(((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 + 8*(a*b - b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 + 2*b*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c) + 8*b^2)/cos(d*x + c)^4) + 4*(3*(3*a*b + b^2)*cos(d*x + c)^2 + 2*b^2)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^3), 1/16*(2*sqrt(-a)*a^2*cos(d*x + c)^3*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 - a^3*b)*cos(d*x + c)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(d*x + c)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 - a^2*b)*cos(d*x + c)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(d*x + c)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c)) + (15*a^2 + 10*a*b + 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^3 + 2*b*cos(d*x + c))*sqrt(-b)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)/((a*b*cos(d*x + c)^2 + b^2)*sin(d*x + c)))*cos(d*x + c)^3 + 2*(3*(3*a*b + b^2)*cos(d*x + c)^2 + 2*b^2)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^3), -1/32*(8*a^(5/2)*arctan(1/4*(8*a^2*cos(d*x + c)^5 - 8*(a^2 - a*b)*cos(d*x + c)^3 + (a^2 - 6*a*b + b^2)*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)/((2*a^3*cos(d*x + c)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(d*x + c)^2)*sin(d*x + c)))*cos(d*x + c)^3 - (15*a^2 + 10*a*b + 3*b^2)*sqrt(b)*cos(d*x + c)^3*log(((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 + 8*(a*b - b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 + 2*b*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c) + 8*b^2)/cos(d*x + c)^4) - 4*(3*(3*a*b + b^2)*cos(d*x + c)^2 + 2*b^2)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^3), -1/16*(4*a^(5/2)*arctan(1/4*(8*a^2*cos(d*x + c)^5 - 8*(a^2 - a*b)*cos(d*x + c)^3 + (a^2 - 6*a*b + b^2)*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)/((2*a^3*cos(d*x + c)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(d*x + c)^2)*sin(d*x + c)))*cos(d*x + c)^3 - (15*a^2 + 10*a*b + 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^3 + 2*b*cos(d*x + c))*sqrt(-b)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)/((a*b*cos(d*x + c)^2 + b^2)*sin(d*x + c)))*cos(d*x + c)^3 - 2*(3*(3*a*b + b^2)*cos(d*x + c)^2 + 2*b^2)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^3)]","B",0
255,1,160,0,0.526743," ","integrate((1+sec(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{\arctan\left(\frac{\sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} \cos\left(x\right)^{3} \sin\left(x\right) + \cos\left(x\right) \sin\left(x\right)}{\cos\left(x\right)^{4} + \cos\left(x\right)^{2} - 1}\right) \cos\left(x\right) - \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right) \cos\left(x\right) + 2 \, \cos\left(x\right) \log\left(\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right) + {\left(\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} + 1\right) - 2 \, \cos\left(x\right) \log\left(\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right) + {\left(\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} + 1\right) + \sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} \sin\left(x\right)}{2 \, \cos\left(x\right)}"," ",0,"1/2*(arctan((sqrt((cos(x)^2 + 1)/cos(x)^2)*cos(x)^3*sin(x) + cos(x)*sin(x))/(cos(x)^4 + cos(x)^2 - 1))*cos(x) - arctan(sin(x)/cos(x))*cos(x) + 2*cos(x)*log(cos(x)^2 + cos(x)*sin(x) + (cos(x)^2 + cos(x)*sin(x))*sqrt((cos(x)^2 + 1)/cos(x)^2) + 1) - 2*cos(x)*log(cos(x)^2 - cos(x)*sin(x) + (cos(x)^2 - cos(x)*sin(x))*sqrt((cos(x)^2 + 1)/cos(x)^2) + 1) + sqrt((cos(x)^2 + 1)/cos(x)^2)*sin(x))/cos(x)","B",0
256,1,131,0,0.574359," ","integrate((1+sec(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \arctan\left(\frac{\sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} \cos\left(x\right)^{3} \sin\left(x\right) + \cos\left(x\right) \sin\left(x\right)}{\cos\left(x\right)^{4} + \cos\left(x\right)^{2} - 1}\right) - \frac{1}{2} \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right) + \frac{1}{2} \, \log\left(\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right) + {\left(\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} + 1\right) - \frac{1}{2} \, \log\left(\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right) + {\left(\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} + 1\right)"," ",0,"1/2*arctan((sqrt((cos(x)^2 + 1)/cos(x)^2)*cos(x)^3*sin(x) + cos(x)*sin(x))/(cos(x)^4 + cos(x)^2 - 1)) - 1/2*arctan(sin(x)/cos(x)) + 1/2*log(cos(x)^2 + cos(x)*sin(x) + (cos(x)^2 + cos(x)*sin(x))*sqrt((cos(x)^2 + 1)/cos(x)^2) + 1) - 1/2*log(cos(x)^2 - cos(x)*sin(x) + (cos(x)^2 - cos(x)*sin(x))*sqrt((cos(x)^2 + 1)/cos(x)^2) + 1)","B",0
257,0,0,0,0.549493," ","integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sec\left(f x + e\right)^{5}}{\sqrt{b \sec\left(f x + e\right)^{2} + a}}, x\right)"," ",0,"integral(sec(f*x + e)^5/sqrt(b*sec(f*x + e)^2 + a), x)","F",0
258,0,0,0,0.480455," ","integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sec\left(f x + e\right)^{3}}{\sqrt{b \sec\left(f x + e\right)^{2} + a}}, x\right)"," ",0,"integral(sec(f*x + e)^3/sqrt(b*sec(f*x + e)^2 + a), x)","F",0
259,0,0,0,0.485888," ","integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sec\left(f x + e\right)}{\sqrt{b \sec\left(f x + e\right)^{2} + a}}, x\right)"," ",0,"integral(sec(f*x + e)/sqrt(b*sec(f*x + e)^2 + a), x)","F",0
260,0,0,0,0.494555," ","integrate(cos(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\cos\left(f x + e\right)}{\sqrt{b \sec\left(f x + e\right)^{2} + a}}, x\right)"," ",0,"integral(cos(f*x + e)/sqrt(b*sec(f*x + e)^2 + a), x)","F",0
261,0,0,0,0.594971," ","integrate(cos(f*x+e)^3/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\cos\left(f x + e\right)^{3}}{\sqrt{b \sec\left(f x + e\right)^{2} + a}}, x\right)"," ",0,"integral(cos(f*x + e)^3/sqrt(b*sec(f*x + e)^2 + a), x)","F",0
262,0,0,0,0.562369," ","integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\cos\left(f x + e\right)^{5}}{\sqrt{b \sec\left(f x + e\right)^{2} + a}}, x\right)"," ",0,"integral(cos(f*x + e)^5/sqrt(b*sec(f*x + e)^2 + a), x)","F",0
263,1,396,0,0.864857," ","integrate(sec(f*x+e)^6/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left(3 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b^{3} f \cos\left(f x + e\right)^{3}}, \frac{{\left(3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b^{3} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[1/32*((3*a^2 - 2*a*b + 3*b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*(3*(a*b - b^2)*cos(f*x + e)^2 - 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^3), 1/16*((3*a^2 - 2*a*b + 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 - 2*(3*(a*b - b^2)*cos(f*x + e)^2 - 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^3)]","A",0
264,1,324,0,0.742531," ","integrate(sec(f*x+e)^4/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, b^{2} f \cos\left(f x + e\right)}, -\frac{{\left(a - b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, b^{2} f \cos\left(f x + e\right)}\right]"," ",0,"[-1/8*((a - b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)), -1/4*((a - b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e))]","B",0
265,1,215,0,0.595593," ","integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, \sqrt{b} f}, \frac{\sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{2 \, b f}\right]"," ",0,"[1/4*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)/(sqrt(b)*f), 1/2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))/(b*f)]","B",0
266,1,408,0,0.640777," ","integrate(1/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, a f}, -\frac{\arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, \sqrt{a} f}\right]"," ",0,"[-1/8*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f), -1/4*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))/(sqrt(a)*f)]","B",0
267,1,502,0,0.650306," ","integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{-a} {\left(a - b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, a^{2} f}, \frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a - b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, a^{2} f}\right]"," ",0,"[1/16*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + sqrt(-a)*(a - b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a^2*f), 1/8*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - (a - b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/(a^2*f)]","B",0
268,1,567,0,1.188195," ","integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, a^{3} f}, -\frac{{\left(3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a^{3} f}\right]"," ",0,"[-1/64*((3*a^2 - 2*a*b + 3*b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*a^2*cos(f*x + e)^3 + 3*(a^2 - a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f), -1/32*((3*a^2 - 2*a*b + 3*b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*a^2*cos(f*x + e)^3 + 3*(a^2 - a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*f)]","A",0
269,1,643,0,1.954018," ","integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} - 14 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, a^{4} f}, -\frac{3 \, {\left(5 \, a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} - 14 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, a^{4} f}\right]"," ",0,"[1/384*(3*(5*a^3 - 3*a^2*b + 3*a*b^2 - 5*b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*a^3*cos(f*x + e)^5 + 10*(a^3 - a^2*b)*cos(f*x + e)^3 + (15*a^3 - 14*a^2*b + 15*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^4*f), -1/192*(3*(5*a^3 - 3*a^2*b + 3*a*b^2 - 5*b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*a^3*cos(f*x + e)^5 + 10*(a^3 - a^2*b)*cos(f*x + e)^3 + (15*a^3 - 14*a^2*b + 15*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^4*f)]","A",0
270,0,0,0,0.493631," ","integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{5}}{b^{2} \sec\left(f x + e\right)^{4} + 2 \, a b \sec\left(f x + e\right)^{2} + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)^5/(b^2*sec(f*x + e)^4 + 2*a*b*sec(f*x + e)^2 + a^2), x)","F",0
271,0,0,0,0.496066," ","integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{3}}{b^{2} \sec\left(f x + e\right)^{4} + 2 \, a b \sec\left(f x + e\right)^{2} + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)^3/(b^2*sec(f*x + e)^4 + 2*a*b*sec(f*x + e)^2 + a^2), x)","F",0
272,0,0,0,0.519877," ","integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)}{b^{2} \sec\left(f x + e\right)^{4} + 2 \, a b \sec\left(f x + e\right)^{2} + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)/(b^2*sec(f*x + e)^4 + 2*a*b*sec(f*x + e)^2 + a^2), x)","F",0
273,0,0,0,0.558600," ","integrate(cos(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)}{b^{2} \sec\left(f x + e\right)^{4} + 2 \, a b \sec\left(f x + e\right)^{2} + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)/(b^2*sec(f*x + e)^4 + 2*a*b*sec(f*x + e)^2 + a^2), x)","F",0
274,0,0,0,0.570145," ","integrate(cos(f*x+e)^3/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{3}}{b^{2} \sec\left(f x + e\right)^{4} + 2 \, a b \sec\left(f x + e\right)^{2} + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^3/(b^2*sec(f*x + e)^4 + 2*a*b*sec(f*x + e)^2 + a^2), x)","F",0
275,0,0,0,0.673014," ","integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{5}}{b^{2} \sec\left(f x + e\right)^{4} + 2 \, a b \sec\left(f x + e\right)^{2} + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^5/(b^2*sec(f*x + e)^4 + 2*a*b*sec(f*x + e)^2 + a^2), x)","F",0
276,1,524,0,0.916663," ","integrate(sec(f*x+e)^6/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(3 \, a^{3} + 2 \, a^{2} b - a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left(a b^{2} + b^{3} + {\left(3 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{3} + {\left(a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)\right)}}, -\frac{{\left({\left(3 \, a^{3} + 2 \, a^{2} b - a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, {\left(a b^{2} + b^{3} + {\left(3 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{3} + {\left(a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[-1/8*(((3*a^3 + 2*a^2*b - a*b^2)*cos(f*x + e)^3 + (3*a^2*b + 2*a*b^2 - b^3)*cos(f*x + e))*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*(a*b^2 + b^3 + (3*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^2*b^3 + a*b^4)*f*cos(f*x + e)^3 + (a*b^4 + b^5)*f*cos(f*x + e)), -1/4*(((3*a^3 + 2*a^2*b - a*b^2)*cos(f*x + e)^3 + (3*a^2*b + 2*a*b^2 - b^3)*cos(f*x + e))*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - 2*(a*b^2 + b^3 + (3*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^2*b^3 + a*b^4)*f*cos(f*x + e)^3 + (a*b^4 + b^5)*f*cos(f*x + e))]","A",0
277,1,410,0,0.641459," ","integrate(sec(f*x+e)^4/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, {\left({\left(a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a b^{3} + b^{4}\right)} f\right)}}, -\frac{2 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{2 \, {\left({\left(a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a b^{3} + b^{4}\right)} f\right)}}\right]"," ",0,"[-1/4*(4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/((a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a*b^3 + b^4)*f), -1/2*(2*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/((a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a*b^3 + b^4)*f)]","B",0
278,1,65,0,0.550080," ","integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right)}{{\left(a^{2} + a b\right)} f \cos\left(f x + e\right)^{2} + {\left(a b + b^{2}\right)} f}"," ",0,"sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e)/((a^2 + a*b)*f*cos(f*x + e)^2 + (a*b + b^2)*f)","B",0
279,1,601,0,0.742980," ","integrate(1/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}, -\frac{4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), -1/4*(4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]","B",0
280,1,699,0,1.178354," ","integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} b - 2 \, a b^{2} - 3 \, b^{3} + {\left(a^{3} - 2 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left({\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{5} + a^{4} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + a^{3} b^{2}\right)} f\right)}}, -\frac{{\left(a^{2} b - 2 \, a b^{2} - 3 \, b^{3} + {\left(a^{3} - 2 \, a^{2} b - 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left({\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{5} + a^{4} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + a^{3} b^{2}\right)} f\right)}}\right]"," ",0,"[1/16*((a^2*b - 2*a*b^2 - 3*b^3 + (a^3 - 2*a^2*b - 3*a*b^2)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*((a^3 + a^2*b)*cos(f*x + e)^3 + (a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^5 + a^4*b)*f*cos(f*x + e)^2 + (a^4*b + a^3*b^2)*f), -1/8*((a^2*b - 2*a*b^2 - 3*b^3 + (a^3 - 2*a^2*b - 3*a*b^2)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*((a^3 + a^2*b)*cos(f*x + e)^3 + (a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^5 + a^4*b)*f*cos(f*x + e)^2 + (a^4*b + a^3*b^2)*f)]","B",0
281,1,811,0,2.522092," ","integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} b - a^{2} b^{2} + 3 \, a b^{3} + 5 \, b^{4} + {\left(a^{4} - a^{3} b + 3 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(3 \, a^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} - 15 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left({\left(a^{6} + a^{5} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b + a^{4} b^{2}\right)} f\right)}}, -\frac{3 \, {\left(a^{3} b - a^{2} b^{2} + 3 \, a b^{3} + 5 \, b^{4} + {\left(a^{4} - a^{3} b + 3 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, {\left(a^{4} + a^{3} b\right)} \cos\left(f x + e\right)^{5} + {\left(3 \, a^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} - 15 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{6} + a^{5} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b + a^{4} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/64*(3*(a^3*b - a^2*b^2 + 3*a*b^3 + 5*b^4 + (a^4 - a^3*b + 3*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*(a^4 + a^3*b)*cos(f*x + e)^5 + (3*a^4 - 2*a^3*b - 5*a^2*b^2)*cos(f*x + e)^3 + (3*a^3*b - 4*a^2*b^2 - 15*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^6 + a^5*b)*f*cos(f*x + e)^2 + (a^5*b + a^4*b^2)*f), -1/32*(3*(a^3*b - a^2*b^2 + 3*a*b^3 + 5*b^4 + (a^4 - a^3*b + 3*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*(a^4 + a^3*b)*cos(f*x + e)^5 + (3*a^4 - 2*a^3*b - 5*a^2*b^2)*cos(f*x + e)^3 + (3*a^3*b - 4*a^2*b^2 - 15*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^6 + a^5*b)*f*cos(f*x + e)^2 + (a^5*b + a^4*b^2)*f)]","A",0
282,1,941,0,7.326239," ","integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 20 \, a b^{4} - 35 \, b^{5} + {\left(5 \, a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 20 \, a^{2} b^{3} - 35 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, {\left(a^{5} + a^{4} b\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(5 \, a^{5} - 2 \, a^{4} b - 7 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{5} - 7 \, a^{4} b + 13 \, a^{3} b^{2} + 35 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{4} b - 17 \, a^{3} b^{2} + 25 \, a^{2} b^{3} + 105 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + a^{5} b^{2}\right)} f\right)}}, -\frac{3 \, {\left(5 \, a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 20 \, a b^{4} - 35 \, b^{5} + {\left(5 \, a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 20 \, a^{2} b^{3} - 35 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, {\left(a^{5} + a^{4} b\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(5 \, a^{5} - 2 \, a^{4} b - 7 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{5} - 7 \, a^{4} b + 13 \, a^{3} b^{2} + 35 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{4} b - 17 \, a^{3} b^{2} + 25 \, a^{2} b^{3} + 105 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, {\left({\left(a^{7} + a^{6} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + a^{5} b^{2}\right)} f\right)}}\right]"," ",0,"[1/384*(3*(5*a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 20*a*b^4 - 35*b^5 + (5*a^5 - 4*a^4*b + 6*a^3*b^2 - 20*a^2*b^3 - 35*a*b^4)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*(a^5 + a^4*b)*cos(f*x + e)^7 + 2*(5*a^5 - 2*a^4*b - 7*a^3*b^2)*cos(f*x + e)^5 + (15*a^5 - 7*a^4*b + 13*a^3*b^2 + 35*a^2*b^3)*cos(f*x + e)^3 + (15*a^4*b - 17*a^3*b^2 + 25*a^2*b^3 + 105*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^2 + (a^6*b + a^5*b^2)*f), -1/192*(3*(5*a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 20*a*b^4 - 35*b^5 + (5*a^5 - 4*a^4*b + 6*a^3*b^2 - 20*a^2*b^3 - 35*a*b^4)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*(a^5 + a^4*b)*cos(f*x + e)^7 + 2*(5*a^5 - 2*a^4*b - 7*a^3*b^2)*cos(f*x + e)^5 + (15*a^5 - 7*a^4*b + 13*a^3*b^2 + 35*a^2*b^3)*cos(f*x + e)^3 + (15*a^4*b - 17*a^3*b^2 + 25*a^2*b^3 + 105*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^2 + (a^6*b + a^5*b^2)*f)]","A",0
283,0,0,0,0.525348," ","integrate(sec(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{5}}{b^{3} \sec\left(f x + e\right)^{6} + 3 \, a b^{2} \sec\left(f x + e\right)^{4} + 3 \, a^{2} b \sec\left(f x + e\right)^{2} + a^{3}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)^5/(b^3*sec(f*x + e)^6 + 3*a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)","F",0
284,0,0,0,0.546299," ","integrate(sec(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{3}}{b^{3} \sec\left(f x + e\right)^{6} + 3 \, a b^{2} \sec\left(f x + e\right)^{4} + 3 \, a^{2} b \sec\left(f x + e\right)^{2} + a^{3}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)^3/(b^3*sec(f*x + e)^6 + 3*a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)","F",0
285,0,0,0,0.595582," ","integrate(sec(f*x+e)/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)}{b^{3} \sec\left(f x + e\right)^{6} + 3 \, a b^{2} \sec\left(f x + e\right)^{4} + 3 \, a^{2} b \sec\left(f x + e\right)^{2} + a^{3}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*sec(f*x + e)/(b^3*sec(f*x + e)^6 + 3*a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)","F",0
286,0,0,0,0.640510," ","integrate(cos(f*x+e)/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)}{b^{3} \sec\left(f x + e\right)^{6} + 3 \, a b^{2} \sec\left(f x + e\right)^{4} + 3 \, a^{2} b \sec\left(f x + e\right)^{2} + a^{3}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)/(b^3*sec(f*x + e)^6 + 3*a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)","F",0
287,0,0,0,1.066158," ","integrate(cos(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{3}}{b^{3} \sec\left(f x + e\right)^{6} + 3 \, a b^{2} \sec\left(f x + e\right)^{4} + 3 \, a^{2} b \sec\left(f x + e\right)^{2} + a^{3}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^3/(b^3*sec(f*x + e)^6 + 3*a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)","F",0
288,0,0,0,0.937591," ","integrate(cos(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{5}}{b^{3} \sec\left(f x + e\right)^{6} + 3 \, a b^{2} \sec\left(f x + e\right)^{4} + 3 \, a^{2} b \sec\left(f x + e\right)^{2} + a^{3}}, x\right)"," ",0,"integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^5/(b^3*sec(f*x + e)^6 + 3*a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)","F",0
289,1,688,0,0.925804," ","integrate(sec(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(2 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{4} b^{3} + 2 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{3} b^{4} + 2 \, a^{2} b^{5} + a b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, {\left({\left(3 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(2 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{4} b^{3} + 2 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{3} b^{4} + 2 \, a^{2} b^{5} + a b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} f\right)}}\right]"," ",0,"[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^3*b + 5*a^2*b^2)*cos(f*x + e)^3 + 2*(2*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^4 + 2*(a^3*b^4 + 2*a^2*b^5 + a*b^6)*f*cos(f*x + e)^2 + (a^2*b^5 + 2*a*b^6 + b^7)*f), 1/6*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - 2*((3*a^3*b + 5*a^2*b^2)*cos(f*x + e)^3 + 2*(2*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^4 + 2*(a^3*b^4 + 2*a^2*b^5 + a*b^6)*f*cos(f*x + e)^2 + (a^2*b^5 + 2*a*b^6 + b^7)*f)]","B",0
290,1,134,0,0.861632," ","integrate(sec(f*x+e)^4/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(2 \, a \cos\left(f x + e\right)^{3} + {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f\right)}}"," ",0,"1/3*(2*a*cos(f*x + e)^3 + (a + 3*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^2*b^2 + 2*a*b^3 + b^4)*f)","A",0
291,1,134,0,0.702758," ","integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left({\left(3 \, a + b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f\right)}}"," ",0,"1/3*((3*a + b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)/((a^4 + 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^2*b^2 + 2*a*b^3 + b^4)*f)","B",0
292,1,881,0,1.440549," ","integrate(1/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, -\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[-1/24*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f), -1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f)]","B",0
293,1,1023,0,3.356579," ","integrate(cos(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} - 9 \, a b^{4} - 5 \, b^{5} + {\left(a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} - 9 \, a^{2} b^{3} - 5 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(3 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(3 \, a^{4} b + 15 \, a^{3} b^{2} + 10 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b^{2} + 22 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 2 \, a^{5} b^{3} + a^{4} b^{4}\right)} f\right)}}, -\frac{3 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} - 9 \, a b^{4} - 5 \, b^{5} + {\left(a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} - 5 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} - 9 \, a^{2} b^{3} - 5 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(3 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(3 \, a^{4} b + 15 \, a^{3} b^{2} + 10 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b^{2} + 22 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{8} + 2 \, a^{7} b + a^{6} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 2 \, a^{5} b^{3} + a^{4} b^{4}\right)} f\right)}}\right]"," ",0,"[1/48*(3*(a^3*b^2 - 3*a^2*b^3 - 9*a*b^4 - 5*b^5 + (a^5 - 3*a^4*b - 9*a^3*b^2 - 5*a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 - 9*a^2*b^3 - 5*a*b^4)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(3*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^5 + 2*(3*a^4*b + 15*a^3*b^2 + 10*a^2*b^3)*cos(f*x + e)^3 + (3*a^3*b^2 + 22*a^2*b^3 + 15*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^4 + 2*(a^7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^2 + (a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f), -1/24*(3*(a^3*b^2 - 3*a^2*b^3 - 9*a*b^4 - 5*b^5 + (a^5 - 3*a^4*b - 9*a^3*b^2 - 5*a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 - 9*a^2*b^3 - 5*a*b^4)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(3*(a^5 + 2*a^4*b + a^3*b^2)*cos(f*x + e)^5 + 2*(3*a^4*b + 15*a^3*b^2 + 10*a^2*b^3)*cos(f*x + e)^3 + (3*a^3*b^2 + 22*a^2*b^3 + 15*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^8 + 2*a^7*b + a^6*b^2)*f*cos(f*x + e)^4 + 2*(a^7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^2 + (a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f)]","B",0
294,1,1187,0,9.317796," ","integrate(cos(f*x+e)^4/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + 18 \, a^{2} b^{4} + 60 \, a b^{5} + 35 \, b^{6} + {\left(3 \, a^{6} - 4 \, a^{5} b + 18 \, a^{4} b^{2} + 60 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{5} b - 4 \, a^{4} b^{2} + 18 \, a^{3} b^{3} + 60 \, a^{2} b^{4} + 35 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(6 \, {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(3 \, a^{6} - a^{5} b - 11 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{5} b - 12 \, a^{4} b^{2} - 99 \, a^{3} b^{3} - 70 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(9 \, a^{4} b^{2} - 15 \, a^{3} b^{3} - 145 \, a^{2} b^{4} - 105 \, a b^{5}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, {\left({\left(a^{9} + 2 \, a^{8} b + a^{7} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{8} b + 2 \, a^{7} b^{2} + a^{6} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b^{2} + 2 \, a^{6} b^{3} + a^{5} b^{4}\right)} f\right)}}, -\frac{3 \, {\left(3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + 18 \, a^{2} b^{4} + 60 \, a b^{5} + 35 \, b^{6} + {\left(3 \, a^{6} - 4 \, a^{5} b + 18 \, a^{4} b^{2} + 60 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{5} b - 4 \, a^{4} b^{2} + 18 \, a^{3} b^{3} + 60 \, a^{2} b^{4} + 35 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(6 \, {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(3 \, a^{6} - a^{5} b - 11 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{5} b - 12 \, a^{4} b^{2} - 99 \, a^{3} b^{3} - 70 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(9 \, a^{4} b^{2} - 15 \, a^{3} b^{3} - 145 \, a^{2} b^{4} - 105 \, a b^{5}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, {\left({\left(a^{9} + 2 \, a^{8} b + a^{7} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{8} b + 2 \, a^{7} b^{2} + a^{6} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} b^{2} + 2 \, a^{6} b^{3} + a^{5} b^{4}\right)} f\right)}}\right]"," ",0,"[-1/192*(3*(3*a^4*b^2 - 4*a^3*b^3 + 18*a^2*b^4 + 60*a*b^5 + 35*b^6 + (3*a^6 - 4*a^5*b + 18*a^4*b^2 + 60*a^3*b^3 + 35*a^2*b^4)*cos(f*x + e)^4 + 2*(3*a^5*b - 4*a^4*b^2 + 18*a^3*b^3 + 60*a^2*b^4 + 35*a*b^5)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(6*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^7 + 3*(3*a^6 - a^5*b - 11*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^5 + 2*(9*a^5*b - 12*a^4*b^2 - 99*a^3*b^3 - 70*a^2*b^4)*cos(f*x + e)^3 + (9*a^4*b^2 - 15*a^3*b^3 - 145*a^2*b^4 - 105*a*b^5)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^9 + 2*a^8*b + a^7*b^2)*f*cos(f*x + e)^4 + 2*(a^8*b + 2*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^2 + (a^7*b^2 + 2*a^6*b^3 + a^5*b^4)*f), -1/96*(3*(3*a^4*b^2 - 4*a^3*b^3 + 18*a^2*b^4 + 60*a*b^5 + 35*b^6 + (3*a^6 - 4*a^5*b + 18*a^4*b^2 + 60*a^3*b^3 + 35*a^2*b^4)*cos(f*x + e)^4 + 2*(3*a^5*b - 4*a^4*b^2 + 18*a^3*b^3 + 60*a^2*b^4 + 35*a*b^5)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(6*(a^6 + 2*a^5*b + a^4*b^2)*cos(f*x + e)^7 + 3*(3*a^6 - a^5*b - 11*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^5 + 2*(9*a^5*b - 12*a^4*b^2 - 99*a^3*b^3 - 70*a^2*b^4)*cos(f*x + e)^3 + (9*a^4*b^2 - 15*a^3*b^3 - 145*a^2*b^4 - 105*a*b^5)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^9 + 2*a^8*b + a^7*b^2)*f*cos(f*x + e)^4 + 2*(a^8*b + 2*a^7*b^2 + a^6*b^3)*f*cos(f*x + e)^2 + (a^7*b^2 + 2*a^6*b^3 + a^5*b^4)*f)]","B",0
295,1,1337,0,26.444684," ","integrate(cos(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{5} b^{2} - a^{4} b^{3} + 2 \, a^{3} b^{4} - 10 \, a^{2} b^{5} - 35 \, a b^{6} - 21 \, b^{7} + {\left(a^{7} - a^{6} b + 2 \, a^{5} b^{2} - 10 \, a^{4} b^{3} - 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - a^{5} b^{2} + 2 \, a^{4} b^{3} - 10 \, a^{3} b^{4} - 35 \, a^{2} b^{5} - 21 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(8 \, {\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} \cos\left(f x + e\right)^{9} + 2 \, {\left(5 \, a^{7} + a^{6} b - 13 \, a^{5} b^{2} - 9 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{7} + 6 \, a^{5} b^{2} + 32 \, a^{4} b^{3} + 21 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(15 \, a^{6} b - 15 \, a^{5} b^{2} + 31 \, a^{4} b^{3} + 287 \, a^{3} b^{4} + 210 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 38 \, a^{3} b^{4} + 420 \, a^{2} b^{5} + 315 \, a b^{6}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{384 \, {\left({\left(a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right)} f\right)}}, -\frac{15 \, {\left(a^{5} b^{2} - a^{4} b^{3} + 2 \, a^{3} b^{4} - 10 \, a^{2} b^{5} - 35 \, a b^{6} - 21 \, b^{7} + {\left(a^{7} - a^{6} b + 2 \, a^{5} b^{2} - 10 \, a^{4} b^{3} - 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - a^{5} b^{2} + 2 \, a^{4} b^{3} - 10 \, a^{3} b^{4} - 35 \, a^{2} b^{5} - 21 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(8 \, {\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} \cos\left(f x + e\right)^{9} + 2 \, {\left(5 \, a^{7} + a^{6} b - 13 \, a^{5} b^{2} - 9 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{7} + 6 \, a^{5} b^{2} + 32 \, a^{4} b^{3} + 21 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(15 \, a^{6} b - 15 \, a^{5} b^{2} + 31 \, a^{4} b^{3} + 287 \, a^{3} b^{4} + 210 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 38 \, a^{3} b^{4} + 420 \, a^{2} b^{5} + 315 \, a b^{6}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, {\left({\left(a^{10} + 2 \, a^{9} b + a^{8} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 2 \, a^{8} b^{2} + a^{7} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 2 \, a^{7} b^{3} + a^{6} b^{4}\right)} f\right)}}\right]"," ",0,"[1/384*(15*(a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - 10*a^2*b^5 - 35*a*b^6 - 21*b^7 + (a^7 - a^6*b + 2*a^5*b^2 - 10*a^4*b^3 - 35*a^3*b^4 - 21*a^2*b^5)*cos(f*x + e)^4 + 2*(a^6*b - a^5*b^2 + 2*a^4*b^3 - 10*a^3*b^4 - 35*a^2*b^5 - 21*a*b^6)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 2*(5*a^7 + a^6*b - 13*a^5*b^2 - 9*a^4*b^3)*cos(f*x + e)^7 + 3*(5*a^7 + 6*a^5*b^2 + 32*a^4*b^3 + 21*a^3*b^4)*cos(f*x + e)^5 + 2*(15*a^6*b - 15*a^5*b^2 + 31*a^4*b^3 + 287*a^3*b^4 + 210*a^2*b^5)*cos(f*x + e)^3 + (15*a^5*b^2 - 20*a^4*b^3 + 38*a^3*b^4 + 420*a^2*b^5 + 315*a*b^6)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*f), -1/192*(15*(a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - 10*a^2*b^5 - 35*a*b^6 - 21*b^7 + (a^7 - a^6*b + 2*a^5*b^2 - 10*a^4*b^3 - 35*a^3*b^4 - 21*a^2*b^5)*cos(f*x + e)^4 + 2*(a^6*b - a^5*b^2 + 2*a^4*b^3 - 10*a^3*b^4 - 35*a^2*b^5 - 21*a*b^6)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(8*(a^7 + 2*a^6*b + a^5*b^2)*cos(f*x + e)^9 + 2*(5*a^7 + a^6*b - 13*a^5*b^2 - 9*a^4*b^3)*cos(f*x + e)^7 + 3*(5*a^7 + 6*a^5*b^2 + 32*a^4*b^3 + 21*a^3*b^4)*cos(f*x + e)^5 + 2*(15*a^6*b - 15*a^5*b^2 + 31*a^4*b^3 + 287*a^3*b^4 + 210*a^2*b^5)*cos(f*x + e)^3 + (15*a^5*b^2 - 20*a^4*b^3 + 38*a^3*b^4 + 420*a^2*b^5 + 315*a*b^6)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^10 + 2*a^9*b + a^8*b^2)*f*cos(f*x + e)^4 + 2*(a^9*b + 2*a^8*b^2 + a^7*b^3)*f*cos(f*x + e)^2 + (a^8*b^2 + 2*a^7*b^3 + a^6*b^4)*f)]","A",0
296,1,1241,0,3.722238," ","integrate(1/(a+b*sec(d*x+c)^2)^(7/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} + a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6} + 3 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(d x + c\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)\right) + 8 \, {\left({\left(45 \, a^{5} b + 60 \, a^{4} b^{2} + 23 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} + {\left(75 \, a^{4} b^{2} + 94 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + {\left(33 \, a^{3} b^{3} + 40 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)}{120 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} + 3 \, {\left(a^{9} b + 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} b^{3} + 3 \, a^{6} b^{4} + 3 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}, -\frac{15 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} + a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6} + 3 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}\right) + 4 \, {\left({\left(45 \, a^{5} b + 60 \, a^{4} b^{2} + 23 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} + {\left(75 \, a^{4} b^{2} + 94 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + {\left(33 \, a^{3} b^{3} + 40 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} + b}{\cos\left(d x + c\right)^{2}}} \sin\left(d x + c\right)}{60 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} + 3 \, {\left(a^{9} b + 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} b^{3} + 3 \, a^{6} b^{4} + 3 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}\right]"," ",0,"[-1/120*(15*((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(d*x + c)^6 + a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6 + 3*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*cos(d*x + c)^4 + 3*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*cos(d*x + c)^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 - a^3*b)*cos(d*x + c)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(d*x + c)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(d*x + c)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 - a^2*b)*cos(d*x + c)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(d*x + c)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c)) + 8*((45*a^5*b + 60*a^4*b^2 + 23*a^3*b^3)*cos(d*x + c)^5 + (75*a^4*b^2 + 94*a^3*b^3 + 35*a^2*b^4)*cos(d*x + c)^3 + (33*a^3*b^3 + 40*a^2*b^4 + 15*a*b^5)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 + 3*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6)*d), -1/60*(15*((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(d*x + c)^6 + a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6 + 3*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*cos(d*x + c)^4 + 3*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*cos(d*x + c)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^5 - 8*(a^2 - a*b)*cos(d*x + c)^3 + (a^2 - 6*a*b + b^2)*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)/((2*a^3*cos(d*x + c)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(d*x + c)^2)*sin(d*x + c))) + 4*((45*a^5*b + 60*a^4*b^2 + 23*a^3*b^3)*cos(d*x + c)^5 + (75*a^4*b^2 + 94*a^3*b^3 + 35*a^2*b^4)*cos(d*x + c)^3 + (33*a^3*b^3 + 40*a^2*b^4 + 15*a*b^5)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 + b)/cos(d*x + c)^2)*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 + 3*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6)*d)]","B",0
297,1,53,0,0.567679," ","integrate(1/(1+sec(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \arctan\left(\frac{\sqrt{\frac{\cos\left(x\right)^{2} + 1}{\cos\left(x\right)^{2}}} \cos\left(x\right)^{3} \sin\left(x\right) + \cos\left(x\right) \sin\left(x\right)}{\cos\left(x\right)^{4} + \cos\left(x\right)^{2} - 1}\right) - \frac{1}{2} \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)"," ",0,"1/2*arctan((sqrt((cos(x)^2 + 1)/cos(x)^2)*cos(x)^3*sin(x) + cos(x)*sin(x))/(cos(x)^4 + cos(x)^2 - 1)) - 1/2*arctan(sin(x)/cos(x))","B",0
298,0,0,0,0.565870," ","integrate((d*sec(f*x+e))^m*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \left(d \sec\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*(d*sec(f*x + e))^m, x)","F",0
299,0,0,0,0.505458," ","integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*sec(f*x + e)^3, x)","F",0
300,0,0,0,0.519407," ","integrate(sec(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right), x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*sec(f*x + e), x)","F",0
301,0,0,0,0.555594," ","integrate(cos(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right), x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e), x)","F",0
302,0,0,0,0.508887," ","integrate(cos(f*x+e)^3*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^3, x)","F",0
303,0,0,0,0.516283," ","integrate(cos(f*x+e)^5*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{5}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^5, x)","F",0
304,0,0,0,0.558439," ","integrate(sec(f*x+e)^6*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{6}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*sec(f*x + e)^6, x)","F",0
305,0,0,0,0.487038," ","integrate(sec(f*x+e)^4*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*sec(f*x + e)^4, x)","F",0
306,0,0,0,0.554685," ","integrate(sec(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*sec(f*x + e)^2, x)","F",0
307,0,0,0,0.472402," ","integrate((a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p, x)","F",0
308,0,0,0,0.497575," ","integrate(cos(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^2, x)","F",0
309,0,0,0,0.503207," ","integrate(cos(f*x+e)^4*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^4, x)","F",0
310,0,0,0,0.578834," ","integrate(cos(f*x+e)^6*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{6}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cos(f*x + e)^6, x)","F",0
311,1,69,0,0.489188," ","integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^5,x, algorithm=""fricas"")","-\frac{12 \, a \cos\left(f x + e\right)^{6} \log\left(-\cos\left(f x + e\right)\right) + 6 \, {\left(2 \, a - b\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, b}{12 \, f \cos\left(f x + e\right)^{6}}"," ",0,"-1/12*(12*a*cos(f*x + e)^6*log(-cos(f*x + e)) + 6*(2*a - b)*cos(f*x + e)^4 - 3*(a - 2*b)*cos(f*x + e)^2 - 2*b)/(f*cos(f*x + e)^6)","A",0
312,1,50,0,0.539883," ","integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\frac{4 \, a \cos\left(f x + e\right)^{4} \log\left(-\cos\left(f x + e\right)\right) + 2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{4 \, f \cos\left(f x + e\right)^{4}}"," ",0,"1/4*(4*a*cos(f*x + e)^4*log(-cos(f*x + e)) + 2*(a - b)*cos(f*x + e)^2 + b)/(f*cos(f*x + e)^4)","A",0
313,1,37,0,0.486891," ","integrate((a+b*sec(f*x+e)^2)*tan(f*x+e),x, algorithm=""fricas"")","-\frac{2 \, a \cos\left(f x + e\right)^{2} \log\left(-\cos\left(f x + e\right)\right) - b}{2 \, f \cos\left(f x + e\right)^{2}}"," ",0,"-1/2*(2*a*cos(f*x + e)^2*log(-cos(f*x + e)) - b)/(f*cos(f*x + e)^2)","A",0
314,1,35,0,0.507118," ","integrate(cot(f*x+e)*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{b \log\left(\cos\left(f x + e\right)^{2}\right) - {\left(a + b\right)} \log\left(-\frac{1}{4} \, \cos\left(f x + e\right)^{2} + \frac{1}{4}\right)}{2 \, f}"," ",0,"-1/2*(b*log(cos(f*x + e)^2) - (a + b)*log(-1/4*cos(f*x + e)^2 + 1/4))/f","A",0
315,1,50,0,0.451616," ","integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right) - a - b}{2 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}"," ",0,"-1/2*(2*(a*cos(f*x + e)^2 - a)*log(1/2*sin(f*x + e)) - a - b)/(f*cos(f*x + e)^2 - f)","A",0
316,1,83,0,0.457946," ","integrate(cot(f*x+e)^5*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{2 \, {\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(a \cos\left(f x + e\right)^{4} - 2 \, a \cos\left(f x + e\right)^{2} + a\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right) - 3 \, a - b}{4 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} + f\right)}}"," ",0,"-1/4*(2*(2*a + b)*cos(f*x + e)^2 - 4*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 + a)*log(1/2*sin(f*x + e)) - 3*a - b)/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)","A",0
317,1,89,0,0.500598," ","integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^6,x, algorithm=""fricas"")","-\frac{105 \, a f x \cos\left(f x + e\right)^{7} - {\left({\left(161 \, a - 15 \, b\right)} \cos\left(f x + e\right)^{6} - {\left(77 \, a - 45 \, b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(7 \, a - 15 \, b\right)} \cos\left(f x + e\right)^{2} + 15 \, b\right)} \sin\left(f x + e\right)}{105 \, f \cos\left(f x + e\right)^{7}}"," ",0,"-1/105*(105*a*f*x*cos(f*x + e)^7 - ((161*a - 15*b)*cos(f*x + e)^6 - (77*a - 45*b)*cos(f*x + e)^4 + 3*(7*a - 15*b)*cos(f*x + e)^2 + 15*b)*sin(f*x + e))/(f*cos(f*x + e)^7)","A",0
318,1,72,0,0.461292," ","integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\frac{15 \, a f x \cos\left(f x + e\right)^{5} - {\left({\left(20 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a - 6 \, b\right)} \cos\left(f x + e\right)^{2} - 3 \, b\right)} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)^{5}}"," ",0,"1/15*(15*a*f*x*cos(f*x + e)^5 - ((20*a - 3*b)*cos(f*x + e)^4 - (5*a - 6*b)*cos(f*x + e)^2 - 3*b)*sin(f*x + e))/(f*cos(f*x + e)^5)","A",0
319,1,53,0,0.490480," ","integrate((a+b*sec(f*x+e)^2)*tan(f*x+e)^2,x, algorithm=""fricas"")","-\frac{3 \, a f x \cos\left(f x + e\right)^{3} - {\left({\left(3 \, a - b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/3*(3*a*f*x*cos(f*x + e)^3 - ((3*a - b)*cos(f*x + e)^2 + b)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
320,1,31,0,0.484868," ","integrate(a+b*sec(f*x+e)^2,x, algorithm=""fricas"")","\frac{a f x \cos\left(f x + e\right) + b \sin\left(f x + e\right)}{f \cos\left(f x + e\right)}"," ",0,"(a*f*x*cos(f*x + e) + b*sin(f*x + e))/(f*cos(f*x + e))","B",0
321,1,34,0,0.493515," ","integrate(cot(f*x+e)^2*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{a f x \sin\left(f x + e\right) + {\left(a + b\right)} \cos\left(f x + e\right)}{f \sin\left(f x + e\right)}"," ",0,"-(a*f*x*sin(f*x + e) + (a + b)*cos(f*x + e))/(f*sin(f*x + e))","A",0
322,1,76,0,0.487009," ","integrate(cot(f*x+e)^4*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(4 \, a + b\right)} \cos\left(f x + e\right)^{3} - 3 \, a \cos\left(f x + e\right) + 3 \, {\left(a f x \cos\left(f x + e\right)^{2} - a f x\right)} \sin\left(f x + e\right)}{3 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}"," ",0,"1/3*((4*a + b)*cos(f*x + e)^3 - 3*a*cos(f*x + e) + 3*(a*f*x*cos(f*x + e)^2 - a*f*x)*sin(f*x + e))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))","B",0
323,1,110,0,0.494495," ","integrate(cot(f*x+e)^6*(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(23 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{5} - 35 \, a \cos\left(f x + e\right)^{3} + 15 \, a \cos\left(f x + e\right) + 15 \, {\left(a f x \cos\left(f x + e\right)^{4} - 2 \, a f x \cos\left(f x + e\right)^{2} + a f x\right)} \sin\left(f x + e\right)}{15 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} + f\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*((23*a + 3*b)*cos(f*x + e)^5 - 35*a*cos(f*x + e)^3 + 15*a*cos(f*x + e) + 15*(a*f*x*cos(f*x + e)^4 - 2*a*f*x*cos(f*x + e)^2 + a*f*x)*sin(f*x + e))/((f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)*sin(f*x + e))","B",0
324,1,99,0,0.562780," ","integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^5,x, algorithm=""fricas"")","-\frac{24 \, a^{2} \cos\left(f x + e\right)^{8} \log\left(-\cos\left(f x + e\right)\right) + 24 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{6} - 6 \, {\left(a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, b^{2}}{24 \, f \cos\left(f x + e\right)^{8}}"," ",0,"-1/24*(24*a^2*cos(f*x + e)^8*log(-cos(f*x + e)) + 24*(a^2 - a*b)*cos(f*x + e)^6 - 6*(a^2 - 4*a*b + b^2)*cos(f*x + e)^4 - 8*(a*b - b^2)*cos(f*x + e)^2 - 3*b^2)/(f*cos(f*x + e)^8)","A",0
325,1,79,0,0.476465," ","integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^3,x, algorithm=""fricas"")","\frac{12 \, a^{2} \cos\left(f x + e\right)^{6} \log\left(-\cos\left(f x + e\right)\right) + 6 \, {\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}}{12 \, f \cos\left(f x + e\right)^{6}}"," ",0,"1/12*(12*a^2*cos(f*x + e)^6*log(-cos(f*x + e)) + 6*(a^2 - 2*a*b)*cos(f*x + e)^4 + 3*(2*a*b - b^2)*cos(f*x + e)^2 + 2*b^2)/(f*cos(f*x + e)^6)","A",0
326,1,53,0,0.457928," ","integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e),x, algorithm=""fricas"")","-\frac{4 \, a^{2} \cos\left(f x + e\right)^{4} \log\left(-\cos\left(f x + e\right)\right) - 4 \, a b \cos\left(f x + e\right)^{2} - b^{2}}{4 \, f \cos\left(f x + e\right)^{4}}"," ",0,"-1/4*(4*a^2*cos(f*x + e)^4*log(-cos(f*x + e)) - 4*a*b*cos(f*x + e)^2 - b^2)/(f*cos(f*x + e)^4)","A",0
327,1,79,0,0.517930," ","integrate(cot(f*x+e)*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} \log\left(\cos\left(f x + e\right)^{2}\right) - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} \log\left(-\frac{1}{4} \, \cos\left(f x + e\right)^{2} + \frac{1}{4}\right) - b^{2}}{2 \, f \cos\left(f x + e\right)^{2}}"," ",0,"-1/2*((2*a*b + b^2)*cos(f*x + e)^2*log(cos(f*x + e)^2) - (a^2 + 2*a*b + b^2)*cos(f*x + e)^2*log(-1/4*cos(f*x + e)^2 + 1/4) - b^2)/(f*cos(f*x + e)^2)","A",0
328,1,100,0,0.460548," ","integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{a^{2} + 2 \, a b + b^{2} - {\left(b^{2} \cos\left(f x + e\right)^{2} - b^{2}\right)} \log\left(\cos\left(f x + e\right)^{2}\right) - {\left({\left(a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} + b^{2}\right)} \log\left(-\frac{1}{4} \, \cos\left(f x + e\right)^{2} + \frac{1}{4}\right)}{2 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}"," ",0,"1/2*(a^2 + 2*a*b + b^2 - (b^2*cos(f*x + e)^2 - b^2)*log(cos(f*x + e)^2) - ((a^2 - b^2)*cos(f*x + e)^2 - a^2 + b^2)*log(-1/4*cos(f*x + e)^2 + 1/4))/(f*cos(f*x + e)^2 - f)","A",0
329,1,97,0,0.496331," ","integrate(cot(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{4 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} - 3 \, a^{2} - 2 \, a b + b^{2} - 4 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{4 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} + f\right)}}"," ",0,"-1/4*(4*(a^2 + a*b)*cos(f*x + e)^2 - 3*a^2 - 2*a*b + b^2 - 4*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*log(1/2*sin(f*x + e)))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)","A",0
330,1,137,0,0.485401," ","integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^6,x, algorithm=""fricas"")","-\frac{315 \, a^{2} f x \cos\left(f x + e\right)^{9} - {\left({\left(483 \, a^{2} - 90 \, a b - 10 \, b^{2}\right)} \cos\left(f x + e\right)^{8} - {\left(231 \, a^{2} - 270 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + 3 \, {\left(21 \, a^{2} - 90 \, a b + 25 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 5 \, {\left(18 \, a b - 19 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 35 \, b^{2}\right)} \sin\left(f x + e\right)}{315 \, f \cos\left(f x + e\right)^{9}}"," ",0,"-1/315*(315*a^2*f*x*cos(f*x + e)^9 - ((483*a^2 - 90*a*b - 10*b^2)*cos(f*x + e)^8 - (231*a^2 - 270*a*b + 5*b^2)*cos(f*x + e)^6 + 3*(21*a^2 - 90*a*b + 25*b^2)*cos(f*x + e)^4 + 5*(18*a*b - 19*b^2)*cos(f*x + e)^2 + 35*b^2)*sin(f*x + e))/(f*cos(f*x + e)^9)","A",0
331,1,113,0,0.503672," ","integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^4,x, algorithm=""fricas"")","\frac{105 \, a^{2} f x \cos\left(f x + e\right)^{7} - {\left(2 \, {\left(70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(35 \, a^{2} - 84 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 6 \, {\left(7 \, a b - 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, b^{2}\right)} \sin\left(f x + e\right)}{105 \, f \cos\left(f x + e\right)^{7}}"," ",0,"1/105*(105*a^2*f*x*cos(f*x + e)^7 - (2*(70*a^2 - 21*a*b - 3*b^2)*cos(f*x + e)^6 - (35*a^2 - 84*a*b + 3*b^2)*cos(f*x + e)^4 - 6*(7*a*b - 4*b^2)*cos(f*x + e)^2 - 15*b^2)*sin(f*x + e))/(f*cos(f*x + e)^7)","A",0
332,1,86,0,0.511932," ","integrate((a+b*sec(f*x+e)^2)^2*tan(f*x+e)^2,x, algorithm=""fricas"")","-\frac{15 \, a^{2} f x \cos\left(f x + e\right)^{5} - {\left({\left(15 \, a^{2} - 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(10 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, b^{2}\right)} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)^{5}}"," ",0,"-1/15*(15*a^2*f*x*cos(f*x + e)^5 - ((15*a^2 - 10*a*b - 2*b^2)*cos(f*x + e)^4 + (10*a*b - b^2)*cos(f*x + e)^2 + 3*b^2)*sin(f*x + e))/(f*cos(f*x + e)^5)","A",0
333,1,58,0,0.535491," ","integrate((a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} f x \cos\left(f x + e\right)^{3} + {\left(2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*(3*a^2*f*x*cos(f*x + e)^3 + (2*(3*a*b + b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
334,1,67,0,0.578694," ","integrate(cot(f*x+e)^2*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{a^{2} f x \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a^{2} + 2 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - b^{2}}{f \cos\left(f x + e\right) \sin\left(f x + e\right)}"," ",0,"-(a^2*f*x*cos(f*x + e)*sin(f*x + e) + (a^2 + 2*a*b + 2*b^2)*cos(f*x + e)^2 - b^2)/(f*cos(f*x + e)*sin(f*x + e))","A",0
335,1,98,0,0.487723," ","integrate(cot(f*x+e)^4*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{2 \, {\left(2 \, a^{2} + a b - b^{2}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a^{2} - b^{2}\right)} \cos\left(f x + e\right) + 3 \, {\left(a^{2} f x \cos\left(f x + e\right)^{2} - a^{2} f x\right)} \sin\left(f x + e\right)}{3 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}"," ",0,"1/3*(2*(2*a^2 + a*b - b^2)*cos(f*x + e)^3 - 3*(a^2 - b^2)*cos(f*x + e) + 3*(a^2*f*x*cos(f*x + e)^2 - a^2*f*x)*sin(f*x + e))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))","B",0
336,1,136,0,0.628103," ","integrate(cot(f*x+e)^6*(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(23 \, a^{2} + 6 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 5 \, {\left(7 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, a^{2} \cos\left(f x + e\right) + 15 \, {\left(a^{2} f x \cos\left(f x + e\right)^{4} - 2 \, a^{2} f x \cos\left(f x + e\right)^{2} + a^{2} f x\right)} \sin\left(f x + e\right)}{15 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} + f\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*((23*a^2 + 6*a*b - 2*b^2)*cos(f*x + e)^5 - 5*(7*a^2 - b^2)*cos(f*x + e)^3 + 15*a^2*cos(f*x + e) + 15*(a^2*f*x*cos(f*x + e)^4 - 2*a^2*f*x*cos(f*x + e)^2 + a^2*f*x)*sin(f*x + e))/((f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)*sin(f*x + e))","B",0
337,1,84,0,0.711823," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} \log\left(a \cos\left(f x + e\right)^{2} + b\right) - 2 \, {\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} \log\left(-\cos\left(f x + e\right)\right) - a b}{2 \, a b^{2} f \cos\left(f x + e\right)^{2}}"," ",0,"-1/2*((a^2 + 2*a*b + b^2)*cos(f*x + e)^2*log(a*cos(f*x + e)^2 + b) - 2*(a^2 + 2*a*b)*cos(f*x + e)^2*log(-cos(f*x + e)) - a*b)/(a*b^2*f*cos(f*x + e)^2)","A",0
338,1,41,0,0.704467," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(a + b\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) - 2 \, a \log\left(-\cos\left(f x + e\right)\right)}{2 \, a b f}"," ",0,"1/2*((a + b)*log(a*cos(f*x + e)^2 + b) - 2*a*log(-cos(f*x + e)))/(a*b*f)","A",0
339,1,21,0,0.528895," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","-\frac{\log\left(a \cos\left(f x + e\right)^{2} + b\right)}{2 \, a f}"," ",0,"-1/2*log(a*cos(f*x + e)^2 + b)/(a*f)","A",0
340,1,42,0,0.537390," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{b \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 2 \, a \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{2 \, {\left(a^{2} + a b\right)} f}"," ",0,"1/2*(b*log(a*cos(f*x + e)^2 + b) + 2*a*log(1/2*sin(f*x + e)))/((a^2 + a*b)*f)","A",0
341,1,126,0,0.939595," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{a^{2} + a b - {\left(b^{2} \cos\left(f x + e\right)^{2} - b^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) - 2 \, {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{2 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)}}"," ",0,"1/2*(a^2 + a*b - (b^2*cos(f*x + e)^2 - b^2)*log(a*cos(f*x + e)^2 + b) - 2*((a^2 + 2*a*b)*cos(f*x + e)^2 - a^2 - 2*a*b)*log(1/2*sin(f*x + e)))/((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f)","A",0
342,1,265,0,0.918104," ","integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, a^{3} + 8 \, a^{2} b + 5 \, a b^{2} - 2 \, {\left(2 \, a^{3} + 5 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(b^{3} \cos\left(f x + e\right)^{4} - 2 \, b^{3} \cos\left(f x + e\right)^{2} + b^{3}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 4 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{4 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}"," ",0,"1/4*(3*a^3 + 8*a^2*b + 5*a*b^2 - 2*(2*a^3 + 5*a^2*b + 3*a*b^2)*cos(f*x + e)^2 + 2*(b^3*cos(f*x + e)^4 - 2*b^3*cos(f*x + e)^2 + b^3)*log(a*cos(f*x + e)^2 + b) + 4*((a^3 + 3*a^2*b + 3*a*b^2)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 - 2*(a^3 + 3*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*log(1/2*sin(f*x + e)))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)","B",0
343,1,373,0,0.538498," ","integrate(tan(f*x+e)^6/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{12 \, b^{2} f x \cos\left(f x + e\right)^{3} - 3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-\frac{a + b}{b}} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - b^{2} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left({\left(3 \, a^{2} + 7 \, a b\right)} \cos\left(f x + e\right)^{2} - a b\right)} \sin\left(f x + e\right)}{12 \, a b^{2} f \cos\left(f x + e\right)^{3}}, -\frac{6 \, b^{2} f x \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{\frac{a + b}{b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a + b}{b}}}{2 \, {\left(a + b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + 2 \, {\left({\left(3 \, a^{2} + 7 \, a b\right)} \cos\left(f x + e\right)^{2} - a b\right)} \sin\left(f x + e\right)}{6 \, a b^{2} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[-1/12*(12*b^2*f*x*cos(f*x + e)^3 - 3*(a^2 + 2*a*b + b^2)*sqrt(-(a + b)/b)*cos(f*x + e)^3*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a*b + 2*b^2)*cos(f*x + e)^3 - b^2*cos(f*x + e))*sqrt(-(a + b)/b)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*((3*a^2 + 7*a*b)*cos(f*x + e)^2 - a*b)*sin(f*x + e))/(a*b^2*f*cos(f*x + e)^3), -1/6*(6*b^2*f*x*cos(f*x + e)^3 + 3*(a^2 + 2*a*b + b^2)*sqrt((a + b)/b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a + b)/b)/((a + b)*cos(f*x + e)*sin(f*x + e)))*cos(f*x + e)^3 + 2*((3*a^2 + 7*a*b)*cos(f*x + e)^2 - a*b)*sin(f*x + e))/(a*b^2*f*cos(f*x + e)^3)]","B",0
344,1,297,0,0.581316," ","integrate(tan(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, b f x \cos\left(f x + e\right) + {\left(a + b\right)} \sqrt{-\frac{a + b}{b}} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - b^{2} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, a \sin\left(f x + e\right)}{4 \, a b f \cos\left(f x + e\right)}, \frac{2 \, b f x \cos\left(f x + e\right) + {\left(a + b\right)} \sqrt{\frac{a + b}{b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a + b}{b}}}{2 \, {\left(a + b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 2 \, a \sin\left(f x + e\right)}{2 \, a b f \cos\left(f x + e\right)}\right]"," ",0,"[1/4*(4*b*f*x*cos(f*x + e) + (a + b)*sqrt(-(a + b)/b)*cos(f*x + e)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a*b + 2*b^2)*cos(f*x + e)^3 - b^2*cos(f*x + e))*sqrt(-(a + b)/b)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*a*sin(f*x + e))/(a*b*f*cos(f*x + e)), 1/2*(2*b*f*x*cos(f*x + e) + (a + b)*sqrt((a + b)/b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a + b)/b)/((a + b)*cos(f*x + e)*sin(f*x + e)))*cos(f*x + e) + 2*a*sin(f*x + e))/(a*b*f*cos(f*x + e))]","B",0
345,1,226,0,0.538825," ","integrate(tan(f*x+e)^2/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, f x - \sqrt{-\frac{a + b}{b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - b^{2} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, a f}, -\frac{2 \, f x + \sqrt{\frac{a + b}{b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a + b}{b}}}{2 \, {\left(a + b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, a f}\right]"," ",0,"[-1/4*(4*f*x - sqrt(-(a + b)/b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a*b + 2*b^2)*cos(f*x + e)^3 - b^2*cos(f*x + e))*sqrt(-(a + b)/b)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a*f), -1/2*(2*f*x + sqrt((a + b)/b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a + b)/b)/((a + b)*cos(f*x + e)*sin(f*x + e))))/(a*f)]","A",0
346,1,231,0,0.591649," ","integrate(1/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, f x + \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, a f}, \frac{2 \, f x + \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, a f}\right]"," ",0,"[1/4*(4*f*x + sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a*f), 1/2*(2*f*x + sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/(a*f)]","A",0
347,1,310,0,0.496087," ","integrate(cot(f*x+e)^2/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a + b\right)} f x \sin\left(f x + e\right) - b \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 4 \, a \cos\left(f x + e\right)}{4 \, {\left(a^{2} + a b\right)} f \sin\left(f x + e\right)}, -\frac{2 \, {\left(a + b\right)} f x \sin\left(f x + e\right) + b \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, a \cos\left(f x + e\right)}{2 \, {\left(a^{2} + a b\right)} f \sin\left(f x + e\right)}\right]"," ",0,"[-1/4*(4*(a + b)*f*x*sin(f*x + e) - b*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 4*a*cos(f*x + e))/((a^2 + a*b)*f*sin(f*x + e)), -1/2*(2*(a + b)*f*x*sin(f*x + e) + b*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 2*a*cos(f*x + e))/((a^2 + a*b)*f*sin(f*x + e))]","B",0
348,1,533,0,0.533284," ","integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(4 \, a^{2} + 7 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(b^{2} \cos\left(f x + e\right)^{2} - b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, {\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right) + 12 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f x \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f x\right)} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)} \sin\left(f x + e\right)}, \frac{2 \, {\left(4 \, a^{2} + 7 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(b^{2} \cos\left(f x + e\right)^{2} - b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, {\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right) + 6 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f x \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f x\right)} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/12*(4*(4*a^2 + 7*a*b)*cos(f*x + e)^3 + 3*(b^2*cos(f*x + e)^2 - b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(a^2 + 2*a*b)*cos(f*x + e) + 12*((a^2 + 2*a*b + b^2)*f*x*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f*x)*sin(f*x + e))/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f)*sin(f*x + e)), 1/6*(2*(4*a^2 + 7*a*b)*cos(f*x + e)^3 + 3*(b^2*cos(f*x + e)^2 - b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*(a^2 + 2*a*b)*cos(f*x + e) + 6*((a^2 + 2*a*b + b^2)*f*x*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f*x)*sin(f*x + e))/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f)*sin(f*x + e))]","B",0
349,1,833,0,0.623011," ","integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(23 \, a^{3} + 66 \, a^{2} b + 58 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 20 \, {\left(7 \, a^{3} + 21 \, a^{2} b + 20 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(b^{3} \cos\left(f x + e\right)^{4} - 2 \, b^{3} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right) + 60 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f x \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f x\right)} \sin\left(f x + e\right)}{60 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(23 \, a^{3} + 66 \, a^{2} b + 58 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(7 \, a^{3} + 21 \, a^{2} b + 20 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left(b^{3} \cos\left(f x + e\right)^{4} - 2 \, b^{3} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right) + 30 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f x \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f x\right)} \sin\left(f x + e\right)}{30 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/60*(4*(23*a^3 + 66*a^2*b + 58*a*b^2)*cos(f*x + e)^5 - 20*(7*a^3 + 21*a^2*b + 20*a*b^2)*cos(f*x + e)^3 - 15*(b^3*cos(f*x + e)^4 - 2*b^3*cos(f*x + e)^2 + b^3)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(a^3 + 3*a^2*b + 3*a*b^2)*cos(f*x + e) + 60*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*x*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*x*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*x)*sin(f*x + e))/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)*sin(f*x + e)), -1/30*(2*(23*a^3 + 66*a^2*b + 58*a*b^2)*cos(f*x + e)^5 - 10*(7*a^3 + 21*a^2*b + 20*a*b^2)*cos(f*x + e)^3 + 15*(b^3*cos(f*x + e)^4 - 2*b^3*cos(f*x + e)^2 + b^3)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(a^3 + 3*a^2*b + 3*a*b^2)*cos(f*x + e) + 30*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*x*cos(f*x + e)^4 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*x*cos(f*x + e)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*x)*sin(f*x + e))/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)*sin(f*x + e))]","B",0
350,1,118,0,0.663524," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{a^{2} b + 2 \, a b^{2} + b^{3} - {\left(a^{2} b - b^{3} + {\left(a^{3} - a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 2 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \log\left(-\cos\left(f x + e\right)\right)}{2 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}"," ",0,"-1/2*(a^2*b + 2*a*b^2 + b^3 - (a^2*b - b^3 + (a^3 - a*b^2)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) + 2*(a^3*cos(f*x + e)^2 + a^2*b)*log(-cos(f*x + e)))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f)","A",0
351,1,53,0,0.537498," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(a \cos\left(f x + e\right)^{2} + b\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + a + b}{2 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}"," ",0,"1/2*((a*cos(f*x + e)^2 + b)*log(a*cos(f*x + e)^2 + b) + a + b)/(a^3*f*cos(f*x + e)^2 + a^2*b*f)","A",0
352,1,52,0,0.530777," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(a \cos\left(f x + e\right)^{2} + b\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + b}{2 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}"," ",0,"-1/2*((a*cos(f*x + e)^2 + b)*log(a*cos(f*x + e)^2 + b) + b)/(a^3*f*cos(f*x + e)^2 + a^2*b*f)","A",0
353,1,138,0,0.720072," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{a b^{2} + b^{3} + {\left(2 \, a b^{2} + b^{3} + {\left(2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 2 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{2 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}"," ",0,"1/2*(a*b^2 + b^3 + (2*a*b^2 + b^3 + (2*a^2*b + a*b^2)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) + 2*(a^3*cos(f*x + e)^2 + a^2*b)*log(1/2*sin(f*x + e)))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)","A",0
354,1,312,0,0.944806," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{a^{3} b + a^{2} b^{2} + a b^{3} + b^{4} + {\left(a^{4} + a^{3} b - a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2} - {\left({\left(3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - 3 \, a b^{3} - b^{4} - {\left(3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) - 2 \, {\left({\left(a^{4} + 3 \, a^{3} b\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - {\left(a^{4} + 2 \, a^{3} b - 3 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{2 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)}}"," ",0,"1/2*(a^3*b + a^2*b^2 + a*b^3 + b^4 + (a^4 + a^3*b - a*b^3 - b^4)*cos(f*x + e)^2 - ((3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - 3*a*b^3 - b^4 - (3*a^2*b^2 - 2*a*b^3 - b^4)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) - 2*((a^4 + 3*a^3*b)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - (a^4 + 2*a^3*b - 3*a^2*b^2)*cos(f*x + e)^2)*log(1/2*sin(f*x + e)))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)","B",0
355,1,557,0,1.556659," ","integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, a^{4} b + 10 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 2 \, a b^{4} + 2 \, b^{5} - 2 \, {\left(2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(3 \, a^{5} + 6 \, a^{4} b - 5 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 4 \, a b^{4} - 4 \, b^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left({\left(4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + 4 \, a b^{4} + b^{5} - {\left(8 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(4 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 4 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} - 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 12 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{4 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}"," ",0,"1/4*(3*a^4*b + 10*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4 + 2*b^5 - 2*(2*a^5 + 6*a^4*b + 4*a^3*b^2 - a*b^4 - b^5)*cos(f*x + e)^4 + (3*a^5 + 6*a^4*b - 5*a^3*b^2 - 8*a^2*b^3 - 4*a*b^4 - 4*b^5)*cos(f*x + e)^2 + 2*((4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + 4*a*b^4 + b^5 - (8*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (4*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) + 4*((a^5 + 4*a^4*b + 6*a^3*b^2)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 - 6*a^2*b^3)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 12*a^2*b^3)*cos(f*x + e)^2)*log(1/2*sin(f*x + e)))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f)","B",0
356,1,514,0,0.580321," ","integrate(tan(f*x+e)^6/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{8 \, a b^{2} f x \cos\left(f x + e\right)^{3} + 8 \, b^{3} f x \cos\left(f x + e\right) + {\left({\left(3 \, a^{3} + a^{2} b - 2 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} b + a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - b^{2} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(2 \, a^{2} b + {\left(3 \, a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{3} + a^{2} b^{3} f \cos\left(f x + e\right)\right)}}, -\frac{4 \, a b^{2} f x \cos\left(f x + e\right)^{3} + 4 \, b^{3} f x \cos\left(f x + e\right) - {\left({\left(3 \, a^{3} + a^{2} b - 2 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} b + a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a + b}{b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a + b}{b}}}{2 \, {\left(a + b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(2 \, a^{2} b + {\left(3 \, a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{3} + a^{2} b^{3} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[-1/8*(8*a*b^2*f*x*cos(f*x + e)^3 + 8*b^3*f*x*cos(f*x + e) + ((3*a^3 + a^2*b - 2*a*b^2)*cos(f*x + e)^3 + (3*a^2*b + a*b^2 - 2*b^3)*cos(f*x + e))*sqrt(-(a + b)/b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a*b + 2*b^2)*cos(f*x + e)^3 - b^2*cos(f*x + e))*sqrt(-(a + b)/b)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(2*a^2*b + (3*a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sin(f*x + e))/(a^3*b^2*f*cos(f*x + e)^3 + a^2*b^3*f*cos(f*x + e)), -1/4*(4*a*b^2*f*x*cos(f*x + e)^3 + 4*b^3*f*x*cos(f*x + e) - ((3*a^3 + a^2*b - 2*a*b^2)*cos(f*x + e)^3 + (3*a^2*b + a*b^2 - 2*b^3)*cos(f*x + e))*sqrt((a + b)/b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a + b)/b)/((a + b)*cos(f*x + e)*sin(f*x + e))) - 2*(2*a^2*b + (3*a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sin(f*x + e))/(a^3*b^2*f*cos(f*x + e)^3 + a^2*b^3*f*cos(f*x + e))]","B",0
357,1,393,0,0.533264," ","integrate(tan(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, a b f x \cos\left(f x + e\right)^{2} + 8 \, b^{2} f x - 4 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b - 2 \, b^{2}\right)} \sqrt{-\frac{a + b}{b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - b^{2} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left(a^{3} b f \cos\left(f x + e\right)^{2} + a^{2} b^{2} f\right)}}, \frac{4 \, a b f x \cos\left(f x + e\right)^{2} + 4 \, b^{2} f x - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b - 2 \, b^{2}\right)} \sqrt{\frac{a + b}{b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a + b}{b}}}{2 \, {\left(a + b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left(a^{3} b f \cos\left(f x + e\right)^{2} + a^{2} b^{2} f\right)}}\right]"," ",0,"[1/8*(8*a*b*f*x*cos(f*x + e)^2 + 8*b^2*f*x - 4*(a^2 + a*b)*cos(f*x + e)*sin(f*x + e) - ((a^2 - 2*a*b)*cos(f*x + e)^2 + a*b - 2*b^2)*sqrt(-(a + b)/b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a*b + 2*b^2)*cos(f*x + e)^3 - b^2*cos(f*x + e))*sqrt(-(a + b)/b)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/(a^3*b*f*cos(f*x + e)^2 + a^2*b^2*f), 1/4*(4*a*b*f*x*cos(f*x + e)^2 + 4*b^2*f*x - 2*(a^2 + a*b)*cos(f*x + e)*sin(f*x + e) - ((a^2 - 2*a*b)*cos(f*x + e)^2 + a*b - 2*b^2)*sqrt((a + b)/b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a + b)/b)/((a + b)*cos(f*x + e)*sin(f*x + e))))/(a^3*b*f*cos(f*x + e)^2 + a^2*b^2*f)]","B",0
358,1,458,0,0.678406," ","integrate(tan(f*x+e)^2/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{2} b + a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 8 \, {\left(a b^{2} + b^{3}\right)} f x - 4 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left({\left(a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}, -\frac{4 \, {\left(a^{2} b + a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 4 \, {\left(a b^{2} + b^{3}\right)} f x - 2 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*(a^2*b + a*b^2)*f*x*cos(f*x + e)^2 + 8*(a*b^2 + b^3)*f*x - 4*(a^2*b + a*b^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/((a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^3*b^2 + a^2*b^3)*f), -1/4*(4*(a^2*b + a*b^2)*f*x*cos(f*x + e)^2 + 4*(a*b^2 + b^3)*f*x - 2*(a^2*b + a*b^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))))/((a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^3*b^2 + a^2*b^3)*f)]","B",0
359,1,435,0,0.559015," ","integrate(1/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} + a b\right)} f x \cos\left(f x + e\right)^{2} - 4 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 8 \, {\left(a b + b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{4 \, {\left(a^{2} + a b\right)} f x \cos\left(f x + e\right)^{2} - 2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 4 \, {\left(a b + b^{2}\right)} f x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(8*(a^2 + a*b)*f*x*cos(f*x + e)^2 - 4*a*b*cos(f*x + e)*sin(f*x + e) + 8*(a*b + b^2)*f*x + ((3*a^2 + 2*a*b)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), 1/4*(4*(a^2 + a*b)*f*x*cos(f*x + e)^2 - 2*a*b*cos(f*x + e)*sin(f*x + e) + 4*(a*b + b^2)*f*x + ((3*a^2 + 2*a*b)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]","B",0
360,1,604,0,0.600495," ","integrate(cot(f*x+e)^2/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a^{3} + a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a b^{2} + 2 \, b^{3} + {\left(5 \, a^{2} b + 2 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 4 \, {\left(2 \, a^{2} b - a b^{2}\right)} \cos\left(f x + e\right) + 8 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f x\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(2 \, a^{3} + a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a b^{2} + 2 \, b^{3} + {\left(5 \, a^{2} b + 2 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, {\left(2 \, a^{2} b - a b^{2}\right)} \cos\left(f x + e\right) + 4 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f x\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/8*(4*(2*a^3 + a*b^2)*cos(f*x + e)^3 - (5*a*b^2 + 2*b^3 + (5*a^2*b + 2*a*b^2)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 4*(2*a^2*b - a*b^2)*cos(f*x + e) + 8*((a^3 + 2*a^2*b + a*b^2)*f*x*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3)*f*x)*sin(f*x + e))/(((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)*sin(f*x + e)), -1/4*(2*(2*a^3 + a*b^2)*cos(f*x + e)^3 + (5*a*b^2 + 2*b^3 + (5*a^2*b + 2*a*b^2)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 2*(2*a^2*b - a*b^2)*cos(f*x + e) + 4*((a^3 + 2*a^2*b + a*b^2)*f*x*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3)*f*x)*sin(f*x + e))/(((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)*sin(f*x + e))]","B",0
361,1,979,0,0.608483," ","integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(8 \, a^{4} + 20 \, a^{3} b + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(3 \, a^{4} + 5 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(7 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} - 7 \, a b^{3} - 2 \, b^{4} - {\left(7 \, a^{2} b^{2} - 5 \, a b^{3} - 2 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, {\left(2 \, a^{3} b + 6 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right) + 24 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f x \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f x\right)} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)} \sin\left(f x + e\right)}, \frac{2 \, {\left(8 \, a^{4} + 20 \, a^{3} b + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(3 \, a^{4} + 5 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(7 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} - 7 \, a b^{3} - 2 \, b^{4} - {\left(7 \, a^{2} b^{2} - 5 \, a b^{3} - 2 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, {\left(2 \, a^{3} b + 6 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right) + 12 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} f x \cos\left(f x + e\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f x\right)} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/24*(4*(8*a^4 + 20*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - 8*(3*a^4 + 5*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^3 + 3*((7*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 7*a*b^3 - 2*b^4 - (7*a^2*b^2 - 5*a*b^3 - 2*b^4)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(2*a^3*b + 6*a^2*b^2 - a*b^3)*cos(f*x + e) + 24*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*x*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*x)*sin(f*x + e))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e)), 1/12*(2*(8*a^4 + 20*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - 4*(3*a^4 + 5*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^3 + 3*((7*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 - 7*a*b^3 - 2*b^4 - (7*a^2*b^2 - 5*a*b^3 - 2*b^4)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*(2*a^3*b + 6*a^2*b^2 - a*b^3)*cos(f*x + e) + 12*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*f*x*cos(f*x + e)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*x)*sin(f*x + e))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e))]","B",0
362,1,1505,0,0.607839," ","integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(46 \, a^{5} + 172 \, a^{4} b + 216 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(70 \, a^{5} + 234 \, a^{4} b + 218 \, a^{3} b^{2} - 216 \, a^{2} b^{3} + 45 \, a b^{4}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(6 \, a^{5} + 10 \, a^{4} b - 20 \, a^{3} b^{2} - 78 \, a^{2} b^{3} + 9 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(9 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \cos\left(f x + e\right)^{6} + 9 \, a b^{4} + 2 \, b^{5} - {\left(18 \, a^{2} b^{3} - 5 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(9 \, a^{2} b^{3} - 16 \, a b^{4} - 4 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(2 \, a^{4} b + 8 \, a^{3} b^{2} + 12 \, a^{2} b^{3} - a b^{4}\right)} \cos\left(f x + e\right) + 120 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f x \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f x\right)} \sin\left(f x + e\right)}{120 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(46 \, a^{5} + 172 \, a^{4} b + 216 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(70 \, a^{5} + 234 \, a^{4} b + 218 \, a^{3} b^{2} - 216 \, a^{2} b^{3} + 45 \, a b^{4}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(6 \, a^{5} + 10 \, a^{4} b - 20 \, a^{3} b^{2} - 78 \, a^{2} b^{3} + 9 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(9 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \cos\left(f x + e\right)^{6} + 9 \, a b^{4} + 2 \, b^{5} - {\left(18 \, a^{2} b^{3} - 5 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(9 \, a^{2} b^{3} - 16 \, a b^{4} - 4 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(2 \, a^{4} b + 8 \, a^{3} b^{2} + 12 \, a^{2} b^{3} - a b^{4}\right)} \cos\left(f x + e\right) + 60 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} f x \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f x\right)} \sin\left(f x + e\right)}{60 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/120*(4*(46*a^5 + 172*a^4*b + 216*a^3*b^2 + 15*a*b^4)*cos(f*x + e)^7 - 4*(70*a^5 + 234*a^4*b + 218*a^3*b^2 - 216*a^2*b^3 + 45*a*b^4)*cos(f*x + e)^5 + 20*(6*a^5 + 10*a^4*b - 20*a^3*b^2 - 78*a^2*b^3 + 9*a*b^4)*cos(f*x + e)^3 - 15*((9*a^2*b^3 + 2*a*b^4)*cos(f*x + e)^6 + 9*a*b^4 + 2*b^5 - (18*a^2*b^3 - 5*a*b^4 - 2*b^5)*cos(f*x + e)^4 + (9*a^2*b^3 - 16*a*b^4 - 4*b^5)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(2*a^4*b + 8*a^3*b^2 + 12*a^2*b^3 - a*b^4)*cos(f*x + e) + 120*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*x*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*x*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f*x)*sin(f*x + e))/(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f)*sin(f*x + e)), -1/60*(2*(46*a^5 + 172*a^4*b + 216*a^3*b^2 + 15*a*b^4)*cos(f*x + e)^7 - 2*(70*a^5 + 234*a^4*b + 218*a^3*b^2 - 216*a^2*b^3 + 45*a*b^4)*cos(f*x + e)^5 + 10*(6*a^5 + 10*a^4*b - 20*a^3*b^2 - 78*a^2*b^3 + 9*a*b^4)*cos(f*x + e)^3 + 15*((9*a^2*b^3 + 2*a*b^4)*cos(f*x + e)^6 + 9*a*b^4 + 2*b^5 - (18*a^2*b^3 - 5*a*b^4 - 2*b^5)*cos(f*x + e)^4 + (9*a^2*b^3 - 16*a*b^4 - 4*b^5)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(2*a^4*b + 8*a^3*b^2 + 12*a^2*b^3 - a*b^4)*cos(f*x + e) + 60*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^6 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*f*x*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*f*x*cos(f*x + e)^2 + (a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f*x)*sin(f*x + e))/(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f)*sin(f*x + e))]","B",0
363,1,116,0,0.540666," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","-\frac{4 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} - a^{2} + 2 \, a b + 3 \, b^{2} + 2 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right)}{4 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}"," ",0,"-1/4*(4*(a^2 + a*b)*cos(f*x + e)^2 - a^2 + 2*a*b + 3*b^2 + 2*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*log(a*cos(f*x + e)^2 + b))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)","A",0
364,1,111,0,0.556224," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{2 \, {\left(a^{2} + 2 \, a b\right)} \cos\left(f x + e\right)^{2} + a b + 3 \, b^{2} + 2 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right)}{4 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}"," ",0,"1/4*(2*(a^2 + 2*a*b)*cos(f*x + e)^2 + a*b + 3*b^2 + 2*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*log(a*cos(f*x + e)^2 + b))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)","A",0
365,1,102,0,0.527418," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","-\frac{4 \, a b \cos\left(f x + e\right)^{2} + 3 \, b^{2} + 2 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right)}{4 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}"," ",0,"-1/4*(4*a*b*cos(f*x + e)^2 + 3*b^2 + 2*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*log(a*cos(f*x + e)^2 + b))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)","A",0
366,1,307,0,1.073747," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{5 \, a^{2} b^{3} + 8 \, a b^{4} + 3 \, b^{5} + 2 \, {\left(3 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 4 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{4 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}"," ",0,"1/4*(5*a^2*b^3 + 8*a*b^4 + 3*b^5 + 2*(3*a^3*b^2 + 5*a^2*b^3 + 2*a*b^4)*cos(f*x + e)^2 + 2*(3*a^2*b^3 + 3*a*b^4 + b^5 + (3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) + 4*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*log(1/2*sin(f*x + e)))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)","B",0
367,1,584,0,1.948920," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + 7 \, a^{2} b^{4} + 10 \, a b^{5} + 3 \, b^{6} + 2 \, {\left(a^{6} + a^{5} b - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(4 \, a^{5} b + 4 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - 6 \, a b^{5} - 3 \, b^{6}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left({\left(6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(6 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(12 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) - 4 \, {\left({\left(a^{6} + 4 \, a^{5} b\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - {\left(a^{6} + 2 \, a^{5} b - 8 \, a^{4} b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} - 4 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{4 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)}}"," ",0,"1/4*(2*a^4*b^2 + 2*a^3*b^3 + 7*a^2*b^4 + 10*a*b^5 + 3*b^6 + 2*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 + (4*a^5*b + 4*a^4*b^2 + 8*a^3*b^3 + 5*a^2*b^4 - 6*a*b^5 - 3*b^6)*cos(f*x + e)^2 - 2*((6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (6*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (12*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) - 4*((a^6 + 4*a^5*b)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - (a^6 + 2*a^5*b - 8*a^4*b^2)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 - 4*a^3*b^3)*cos(f*x + e)^2)*log(1/2*sin(f*x + e)))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)","B",0
368,1,859,0,3.170424," ","integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{3 \, a^{5} b^{2} + 12 \, a^{4} b^{3} + 9 \, a^{3} b^{4} + 9 \, a^{2} b^{5} + 12 \, a b^{6} + 3 \, b^{7} - 2 \, {\left(2 \, a^{7} + 7 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - 2 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(3 \, a^{7} + 4 \, a^{6} b - 19 \, a^{5} b^{2} - 20 \, a^{4} b^{3} - 20 \, a^{3} b^{4} - 19 \, a^{2} b^{5} + 4 \, a b^{6} + 3 \, b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{6} b + 10 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{2} b^{5} - 10 \, a b^{6} - 3 \, b^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left({\left(10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(10 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(10 \, a^{4} b^{3} - 35 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(10 \, a^{3} b^{4} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(a \cos\left(f x + e\right)^{2} + b\right) + 4 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 10 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 10 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 10 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(f x + e\right)\right)}{4 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)}}"," ",0,"1/4*(3*a^5*b^2 + 12*a^4*b^3 + 9*a^3*b^4 + 9*a^2*b^5 + 12*a*b^6 + 3*b^7 - 2*(2*a^7 + 7*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 7*a^2*b^5 - 2*a*b^6)*cos(f*x + e)^6 + (3*a^7 + 4*a^6*b - 19*a^5*b^2 - 20*a^4*b^3 - 20*a^3*b^4 - 19*a^2*b^5 + 4*a*b^6 + 3*b^7)*cos(f*x + e)^4 + 2*(3*a^6*b + 10*a^5*b^2 + 2*a^4*b^3 - 2*a^2*b^5 - 10*a*b^6 - 3*b^7)*cos(f*x + e)^2 + 2*((10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(10*a^4*b^3 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (10*a^4*b^3 - 35*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(10*a^3*b^4 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*log(a*cos(f*x + e)^2 + b) + 4*((a^7 + 5*a^6*b + 10*a^5*b^2)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 10*a^4*b^3)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 35*a^4*b^3 + 10*a^3*b^4)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 10*a^3*b^4)*cos(f*x + e)^2)*log(1/2*sin(f*x + e)))/((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f)","B",0
369,1,664,0,0.637167," ","integrate(tan(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{32 \, a^{2} b^{2} f x \cos\left(f x + e\right)^{4} + 64 \, a b^{3} f x \cos\left(f x + e\right)^{2} + 32 \, b^{4} f x - {\left({\left(3 \, a^{4} - 4 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} - 4 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{a + b}{b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - b^{2} \cos\left(f x + e\right)\right)} \sqrt{-\frac{a + b}{b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left(3 \, {\left(a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{3} b + a^{2} b^{2} - 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left(a^{5} b^{2} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{3} f \cos\left(f x + e\right)^{2} + a^{3} b^{4} f\right)}}, -\frac{16 \, a^{2} b^{2} f x \cos\left(f x + e\right)^{4} + 32 \, a b^{3} f x \cos\left(f x + e\right)^{2} + 16 \, b^{4} f x + {\left({\left(3 \, a^{4} - 4 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} - 4 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a + b}{b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{a + b}{b}}}{2 \, {\left(a + b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(3 \, {\left(a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{3} b + a^{2} b^{2} - 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left(a^{5} b^{2} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{3} f \cos\left(f x + e\right)^{2} + a^{3} b^{4} f\right)}}\right]"," ",0,"[-1/32*(32*a^2*b^2*f*x*cos(f*x + e)^4 + 64*a*b^3*f*x*cos(f*x + e)^2 + 32*b^4*f*x - ((3*a^4 - 4*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 - 4*a*b^3 + 8*b^4 + 2*(3*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(-(a + b)/b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a*b + 2*b^2)*cos(f*x + e)^3 - b^2*cos(f*x + e))*sqrt(-(a + b)/b)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) + 4*(3*(a^4 - a^3*b - 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^3*b + a^2*b^2 - 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^5*b^2*f*cos(f*x + e)^4 + 2*a^4*b^3*f*cos(f*x + e)^2 + a^3*b^4*f), -1/16*(16*a^2*b^2*f*x*cos(f*x + e)^4 + 32*a*b^3*f*x*cos(f*x + e)^2 + 16*b^4*f*x + ((3*a^4 - 4*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 - 4*a*b^3 + 8*b^4 + 2*(3*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt((a + b)/b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt((a + b)/b)/((a + b)*cos(f*x + e)*sin(f*x + e))) + 2*(3*(a^4 - a^3*b - 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^3*b + a^2*b^2 - 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/(a^5*b^2*f*cos(f*x + e)^4 + 2*a^4*b^3*f*cos(f*x + e)^2 + a^3*b^4*f)]","B",0
370,1,763,0,0.576501," ","integrate(tan(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{3} b^{2} + a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 64 \, {\left(a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 32 \, {\left(a b^{4} + b^{5}\right)} f x + {\left({\left(a^{4} - 4 \, a^{3} b - 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 4 \, a b^{3} - 8 \, b^{4} + 2 \, {\left(a^{3} b - 4 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left({\left(a^{4} b + 7 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}, \frac{16 \, {\left(a^{3} b^{2} + a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a b^{4} + b^{5}\right)} f x - {\left({\left(a^{4} - 4 \, a^{3} b - 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 4 \, a b^{3} - 8 \, b^{4} + 2 \, {\left(a^{3} b - 4 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left({\left(a^{4} b + 7 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}\right]"," ",0,"[1/32*(32*(a^3*b^2 + a^2*b^3)*f*x*cos(f*x + e)^4 + 64*(a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^2 + 32*(a*b^4 + b^5)*f*x + ((a^4 - 4*a^3*b - 8*a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 - 4*a*b^3 - 8*b^4 + 2*(a^3*b - 4*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*((a^4*b + 7*a^3*b^2 + 6*a^2*b^3)*cos(f*x + e)^3 - (a^3*b^2 - 3*a^2*b^3 - 4*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^4*b^4 + a^3*b^5)*f), 1/16*(16*(a^3*b^2 + a^2*b^3)*f*x*cos(f*x + e)^4 + 32*(a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^2 + 16*(a*b^4 + b^5)*f*x - ((a^4 - 4*a^3*b - 8*a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 - 4*a*b^3 - 8*b^4 + 2*(a^3*b - 4*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - 2*((a^4*b + 7*a^3*b^2 + 6*a^2*b^3)*cos(f*x + e)^3 - (a^3*b^2 - 3*a^2*b^3 - 4*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^4*b^4 + a^3*b^5)*f)]","B",0
371,1,860,0,0.550991," ","integrate(tan(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{32 \, {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 64 \, {\left(a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 32 \, {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} f x + {\left({\left(3 \, a^{4} + 12 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 12 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 12 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a b - b^{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b - b^{2}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left({\left(5 \, a^{4} b + 11 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 4 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b^{2} + 2 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{3} + 2 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}, -\frac{16 \, {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} f x + {\left({\left(3 \, a^{4} + 12 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} + 12 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(3 \, a^{3} b + 12 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a b + b^{2}} \arctan\left(\frac{{\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b}{2 \, \sqrt{a b + b^{2}} \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left({\left(5 \, a^{4} b + 11 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 4 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} b + 2 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b^{2} + 2 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{3} + 2 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}\right]"," ",0,"[-1/32*(32*(a^4*b + 2*a^3*b^2 + a^2*b^3)*f*x*cos(f*x + e)^4 + 64*(a^3*b^2 + 2*a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^2 + 32*(a^2*b^3 + 2*a*b^4 + b^5)*f*x + ((3*a^4 + 12*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 12*a*b^3 + 8*b^4 + 2*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(-a*b - b^2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b - b^2)*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*((5*a^4*b + 11*a^3*b^2 + 6*a^2*b^3)*cos(f*x + e)^3 + (3*a^3*b^2 + 7*a^2*b^3 + 4*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*f), -1/16*(16*(a^4*b + 2*a^3*b^2 + a^2*b^3)*f*x*cos(f*x + e)^4 + 32*(a^3*b^2 + 2*a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^2 + 16*(a^2*b^3 + 2*a*b^4 + b^5)*f*x + ((3*a^4 + 12*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 3*a^2*b^2 + 12*a*b^3 + 8*b^4 + 2*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(a*b + b^2)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)/(sqrt(a*b + b^2)*cos(f*x + e)*sin(f*x + e))) - 2*((5*a^4*b + 11*a^3*b^2 + 6*a^2*b^3)*cos(f*x + e)^3 + (3*a^3*b^2 + 7*a^2*b^3 + 4*a*b^4)*cos(f*x + e))*sin(f*x + e))/((a^7*b + 2*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^6*b^2 + 2*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*f)]","B",0
372,1,819,0,0.625024," ","integrate(1/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 64 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 32 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 20 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(7 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, \frac{16 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 20 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(7 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[1/32*(32*(a^4 + 2*a^3*b + a^2*b^2)*f*x*cos(f*x + e)^4 + 64*(a^3*b + 2*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^2 + 32*(a^2*b^2 + 2*a*b^3 + b^4)*f*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 20*a*b^3 + 8*b^4 + 2*(15*a^3*b + 20*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(3*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (7*a^2*b^2 + 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f), 1/16*(16*(a^4 + 2*a^3*b + a^2*b^2)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 2*a^2*b^2 + a*b^3)*f*x*cos(f*x + e)^2 + 16*(a^2*b^2 + 2*a*b^3 + b^4)*f*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 20*a*b^3 + 8*b^4 + 2*(15*a^3*b + 20*a^2*b^2 + 8*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 2*(3*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (7*a^2*b^2 + 4*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f)]","B",0
373,1,1060,0,0.684242," ","integrate(cot(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{5} + 13 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{5} + 4 \, {\left(16 \, a^{4} b - 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 4 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(35 \, a^{2} b^{3} + 28 \, a b^{4} + 8 \, b^{5} + {\left(35 \, a^{4} b + 28 \, a^{3} b^{2} + 8 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(35 \, a^{3} b^{2} + 28 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 4 \, {\left(8 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \cos\left(f x + e\right) + 32 \, {\left({\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f x\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(8 \, a^{5} + 13 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(16 \, a^{4} b - 13 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 4 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(35 \, a^{2} b^{3} + 28 \, a b^{4} + 8 \, b^{5} + {\left(35 \, a^{4} b + 28 \, a^{3} b^{2} + 8 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(35 \, a^{3} b^{2} + 28 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, {\left(8 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \cos\left(f x + e\right) + 16 \, {\left({\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f x\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/32*(4*(8*a^5 + 13*a^3*b^2 + 6*a^2*b^3)*cos(f*x + e)^5 + 4*(16*a^4*b - 13*a^3*b^2 + 5*a^2*b^3 + 4*a*b^4)*cos(f*x + e)^3 - (35*a^2*b^3 + 28*a*b^4 + 8*b^5 + (35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cos(f*x + e)^4 + 2*(35*a^3*b^2 + 28*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 4*(8*a^3*b^2 - 11*a^2*b^3 - 4*a*b^4)*cos(f*x + e) + 32*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*f*x*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^2 + (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f*x)*sin(f*x + e))/(((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)*sin(f*x + e)), -1/16*(2*(8*a^5 + 13*a^3*b^2 + 6*a^2*b^3)*cos(f*x + e)^5 + 2*(16*a^4*b - 13*a^3*b^2 + 5*a^2*b^3 + 4*a*b^4)*cos(f*x + e)^3 + (35*a^2*b^3 + 28*a*b^4 + 8*b^5 + (35*a^4*b + 28*a^3*b^2 + 8*a^2*b^3)*cos(f*x + e)^4 + 2*(35*a^3*b^2 + 28*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 2*(8*a^3*b^2 - 11*a^2*b^3 - 4*a*b^4)*cos(f*x + e) + 16*((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*f*x*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*x*cos(f*x + e)^2 + (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f*x)*sin(f*x + e))/(((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)*sin(f*x + e))]","B",0
374,1,1649,0,0.877177," ","integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(32 \, a^{6} + 104 \, a^{5} b + 51 \, a^{3} b^{3} + 18 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(24 \, a^{6} + 32 \, a^{5} b - 208 \, a^{4} b^{2} + 102 \, a^{3} b^{3} - 9 \, a^{2} b^{4} - 12 \, a b^{5}\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(48 \, a^{5} b + 160 \, a^{4} b^{2} - 155 \, a^{3} b^{3} + 72 \, a^{2} b^{4} + 24 \, a b^{5}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - 63 \, a^{2} b^{4} - 36 \, a b^{5} - 8 \, b^{6} - {\left(63 \, a^{4} b^{2} - 90 \, a^{3} b^{3} - 64 \, a^{2} b^{4} - 16 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(126 \, a^{3} b^{3} + 9 \, a^{2} b^{4} - 20 \, a b^{5} - 8 \, b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, {\left(8 \, a^{4} b^{2} + 32 \, a^{3} b^{3} - 15 \, a^{2} b^{4} - 4 \, a b^{5}\right)} \cos\left(f x + e\right) + 96 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} f x \cos\left(f x + e\right)^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} f x \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} f x \cos\left(f x + e\right)^{2} - {\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f x\right)} \sin\left(f x + e\right)}{96 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)} \sin\left(f x + e\right)}, \frac{2 \, {\left(32 \, a^{6} + 104 \, a^{5} b + 51 \, a^{3} b^{3} + 18 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(24 \, a^{6} + 32 \, a^{5} b - 208 \, a^{4} b^{2} + 102 \, a^{3} b^{3} - 9 \, a^{2} b^{4} - 12 \, a b^{5}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(48 \, a^{5} b + 160 \, a^{4} b^{2} - 155 \, a^{3} b^{3} + 72 \, a^{2} b^{4} + 24 \, a b^{5}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - 63 \, a^{2} b^{4} - 36 \, a b^{5} - 8 \, b^{6} - {\left(63 \, a^{4} b^{2} - 90 \, a^{3} b^{3} - 64 \, a^{2} b^{4} - 16 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(126 \, a^{3} b^{3} + 9 \, a^{2} b^{4} - 20 \, a b^{5} - 8 \, b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, {\left(8 \, a^{4} b^{2} + 32 \, a^{3} b^{3} - 15 \, a^{2} b^{4} - 4 \, a b^{5}\right)} \cos\left(f x + e\right) + 48 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} f x \cos\left(f x + e\right)^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} f x \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} f x \cos\left(f x + e\right)^{2} - {\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f x\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/96*(4*(32*a^6 + 104*a^5*b + 51*a^3*b^3 + 18*a^2*b^4)*cos(f*x + e)^7 - 4*(24*a^6 + 32*a^5*b - 208*a^4*b^2 + 102*a^3*b^3 - 9*a^2*b^4 - 12*a*b^5)*cos(f*x + e)^5 - 4*(48*a^5*b + 160*a^4*b^2 - 155*a^3*b^3 + 72*a^2*b^4 + 24*a*b^5)*cos(f*x + e)^3 + 3*((63*a^4*b^2 + 36*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 - 63*a^2*b^4 - 36*a*b^5 - 8*b^6 - (63*a^4*b^2 - 90*a^3*b^3 - 64*a^2*b^4 - 16*a*b^5)*cos(f*x + e)^4 - (126*a^3*b^3 + 9*a^2*b^4 - 20*a*b^5 - 8*b^6)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(8*a^4*b^2 + 32*a^3*b^3 - 15*a^2*b^4 - 4*a*b^5)*cos(f*x + e) + 96*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*x*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*x*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*x*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*x)*sin(f*x + e))/(((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)*sin(f*x + e)), 1/48*(2*(32*a^6 + 104*a^5*b + 51*a^3*b^3 + 18*a^2*b^4)*cos(f*x + e)^7 - 2*(24*a^6 + 32*a^5*b - 208*a^4*b^2 + 102*a^3*b^3 - 9*a^2*b^4 - 12*a*b^5)*cos(f*x + e)^5 - 2*(48*a^5*b + 160*a^4*b^2 - 155*a^3*b^3 + 72*a^2*b^4 + 24*a*b^5)*cos(f*x + e)^3 + 3*((63*a^4*b^2 + 36*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 - 63*a^2*b^4 - 36*a*b^5 - 8*b^6 - (63*a^4*b^2 - 90*a^3*b^3 - 64*a^2*b^4 - 16*a*b^5)*cos(f*x + e)^4 - (126*a^3*b^3 + 9*a^2*b^4 - 20*a*b^5 - 8*b^6)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*(8*a^4*b^2 + 32*a^3*b^3 - 15*a^2*b^4 - 4*a*b^5)*cos(f*x + e) + 48*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*f*x*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*x*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*x*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*x)*sin(f*x + e))/(((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)*sin(f*x + e))]","B",0
375,1,2229,0,0.973769," ","integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(184 \, a^{7} + 848 \, a^{6} b + 1384 \, a^{5} b^{2} + 315 \, a^{3} b^{4} + 90 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{9} - 4 \, {\left(280 \, a^{7} + 1032 \, a^{6} b + 864 \, a^{5} b^{2} - 2768 \, a^{4} b^{3} + 945 \, a^{3} b^{4} - 15 \, a^{2} b^{5} - 60 \, a b^{6}\right)} \cos\left(f x + e\right)^{7} + 4 \, {\left(120 \, a^{7} + 40 \, a^{6} b - 1416 \, a^{5} b^{2} - 4272 \, a^{4} b^{3} + 2329 \, a^{3} b^{4} - 585 \, a^{2} b^{5} - 180 \, a b^{6}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(48 \, a^{6} b + 184 \, a^{5} b^{2} + 200 \, a^{4} b^{3} - 575 \, a^{3} b^{4} + 153 \, a^{2} b^{5} + 36 \, a b^{6}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(99 \, a^{4} b^{3} + 44 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + 99 \, a^{2} b^{5} + 44 \, a b^{6} + 8 \, b^{7} - 2 \, {\left(99 \, a^{4} b^{3} - 55 \, a^{3} b^{4} - 36 \, a^{2} b^{5} - 8 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(99 \, a^{4} b^{3} - 352 \, a^{3} b^{4} - 69 \, a^{2} b^{5} + 12 \, a b^{6} + 8 \, b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(99 \, a^{3} b^{4} - 55 \, a^{2} b^{5} - 36 \, a b^{6} - 8 \, b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(f x + e\right) + b^{2}}{a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(8 \, a^{5} b^{2} + 40 \, a^{4} b^{3} + 80 \, a^{3} b^{4} - 19 \, a^{2} b^{5} - 4 \, a b^{6}\right)} \cos\left(f x + e\right) + 480 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f x \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f x \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f x \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f x\right)} \sin\left(f x + e\right)}{480 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(184 \, a^{7} + 848 \, a^{6} b + 1384 \, a^{5} b^{2} + 315 \, a^{3} b^{4} + 90 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{9} - 2 \, {\left(280 \, a^{7} + 1032 \, a^{6} b + 864 \, a^{5} b^{2} - 2768 \, a^{4} b^{3} + 945 \, a^{3} b^{4} - 15 \, a^{2} b^{5} - 60 \, a b^{6}\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(120 \, a^{7} + 40 \, a^{6} b - 1416 \, a^{5} b^{2} - 4272 \, a^{4} b^{3} + 2329 \, a^{3} b^{4} - 585 \, a^{2} b^{5} - 180 \, a b^{6}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(48 \, a^{6} b + 184 \, a^{5} b^{2} + 200 \, a^{4} b^{3} - 575 \, a^{3} b^{4} + 153 \, a^{2} b^{5} + 36 \, a b^{6}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(99 \, a^{4} b^{3} + 44 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + 99 \, a^{2} b^{5} + 44 \, a b^{6} + 8 \, b^{7} - 2 \, {\left(99 \, a^{4} b^{3} - 55 \, a^{3} b^{4} - 36 \, a^{2} b^{5} - 8 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(99 \, a^{4} b^{3} - 352 \, a^{3} b^{4} - 69 \, a^{2} b^{5} + 12 \, a b^{6} + 8 \, b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(99 \, a^{3} b^{4} - 55 \, a^{2} b^{5} - 36 \, a b^{6} - 8 \, b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(8 \, a^{5} b^{2} + 40 \, a^{4} b^{3} + 80 \, a^{3} b^{4} - 19 \, a^{2} b^{5} - 4 \, a b^{6}\right)} \cos\left(f x + e\right) + 240 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f x \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} f x \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} f x \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} f x \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f x\right)} \sin\left(f x + e\right)}{240 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/480*(4*(184*a^7 + 848*a^6*b + 1384*a^5*b^2 + 315*a^3*b^4 + 90*a^2*b^5)*cos(f*x + e)^9 - 4*(280*a^7 + 1032*a^6*b + 864*a^5*b^2 - 2768*a^4*b^3 + 945*a^3*b^4 - 15*a^2*b^5 - 60*a*b^6)*cos(f*x + e)^7 + 4*(120*a^7 + 40*a^6*b - 1416*a^5*b^2 - 4272*a^4*b^3 + 2329*a^3*b^4 - 585*a^2*b^5 - 180*a*b^6)*cos(f*x + e)^5 + 20*(48*a^6*b + 184*a^5*b^2 + 200*a^4*b^3 - 575*a^3*b^4 + 153*a^2*b^5 + 36*a*b^6)*cos(f*x + e)^3 - 15*((99*a^4*b^3 + 44*a^3*b^4 + 8*a^2*b^5)*cos(f*x + e)^8 + 99*a^2*b^5 + 44*a*b^6 + 8*b^7 - 2*(99*a^4*b^3 - 55*a^3*b^4 - 36*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^6 + (99*a^4*b^3 - 352*a^3*b^4 - 69*a^2*b^5 + 12*a*b^6 + 8*b^7)*cos(f*x + e)^4 + 2*(99*a^3*b^4 - 55*a^2*b^5 - 36*a*b^6 - 8*b^7)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(8*a^5*b^2 + 40*a^4*b^3 + 80*a^3*b^4 - 19*a^2*b^5 - 4*a*b^6)*cos(f*x + e) + 480*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*x*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*x*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*x*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*x*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f*x)*sin(f*x + e))/(((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f)*sin(f*x + e)), -1/240*(2*(184*a^7 + 848*a^6*b + 1384*a^5*b^2 + 315*a^3*b^4 + 90*a^2*b^5)*cos(f*x + e)^9 - 2*(280*a^7 + 1032*a^6*b + 864*a^5*b^2 - 2768*a^4*b^3 + 945*a^3*b^4 - 15*a^2*b^5 - 60*a*b^6)*cos(f*x + e)^7 + 2*(120*a^7 + 40*a^6*b - 1416*a^5*b^2 - 4272*a^4*b^3 + 2329*a^3*b^4 - 585*a^2*b^5 - 180*a*b^6)*cos(f*x + e)^5 + 10*(48*a^6*b + 184*a^5*b^2 + 200*a^4*b^3 - 575*a^3*b^4 + 153*a^2*b^5 + 36*a*b^6)*cos(f*x + e)^3 + 15*((99*a^4*b^3 + 44*a^3*b^4 + 8*a^2*b^5)*cos(f*x + e)^8 + 99*a^2*b^5 + 44*a*b^6 + 8*b^7 - 2*(99*a^4*b^3 - 55*a^3*b^4 - 36*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^6 + (99*a^4*b^3 - 352*a^3*b^4 - 69*a^2*b^5 + 12*a*b^6 + 8*b^7)*cos(f*x + e)^4 + 2*(99*a^3*b^4 - 55*a^2*b^5 - 36*a*b^6 - 8*b^7)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(8*a^5*b^2 + 40*a^4*b^3 + 80*a^3*b^4 - 19*a^2*b^5 - 4*a*b^6)*cos(f*x + e) + 240*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f*x*cos(f*x + e)^8 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*f*x*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*f*x*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*f*x*cos(f*x + e)^2 + (a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f*x)*sin(f*x + e))/(((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f)*sin(f*x + e))]","B",0
376,1,456,0,3.277666," ","integrate((a+b*sec(f*x+e)^2)^(1/2)*tan(f*x+e)^5,x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{a} b^{2} \cos\left(f x + e\right)^{4} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 8 \, {\left({\left(2 \, a^{2} + 10 \, a b - 15 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a b - 10 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, b^{2} f \cos\left(f x + e\right)^{4}}, \frac{15 \, \sqrt{-a} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) \cos\left(f x + e\right)^{4} - 4 \, {\left({\left(2 \, a^{2} + 10 \, a b - 15 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a b - 10 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, b^{2} f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/120*(15*sqrt(a)*b^2*cos(f*x + e)^4*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 8*((2*a^2 + 10*a*b - 15*b^2)*cos(f*x + e)^4 - (a*b - 10*b^2)*cos(f*x + e)^2 - 3*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b^2*f*cos(f*x + e)^4), 1/60*(15*sqrt(-a)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))*cos(f*x + e)^4 - 4*((2*a^2 + 10*a*b - 15*b^2)*cos(f*x + e)^4 - (a*b - 10*b^2)*cos(f*x + e)^2 - 3*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b^2*f*cos(f*x + e)^4)]","B",0
377,1,386,0,1.241654," ","integrate((a+b*sec(f*x+e)^2)^(1/2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a} b \cos\left(f x + e\right)^{2} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, {\left({\left(a - 3 \, b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, b f \cos\left(f x + e\right)^{2}}, -\frac{3 \, \sqrt{-a} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - 3 \, b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, b f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/24*(3*sqrt(a)*b*cos(f*x + e)^2*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*((a - 3*b)*cos(f*x + e)^2 + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b*f*cos(f*x + e)^2), -1/12*(3*sqrt(-a)*b*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))*cos(f*x + e)^2 - 4*((a - 3*b)*cos(f*x + e)^2 + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b*f*cos(f*x + e)^2)]","B",0
378,1,312,0,0.690866," ","integrate((a+b*sec(f*x+e)^2)^(1/2)*tan(f*x+e),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, f}, \frac{\sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + 4 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f}\right]"," ",0,"[1/8*(sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/f, 1/4*(sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + 4*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/f]","B",0
379,1,963,0,0.925190," ","integrate(cot(f*x+e)*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 2 \, \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, f}, \frac{4 \, \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, f}, -\frac{\sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{4 \, f}, -\frac{\sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 2 \, \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right)}{4 \, f}\right]"," ",0,"[1/8*(sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 2*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/f, 1/8*(4*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/f, -1/4*(sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/f, -1/4*(sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 2*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)))/f]","B",0
380,1,1342,0,1.274664," ","integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}, \frac{4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} - 2 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}, \frac{4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + 2 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}, \frac{2 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right)}{4 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)}}\right]"," ",0,"[1/8*(4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + ((2*a + b)*cos(f*x + e)^2 - 2*a - b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f), 1/8*(4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 - 2*((2*a + b)*cos(f*x + e)^2 - 2*a - b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f), 1/8*(4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + 2*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + ((2*a + b)*cos(f*x + e)^2 - 2*a - b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f), 1/4*(2*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((2*a + b)*cos(f*x + e)^2 - 2*a - b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)))/((a + b)*f*cos(f*x + e)^2 - (a + b)*f)]","B",0
381,1,1953,0,2.712395," ","integrate(cot(f*x+e)^5*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + {\left({\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(6 \, a^{2} + 11 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 7 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, \frac{{\left({\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 2 \, {\left({\left(6 \, a^{2} + 11 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 7 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, -\frac{8 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(6 \, a^{2} + 11 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 7 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}, -\frac{4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 12 \, a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(6 \, a^{2} + 11 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 7 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)}}\right]"," ",0,"[1/32*(4*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + ((8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 + 12*a*b + 3*b^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((6*a^2 + 11*a*b + 5*b^2)*cos(f*x + e)^4 - (4*a^2 + 7*a*b + 3*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f), 1/16*(((8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 + 12*a*b + 3*b^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 2*((6*a^2 + 11*a*b + 5*b^2)*cos(f*x + e)^4 - (4*a^2 + 7*a*b + 3*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f), -1/32*(8*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 + 12*a*b + 3*b^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((6*a^2 + 11*a*b + 5*b^2)*cos(f*x + e)^4 - (4*a^2 + 7*a*b + 3*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f), -1/16*(4*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 12*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 + 12*a*b + 3*b^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((6*a^2 + 11*a*b + 5*b^2)*cos(f*x + e)^4 - (4*a^2 + 7*a*b + 3*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f)]","B",0
382,1,1775,0,7.636858," ","integrate((a+b*sec(f*x+e)^2)^(1/2)*tan(f*x+e)^6,x, algorithm=""fricas"")","\left[\frac{24 \, \sqrt{-a} b^{3} \cos\left(f x + e\right)^{5} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 3 \, {\left(a^{3} + 5 \, a^{2} b + 15 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{b} \cos\left(f x + e\right)^{5} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{2} b + 14 \, a b^{2} - 33 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, b^{3} - 2 \, {\left(a b^{2} - 13 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, b^{3} f \cos\left(f x + e\right)^{5}}, \frac{12 \, \sqrt{-a} b^{3} \cos\left(f x + e\right)^{5} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 3 \, {\left(a^{3} + 5 \, a^{2} b + 15 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - 2 \, {\left({\left(3 \, a^{2} b + 14 \, a b^{2} - 33 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, b^{3} - 2 \, {\left(a b^{2} - 13 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, b^{3} f \cos\left(f x + e\right)^{5}}, \frac{48 \, \sqrt{a} b^{3} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - 3 \, {\left(a^{3} + 5 \, a^{2} b + 15 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{b} \cos\left(f x + e\right)^{5} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{2} b + 14 \, a b^{2} - 33 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, b^{3} - 2 \, {\left(a b^{2} - 13 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, b^{3} f \cos\left(f x + e\right)^{5}}, \frac{24 \, \sqrt{a} b^{3} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} + 3 \, {\left(a^{3} + 5 \, a^{2} b + 15 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - 2 \, {\left({\left(3 \, a^{2} b + 14 \, a b^{2} - 33 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, b^{3} - 2 \, {\left(a b^{2} - 13 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, b^{3} f \cos\left(f x + e\right)^{5}}\right]"," ",0,"[1/192*(24*sqrt(-a)*b^3*cos(f*x + e)^5*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 3*(a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*sqrt(b)*cos(f*x + e)^5*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^2*b + 14*a*b^2 - 33*b^3)*cos(f*x + e)^4 - 8*b^3 - 2*(a*b^2 - 13*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^5), 1/96*(12*sqrt(-a)*b^3*cos(f*x + e)^5*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 3*(a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^5 - 2*((3*a^2*b + 14*a*b^2 - 33*b^3)*cos(f*x + e)^4 - 8*b^3 - 2*(a*b^2 - 13*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^5), 1/192*(48*sqrt(a)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^5 - 3*(a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*sqrt(b)*cos(f*x + e)^5*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^2*b + 14*a*b^2 - 33*b^3)*cos(f*x + e)^4 - 8*b^3 - 2*(a*b^2 - 13*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^5), 1/96*(24*sqrt(a)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^5 + 3*(a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^5 - 2*((3*a^2*b + 14*a*b^2 - 33*b^3)*cos(f*x + e)^4 - 8*b^3 - 2*(a*b^2 - 13*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^5)]","A",0
383,1,1621,0,2.195616," ","integrate((a+b*sec(f*x+e)^2)^(1/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a} b^{2} \cos\left(f x + e\right)^{3} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(a^{2} + 6 \, a b - 3 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b^{2} f \cos\left(f x + e\right)^{3}}, \frac{2 \, \sqrt{-a} b^{2} \cos\left(f x + e\right)^{3} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(a^{2} + 6 \, a b - 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + 2 \, {\left({\left(a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b^{2} f \cos\left(f x + e\right)^{3}}, -\frac{8 \, \sqrt{a} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + {\left(a^{2} + 6 \, a b - 3 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b^{2} f \cos\left(f x + e\right)^{3}}, -\frac{4 \, \sqrt{a} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + {\left(a^{2} + 6 \, a b - 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - 2 \, {\left({\left(a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b^{2} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[1/32*(4*sqrt(-a)*b^2*cos(f*x + e)^3*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (a^2 + 6*a*b - 3*b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*((a*b - 5*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^3), 1/16*(2*sqrt(-a)*b^2*cos(f*x + e)^3*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (a^2 + 6*a*b - 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 2*((a*b - 5*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^3), -1/32*(8*sqrt(a)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^3 + (a^2 + 6*a*b - 3*b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((a*b - 5*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^3), -1/16*(4*sqrt(a)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^3 + (a^2 + 6*a*b - 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 - 2*((a*b - 5*b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^3)]","B",0
384,1,1471,0,1.153587," ","integrate((a+b*sec(f*x+e)^2)^(1/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} b \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, b f \cos\left(f x + e\right)}, \frac{2 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + \sqrt{-a} b \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, b f \cos\left(f x + e\right)}, \frac{2 \, \sqrt{a} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, b f \cos\left(f x + e\right)}, \frac{\sqrt{a} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + {\left(a - b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, b f \cos\left(f x + e\right)}\right]"," ",0,"[1/8*(sqrt(-a)*b*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (a - b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)), 1/8*(2*(a - b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + sqrt(-a)*b*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)), 1/8*(2*sqrt(a)*b*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (a - b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)), 1/4*(sqrt(a)*b*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) + (a - b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e))]","B",0
385,1,1227,0,0.806118," ","integrate((a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 2 \, \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, f}, \frac{4 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, f}\right]"," ",0,"[1/8*(sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 2*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, 1/8*(4*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/f]","B",0
386,1,499,0,0.686694," ","integrate(cot(f*x+e)^2*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{8 \, f \sin\left(f x + e\right)}, \frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{4 \, f \sin\left(f x + e\right)}\right]"," ",0,"[1/8*(sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e)), 1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e))]","B",0
387,1,629,0,1.243034," ","integrate(cot(f*x+e)^4*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left({\left(4 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(4 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{2} - {\left(a + b\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/24*(3*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*((4*a + 3*b)*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^2 - (a + b)*f)*sin(f*x + e)), -1/12*(3*((a + b)*cos(f*x + e)^2 - a - b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((4*a + 3*b)*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^2 - (a + b)*f)*sin(f*x + e))]","B",0
388,1,849,0,3.950681," ","integrate(cot(f*x+e)^6*(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, {\left({\left(23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(35 \, a^{2} + 59 \, a b + 20 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 25 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(35 \, a^{2} + 59 \, a b + 20 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 25 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} + 2 \, a b + b^{2}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/120*(15*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*((23*a^2 + 40*a*b + 15*b^2)*cos(f*x + e)^5 - (35*a^2 + 59*a*b + 20*b^2)*cos(f*x + e)^3 + (15*a^2 + 25*a*b + 8*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f)*sin(f*x + e)), 1/60*(15*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((23*a^2 + 40*a*b + 15*b^2)*cos(f*x + e)^5 - (35*a^2 + 59*a*b + 20*b^2)*cos(f*x + e)^3 + (15*a^2 + 25*a*b + 8*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2 + (a^2 + 2*a*b + b^2)*f)*sin(f*x + e))]","B",0
389,1,527,0,12.892437," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^5,x, algorithm=""fricas"")","\left[\frac{105 \, a^{\frac{3}{2}} b^{2} \cos\left(f x + e\right)^{6} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 8 \, {\left(2 \, {\left(3 \, a^{3} + 21 \, a^{2} b - 70 \, a b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(3 \, a^{2} b - 84 \, a b^{2} + 35 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 15 \, b^{3} - 6 \, {\left(4 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{840 \, b^{2} f \cos\left(f x + e\right)^{6}}, \frac{105 \, \sqrt{-a} a b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) \cos\left(f x + e\right)^{6} - 4 \, {\left(2 \, {\left(3 \, a^{3} + 21 \, a^{2} b - 70 \, a b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(3 \, a^{2} b - 84 \, a b^{2} + 35 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 15 \, b^{3} - 6 \, {\left(4 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{420 \, b^{2} f \cos\left(f x + e\right)^{6}}\right]"," ",0,"[1/840*(105*a^(3/2)*b^2*cos(f*x + e)^6*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 8*(2*(3*a^3 + 21*a^2*b - 70*a*b^2)*cos(f*x + e)^6 - (3*a^2*b - 84*a*b^2 + 35*b^3)*cos(f*x + e)^4 - 15*b^3 - 6*(4*a*b^2 - 7*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b^2*f*cos(f*x + e)^6), 1/420*(105*sqrt(-a)*a*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))*cos(f*x + e)^6 - 4*(2*(3*a^3 + 21*a^2*b - 70*a*b^2)*cos(f*x + e)^6 - (3*a^2*b - 84*a*b^2 + 35*b^3)*cos(f*x + e)^4 - 15*b^3 - 6*(4*a*b^2 - 7*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b^2*f*cos(f*x + e)^6)]","B",0
390,1,443,0,3.586942," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\left[\frac{15 \, a^{\frac{3}{2}} b \cos\left(f x + e\right)^{4} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, {\left({\left(3 \, a^{2} - 20 \, a b\right)} \cos\left(f x + e\right)^{4} + {\left(6 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, b f \cos\left(f x + e\right)^{4}}, -\frac{15 \, \sqrt{-a} a b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) \cos\left(f x + e\right)^{4} - 4 \, {\left({\left(3 \, a^{2} - 20 \, a b\right)} \cos\left(f x + e\right)^{4} + {\left(6 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, b f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/120*(15*a^(3/2)*b*cos(f*x + e)^4*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*((3*a^2 - 20*a*b)*cos(f*x + e)^4 + (6*a*b - 5*b^2)*cos(f*x + e)^2 + 3*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b*f*cos(f*x + e)^4), -1/60*(15*sqrt(-a)*a*b*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))*cos(f*x + e)^4 - 4*((3*a^2 - 20*a*b)*cos(f*x + e)^4 + (6*a*b - 5*b^2)*cos(f*x + e)^2 + 3*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(b*f*cos(f*x + e)^4)]","B",0
391,1,373,0,1.057655," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e),x, algorithm=""fricas"")","\left[\frac{3 \, a^{\frac{3}{2}} \cos\left(f x + e\right)^{2} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, {\left(4 \, a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, f \cos\left(f x + e\right)^{2}}, \frac{3 \, \sqrt{-a} a \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) \cos\left(f x + e\right)^{2} + 4 \, {\left(4 \, a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/24*(3*a^(3/2)*cos(f*x + e)^2*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*(4*a*cos(f*x + e)^2 + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^2), 1/12*(3*sqrt(-a)*a*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))*cos(f*x + e)^2 + 4*(4*a*cos(f*x + e)^2 + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^2)]","B",0
392,1,1075,0,1.048855," ","integrate(cot(f*x+e)*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{a^{\frac{3}{2}} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 2 \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 8 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, f}, \frac{4 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + a^{\frac{3}{2}} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, f}, -\frac{\sqrt{-a} a \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f}, -\frac{\sqrt{-a} a \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 2 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) - 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f}\right]"," ",0,"[1/8*(a^(3/2)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 2*(a + b)^(3/2)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 8*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/f, 1/8*(4*(a + b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + a^(3/2)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/f, -1/4*(sqrt(-a)*a*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - (a + b)^(3/2)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/f, -1/4*(sqrt(-a)*a*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 2*(a + b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) - 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/f]","B",0
393,1,1300,0,1.031673," ","integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - {\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, \frac{4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} - 2 \, {\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, \frac{4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + 2 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, \frac{2 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right)}{4 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}\right]"," ",0,"[1/8*(4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a*cos(f*x + e)^2 - a)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - ((2*a - b)*cos(f*x + e)^2 - 2*a + b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/(f*cos(f*x + e)^2 - f), 1/8*(4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 - 2*((2*a - b)*cos(f*x + e)^2 - 2*a + b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + (a*cos(f*x + e)^2 - a)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/(f*cos(f*x + e)^2 - f), 1/8*(4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + 2*(a*cos(f*x + e)^2 - a)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((2*a - b)*cos(f*x + e)^2 - 2*a + b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/(f*cos(f*x + e)^2 - f), 1/4*(2*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a*cos(f*x + e)^2 - a)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((2*a - b)*cos(f*x + e)^2 - 2*a + b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)))/(f*cos(f*x + e)^2 - f)]","B",0
394,1,1801,0,3.156603," ","integrate(cot(f*x+e)^5*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - {\left({\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 4 \, a b - b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(6 \, a^{2} + 7 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2} + {\left(a + b\right)} f\right)}}, \frac{{\left({\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 4 \, a b - b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 2 \, {\left({\left(6 \, a^{2} + 7 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2} + {\left(a + b\right)} f\right)}}, -\frac{8 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + {\left({\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 4 \, a b - b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(6 \, a^{2} + 7 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2} + {\left(a + b\right)} f\right)}}, -\frac{4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 4 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 4 \, a b - b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(6 \, a^{2} + 7 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{2} + 3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2} + {\left(a + b\right)} f\right)}}\right]"," ",0,"[1/32*(4*((a^2 + a*b)*cos(f*x + e)^4 - 2*(a^2 + a*b)*cos(f*x + e)^2 + a^2 + a*b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - ((8*a^2 + 4*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 4*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 + 4*a*b - b^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((6*a^2 + 7*a*b + b^2)*cos(f*x + e)^4 - (4*a^2 + 3*a*b - b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^4 - 2*(a + b)*f*cos(f*x + e)^2 + (a + b)*f), 1/16*(((8*a^2 + 4*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 4*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 + 4*a*b - b^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((a^2 + a*b)*cos(f*x + e)^4 - 2*(a^2 + a*b)*cos(f*x + e)^2 + a^2 + a*b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 2*((6*a^2 + 7*a*b + b^2)*cos(f*x + e)^4 - (4*a^2 + 3*a*b - b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^4 - 2*(a + b)*f*cos(f*x + e)^2 + (a + b)*f), -1/32*(8*((a^2 + a*b)*cos(f*x + e)^4 - 2*(a^2 + a*b)*cos(f*x + e)^2 + a^2 + a*b)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + ((8*a^2 + 4*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 4*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 + 4*a*b - b^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((6*a^2 + 7*a*b + b^2)*cos(f*x + e)^4 - (4*a^2 + 3*a*b - b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^4 - 2*(a + b)*f*cos(f*x + e)^2 + (a + b)*f), -1/16*(4*((a^2 + a*b)*cos(f*x + e)^4 - 2*(a^2 + a*b)*cos(f*x + e)^2 + a^2 + a*b)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^2 + 4*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 4*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 + 4*a*b - b^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((6*a^2 + 7*a*b + b^2)*cos(f*x + e)^4 - (4*a^2 + 3*a*b - b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a + b)*f*cos(f*x + e)^4 - 2*(a + b)*f*cos(f*x + e)^2 + (a + b)*f)]","B",0
395,1,1973,0,25.227941," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^6,x, algorithm=""fricas"")","\left[\frac{192 \, \sqrt{-a} a b^{3} \cos\left(f x + e\right)^{7} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 3 \, {\left(3 \, a^{4} + 20 \, a^{3} b + 90 \, a^{2} b^{2} - 60 \, a b^{3} - 5 \, b^{4}\right)} \sqrt{b} \cos\left(f x + e\right)^{7} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(9 \, a^{3} b + 57 \, a^{2} b^{2} - 337 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(3 \, a^{2} b^{2} - 122 \, a b^{3} + 59 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 48 \, b^{4} - 8 \, {\left(9 \, a b^{3} - 17 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{1536 \, b^{3} f \cos\left(f x + e\right)^{7}}, \frac{96 \, \sqrt{-a} a b^{3} \cos\left(f x + e\right)^{7} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 3 \, {\left(3 \, a^{4} + 20 \, a^{3} b + 90 \, a^{2} b^{2} - 60 \, a b^{3} - 5 \, b^{4}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{7} - 2 \, {\left({\left(9 \, a^{3} b + 57 \, a^{2} b^{2} - 337 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(3 \, a^{2} b^{2} - 122 \, a b^{3} + 59 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 48 \, b^{4} - 8 \, {\left(9 \, a b^{3} - 17 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{768 \, b^{3} f \cos\left(f x + e\right)^{7}}, \frac{384 \, a^{\frac{3}{2}} b^{3} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{7} - 3 \, {\left(3 \, a^{4} + 20 \, a^{3} b + 90 \, a^{2} b^{2} - 60 \, a b^{3} - 5 \, b^{4}\right)} \sqrt{b} \cos\left(f x + e\right)^{7} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(9 \, a^{3} b + 57 \, a^{2} b^{2} - 337 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(3 \, a^{2} b^{2} - 122 \, a b^{3} + 59 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 48 \, b^{4} - 8 \, {\left(9 \, a b^{3} - 17 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{1536 \, b^{3} f \cos\left(f x + e\right)^{7}}, \frac{192 \, a^{\frac{3}{2}} b^{3} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{7} + 3 \, {\left(3 \, a^{4} + 20 \, a^{3} b + 90 \, a^{2} b^{2} - 60 \, a b^{3} - 5 \, b^{4}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{7} - 2 \, {\left({\left(9 \, a^{3} b + 57 \, a^{2} b^{2} - 337 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(3 \, a^{2} b^{2} - 122 \, a b^{3} + 59 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 48 \, b^{4} - 8 \, {\left(9 \, a b^{3} - 17 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{768 \, b^{3} f \cos\left(f x + e\right)^{7}}\right]"," ",0,"[1/1536*(192*sqrt(-a)*a*b^3*cos(f*x + e)^7*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqrt(b)*cos(f*x + e)^7*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7), 1/768*(96*sqrt(-a)*a*b^3*cos(f*x + e)^7*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^7 - 2*((9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7), 1/1536*(384*a^(3/2)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^7 - 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqrt(b)*cos(f*x + e)^7*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7), 1/768*(192*a^(3/2)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^7 + 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^7 - 2*((9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7)]","A",0
396,1,1777,0,7.523168," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\left[\frac{24 \, \sqrt{-a} a b^{2} \cos\left(f x + e\right)^{5} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 3 \, {\left(a^{3} + 9 \, a^{2} b - 9 \, a b^{2} - b^{3}\right)} \sqrt{b} \cos\left(f x + e\right)^{5} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(3 \, a^{2} b - 38 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, b^{3} + 14 \, {\left(a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, b^{2} f \cos\left(f x + e\right)^{5}}, \frac{12 \, \sqrt{-a} a b^{2} \cos\left(f x + e\right)^{5} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 3 \, {\left(a^{3} + 9 \, a^{2} b - 9 \, a b^{2} - b^{3}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} + 2 \, {\left({\left(3 \, a^{2} b - 38 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, b^{3} + 14 \, {\left(a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, b^{2} f \cos\left(f x + e\right)^{5}}, -\frac{48 \, a^{\frac{3}{2}} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} + 3 \, {\left(a^{3} + 9 \, a^{2} b - 9 \, a b^{2} - b^{3}\right)} \sqrt{b} \cos\left(f x + e\right)^{5} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{2} b - 38 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, b^{3} + 14 \, {\left(a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{192 \, b^{2} f \cos\left(f x + e\right)^{5}}, -\frac{24 \, a^{\frac{3}{2}} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} + 3 \, {\left(a^{3} + 9 \, a^{2} b - 9 \, a b^{2} - b^{3}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - 2 \, {\left({\left(3 \, a^{2} b - 38 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, b^{3} + 14 \, {\left(a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{96 \, b^{2} f \cos\left(f x + e\right)^{5}}\right]"," ",0,"[1/192*(24*sqrt(-a)*a*b^2*cos(f*x + e)^5*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 3*(a^3 + 9*a^2*b - 9*a*b^2 - b^3)*sqrt(b)*cos(f*x + e)^5*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*((3*a^2*b - 38*a*b^2 + 3*b^3)*cos(f*x + e)^4 + 8*b^3 + 14*(a*b^2 - b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^5), 1/96*(12*sqrt(-a)*a*b^2*cos(f*x + e)^5*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 3*(a^3 + 9*a^2*b - 9*a*b^2 - b^3)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^5 + 2*((3*a^2*b - 38*a*b^2 + 3*b^3)*cos(f*x + e)^4 + 8*b^3 + 14*(a*b^2 - b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^5), -1/192*(48*a^(3/2)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^5 + 3*(a^3 + 9*a^2*b - 9*a*b^2 - b^3)*sqrt(b)*cos(f*x + e)^5*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^2*b - 38*a*b^2 + 3*b^3)*cos(f*x + e)^4 + 8*b^3 + 14*(a*b^2 - b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^5), -1/96*(24*a^(3/2)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^5 + 3*(a^3 + 9*a^2*b - 9*a*b^2 - b^3)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^5 - 2*((3*a^2*b - 38*a*b^2 + 3*b^3)*cos(f*x + e)^4 + 8*b^3 + 14*(a*b^2 - b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^2*f*cos(f*x + e)^5)]","A",0
397,1,1627,0,2.138465," ","integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-a} a b \cos\left(f x + e\right)^{3} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(5 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b f \cos\left(f x + e\right)^{3}}, \frac{2 \, \sqrt{-a} a b \cos\left(f x + e\right)^{3} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + 2 \, {\left({\left(5 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b f \cos\left(f x + e\right)^{3}}, \frac{8 \, a^{\frac{3}{2}} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(5 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, b f \cos\left(f x + e\right)^{3}}, \frac{4 \, a^{\frac{3}{2}} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} - 6 \, a b - b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + 2 \, {\left({\left(5 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, b f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[1/32*(4*sqrt(-a)*a*b*cos(f*x + e)^3*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (3*a^2 - 6*a*b - b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*((5*a*b - b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)^3), 1/16*(2*sqrt(-a)*a*b*cos(f*x + e)^3*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + (3*a^2 - 6*a*b - b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 2*((5*a*b - b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)^3), 1/32*(8*a^(3/2)*b*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^3 - (3*a^2 - 6*a*b - b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*((5*a*b - b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)^3), 1/16*(4*a^(3/2)*b*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^3 + (3*a^2 - 6*a*b - b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 2*((5*a*b - b^2)*cos(f*x + e)^2 + 2*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b*f*cos(f*x + e)^3)]","B",0
398,1,1457,0,1.120158," ","integrate((a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + {\left(3 \, a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, \frac{2 \, {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + \sqrt{-a} a \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{2 \, a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, -\frac{a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, f \cos\left(f x + e\right)}\right]"," ",0,"[1/8*(sqrt(-a)*a*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + (3*a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), 1/8*(2*(3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + sqrt(-a)*a*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/8*(2*a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a + b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/4*(a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 2*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e))]","B",0
399,1,1446,0,1.209994," ","integrate(cot(f*x+e)^2*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 2 \, b^{\frac{3}{2}} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 8 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{8 \, f \sin\left(f x + e\right)}, \frac{4 \, \sqrt{-b} b \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + \sqrt{-a} a \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{8 \, f \sin\left(f x + e\right)}, \frac{a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + b^{\frac{3}{2}} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{4 \, f \sin\left(f x + e\right)}, \frac{a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, \sqrt{-b} b \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left(a + b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{4 \, f \sin\left(f x + e\right)}\right]"," ",0,"[1/8*(sqrt(-a)*a*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 2*b^(3/2)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 8*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e)), 1/8*(4*sqrt(-b)*b*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) + sqrt(-a)*a*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e)), 1/4*(a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) + b^(3/2)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e)), 1/4*(a^(3/2)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) + 2*sqrt(-b)*b*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) - 4*(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e))]","B",0
400,1,597,0,1.539972," ","integrate(cot(f*x+e)^4*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left(4 \, a \cos\left(f x + e\right)^{3} - {\left(3 \, a - b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left(4 \, a \cos\left(f x + e\right)^{3} - {\left(3 \, a - b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/24*(3*(a*cos(f*x + e)^2 - a)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*(4*a*cos(f*x + e)^3 - (3*a - b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^2 - f)*sin(f*x + e)), -1/12*(3*(a*cos(f*x + e)^2 - a)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*(4*a*cos(f*x + e)^3 - (3*a - b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))]","B",0
401,1,767,0,5.495695," ","integrate(cot(f*x+e)^6*(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, {\left({\left(23 \, a^{2} + 20 \, a b\right)} \cos\left(f x + e\right)^{5} - {\left(35 \, a^{2} + 24 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2} + {\left(a + b\right)} f\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(23 \, a^{2} + 20 \, a b\right)} \cos\left(f x + e\right)^{5} - {\left(35 \, a^{2} + 24 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a + b\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2} + {\left(a + b\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/120*(15*((a^2 + a*b)*cos(f*x + e)^4 - 2*(a^2 + a*b)*cos(f*x + e)^2 + a^2 + a*b)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*((23*a^2 + 20*a*b)*cos(f*x + e)^5 - (35*a^2 + 24*a*b - 5*b^2)*cos(f*x + e)^3 + (15*a^2 + 10*a*b - 2*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^4 - 2*(a + b)*f*cos(f*x + e)^2 + (a + b)*f)*sin(f*x + e)), 1/60*(15*((a^2 + a*b)*cos(f*x + e)^4 - 2*(a^2 + a*b)*cos(f*x + e)^2 + a^2 + a*b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((23*a^2 + 20*a*b)*cos(f*x + e)^5 - (35*a^2 + 24*a*b - 5*b^2)*cos(f*x + e)^3 + (15*a^2 + 10*a*b - 2*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a + b)*f*cos(f*x + e)^4 - 2*(a + b)*f*cos(f*x + e)^2 + (a + b)*f)*sin(f*x + e))]","B",0
402,1,410,0,1.147765," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a} b^{2} \cos\left(f x + e\right)^{2} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 8 \, {\left(2 \, {\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, a b^{2} f \cos\left(f x + e\right)^{2}}, \frac{3 \, \sqrt{-a} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) \cos\left(f x + e\right)^{2} - 4 \, {\left(2 \, {\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, a b^{2} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/24*(3*sqrt(a)*b^2*cos(f*x + e)^2*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 8*(2*(a^2 + 3*a*b)*cos(f*x + e)^2 - a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*b^2*f*cos(f*x + e)^2), 1/12*(3*sqrt(-a)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))*cos(f*x + e)^2 - 4*(2*(a^2 + 3*a*b)*cos(f*x + e)^2 - a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*b^2*f*cos(f*x + e)^2)]","B",0
403,1,328,0,0.693233," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} b \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, a b f}, -\frac{\sqrt{-a} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, a b f}\right]"," ",0,"[1/8*(sqrt(a)*b*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*b*f), -1/4*(sqrt(-a)*b*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*b*f)]","B",0
404,1,261,0,0.580680," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, \sqrt{a} f}, \frac{\sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right)}{4 \, a f}\right]"," ",0,"[1/8*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(sqrt(a)*f), 1/4*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2))/(a*f)]","B",0
405,1,1015,0,0.770359," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a + b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 2 \, \sqrt{a + b} a \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(a^{2} + a b\right)} f}, \frac{4 \, a \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + {\left(a + b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left(a^{2} + a b\right)} f}, -\frac{\sqrt{-a} {\left(a + b\right)} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - \sqrt{a + b} a \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left(a^{2} + a b\right)} f}, -\frac{\sqrt{-a} {\left(a + b\right)} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 2 \, a \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right)}{4 \, {\left(a^{2} + a b\right)} f}\right]"," ",0,"[1/8*((a + b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 2*sqrt(a + b)*a*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^2 + a*b)*f), 1/8*(4*a*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + (a + b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/((a^2 + a*b)*f), -1/4*(sqrt(-a)*(a + b)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - sqrt(a + b)*a*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^2 + a*b)*f), -1/4*(sqrt(-a)*(a + b)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 2*a*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)))/((a^2 + a*b)*f)]","B",0
406,1,1550,0,1.019276," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} + a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 3 \, a b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)}}, \frac{4 \, {\left(a^{2} + a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} - 2 \, {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 3 \, a b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)}}, \frac{4 \, {\left(a^{2} + a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + 2 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 3 \, a b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)}}, \frac{2 \, {\left(a^{2} + a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 3 \, a b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right)}{4 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(4*(a^2 + a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + ((a^2 + 2*a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + ((2*a^2 + 3*a*b)*cos(f*x + e)^2 - 2*a^2 - 3*a*b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f), 1/8*(4*(a^2 + a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 - 2*((2*a^2 + 3*a*b)*cos(f*x + e)^2 - 2*a^2 - 3*a*b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + ((a^2 + 2*a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f), 1/8*(4*(a^2 + a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + 2*((a^2 + 2*a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + ((2*a^2 + 3*a*b)*cos(f*x + e)^2 - 2*a^2 - 3*a*b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f), 1/4*(2*(a^2 + a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + ((a^2 + 2*a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((2*a^2 + 3*a*b)*cos(f*x + e)^2 - 2*a^2 - 3*a*b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)))/((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f)]","B",0
407,1,2257,0,2.373297," ","integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + {\left({\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2} - 2 \, {\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(3 \, {\left(2 \, a^{3} + 5 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{3} + 11 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, \frac{{\left({\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2} - 2 \, {\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 2 \, {\left(3 \, {\left(2 \, a^{3} + 5 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{3} + 11 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, -\frac{8 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2} - 2 \, {\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left(3 \, {\left(2 \, a^{3} + 5 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{3} + 11 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, -\frac{4 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2} - 2 \, {\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left(3 \, {\left(2 \, a^{3} + 5 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{3} + 11 \, a^{2} b + 7 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[1/32*(4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + ((8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^4 + 8*a^3 + 20*a^2*b + 15*a*b^2 - 2*(8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*(3*(2*a^3 + 5*a^2*b + 3*a*b^2)*cos(f*x + e)^4 - (4*a^3 + 11*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f), 1/16*(((8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^4 + 8*a^3 + 20*a^2*b + 15*a*b^2 - 2*(8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 2*(3*(2*a^3 + 5*a^2*b + 3*a*b^2)*cos(f*x + e)^4 - (4*a^3 + 11*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f), -1/32*(8*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^4 + 8*a^3 + 20*a^2*b + 15*a*b^2 - 2*(8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*(3*(2*a^3 + 5*a^2*b + 3*a*b^2)*cos(f*x + e)^4 - (4*a^3 + 11*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f), -1/16*(4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^4 + 8*a^3 + 20*a^2*b + 15*a*b^2 - 2*(8*a^3 + 20*a^2*b + 15*a*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*(3*(2*a^3 + 5*a^2*b + 3*a*b^2)*cos(f*x + e)^4 - (4*a^3 + 11*a^2*b + 7*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)]","B",0
408,1,1673,0,2.848006," ","integrate(tan(f*x+e)^6/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-a} b^{3} \cos\left(f x + e\right)^{3} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(3 \, a^{3} + 10 \, a^{2} b + 15 \, a b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left(2 \, a b^{2} - 3 \, {\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a b^{3} f \cos\left(f x + e\right)^{3}}, -\frac{2 \, \sqrt{-a} b^{3} \cos\left(f x + e\right)^{3} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(3 \, a^{3} + 10 \, a^{2} b + 15 \, a b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - 2 \, {\left(2 \, a b^{2} - 3 \, {\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, a b^{3} f \cos\left(f x + e\right)^{3}}, \frac{8 \, \sqrt{a} b^{3} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} + 10 \, a^{2} b + 15 \, a b^{2}\right)} \sqrt{b} \cos\left(f x + e\right)^{3} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(2 \, a b^{2} - 3 \, {\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, a b^{3} f \cos\left(f x + e\right)^{3}}, \frac{4 \, \sqrt{a} b^{3} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} + 10 \, a^{2} b + 15 \, a b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} + 2 \, {\left(2 \, a b^{2} - 3 \, {\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, a b^{3} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[-1/32*(4*sqrt(-a)*b^3*cos(f*x + e)^3*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (3*a^3 + 10*a^2*b + 15*a*b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*(2*a*b^2 - 3*(a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^3*f*cos(f*x + e)^3), -1/16*(2*sqrt(-a)*b^3*cos(f*x + e)^3*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (3*a^3 + 10*a^2*b + 15*a*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 - 2*(2*a*b^2 - 3*(a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^3*f*cos(f*x + e)^3), 1/32*(8*sqrt(a)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^3 + (3*a^3 + 10*a^2*b + 15*a*b^2)*sqrt(b)*cos(f*x + e)^3*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*(2*a*b^2 - 3*(a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^3*f*cos(f*x + e)^3), 1/16*(4*sqrt(a)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^3 + (3*a^3 + 10*a^2*b + 15*a*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 2*(2*a*b^2 - 3*(a^2*b + 3*a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^3*f*cos(f*x + e)^3)]","B",0
409,1,1507,0,1.132529," ","integrate(tan(f*x+e)^4/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} b^{2} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left(a^{2} + 3 \, a b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, a b^{2} f \cos\left(f x + e\right)}, -\frac{\sqrt{-a} b^{2} \cos\left(f x + e\right) \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 2 \, {\left(a^{2} + 3 \, a b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, a b^{2} f \cos\left(f x + e\right)}, -\frac{2 \, \sqrt{a} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(a^{2} + 3 \, a b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, a b^{2} f \cos\left(f x + e\right)}, -\frac{\sqrt{a} b^{2} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + {\left(a^{2} + 3 \, a b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, a b^{2} f \cos\left(f x + e\right)}\right]"," ",0,"[-1/8*(sqrt(-a)*b^2*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - (a^2 + 3*a*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^2*f*cos(f*x + e)), -1/8*(sqrt(-a)*b^2*cos(f*x + e)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 2*(a^2 + 3*a*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^2*f*cos(f*x + e)), -1/8*(2*sqrt(a)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (a^2 + 3*a*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^2*f*cos(f*x + e)), -1/4*(sqrt(a)*b^2*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) + (a^2 + 3*a*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 2*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b^2*f*cos(f*x + e))]","B",0
410,1,1259,0,0.907431," ","integrate(tan(f*x+e)^2/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} b \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 2 \, a \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, a b f}, \frac{4 \, a \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{-a} b \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, a b f}, \frac{\sqrt{a} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + a \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, a b f}, \frac{\sqrt{a} b \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 2 \, a \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, a b f}\right]"," ",0,"[-1/8*(sqrt(-a)*b*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 2*a*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/(a*b*f), 1/8*(4*a*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - sqrt(-a)*b*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a*b*f), 1/4*(sqrt(a)*b*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + a*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/(a*b*f), 1/4*(sqrt(a)*b*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 2*a*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/(a*b*f)]","B",0
411,1,408,0,0.725606," ","integrate(1/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, a f}, -\frac{\arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, \sqrt{a} f}\right]"," ",0,"[-1/8*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*f), -1/4*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))/(sqrt(a)*f)]","B",0
412,1,525,0,0.715929," ","integrate(cot(f*x+e)^2/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} {\left(a + b\right)} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{8 \, {\left(a^{2} + a b\right)} f \sin\left(f x + e\right)}, \frac{{\left(a + b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{4 \, {\left(a^{2} + a b\right)} f \sin\left(f x + e\right)}\right]"," ",0,"[-1/8*(sqrt(-a)*(a + b)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/((a^2 + a*b)*f*sin(f*x + e)), 1/4*((a + b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/((a^2 + a*b)*f*sin(f*x + e))]","B",0
413,1,723,0,1.189332," ","integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, {\left(2 \, {\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left(2 \, {\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/24*(3*((a^2 + 2*a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*(2*(2*a^2 + 3*a*b)*cos(f*x + e)^3 - (3*a^2 + 5*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f)*sin(f*x + e)), -1/12*(3*((a^2 + 2*a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*(2*(2*a^2 + 3*a*b)*cos(f*x + e)^3 - (3*a^2 + 5*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^3 + 2*a^2*b + a*b^2)*f*cos(f*x + e)^2 - (a^3 + 2*a^2*b + a*b^2)*f)*sin(f*x + e))]","B",0
414,1,987,0,3.438006," ","integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left({\left(23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(35 \, a^{3} + 94 \, a^{2} b + 75 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} + 40 \, a^{2} b + 33 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(35 \, a^{3} + 94 \, a^{2} b + 75 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{3} + 40 \, a^{2} b + 33 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/120*(15*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*((23*a^3 + 60*a^2*b + 45*a*b^2)*cos(f*x + e)^5 - (35*a^3 + 94*a^2*b + 75*a*b^2)*cos(f*x + e)^3 + (15*a^3 + 40*a^2*b + 33*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)*sin(f*x + e)), 1/60*(15*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((23*a^3 + 60*a^2*b + 45*a*b^2)*cos(f*x + e)^5 - (35*a^3 + 94*a^2*b + 75*a*b^2)*cos(f*x + e)^3 + (15*a^3 + 40*a^2*b + 33*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)*sin(f*x + e))]","B",0
415,1,458,0,1.421878," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, {\left(a^{2} b + {\left(2 \, a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}, \frac{{\left(a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + 4 \, {\left(a^{2} b + {\left(2 \, a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}\right]"," ",0,"[1/8*((a*b^2*cos(f*x + e)^2 + b^3)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*(a^2*b + (2*a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f), 1/4*((a*b^2*cos(f*x + e)^2 + b^3)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + 4*(a^2*b + (2*a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f)]","B",0
416,1,417,0,0.814186," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{2} + a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} - {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left(a^{3} b f \cos\left(f x + e\right)^{2} + a^{2} b^{2} f\right)}}, -\frac{4 \, {\left(a^{2} + a b\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right)}{4 \, {\left(a^{3} b f \cos\left(f x + e\right)^{2} + a^{2} b^{2} f\right)}}\right]"," ",0,"[-1/8*(8*(a^2 + a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 - (a*b*cos(f*x + e)^2 + b^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/(a^3*b*f*cos(f*x + e)^2 + a^2*b^2*f), -1/4*(4*(a^2 + a*b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)))/(a^3*b*f*cos(f*x + e)^2 + a^2*b^2*f)]","B",0
417,1,392,0,0.623650," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}, \frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right)}{4 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}\right]"," ",0,"[1/8*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a*cos(f*x + e)^2 + b)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/(a^3*f*cos(f*x + e)^2 + a^2*b*f), 1/4*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a*cos(f*x + e)^2 + b)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)))/(a^3*f*cos(f*x + e)^2 + a^2*b*f)]","B",0
418,1,1569,0,1.217764," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} - {\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 2 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}, -\frac{8 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} - 4 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) - {\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}{8 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}, -\frac{4 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}, -\frac{4 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 2 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right)}{4 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 - (a^2*b + 2*a*b^2 + b^3 + (a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 2*(a^3*cos(f*x + e)^2 + a^2*b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f), -1/8*(8*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 - 4*(a^3*cos(f*x + e)^2 + a^2*b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) - (a^2*b + 2*a*b^2 + b^3 + (a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f), -1/4*(4*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3 + (a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - (a^3*cos(f*x + e)^2 + a^2*b)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f), -1/4*(4*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)^2 + (a^2*b + 2*a*b^2 + b^3 + (a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 2*(a^3*cos(f*x + e)^2 + a^2*b)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)))/((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)]","B",0
419,1,2347,0,2.445488," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - 3 \, a b^{3} - b^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + {\left({\left(2 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} - {\left(2 \, a^{4} + 3 \, a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(a^{4} + a^{3} b + 2 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)}}, -\frac{2 \, {\left({\left(2 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} - {\left(2 \, a^{4} + 3 \, a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) - {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - 3 \, a b^{3} - b^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 4 \, {\left({\left(a^{4} + a^{3} b + 2 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)}}, \frac{2 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - 3 \, a b^{3} - b^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + {\left({\left(2 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} - {\left(2 \, a^{4} + 3 \, a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(a^{4} + a^{3} b + 2 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)}}, \frac{{\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - 3 \, a b^{3} - b^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(2 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} - {\left(2 \, a^{4} + 3 \, a^{3} b - 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(a^{4} + a^{3} b + 2 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)}}\right]"," ",0,"[1/8*(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - 3*a*b^3 - b^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + ((2*a^4 + 5*a^3*b)*cos(f*x + e)^4 - 2*a^3*b - 5*a^2*b^2 - (2*a^4 + 3*a^3*b - 5*a^2*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((a^4 + a^3*b + 2*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 + (a^3*b - a^2*b^2 - 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f), -1/8*(2*((2*a^4 + 5*a^3*b)*cos(f*x + e)^4 - 2*a^3*b - 5*a^2*b^2 - (2*a^4 + 3*a^3*b - 5*a^2*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) - ((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - 3*a*b^3 - b^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 4*((a^4 + a^3*b + 2*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 + (a^3*b - a^2*b^2 - 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f), 1/8*(2*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - 3*a*b^3 - b^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + ((2*a^4 + 5*a^3*b)*cos(f*x + e)^4 - 2*a^3*b - 5*a^2*b^2 - (2*a^4 + 3*a^3*b - 5*a^2*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((a^4 + a^3*b + 2*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 + (a^3*b - a^2*b^2 - 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f), 1/4*(((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - 3*a*b^3 - b^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((2*a^4 + 5*a^3*b)*cos(f*x + e)^4 - 2*a^3*b - 5*a^2*b^2 - (2*a^4 + 3*a^3*b - 5*a^2*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((a^4 + a^3*b + 2*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^4 + (a^3*b - a^2*b^2 - 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)]","B",0
420,1,3501,0,8.708771," ","integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + {\left({\left(8 \, a^{5} + 28 \, a^{4} b + 35 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{6} + 8 \, a^{4} b + 28 \, a^{3} b^{2} + 35 \, a^{2} b^{3} - {\left(16 \, a^{5} + 48 \, a^{4} b + 42 \, a^{3} b^{2} - 35 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, a^{5} + 12 \, a^{4} b - 21 \, a^{3} b^{2} - 70 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(6 \, a^{5} + 19 \, a^{4} b + 13 \, a^{3} b^{2} + 8 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(4 \, a^{5} + 9 \, a^{4} b - 8 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + 16 \, a b^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{4} b + 15 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, \frac{{\left({\left(8 \, a^{5} + 28 \, a^{4} b + 35 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{6} + 8 \, a^{4} b + 28 \, a^{3} b^{2} + 35 \, a^{2} b^{3} - {\left(16 \, a^{5} + 48 \, a^{4} b + 42 \, a^{3} b^{2} - 35 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, a^{5} + 12 \, a^{4} b - 21 \, a^{3} b^{2} - 70 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 2 \, {\left({\left(6 \, a^{5} + 19 \, a^{4} b + 13 \, a^{3} b^{2} + 8 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(4 \, a^{5} + 9 \, a^{4} b - 8 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + 16 \, a b^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{4} b + 15 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, -\frac{8 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{5} + 28 \, a^{4} b + 35 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{6} + 8 \, a^{4} b + 28 \, a^{3} b^{2} + 35 \, a^{2} b^{3} - {\left(16 \, a^{5} + 48 \, a^{4} b + 42 \, a^{3} b^{2} - 35 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, a^{5} + 12 \, a^{4} b - 21 \, a^{3} b^{2} - 70 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(6 \, a^{5} + 19 \, a^{4} b + 13 \, a^{3} b^{2} + 8 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(4 \, a^{5} + 9 \, a^{4} b - 8 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + 16 \, a b^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{4} b + 15 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{32 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, -\frac{4 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - {\left({\left(8 \, a^{5} + 28 \, a^{4} b + 35 \, a^{3} b^{2}\right)} \cos\left(f x + e\right)^{6} + 8 \, a^{4} b + 28 \, a^{3} b^{2} + 35 \, a^{2} b^{3} - {\left(16 \, a^{5} + 48 \, a^{4} b + 42 \, a^{3} b^{2} - 35 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, a^{5} + 12 \, a^{4} b - 21 \, a^{3} b^{2} - 70 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(6 \, a^{5} + 19 \, a^{4} b + 13 \, a^{3} b^{2} + 8 \, a^{2} b^{3} + 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(4 \, a^{5} + 9 \, a^{4} b - 8 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + 16 \, a b^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{4} b + 15 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - 8 \, a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[1/32*(4*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + ((8*a^5 + 28*a^4*b + 35*a^3*b^2)*cos(f*x + e)^6 + 8*a^4*b + 28*a^3*b^2 + 35*a^2*b^3 - (16*a^5 + 48*a^4*b + 42*a^3*b^2 - 35*a^2*b^3)*cos(f*x + e)^4 + (8*a^5 + 12*a^4*b - 21*a^3*b^2 - 70*a^2*b^3)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((6*a^5 + 19*a^4*b + 13*a^3*b^2 + 8*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^6 - (4*a^5 + 9*a^4*b - 8*a^3*b^2 + 3*a^2*b^3 + 16*a*b^4)*cos(f*x + e)^4 - (4*a^4*b + 15*a^3*b^2 + 3*a^2*b^3 - 8*a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f), 1/16*(((8*a^5 + 28*a^4*b + 35*a^3*b^2)*cos(f*x + e)^6 + 8*a^4*b + 28*a^3*b^2 + 35*a^2*b^3 - (16*a^5 + 48*a^4*b + 42*a^3*b^2 - 35*a^2*b^3)*cos(f*x + e)^4 + (8*a^5 + 12*a^4*b - 21*a^3*b^2 - 70*a^2*b^3)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 2*((6*a^5 + 19*a^4*b + 13*a^3*b^2 + 8*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^6 - (4*a^5 + 9*a^4*b - 8*a^3*b^2 + 3*a^2*b^3 + 16*a*b^4)*cos(f*x + e)^4 - (4*a^4*b + 15*a^3*b^2 + 3*a^2*b^3 - 8*a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f), -1/32*(8*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^5 + 28*a^4*b + 35*a^3*b^2)*cos(f*x + e)^6 + 8*a^4*b + 28*a^3*b^2 + 35*a^2*b^3 - (16*a^5 + 48*a^4*b + 42*a^3*b^2 - 35*a^2*b^3)*cos(f*x + e)^4 + (8*a^5 + 12*a^4*b - 21*a^3*b^2 - 70*a^2*b^3)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((6*a^5 + 19*a^4*b + 13*a^3*b^2 + 8*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^6 - (4*a^5 + 9*a^4*b - 8*a^3*b^2 + 3*a^2*b^3 + 16*a*b^4)*cos(f*x + e)^4 - (4*a^4*b + 15*a^3*b^2 + 3*a^2*b^3 - 8*a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f), -1/16*(4*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - ((8*a^5 + 28*a^4*b + 35*a^3*b^2)*cos(f*x + e)^6 + 8*a^4*b + 28*a^3*b^2 + 35*a^2*b^3 - (16*a^5 + 48*a^4*b + 42*a^3*b^2 - 35*a^2*b^3)*cos(f*x + e)^4 + (8*a^5 + 12*a^4*b - 21*a^3*b^2 - 70*a^2*b^3)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((6*a^5 + 19*a^4*b + 13*a^3*b^2 + 8*a^2*b^3 + 8*a*b^4)*cos(f*x + e)^6 - (4*a^5 + 9*a^4*b - 8*a^3*b^2 + 3*a^2*b^3 + 16*a*b^4)*cos(f*x + e)^4 - (4*a^4*b + 15*a^3*b^2 + 3*a^2*b^3 - 8*a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f)]","B",0
421,1,1895,0,3.100455," ","integrate(tan(f*x+e)^6/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a b^{3} \cos\left(f x + e\right)^{3} + b^{4} \cos\left(f x + e\right)\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - {\left({\left(3 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left(a^{2} b^{2} + {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left(a^{3} b^{3} f \cos\left(f x + e\right)^{3} + a^{2} b^{4} f \cos\left(f x + e\right)\right)}}, -\frac{2 \, {\left({\left(3 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + {\left(a b^{3} \cos\left(f x + e\right)^{3} + b^{4} \cos\left(f x + e\right)\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 4 \, {\left(a^{2} b^{2} + {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left(a^{3} b^{3} f \cos\left(f x + e\right)^{3} + a^{2} b^{4} f \cos\left(f x + e\right)\right)}}, \frac{2 \, {\left(a b^{3} \cos\left(f x + e\right)^{3} + b^{4} \cos\left(f x + e\right)\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + {\left({\left(3 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(a^{2} b^{2} + {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left(a^{3} b^{3} f \cos\left(f x + e\right)^{3} + a^{2} b^{4} f \cos\left(f x + e\right)\right)}}, \frac{{\left(a b^{3} \cos\left(f x + e\right)^{3} + b^{4} \cos\left(f x + e\right)\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - {\left({\left(3 \, a^{4} + 5 \, a^{3} b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + 2 \, {\left(a^{2} b^{2} + {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{4 \, {\left(a^{3} b^{3} f \cos\left(f x + e\right)^{3} + a^{2} b^{4} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[-1/8*((a*b^3*cos(f*x + e)^3 + b^4*cos(f*x + e))*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - ((3*a^4 + 5*a^3*b)*cos(f*x + e)^3 + (3*a^3*b + 5*a^2*b^2)*cos(f*x + e))*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*(a^2*b^2 + (3*a^3*b + 4*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*b^3*f*cos(f*x + e)^3 + a^2*b^4*f*cos(f*x + e)), -1/8*(2*((3*a^4 + 5*a^3*b)*cos(f*x + e)^3 + (3*a^3*b + 5*a^2*b^2)*cos(f*x + e))*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + (a*b^3*cos(f*x + e)^3 + b^4*cos(f*x + e))*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 4*(a^2*b^2 + (3*a^3*b + 4*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*b^3*f*cos(f*x + e)^3 + a^2*b^4*f*cos(f*x + e)), 1/8*(2*(a*b^3*cos(f*x + e)^3 + b^4*cos(f*x + e))*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + ((3*a^4 + 5*a^3*b)*cos(f*x + e)^3 + (3*a^3*b + 5*a^2*b^2)*cos(f*x + e))*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*(a^2*b^2 + (3*a^3*b + 4*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*b^3*f*cos(f*x + e)^3 + a^2*b^4*f*cos(f*x + e)), 1/4*((a*b^3*cos(f*x + e)^3 + b^4*cos(f*x + e))*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - ((3*a^4 + 5*a^3*b)*cos(f*x + e)^3 + (3*a^3*b + 5*a^2*b^2)*cos(f*x + e))*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + 2*(a^2*b^2 + (3*a^3*b + 4*a^2*b^2 + 2*a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^3*b^3*f*cos(f*x + e)^3 + a^2*b^4*f*cos(f*x + e))]","B",0
422,1,1655,0,2.217899," ","integrate(tan(f*x+e)^4/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 2 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}, -\frac{8 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - 4 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + {\left(a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}, -\frac{4 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}, -\frac{4 \, {\left(a^{2} b + a b^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, {\left(a^{3} \cos\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{2} + a^{2} b^{3} f\right)}}\right]"," ",0,"[-1/8*(8*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a*b^2*cos(f*x + e)^2 + b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 2*(a^3*cos(f*x + e)^2 + a^2*b)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f), -1/8*(8*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - 4*(a^3*cos(f*x + e)^2 + a^2*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + (a*b^2*cos(f*x + e)^2 + b^3)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f), -1/4*(4*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a*b^2*cos(f*x + e)^2 + b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - (a^3*cos(f*x + e)^2 + a^2*b)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f), -1/4*(4*(a^2*b + a*b^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a*b^2*cos(f*x + e)^2 + b^3)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 2*(a^3*cos(f*x + e)^2 + a^2*b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/(a^3*b^2*f*cos(f*x + e)^2 + a^2*b^3*f)]","B",0
423,1,548,0,1.087855," ","integrate(tan(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}, \frac{4 \, a \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + b\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, {\left(a^{3} f \cos\left(f x + e\right)^{2} + a^{2} b f\right)}}\right]"," ",0,"[1/8*(8*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - (a*cos(f*x + e)^2 + b)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/(a^3*f*cos(f*x + e)^2 + a^2*b*f), 1/4*(4*a*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + (a*cos(f*x + e)^2 + b)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/(a^3*f*cos(f*x + e)^2 + a^2*b*f)]","B",0
424,1,601,0,0.909105," ","integrate(1/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}, -\frac{4 \, a b \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f), -1/4*(4*a*b*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + ((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))))/((a^4 + a^3*b)*f*cos(f*x + e)^2 + (a^3*b + a^2*b^2)*f)]","B",0
425,1,741,0,1.251595," ","integrate(cot(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left({\left(a^{3} + a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b - a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)} \sin\left(f x + e\right)}, \frac{{\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(a^{3} + a b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b - a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left({\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/8*((a^2*b + 2*a*b^2 + b^3 + (a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*((a^3 + a*b^2)*cos(f*x + e)^3 + (a^2*b - a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)*sin(f*x + e)), 1/4*((a^2*b + 2*a*b^2 + b^3 + (a^3 + 2*a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((a^3 + a*b^2)*cos(f*x + e)^3 + (a^2*b - a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^5 + 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*f)*sin(f*x + e))]","B",0
426,1,1061,0,3.818335," ","integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - 3 \, a b^{3} - b^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, {\left({\left(4 \, a^{4} + 9 \, a^{3} b + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - {\left(3 \, a^{4} + 4 \, a^{3} b - 9 \, a^{2} b^{2} + 6 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left({\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} b - 3 \, a^{2} b^{2} - 3 \, a b^{3} - b^{4} - {\left(a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(4 \, a^{4} + 9 \, a^{3} b + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - {\left(3 \, a^{4} + 4 \, a^{3} b - 9 \, a^{2} b^{2} + 6 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/24*(3*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - 3*a*b^3 - b^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*((4*a^4 + 9*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - (3*a^4 + 4*a^3*b - 9*a^2*b^2 + 6*a*b^3)*cos(f*x + e)^3 - (3*a^3*b + 8*a^2*b^2 - 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e)), -1/12*(3*((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e)^4 - a^3*b - 3*a^2*b^2 - 3*a*b^3 - b^4 - (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((4*a^4 + 9*a^3*b + 3*a*b^3)*cos(f*x + e)^5 - (3*a^4 + 4*a^3*b - 9*a^2*b^2 + 6*a*b^3)*cos(f*x + e)^3 - (3*a^3*b + 8*a^2*b^2 - 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*f*cos(f*x + e)^4 - (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*f*cos(f*x + e)^2 - (a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*f)*sin(f*x + e))]","B",0
427,1,1517,0,11.852968," ","integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left({\left(23 \, a^{5} + 80 \, a^{4} b + 90 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(f x + e\right)^{7} - {\left(35 \, a^{5} + 106 \, a^{4} b + 80 \, a^{3} b^{2} - 90 \, a^{2} b^{3} + 45 \, a b^{4}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{5} + 20 \, a^{4} b - 56 \, a^{3} b^{2} - 160 \, a^{2} b^{3} + 45 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{4} b + 55 \, a^{3} b^{2} + 73 \, a^{2} b^{3} - 15 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{6} + a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5} - {\left(2 \, a^{5} + 7 \, a^{4} b + 8 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 2 \, a b^{4} - b^{5}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{5} + 2 \, a^{4} b - 2 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 7 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(23 \, a^{5} + 80 \, a^{4} b + 90 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(f x + e\right)^{7} - {\left(35 \, a^{5} + 106 \, a^{4} b + 80 \, a^{3} b^{2} - 90 \, a^{2} b^{3} + 45 \, a b^{4}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{5} + 20 \, a^{4} b - 56 \, a^{3} b^{2} - 160 \, a^{2} b^{3} + 45 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{4} b + 55 \, a^{3} b^{2} + 73 \, a^{2} b^{3} - 15 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{7} + 7 \, a^{6} b + 8 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{7} + 2 \, a^{6} b - 2 \, a^{5} b^{2} - 8 \, a^{4} b^{3} - 7 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/120*(15*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*((23*a^5 + 80*a^4*b + 90*a^3*b^2 + 15*a*b^4)*cos(f*x + e)^7 - (35*a^5 + 106*a^4*b + 80*a^3*b^2 - 90*a^2*b^3 + 45*a*b^4)*cos(f*x + e)^5 + (15*a^5 + 20*a^4*b - 56*a^3*b^2 - 160*a^2*b^3 + 45*a*b^4)*cos(f*x + e)^3 + (15*a^4*b + 55*a^3*b^2 + 73*a^2*b^3 - 15*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f)*sin(f*x + e)), 1/60*(15*((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(f*x + e)^6 + a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5 - (2*a^5 + 7*a^4*b + 8*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 - b^5)*cos(f*x + e)^4 + (a^5 + 2*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 7*a*b^4 - 2*b^5)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((23*a^5 + 80*a^4*b + 90*a^3*b^2 + 15*a*b^4)*cos(f*x + e)^7 - (35*a^5 + 106*a^4*b + 80*a^3*b^2 - 90*a^2*b^3 + 45*a*b^4)*cos(f*x + e)^5 + (15*a^5 + 20*a^4*b - 56*a^3*b^2 - 160*a^2*b^3 + 45*a*b^4)*cos(f*x + e)^3 + (15*a^4*b + 55*a^3*b^2 + 73*a^2*b^3 - 15*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^6 - (2*a^7 + 7*a^6*b + 8*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - a^2*b^5)*f*cos(f*x + e)^4 + (a^7 + 2*a^6*b - 2*a^5*b^2 - 8*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5)*f*cos(f*x + e)^2 + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f)*sin(f*x + e))]","B",0
428,1,564,0,1.761972," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b^{2} \cos\left(f x + e\right)^{4} + 2 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 8 \, {\left(2 \, {\left(a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left(a^{5} b^{2} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{3} f \cos\left(f x + e\right)^{2} + a^{3} b^{4} f\right)}}, \frac{3 \, {\left(a^{2} b^{2} \cos\left(f x + e\right)^{4} + 2 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 4 \, {\left(2 \, {\left(a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(a^{5} b^{2} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{3} f \cos\left(f x + e\right)^{2} + a^{3} b^{4} f\right)}}\right]"," ",0,"[1/24*(3*(a^2*b^2*cos(f*x + e)^4 + 2*a*b^3*cos(f*x + e)^2 + b^4)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 8*(2*(a^4 - a^3*b - 2*a^2*b^2)*cos(f*x + e)^4 + 3*(a^3*b - a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^5*b^2*f*cos(f*x + e)^4 + 2*a^4*b^3*f*cos(f*x + e)^2 + a^3*b^4*f), 1/12*(3*(a^2*b^2*cos(f*x + e)^4 + 2*a*b^3*cos(f*x + e)^2 + b^4)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 4*(2*(a^4 - a^3*b - 2*a^2*b^2)*cos(f*x + e)^4 + 3*(a^3*b - a*b^3)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^5*b^2*f*cos(f*x + e)^4 + 2*a^4*b^3*f*cos(f*x + e)^2 + a^3*b^4*f)]","B",0
429,1,522,0,1.497295," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b \cos\left(f x + e\right)^{4} + 2 \, a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 8 \, {\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} + {\left(a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left(a^{5} b f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{2} f \cos\left(f x + e\right)^{2} + a^{3} b^{3} f\right)}}, -\frac{3 \, {\left(a^{2} b \cos\left(f x + e\right)^{4} + 2 \, a b^{2} \cos\left(f x + e\right)^{2} + b^{3}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + 4 \, {\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} + {\left(a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(a^{5} b f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{2} f \cos\left(f x + e\right)^{2} + a^{3} b^{3} f\right)}}\right]"," ",0,"[1/24*(3*(a^2*b*cos(f*x + e)^4 + 2*a*b^2*cos(f*x + e)^2 + b^3)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 8*(3*a*b^2*cos(f*x + e)^2 + (a^3 + 4*a^2*b)*cos(f*x + e)^4)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^5*b*f*cos(f*x + e)^4 + 2*a^4*b^2*f*cos(f*x + e)^2 + a^3*b^3*f), -1/12*(3*(a^2*b*cos(f*x + e)^4 + 2*a*b^2*cos(f*x + e)^2 + b^3)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + 4*(3*a*b^2*cos(f*x + e)^2 + (a^3 + 4*a^2*b)*cos(f*x + e)^4)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^5*b*f*cos(f*x + e)^4 + 2*a^4*b^2*f*cos(f*x + e)^2 + a^3*b^3*f)]","B",0
430,1,494,0,1.373627," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 8 \, {\left(4 \, a^{2} \cos\left(f x + e\right)^{4} + 3 \, a b \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}, \frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + 4 \, {\left(4 \, a^{2} \cos\left(f x + e\right)^{4} + 3 \, a b \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}\right]"," ",0,"[1/24*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 8*(4*a^2*cos(f*x + e)^4 + 3*a*b*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f), 1/12*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + 4*(4*a^2*cos(f*x + e)^4 + 3*a*b*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)]","B",0
431,1,2279,0,2.614165," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 6 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 8 \, {\left({\left(7 \, a^{4} b + 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}, \frac{12 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 3 \, {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 8 \, {\left({\left(7 \, a^{4} b + 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}, -\frac{3 \, {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 3 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(7 \, a^{4} b + 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}, -\frac{3 \, {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 6 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 4 \, {\left({\left(7 \, a^{4} b + 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)}}\right]"," ",0,"[1/24*(3*(a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 6*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 8*((7*a^4*b + 11*a^3*b^2 + 4*a^2*b^3)*cos(f*x + e)^4 + 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f), 1/24*(12*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 3*(a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 8*((7*a^4*b + 11*a^3*b^2 + 4*a^2*b^3)*cos(f*x + e)^4 + 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f), -1/12*(3*(a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 3*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((7*a^4*b + 11*a^3*b^2 + 4*a^2*b^3)*cos(f*x + e)^4 + 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f), -1/12*(3*(a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 6*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 4*((7*a^4*b + 11*a^3*b^2 + 4*a^2*b^3)*cos(f*x + e)^4 + 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)]","B",0
432,1,3507,0,8.705748," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 3 \, {\left({\left(2 \, a^{6} + 7 \, a^{5} b\right)} \cos\left(f x + e\right)^{6} - 2 \, a^{4} b^{2} - 7 \, a^{3} b^{3} - {\left(2 \, a^{6} + 3 \, a^{5} b - 14 \, a^{4} b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{5} b + 12 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(3 \, a^{6} + 3 \, a^{5} b + 20 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} + 2 \, {\left(3 \, a^{5} b - 7 \, a^{4} b^{2} - 5 \, a^{3} b^{3} + 8 \, a^{2} b^{4} + 3 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} - 8 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)}}, -\frac{6 \, {\left({\left(2 \, a^{6} + 7 \, a^{5} b\right)} \cos\left(f x + e\right)^{6} - 2 \, a^{4} b^{2} - 7 \, a^{3} b^{3} - {\left(2 \, a^{6} + 3 \, a^{5} b - 14 \, a^{4} b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{5} b + 12 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) - 3 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 4 \, {\left({\left(3 \, a^{6} + 3 \, a^{5} b + 20 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} + 2 \, {\left(3 \, a^{5} b - 7 \, a^{4} b^{2} - 5 \, a^{3} b^{3} + 8 \, a^{2} b^{4} + 3 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} - 8 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)}}, \frac{6 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) + 3 \, {\left({\left(2 \, a^{6} + 7 \, a^{5} b\right)} \cos\left(f x + e\right)^{6} - 2 \, a^{4} b^{2} - 7 \, a^{3} b^{3} - {\left(2 \, a^{6} + 3 \, a^{5} b - 14 \, a^{4} b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{5} b + 12 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(3 \, a^{6} + 3 \, a^{5} b + 20 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} + 2 \, {\left(3 \, a^{5} b - 7 \, a^{4} b^{2} - 5 \, a^{3} b^{3} + 8 \, a^{2} b^{4} + 3 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} - 8 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 3 \, {\left({\left(2 \, a^{6} + 7 \, a^{5} b\right)} \cos\left(f x + e\right)^{6} - 2 \, a^{4} b^{2} - 7 \, a^{3} b^{3} - {\left(2 \, a^{6} + 3 \, a^{5} b - 14 \, a^{4} b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, a^{5} b + 12 \, a^{4} b^{2} - 7 \, a^{3} b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(3 \, a^{6} + 3 \, a^{5} b + 20 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 8 \, a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} + 2 \, {\left(3 \, a^{5} b - 7 \, a^{4} b^{2} - 5 \, a^{3} b^{3} + 8 \, a^{2} b^{4} + 3 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} - 8 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)}}\right]"," ",0,"[1/24*(3*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 3*((2*a^6 + 7*a^5*b)*cos(f*x + e)^6 - 2*a^4*b^2 - 7*a^3*b^3 - (2*a^6 + 3*a^5*b - 14*a^4*b^2)*cos(f*x + e)^4 - (4*a^5*b + 12*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((3*a^6 + 3*a^5*b + 20*a^4*b^2 + 28*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 + 2*(3*a^5*b - 7*a^4*b^2 - 5*a^3*b^3 + 8*a^2*b^4 + 3*a*b^5)*cos(f*x + e)^4 + 3*(a^4*b^2 - 5*a^3*b^3 - 8*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f), -1/24*(6*((2*a^6 + 7*a^5*b)*cos(f*x + e)^6 - 2*a^4*b^2 - 7*a^3*b^3 - (2*a^6 + 3*a^5*b - 14*a^4*b^2)*cos(f*x + e)^4 - (4*a^5*b + 12*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) - 3*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 - 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 4*((3*a^6 + 3*a^5*b + 20*a^4*b^2 + 28*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 + 2*(3*a^5*b - 7*a^4*b^2 - 5*a^3*b^3 + 8*a^2*b^4 + 3*a*b^5)*cos(f*x + e)^4 + 3*(a^4*b^2 - 5*a^3*b^3 - 8*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f), 1/24*(6*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) + 3*((2*a^6 + 7*a^5*b)*cos(f*x + e)^6 - 2*a^4*b^2 - 7*a^3*b^3 - (2*a^6 + 3*a^5*b - 14*a^4*b^2)*cos(f*x + e)^4 - (4*a^5*b + 12*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 + 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((3*a^6 + 3*a^5*b + 20*a^4*b^2 + 28*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 + 2*(3*a^5*b - 7*a^4*b^2 - 5*a^3*b^3 + 8*a^2*b^4 + 3*a*b^5)*cos(f*x + e)^4 + 3*(a^4*b^2 - 5*a^3*b^3 - 8*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f), 1/12*(3*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 3*((2*a^6 + 7*a^5*b)*cos(f*x + e)^6 - 2*a^4*b^2 - 7*a^3*b^3 - (2*a^6 + 3*a^5*b - 14*a^4*b^2)*cos(f*x + e)^4 - (4*a^5*b + 12*a^4*b^2 - 7*a^3*b^3)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((3*a^6 + 3*a^5*b + 20*a^4*b^2 + 28*a^3*b^3 + 8*a^2*b^4)*cos(f*x + e)^6 + 2*(3*a^5*b - 7*a^4*b^2 - 5*a^3*b^3 + 8*a^2*b^4 + 3*a*b^5)*cos(f*x + e)^4 + 3*(a^4*b^2 - 5*a^3*b^3 - 8*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)]","B",0
433,1,4751,0,33.455718," ","integrate(cot(f*x+e)^5/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{12 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) + 3 \, {\left({\left(8 \, a^{7} + 36 \, a^{6} b + 63 \, a^{5} b^{2}\right)} \cos\left(f x + e\right)^{8} + 8 \, a^{5} b^{2} + 36 \, a^{4} b^{3} + 63 \, a^{3} b^{4} - 2 \, {\left(8 \, a^{7} + 28 \, a^{6} b + 27 \, a^{5} b^{2} - 63 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{7} + 4 \, a^{6} b - 73 \, a^{5} b^{2} - 216 \, a^{4} b^{3} + 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{6} b + 28 \, a^{5} b^{2} + 27 \, a^{4} b^{3} - 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(18 \, a^{7} + 69 \, a^{6} b + 51 \, a^{5} b^{2} + 104 \, a^{4} b^{3} + 136 \, a^{3} b^{4} + 32 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} - {\left(12 \, a^{7} + 21 \, a^{6} b - 93 \, a^{5} b^{2} + 106 \, a^{4} b^{3} + 176 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 24 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} - {\left(24 \, a^{6} b + 96 \, a^{5} b^{2} - 83 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{5} b^{2} + 19 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 40 \, a^{2} b^{5} - 8 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{96 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)}}, \frac{3 \, {\left({\left(8 \, a^{7} + 36 \, a^{6} b + 63 \, a^{5} b^{2}\right)} \cos\left(f x + e\right)^{8} + 8 \, a^{5} b^{2} + 36 \, a^{4} b^{3} + 63 \, a^{3} b^{4} - 2 \, {\left(8 \, a^{7} + 28 \, a^{6} b + 27 \, a^{5} b^{2} - 63 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{7} + 4 \, a^{6} b - 73 \, a^{5} b^{2} - 216 \, a^{4} b^{3} + 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{6} b + 28 \, a^{5} b^{2} + 27 \, a^{4} b^{3} - 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 6 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} + 256 \, a^{3} b \cos\left(f x + e\right)^{6} + 160 \, a^{2} b^{2} \cos\left(f x + e\right)^{4} + 32 \, a b^{3} \cos\left(f x + e\right)^{2} + b^{4} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{8} + 24 \, a^{2} b \cos\left(f x + e\right)^{6} + 10 \, a b^{2} \cos\left(f x + e\right)^{4} + b^{3} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right) - 2 \, {\left({\left(18 \, a^{7} + 69 \, a^{6} b + 51 \, a^{5} b^{2} + 104 \, a^{4} b^{3} + 136 \, a^{3} b^{4} + 32 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} - {\left(12 \, a^{7} + 21 \, a^{6} b - 93 \, a^{5} b^{2} + 106 \, a^{4} b^{3} + 176 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 24 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} - {\left(24 \, a^{6} b + 96 \, a^{5} b^{2} - 83 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{5} b^{2} + 19 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 40 \, a^{2} b^{5} - 8 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{48 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)}}, -\frac{24 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 3 \, {\left({\left(8 \, a^{7} + 36 \, a^{6} b + 63 \, a^{5} b^{2}\right)} \cos\left(f x + e\right)^{8} + 8 \, a^{5} b^{2} + 36 \, a^{4} b^{3} + 63 \, a^{3} b^{4} - 2 \, {\left(8 \, a^{7} + 28 \, a^{6} b + 27 \, a^{5} b^{2} - 63 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{7} + 4 \, a^{6} b - 73 \, a^{5} b^{2} - 216 \, a^{4} b^{3} + 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{6} b + 28 \, a^{5} b^{2} + 27 \, a^{4} b^{3} - 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{2 \, {\left({\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{4} + b \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(18 \, a^{7} + 69 \, a^{6} b + 51 \, a^{5} b^{2} + 104 \, a^{4} b^{3} + 136 \, a^{3} b^{4} + 32 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} - {\left(12 \, a^{7} + 21 \, a^{6} b - 93 \, a^{5} b^{2} + 106 \, a^{4} b^{3} + 176 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 24 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} - {\left(24 \, a^{6} b + 96 \, a^{5} b^{2} - 83 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{5} b^{2} + 19 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 40 \, a^{2} b^{5} - 8 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{96 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)}}, -\frac{12 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{4} + 8 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b \cos\left(f x + e\right)^{2} + a b^{2}\right)}}\right) - 3 \, {\left({\left(8 \, a^{7} + 36 \, a^{6} b + 63 \, a^{5} b^{2}\right)} \cos\left(f x + e\right)^{8} + 8 \, a^{5} b^{2} + 36 \, a^{4} b^{3} + 63 \, a^{3} b^{4} - 2 \, {\left(8 \, a^{7} + 28 \, a^{6} b + 27 \, a^{5} b^{2} - 63 \, a^{4} b^{3}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{7} + 4 \, a^{6} b - 73 \, a^{5} b^{2} - 216 \, a^{4} b^{3} + 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(8 \, a^{6} b + 28 \, a^{5} b^{2} + 27 \, a^{4} b^{3} - 63 \, a^{3} b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-a - b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)}}\right) + 2 \, {\left({\left(18 \, a^{7} + 69 \, a^{6} b + 51 \, a^{5} b^{2} + 104 \, a^{4} b^{3} + 136 \, a^{3} b^{4} + 32 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} - {\left(12 \, a^{7} + 21 \, a^{6} b - 93 \, a^{5} b^{2} + 106 \, a^{4} b^{3} + 176 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 24 \, a b^{6}\right)} \cos\left(f x + e\right)^{6} - {\left(24 \, a^{6} b + 96 \, a^{5} b^{2} - 83 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{5} b^{2} + 19 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 40 \, a^{2} b^{5} - 8 \, a b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{48 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)}}\right]"," ",0,"[1/96*(12*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) + 3*((8*a^7 + 36*a^6*b + 63*a^5*b^2)*cos(f*x + e)^8 + 8*a^5*b^2 + 36*a^4*b^3 + 63*a^3*b^4 - 2*(8*a^7 + 28*a^6*b + 27*a^5*b^2 - 63*a^4*b^3)*cos(f*x + e)^6 + (8*a^7 + 4*a^6*b - 73*a^5*b^2 - 216*a^4*b^3 + 63*a^3*b^4)*cos(f*x + e)^4 + 2*(8*a^6*b + 28*a^5*b^2 + 27*a^4*b^3 - 63*a^3*b^4)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((18*a^7 + 69*a^6*b + 51*a^5*b^2 + 104*a^4*b^3 + 136*a^3*b^4 + 32*a^2*b^5)*cos(f*x + e)^8 - (12*a^7 + 21*a^6*b - 93*a^5*b^2 + 106*a^4*b^3 + 176*a^3*b^4 - 56*a^2*b^5 - 24*a*b^6)*cos(f*x + e)^6 - (24*a^6*b + 96*a^5*b^2 - 83*a^4*b^3 + 5*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*cos(f*x + e)^4 - 3*(4*a^5*b^2 + 19*a^4*b^3 - 17*a^3*b^4 - 40*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f), 1/48*(3*((8*a^7 + 36*a^6*b + 63*a^5*b^2)*cos(f*x + e)^8 + 8*a^5*b^2 + 36*a^4*b^3 + 63*a^3*b^4 - 2*(8*a^7 + 28*a^6*b + 27*a^5*b^2 - 63*a^4*b^3)*cos(f*x + e)^6 + (8*a^7 + 4*a^6*b - 73*a^5*b^2 - 216*a^4*b^3 + 63*a^3*b^4)*cos(f*x + e)^4 + 2*(8*a^6*b + 28*a^5*b^2 + 27*a^4*b^3 - 63*a^3*b^4)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 6*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(a)*log(128*a^4*cos(f*x + e)^8 + 256*a^3*b*cos(f*x + e)^6 + 160*a^2*b^2*cos(f*x + e)^4 + 32*a*b^3*cos(f*x + e)^2 + b^4 + 8*(16*a^3*cos(f*x + e)^8 + 24*a^2*b*cos(f*x + e)^6 + 10*a*b^2*cos(f*x + e)^4 + b^3*cos(f*x + e)^2)*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)) - 2*((18*a^7 + 69*a^6*b + 51*a^5*b^2 + 104*a^4*b^3 + 136*a^3*b^4 + 32*a^2*b^5)*cos(f*x + e)^8 - (12*a^7 + 21*a^6*b - 93*a^5*b^2 + 106*a^4*b^3 + 176*a^3*b^4 - 56*a^2*b^5 - 24*a*b^6)*cos(f*x + e)^6 - (24*a^6*b + 96*a^5*b^2 - 83*a^4*b^3 + 5*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*cos(f*x + e)^4 - 3*(4*a^5*b^2 + 19*a^4*b^3 - 17*a^3*b^4 - 40*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f), -1/96*(24*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 3*((8*a^7 + 36*a^6*b + 63*a^5*b^2)*cos(f*x + e)^8 + 8*a^5*b^2 + 36*a^4*b^3 + 63*a^3*b^4 - 2*(8*a^7 + 28*a^6*b + 27*a^5*b^2 - 63*a^4*b^3)*cos(f*x + e)^6 + (8*a^7 + 4*a^6*b - 73*a^5*b^2 - 216*a^4*b^3 + 63*a^3*b^4)*cos(f*x + e)^4 + 2*(8*a^6*b + 28*a^5*b^2 + 27*a^4*b^3 - 63*a^3*b^4)*cos(f*x + e)^2)*sqrt(a + b)*log(2*((8*a^2 + 8*a*b + b^2)*cos(f*x + e)^4 + 2*(4*a*b + 3*b^2)*cos(f*x + e)^2 + b^2 - 4*((2*a + b)*cos(f*x + e)^4 + b*cos(f*x + e)^2)*sqrt(a + b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((18*a^7 + 69*a^6*b + 51*a^5*b^2 + 104*a^4*b^3 + 136*a^3*b^4 + 32*a^2*b^5)*cos(f*x + e)^8 - (12*a^7 + 21*a^6*b - 93*a^5*b^2 + 106*a^4*b^3 + 176*a^3*b^4 - 56*a^2*b^5 - 24*a*b^6)*cos(f*x + e)^6 - (24*a^6*b + 96*a^5*b^2 - 83*a^4*b^3 + 5*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*cos(f*x + e)^4 - 3*(4*a^5*b^2 + 19*a^4*b^3 - 17*a^3*b^4 - 40*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f), -1/48*(12*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(-a)*arctan(1/4*(8*a^2*cos(f*x + e)^4 + 8*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(2*a^3*cos(f*x + e)^4 + 3*a^2*b*cos(f*x + e)^2 + a*b^2)) - 3*((8*a^7 + 36*a^6*b + 63*a^5*b^2)*cos(f*x + e)^8 + 8*a^5*b^2 + 36*a^4*b^3 + 63*a^3*b^4 - 2*(8*a^7 + 28*a^6*b + 27*a^5*b^2 - 63*a^4*b^3)*cos(f*x + e)^6 + (8*a^7 + 4*a^6*b - 73*a^5*b^2 - 216*a^4*b^3 + 63*a^3*b^4)*cos(f*x + e)^4 + 2*(8*a^6*b + 28*a^5*b^2 + 27*a^4*b^3 - 63*a^3*b^4)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((2*a + b)*cos(f*x + e)^2 + b)*sqrt(-a - b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 + a*b)*cos(f*x + e)^2 + a*b + b^2)) + 2*((18*a^7 + 69*a^6*b + 51*a^5*b^2 + 104*a^4*b^3 + 136*a^3*b^4 + 32*a^2*b^5)*cos(f*x + e)^8 - (12*a^7 + 21*a^6*b - 93*a^5*b^2 + 106*a^4*b^3 + 176*a^3*b^4 - 56*a^2*b^5 - 24*a*b^6)*cos(f*x + e)^6 - (24*a^6*b + 96*a^5*b^2 - 83*a^4*b^3 + 5*a^3*b^4 + 208*a^2*b^5 + 48*a*b^6)*cos(f*x + e)^4 - 3*(4*a^5*b^2 + 19*a^4*b^3 - 17*a^3*b^4 - 40*a^2*b^5 - 8*a*b^6)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f)]","B",0
434,1,2035,0,3.884120," ","integrate(tan(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} b^{3} \cos\left(f x + e\right)^{4} + 2 \, a b^{4} \cos\left(f x + e\right)^{2} + b^{5}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 6 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 8 \, {\left({\left(3 \, a^{4} b - a^{3} b^{2} - 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, a^{3} b^{2} + a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left(a^{5} b^{3} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{4} f \cos\left(f x + e\right)^{2} + a^{3} b^{5} f\right)}}, \frac{12 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - 3 \, {\left(a^{2} b^{3} \cos\left(f x + e\right)^{4} + 2 \, a b^{4} \cos\left(f x + e\right)^{2} + b^{5}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left({\left(3 \, a^{4} b - a^{3} b^{2} - 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, a^{3} b^{2} + a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left(a^{5} b^{3} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{4} f \cos\left(f x + e\right)^{2} + a^{3} b^{5} f\right)}}, \frac{3 \, {\left(a^{2} b^{3} \cos\left(f x + e\right)^{4} + 2 \, a b^{4} \cos\left(f x + e\right)^{2} + b^{5}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 3 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(3 \, a^{4} b - a^{3} b^{2} - 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, a^{3} b^{2} + a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left(a^{5} b^{3} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{4} f \cos\left(f x + e\right)^{2} + a^{3} b^{5} f\right)}}, \frac{3 \, {\left(a^{2} b^{3} \cos\left(f x + e\right)^{4} + 2 \, a b^{4} \cos\left(f x + e\right)^{2} + b^{5}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 6 \, {\left(a^{5} \cos\left(f x + e\right)^{4} + 2 \, a^{4} b \cos\left(f x + e\right)^{2} + a^{3} b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left({\left(3 \, a^{4} b - a^{3} b^{2} - 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, a^{3} b^{2} + a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left(a^{5} b^{3} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b^{4} f \cos\left(f x + e\right)^{2} + a^{3} b^{5} f\right)}}\right]"," ",0,"[-1/24*(3*(a^2*b^3*cos(f*x + e)^4 + 2*a*b^4*cos(f*x + e)^2 + b^5)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 6*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 8*((3*a^4*b - a^3*b^2 - 4*a^2*b^3)*cos(f*x + e)^3 + (4*a^3*b^2 + a^2*b^3 - 3*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*b^3*f*cos(f*x + e)^4 + 2*a^4*b^4*f*cos(f*x + e)^2 + a^3*b^5*f), 1/24*(12*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - 3*(a^2*b^3*cos(f*x + e)^4 + 2*a*b^4*cos(f*x + e)^2 + b^5)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*((3*a^4*b - a^3*b^2 - 4*a^2*b^3)*cos(f*x + e)^3 + (4*a^3*b^2 + a^2*b^3 - 3*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*b^3*f*cos(f*x + e)^4 + 2*a^4*b^4*f*cos(f*x + e)^2 + a^3*b^5*f), 1/12*(3*(a^2*b^3*cos(f*x + e)^4 + 2*a*b^4*cos(f*x + e)^2 + b^5)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 3*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((3*a^4*b - a^3*b^2 - 4*a^2*b^3)*cos(f*x + e)^3 + (4*a^3*b^2 + a^2*b^3 - 3*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*b^3*f*cos(f*x + e)^4 + 2*a^4*b^4*f*cos(f*x + e)^2 + a^3*b^5*f), 1/12*(3*(a^2*b^3*cos(f*x + e)^4 + 2*a*b^4*cos(f*x + e)^2 + b^5)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 6*(a^5*cos(f*x + e)^4 + 2*a^4*b*cos(f*x + e)^2 + a^3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - 4*((3*a^4*b - a^3*b^2 - 4*a^2*b^3)*cos(f*x + e)^3 + (4*a^3*b^2 + a^2*b^3 - 3*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*b^3*f*cos(f*x + e)^4 + 2*a^4*b^4*f*cos(f*x + e)^2 + a^3*b^5*f)]","B",0
435,1,661,0,1.614688," ","integrate(tan(f*x+e)^4/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(4 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}, -\frac{3 \, {\left(a^{2} \cos\left(f x + e\right)^{4} + 2 \, a b \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(4 \, a^{2} \cos\left(f x + e\right)^{3} - {\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 2 \, a^{4} b f \cos\left(f x + e\right)^{2} + a^{3} b^{2} f\right)}}\right]"," ",0,"[-1/24*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(4*a^2*cos(f*x + e)^3 - (a^2 - 3*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f), -1/12*(3*(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(4*a^2*cos(f*x + e)^3 - (a^2 - 3*a*b)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 2*a^4*b*f*cos(f*x + e)^2 + a^3*b^2*f)]","B",0
436,1,773,0,1.538986," ","integrate(tan(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{4} + a b^{2} + b^{3} + 2 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left({\left(3 \, a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(2 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{6} + a^{5} b\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b + a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} + a^{3} b^{3}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{3} + a^{2} b\right)} \cos\left(f x + e\right)^{4} + a b^{2} + b^{3} + 2 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left({\left(3 \, a^{3} + 4 \, a^{2} b\right)} \cos\left(f x + e\right)^{3} + {\left(2 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{6} + a^{5} b\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b + a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} + a^{3} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/24*(3*((a^3 + a^2*b)*cos(f*x + e)^4 + a*b^2 + b^3 + 2*(a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*((3*a^3 + 4*a^2*b)*cos(f*x + e)^3 + (2*a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^6 + a^5*b)*f*cos(f*x + e)^4 + 2*(a^5*b + a^4*b^2)*f*cos(f*x + e)^2 + (a^4*b^2 + a^3*b^3)*f), 1/12*(3*((a^3 + a^2*b)*cos(f*x + e)^4 + a*b^2 + b^3 + 2*(a^2*b + a*b^2)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*((3*a^3 + 4*a^2*b)*cos(f*x + e)^3 + (2*a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^6 + a^5*b)*f*cos(f*x + e)^4 + 2*(a^5*b + a^4*b^2)*f*cos(f*x + e)^2 + (a^4*b^2 + a^3*b^3)*f)]","B",0
437,1,881,0,1.358514," ","integrate(1/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{24 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, -\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} + 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[-1/24*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f), -1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(f*x + e)^4 + a^2*b^2 + 2*a*b^3 + b^4 + 2*(a^3*b + 2*a^2*b^2 + a*b^3)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*(3*a^3*b + 2*a^2*b^2)*cos(f*x + e)^3 + (5*a^2*b^2 + 3*a*b^3)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^7 + 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b + 2*a^5*b^2 + a^4*b^3)*f*cos(f*x + e)^2 + (a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f)]","B",0
438,1,1097,0,4.096240," ","integrate(cot(f*x+e)^2/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left({\left(3 \, a^{5} + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(6 \, a^{4} b - 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)} \sin\left(f x + e\right)}, \frac{3 \, {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(3 \, a^{5} + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(6 \, a^{4} b - 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{3} b^{2} - 8 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/24*(3*(a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*((3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cos(f*x + e)^5 + (6*a^4*b - 9*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*cos(f*x + e)^3 + (3*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)*sin(f*x + e)), 1/12*(3*(a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(f*x + e)^4 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((3*a^5 + 9*a^3*b^2 + 4*a^2*b^3)*cos(f*x + e)^5 + (6*a^4*b - 9*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*cos(f*x + e)^3 + (3*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*f*cos(f*x + e)^4 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*f*cos(f*x + e)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*f)*sin(f*x + e))]","B",0
439,1,1579,0,13.454362," ","integrate(cot(f*x+e)^4/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) - 8 \, {\left(4 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{7} - 3 \, {\left(a^{6} + a^{5} b - 8 \, a^{4} b^{2} + 8 \, a^{3} b^{3} - a^{2} b^{4} - a b^{5}\right)} \cos\left(f x + e\right)^{5} - 6 \, {\left(a^{5} b + 3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{4} b^{2} + 11 \, a^{3} b^{3} - 11 \, a^{2} b^{4} - 3 \, a b^{5}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left({\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{6} - a^{4} b^{2} - 4 \, a^{3} b^{3} - 6 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6} - {\left(a^{6} + 2 \, a^{5} b - 2 \, a^{4} b^{2} - 8 \, a^{3} b^{3} - 7 \, a^{2} b^{4} - 2 \, a b^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b + 7 \, a^{4} b^{2} + 8 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 2 \, a b^{5} - b^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left(4 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(f x + e\right)^{7} - 3 \, {\left(a^{6} + a^{5} b - 8 \, a^{4} b^{2} + 8 \, a^{3} b^{3} - a^{2} b^{4} - a b^{5}\right)} \cos\left(f x + e\right)^{5} - 6 \, {\left(a^{5} b + 3 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{4} b^{2} + 11 \, a^{3} b^{3} - 11 \, a^{2} b^{4} - 3 \, a b^{5}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{9} + 2 \, a^{8} b - 2 \, a^{7} b^{2} - 8 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 2 \, a^{4} b^{5}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{8} b + 7 \, a^{7} b^{2} + 8 \, a^{6} b^{3} + 2 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 4 \, a^{4} b^{5} + a^{3} b^{6}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/24*(3*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) - 8*(4*(a^6 + 3*a^5*b + 3*a^3*b^3 + a^2*b^4)*cos(f*x + e)^7 - 3*(a^6 + a^5*b - 8*a^4*b^2 + 8*a^3*b^3 - a^2*b^4 - a*b^5)*cos(f*x + e)^5 - 6*(a^5*b + 3*a^4*b^2 - 4*a^3*b^3 + 3*a^2*b^4 + a*b^5)*cos(f*x + e)^3 - (3*a^4*b^2 + 11*a^3*b^3 - 11*a^2*b^4 - 3*a*b^5)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)*sin(f*x + e)), -1/12*(3*((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(f*x + e)^6 - a^4*b^2 - 4*a^3*b^3 - 6*a^2*b^4 - 4*a*b^5 - b^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*cos(f*x + e)^4 - (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*(4*(a^6 + 3*a^5*b + 3*a^3*b^3 + a^2*b^4)*cos(f*x + e)^7 - 3*(a^6 + a^5*b - 8*a^4*b^2 + 8*a^3*b^3 - a^2*b^4 - a*b^5)*cos(f*x + e)^5 - 6*(a^5*b + 3*a^4*b^2 - 4*a^3*b^3 + 3*a^2*b^4 + a*b^5)*cos(f*x + e)^3 - (3*a^4*b^2 + 11*a^3*b^3 - 11*a^2*b^4 - 3*a*b^5)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*f*cos(f*x + e)^6 - (a^9 + 2*a^8*b - 2*a^7*b^2 - 8*a^6*b^3 - 7*a^5*b^4 - 2*a^4*b^5)*f*cos(f*x + e)^4 - (2*a^8*b + 7*a^7*b^2 + 8*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - a^3*b^6)*f*cos(f*x + e)^2 - (a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 + 4*a^4*b^5 + a^3*b^6)*f)*sin(f*x + e))]","B",0
440,1,2059,0,43.486045," ","integrate(cot(f*x+e)^6/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - a^{3} b\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - a^{2} b\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) \sin\left(f x + e\right) + 8 \, {\left({\left(23 \, a^{7} + 100 \, a^{6} b + 150 \, a^{5} b^{2} + 75 \, a^{3} b^{4} + 20 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{9} - {\left(35 \, a^{7} + 118 \, a^{6} b + 75 \, a^{5} b^{2} - 300 \, a^{4} b^{3} + 225 \, a^{3} b^{4} - 10 \, a^{2} b^{5} - 15 \, a b^{6}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{7} - 59 \, a^{5} b^{2} - 150 \, a^{4} b^{3} + 125 \, a^{3} b^{4} - 50 \, a^{2} b^{5} - 15 \, a b^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(30 \, a^{6} b + 105 \, a^{5} b^{2} + 92 \, a^{4} b^{3} - 350 \, a^{3} b^{4} + 190 \, a^{2} b^{5} + 45 \, a b^{6}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{5} b^{2} + 70 \, a^{4} b^{3} + 128 \, a^{3} b^{4} - 70 \, a^{2} b^{5} - 15 \, a b^{6}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)} \sin\left(f x + e\right)}, \frac{15 \, {\left({\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(f x + e\right)^{8} + a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7} - 2 \, {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{7} + a^{6} b - 9 \, a^{5} b^{2} - 25 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + a b^{6} + b^{7}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, a b^{6} - b^{7}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, a^{3} \cos\left(f x + e\right)^{4} - a^{2} b + a b^{2} - {\left(a^{3} - 3 \, a^{2} b\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(23 \, a^{7} + 100 \, a^{6} b + 150 \, a^{5} b^{2} + 75 \, a^{3} b^{4} + 20 \, a^{2} b^{5}\right)} \cos\left(f x + e\right)^{9} - {\left(35 \, a^{7} + 118 \, a^{6} b + 75 \, a^{5} b^{2} - 300 \, a^{4} b^{3} + 225 \, a^{3} b^{4} - 10 \, a^{2} b^{5} - 15 \, a b^{6}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{7} - 59 \, a^{5} b^{2} - 150 \, a^{4} b^{3} + 125 \, a^{3} b^{4} - 50 \, a^{2} b^{5} - 15 \, a b^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(30 \, a^{6} b + 105 \, a^{5} b^{2} + 92 \, a^{4} b^{3} - 350 \, a^{3} b^{4} + 190 \, a^{2} b^{5} + 45 \, a b^{6}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{5} b^{2} + 70 \, a^{4} b^{3} + 128 \, a^{3} b^{4} - 70 \, a^{2} b^{5} - 15 \, a b^{6}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left({\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{10} + 4 \, a^{9} b + 5 \, a^{8} b^{2} - 5 \, a^{6} b^{4} - 4 \, a^{5} b^{5} - a^{4} b^{6}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{10} + a^{9} b - 9 \, a^{8} b^{2} - 25 \, a^{7} b^{3} - 25 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{9} b + 4 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{5} b^{5} - 4 \, a^{4} b^{6} - a^{3} b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{8} b^{2} + 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} + 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/120*(15*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))*sin(f*x + e) + 8*((23*a^7 + 100*a^6*b + 150*a^5*b^2 + 75*a^3*b^4 + 20*a^2*b^5)*cos(f*x + e)^9 - (35*a^7 + 118*a^6*b + 75*a^5*b^2 - 300*a^4*b^3 + 225*a^3*b^4 - 10*a^2*b^5 - 15*a*b^6)*cos(f*x + e)^7 + 3*(5*a^7 - 59*a^5*b^2 - 150*a^4*b^3 + 125*a^3*b^4 - 50*a^2*b^5 - 15*a*b^6)*cos(f*x + e)^5 + (30*a^6*b + 105*a^5*b^2 + 92*a^4*b^3 - 350*a^3*b^4 + 190*a^2*b^5 + 45*a*b^6)*cos(f*x + e)^3 + (15*a^5*b^2 + 70*a^4*b^3 + 128*a^3*b^4 - 70*a^2*b^5 - 15*a*b^6)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f)*sin(f*x + e)), 1/60*(15*((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(f*x + e)^8 + a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7 - 2*(a^7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(f*x + e)^6 + (a^7 + a^6*b - 9*a^5*b^2 - 25*a^4*b^3 - 25*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cos(f*x + e)^4 + 2*(a^6*b + 4*a^5*b^2 + 5*a^4*b^3 - 5*a^2*b^5 - 4*a*b^6 - b^7)*cos(f*x + e)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*sin(f*x + e) - 4*((23*a^7 + 100*a^6*b + 150*a^5*b^2 + 75*a^3*b^4 + 20*a^2*b^5)*cos(f*x + e)^9 - (35*a^7 + 118*a^6*b + 75*a^5*b^2 - 300*a^4*b^3 + 225*a^3*b^4 - 10*a^2*b^5 - 15*a*b^6)*cos(f*x + e)^7 + 3*(5*a^7 - 59*a^5*b^2 - 150*a^4*b^3 + 125*a^3*b^4 - 50*a^2*b^5 - 15*a*b^6)*cos(f*x + e)^5 + (30*a^6*b + 105*a^5*b^2 + 92*a^4*b^3 - 350*a^3*b^4 + 190*a^2*b^5 + 45*a*b^6)*cos(f*x + e)^3 + (15*a^5*b^2 + 70*a^4*b^3 + 128*a^3*b^4 - 70*a^2*b^5 - 15*a*b^6)*cos(f*x + e))*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(((a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*f*cos(f*x + e)^8 - 2*(a^10 + 4*a^9*b + 5*a^8*b^2 - 5*a^6*b^4 - 4*a^5*b^5 - a^4*b^6)*f*cos(f*x + e)^6 + (a^10 + a^9*b - 9*a^8*b^2 - 25*a^7*b^3 - 25*a^6*b^4 - 9*a^5*b^5 + a^4*b^6 + a^3*b^7)*f*cos(f*x + e)^4 + 2*(a^9*b + 4*a^8*b^2 + 5*a^7*b^3 - 5*a^5*b^5 - 4*a^4*b^6 - a^3*b^7)*f*cos(f*x + e)^2 + (a^8*b^2 + 5*a^7*b^3 + 10*a^6*b^4 + 10*a^5*b^5 + 5*a^4*b^6 + a^3*b^7)*f)*sin(f*x + e))]","B",0
441,0,0,0,0.975634," ","integrate((a+b*sec(f*x+e)^2)^p*(d*tan(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*(d*tan(f*x + e))^m, x)","F",0
442,0,0,0,0.529257," ","integrate((a+b*sec(f*x+e)^2)^p*tan(f*x+e)^5,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{5}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*tan(f*x + e)^5, x)","F",0
443,0,0,0,0.507121," ","integrate((a+b*sec(f*x+e)^2)^p*tan(f*x+e)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*tan(f*x + e)^3, x)","F",0
444,0,0,0,0.509645," ","integrate((a+b*sec(f*x+e)^2)^p*tan(f*x+e),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right), x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*tan(f*x + e), x)","F",0
445,0,0,0,0.524789," ","integrate(cot(f*x+e)*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right), x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cot(f*x + e), x)","F",0
446,0,0,0,0.531533," ","integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cot(f*x + e)^3, x)","F",0
447,0,0,0,0.589966," ","integrate((a+b*sec(f*x+e)^2)^p*tan(f*x+e)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*tan(f*x + e)^4, x)","F",0
448,0,0,0,0.528231," ","integrate((a+b*sec(f*x+e)^2)^p*tan(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*tan(f*x + e)^2, x)","F",0
449,0,0,0,0.488561," ","integrate((a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p, x)","F",0
450,0,0,0,0.588274," ","integrate(cot(f*x+e)^2*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cot(f*x + e)^2, x)","F",0
451,0,0,0,0.548309," ","integrate(cot(f*x+e)^4*(a+b*sec(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sec(f*x + e)^2 + a)^p*cot(f*x + e)^4, x)","F",0
452,1,81,0,0.732058," ","integrate((a+b*sec(f*x+e)^3)*tan(f*x+e)^5,x, algorithm=""fricas"")","-\frac{420 \, a \cos\left(f x + e\right)^{7} \log\left(-\cos\left(f x + e\right)\right) + 420 \, a \cos\left(f x + e\right)^{5} - 140 \, b \cos\left(f x + e\right)^{4} - 105 \, a \cos\left(f x + e\right)^{3} + 168 \, b \cos\left(f x + e\right)^{2} - 60 \, b}{420 \, f \cos\left(f x + e\right)^{7}}"," ",0,"-1/420*(420*a*cos(f*x + e)^7*log(-cos(f*x + e)) + 420*a*cos(f*x + e)^5 - 140*b*cos(f*x + e)^4 - 105*a*cos(f*x + e)^3 + 168*b*cos(f*x + e)^2 - 60*b)/(f*cos(f*x + e)^7)","A",0
453,1,59,0,0.598014," ","integrate((a+b*sec(f*x+e)^3)*tan(f*x+e)^3,x, algorithm=""fricas"")","\frac{30 \, a \cos\left(f x + e\right)^{5} \log\left(-\cos\left(f x + e\right)\right) + 15 \, a \cos\left(f x + e\right)^{3} - 10 \, b \cos\left(f x + e\right)^{2} + 6 \, b}{30 \, f \cos\left(f x + e\right)^{5}}"," ",0,"1/30*(30*a*cos(f*x + e)^5*log(-cos(f*x + e)) + 15*a*cos(f*x + e)^3 - 10*b*cos(f*x + e)^2 + 6*b)/(f*cos(f*x + e)^5)","A",0
454,1,37,0,0.521976," ","integrate((a+b*sec(f*x+e)^3)*tan(f*x+e),x, algorithm=""fricas"")","-\frac{3 \, a \cos\left(f x + e\right)^{3} \log\left(-\cos\left(f x + e\right)\right) - b}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/3*(3*a*cos(f*x + e)^3*log(-cos(f*x + e)) - b)/(f*cos(f*x + e)^3)","A",0
455,1,61,0,0.461892," ","integrate(cot(f*x+e)*(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","\frac{{\left(a - b\right)} \cos\left(f x + e\right) \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left(a + b\right)} \cos\left(f x + e\right) \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 2 \, b}{2 \, f \cos\left(f x + e\right)}"," ",0,"1/2*((a - b)*cos(f*x + e)*log(1/2*cos(f*x + e) + 1/2) + (a + b)*cos(f*x + e)*log(-1/2*cos(f*x + e) + 1/2) + 2*b)/(f*cos(f*x + e))","A",0
456,1,99,0,0.539513," ","integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","\frac{2 \, b \cos\left(f x + e\right) - {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - {\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 2 \, a}{4 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}"," ",0,"1/4*(2*b*cos(f*x + e) - ((2*a + b)*cos(f*x + e)^2 - 2*a - b)*log(1/2*cos(f*x + e) + 1/2) - ((2*a - b)*cos(f*x + e)^2 - 2*a + b)*log(-1/2*cos(f*x + e) + 1/2) + 2*a)/(f*cos(f*x + e)^2 - f)","A",0
457,1,4427,0,1.672262," ","integrate(tan(f*x+e)^5/(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","-\frac{2 \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b f \cos\left(f x + e\right) \log\left(\frac{1}{36} \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{3} f^{2} - {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{3} f + 4 \, a^{2} b + 5 \, b^{3} + {\left(a^{3} + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right) - {\left({\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b f \cos\left(f x + e\right) + 3 \, \sqrt{\frac{1}{3}} a b f \sqrt{-\frac{{\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{2} f^{2} - 12 \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{2} f + 288 \, a^{2} + 36 \, b^{2}}{a^{2} b^{2} f^{2}}} \cos\left(f x + e\right) - 18 \, b \cos\left(f x + e\right)\right)} \log\left(\frac{1}{36} \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{3} f^{2} - {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{3} f + 4 \, a^{2} b + 5 \, b^{3} - \frac{1}{12} \, \sqrt{\frac{1}{3}} {\left({\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a^{2} b^{3} f^{2} + 18 \, a b^{3} f\right)} \sqrt{-\frac{{\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{2} f^{2} - 12 \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{2} f + 288 \, a^{2} + 36 \, b^{2}}{a^{2} b^{2} f^{2}}} - 2 \, {\left(a^{3} + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right) - {\left({\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b f \cos\left(f x + e\right) - 3 \, \sqrt{\frac{1}{3}} a b f \sqrt{-\frac{{\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{2} f^{2} - 12 \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{2} f + 288 \, a^{2} + 36 \, b^{2}}{a^{2} b^{2} f^{2}}} \cos\left(f x + e\right) - 18 \, b \cos\left(f x + e\right)\right)} \log\left(-\frac{1}{36} \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{3} f^{2} + {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{3} f - 4 \, a^{2} b - 5 \, b^{3} - \frac{1}{12} \, \sqrt{\frac{1}{3}} {\left({\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a^{2} b^{3} f^{2} + 18 \, a b^{3} f\right)} \sqrt{-\frac{{\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)}^{2} a^{2} b^{2} f^{2} - 12 \, {\left(\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{1}{a^{2} f^{2}} - \frac{2 \, a^{2} + b^{2}}{a^{2} b^{2} f^{2}}\right)}}{{\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{27 \, a^{3} f^{3}} + \frac{a^{2} + 8 \, b^{2}}{54 \, a b^{4} f^{3}} + \frac{2 \, a^{2} + b^{2}}{18 \, a^{3} b^{2} f^{3}} - \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{54 \, a^{3} b^{4} f^{3}}\right)}^{\frac{1}{3}} + \frac{6}{a f}\right)} a b^{2} f + 288 \, a^{2} + 36 \, b^{2}}{a^{2} b^{2} f^{2}}} + 2 \, {\left(a^{3} + 8 \, a b^{2}\right)} \cos\left(f x + e\right)\right) - 36 \, a}{36 \, a b f \cos\left(f x + e\right)}"," ",0,"-1/36*(2*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b*f*cos(f*x + e)*log(1/36*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^3*f^2 - ((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^3*f + 4*a^2*b + 5*b^3 + (a^3 + 8*a*b^2)*cos(f*x + e)) - (((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b*f*cos(f*x + e) + 3*sqrt(1/3)*a*b*f*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2))*cos(f*x + e) - 18*b*cos(f*x + e))*log(1/36*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^3*f^2 - ((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^3*f + 4*a^2*b + 5*b^3 - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a^2*b^3*f^2 + 18*a*b^3*f)*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2)) - 2*(a^3 + 8*a*b^2)*cos(f*x + e)) - (((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b*f*cos(f*x + e) - 3*sqrt(1/3)*a*b*f*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2))*cos(f*x + e) - 18*b*cos(f*x + e))*log(-1/36*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^3*f^2 + ((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^3*f - 4*a^2*b - 5*b^3 - 1/12*sqrt(1/3)*(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a^2*b^3*f^2 + 18*a*b^3*f)*sqrt(-(((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))^2*a^2*b^2*f^2 - 12*((-I*sqrt(3) + 1)*(1/(a^2*f^2) - (2*a^2 + b^2)/(a^2*b^2*f^2))/(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 9*(I*sqrt(3) + 1)*(-1/27/(a^3*f^3) + 1/54*(a^2 + 8*b^2)/(a*b^4*f^3) + 1/18*(2*a^2 + b^2)/(a^3*b^2*f^3) - 1/54*(a^4 - 2*a^2*b^2 + b^4)/(a^3*b^4*f^3))^(1/3) + 6/(a*f))*a*b^2*f + 288*a^2 + 36*b^2)/(a^2*b^2*f^2)) + 2*(a^3 + 8*a*b^2)*cos(f*x + e)) - 36*a)/(a*b*f*cos(f*x + e))","C",0
458,1,2278,0,36.119905," ","integrate(tan(f*x+e)^3/(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","-\frac{6 \, \sqrt{\frac{1}{3}} a f \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} f^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 4}{a^{2} f^{2}}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{3}} \sqrt{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} b^{2} f^{2} + 4 \, a^{2} \cos\left(f x + e\right)^{2} - 4 \, a b \cos\left(f x + e\right) - 2 \, {\left(a^{2} b f \cos\left(f x + e\right) - 2 \, a b^{2} f\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} + 4 \, b^{2}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a b f^{2} + 2 \, b f\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} f^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 4}{a^{2} f^{2}}} + \sqrt{\frac{1}{3}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} b^{2} f^{3} - 8 \, a b f \cos\left(f x + e\right) + 4 \, b^{2} f - 4 \, {\left(a^{2} b f^{2} \cos\left(f x + e\right) - a b^{2} f^{2}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} f^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 4}{a^{2} f^{2}}}}{8 \, a}\right) - 6 \, \sqrt{\frac{1}{3}} a f \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} f^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 4}{a^{2} f^{2}}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{3}} \sqrt{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} b^{2} f^{2} + 4 \, a^{2} \cos\left(f x + e\right)^{2} - 4 \, a b \cos\left(f x + e\right) - 2 \, {\left(a^{2} b f \cos\left(f x + e\right) - 2 \, a b^{2} f\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} + 4 \, b^{2}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a b f^{2} + 2 \, b f\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} f^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 4}{a^{2} f^{2}}} - \sqrt{\frac{1}{3}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} b^{2} f^{3} - 8 \, a b f \cos\left(f x + e\right) + 4 \, b^{2} f - 4 \, {\left(a^{2} b f^{2} \cos\left(f x + e\right) - a b^{2} f^{2}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} f^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 4}{a^{2} f^{2}}}}{8 \, a}\right) + {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f \log\left(\frac{1}{4} \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} b^{2} f^{2} + a^{2} \cos\left(f x + e\right)^{2} + 2 \, a b \cos\left(f x + e\right) + {\left(a^{2} b f \cos\left(f x + e\right) + a b^{2} f\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} + b^{2}\right) - {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} a f + 6\right)} \log\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)}^{2} a^{2} b^{2} f^{2} + 4 \, a^{2} \cos\left(f x + e\right)^{2} - 4 \, a b \cos\left(f x + e\right) - 2 \, {\left(a^{2} b f \cos\left(f x + e\right) - 2 \, a b^{2} f\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3} f^{3}} + \frac{1}{54 \, a b^{2} f^{3}} - \frac{a^{2} - b^{2}}{54 \, a^{3} b^{2} f^{3}}\right)}^{\frac{1}{3}} - \frac{2}{a f}\right)} + 4 \, b^{2}\right)}{12 \, a f}"," ",0,"-1/12*(6*sqrt(1/3)*a*f*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*f^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 4)/(a^2*f^2))*arctan(-1/8*(2*sqrt(1/3)*sqrt((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*b^2*f^2 + 4*a^2*cos(f*x + e)^2 - 4*a*b*cos(f*x + e) - 2*(a^2*b*f*cos(f*x + e) - 2*a*b^2*f)*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f)) + 4*b^2)*((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*b*f^2 + 2*b*f)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*f^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 4)/(a^2*f^2)) + sqrt(1/3)*((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*b^2*f^3 - 8*a*b*f*cos(f*x + e) + 4*b^2*f - 4*(a^2*b*f^2*cos(f*x + e) - a*b^2*f^2)*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f)))*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*f^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 4)/(a^2*f^2)))/a) - 6*sqrt(1/3)*a*f*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*f^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 4)/(a^2*f^2))*arctan(-1/8*(2*sqrt(1/3)*sqrt((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*b^2*f^2 + 4*a^2*cos(f*x + e)^2 - 4*a*b*cos(f*x + e) - 2*(a^2*b*f*cos(f*x + e) - 2*a*b^2*f)*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f)) + 4*b^2)*((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*b*f^2 + 2*b*f)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*f^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 4)/(a^2*f^2)) - sqrt(1/3)*((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*b^2*f^3 - 8*a*b*f*cos(f*x + e) + 4*b^2*f - 4*(a^2*b*f^2*cos(f*x + e) - a*b^2*f^2)*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f)))*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*f^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 4)/(a^2*f^2)))/a) + (3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f*log(1/4*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*b^2*f^2 + a^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + (a^2*b*f*cos(f*x + e) + a*b^2*f)*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f)) + b^2) - ((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))*a*f + 6)*log((3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f))^2*a^2*b^2*f^2 + 4*a^2*cos(f*x + e)^2 - 4*a*b*cos(f*x + e) - 2*(a^2*b*f*cos(f*x + e) - 2*a*b^2*f)*(3*(I*sqrt(3) + 1)*(-1/54/(a^3*f^3) + 1/54/(a*b^2*f^3) - 1/54*(a^2 - b^2)/(a^3*b^2*f^3))^(1/3) - 2/(a*f)) + 4*b^2))/(a*f)","C",0
459,1,21,0,0.500101," ","integrate(tan(f*x+e)/(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","-\frac{\log\left(a \cos\left(f x + e\right)^{3} + b\right)}{3 \, a f}"," ",0,"-1/3*log(a*cos(f*x + e)^3 + b)/(a*f)","A",0
460,1,6482,0,1.661214," ","integrate(cot(f*x+e)/(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{3} - a b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f \log\left(\frac{1}{2} \, {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} a b^{2} f - \frac{1}{36} \, {\left(a^{4} - a^{2} b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} - a b \cos\left(f x + e\right) + 2 \, b^{2}\right) - {\left({\left(a^{3} - a b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f + 3 \, \sqrt{\frac{1}{3}} {\left(a^{3} - a b^{2}\right)} f \sqrt{-\frac{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} - 144 \, a^{2} b^{2} + 36 \, b^{4} - 12 \, {\left(a^{3} b^{2} - a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f}{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} f^{2}}} - 18 \, b^{2}\right)} \log\left(\frac{1}{2} \, {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} a b^{2} f - \frac{1}{36} \, {\left(a^{4} - a^{2} b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} + \frac{1}{12} \, \sqrt{\frac{1}{3}} {\left(a^{4} - a^{2} b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f^{2} \sqrt{-\frac{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} - 144 \, a^{2} b^{2} + 36 \, b^{4} - 12 \, {\left(a^{3} b^{2} - a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f}{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} f^{2}}} + 2 \, a b \cos\left(f x + e\right) + 2 \, b^{2}\right) - {\left({\left(a^{3} - a b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f - 3 \, \sqrt{\frac{1}{3}} {\left(a^{3} - a b^{2}\right)} f \sqrt{-\frac{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} - 144 \, a^{2} b^{2} + 36 \, b^{4} - 12 \, {\left(a^{3} b^{2} - a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f}{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} f^{2}}} - 18 \, b^{2}\right)} \log\left(-\frac{1}{2} \, {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} a b^{2} f + \frac{1}{36} \, {\left(a^{4} - a^{2} b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} + \frac{1}{12} \, \sqrt{\frac{1}{3}} {\left(a^{4} - a^{2} b^{2}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f^{2} \sqrt{-\frac{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)}^{2} f^{2} - 144 \, a^{2} b^{2} + 36 \, b^{4} - 12 \, {\left(a^{3} b^{2} - a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{{\left(a^{3} f - a b^{2} f\right)}^{2}} + \frac{b^{2}}{a^{4} f^{2} - a^{2} b^{2} f^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}}} + 9 \, {\left(-\frac{b^{6}}{27 \, {\left(a^{3} f - a b^{2} f\right)}^{3}} - \frac{b^{4}}{18 \, {\left(a^{4} f^{2} - a^{2} b^{2} f^{2}\right)} {\left(a^{3} f - a b^{2} f\right)}} - \frac{b^{2}}{54 \, {\left(a^{5} f^{3} - a^{3} b^{2} f^{3}\right)}} + \frac{b^{2}}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{6 \, b^{2}}{a^{3} f - a b^{2} f}\right)} f}{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} f^{2}}} - 2 \, a b \cos\left(f x + e\right) - 2 \, b^{2}\right) - 18 \, {\left(a^{2} + a b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 18 \, {\left(a^{2} - a b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{36 \, {\left(a^{3} - a b^{2}\right)} f}"," ",0,"-1/36*(2*(a^3 - a*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f*log(1/2*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*a*b^2*f - 1/36*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - a*b*cos(f*x + e) + 2*b^2) - ((a^3 - a*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f + 3*sqrt(1/3)*(a^3 - a*b^2)*f*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144*a^2*b^2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) - 18*b^2)*log(1/2*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*a*b^2*f - 1/36*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 + 1/12*sqrt(1/3)*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f^2*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144*a^2*b^2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) + 2*a*b*cos(f*x + e) + 2*b^2) - ((a^3 - a*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f - 3*sqrt(1/3)*(a^3 - a*b^2)*f*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144*a^2*b^2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) - 18*b^2)*log(-1/2*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*a*b^2*f + 1/36*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 + 1/12*sqrt(1/3)*(a^4 - a^2*b^2)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f^2*sqrt(-((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))^2*f^2 - 144*a^2*b^2 + 36*b^4 - 12*(a^3*b^2 - a*b^4)*((b^4/(a^3*f - a*b^2*f)^2 + b^2/(a^4*f^2 - a^2*b^2*f^2))*(-I*sqrt(3) + 1)/(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3) + 9*(-1/27*b^6/(a^3*f - a*b^2*f)^3 - 1/18*b^4/((a^4*f^2 - a^2*b^2*f^2)*(a^3*f - a*b^2*f)) - 1/54*b^2/(a^5*f^3 - a^3*b^2*f^3) + 1/54*b^2/((a^2 - b^2)^2*a*f^3))^(1/3)*(I*sqrt(3) + 1) + 6*b^2/(a^3*f - a*b^2*f))*f)/((a^6 - 2*a^4*b^2 + a^2*b^4)*f^2)) - 2*a*b*cos(f*x + e) - 2*b^2) - 18*(a^2 + a*b)*log(1/2*cos(f*x + e) + 1/2) - 18*(a^2 - a*b)*log(-1/2*cos(f*x + e) + 1/2))/((a^3 - a*b^2)*f)","C",0
461,1,10746,0,3.844685," ","integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^3),x, algorithm=""fricas"")","\frac{18 \, a^{4} - 18 \, a^{2} b^{2} + 2 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} \log\left(\frac{1}{12} \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} + 2 \, a^{2} b^{2} + 7 \, b^{4} + \frac{1}{6} \, {\left(a^{5} + 16 \, a^{3} b^{2} + 10 \, a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f - {\left(a^{3} b + 8 \, a b^{3}\right)} \cos\left(f x + e\right)\right) - 18 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(f x + e\right) + {\left(36 \, a^{2} b^{2} + 18 \, b^{4} - 18 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(f x + e\right)^{2} - {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} + 3 \, \sqrt{\frac{1}{3}} {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} \sqrt{\frac{288 \, a^{4} b^{4} + 720 \, a^{2} b^{6} - 36 \, b^{8} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} - 12 \, {\left(2 \, a^{7} b^{2} - 3 \, a^{5} b^{4} + a b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} f^{2}}}\right)} \log\left(\frac{1}{12} \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} + 2 \, a^{2} b^{2} + 7 \, b^{4} + \frac{1}{6} \, {\left(a^{5} + 16 \, a^{3} b^{2} + 10 \, a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f + \frac{1}{4} \, \sqrt{\frac{1}{3}} {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f^{2} - 2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} \sqrt{\frac{288 \, a^{4} b^{4} + 720 \, a^{2} b^{6} - 36 \, b^{8} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} - 12 \, {\left(2 \, a^{7} b^{2} - 3 \, a^{5} b^{4} + a b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} f^{2}}} + 2 \, {\left(a^{3} b + 8 \, a b^{3}\right)} \cos\left(f x + e\right)\right) + {\left(36 \, a^{2} b^{2} + 18 \, b^{4} - 18 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(f x + e\right)^{2} - {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} - 3 \, \sqrt{\frac{1}{3}} {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} \sqrt{\frac{288 \, a^{4} b^{4} + 720 \, a^{2} b^{6} - 36 \, b^{8} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} - 12 \, {\left(2 \, a^{7} b^{2} - 3 \, a^{5} b^{4} + a b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} f^{2}}}\right)} \log\left(-\frac{1}{12} \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} - 2 \, a^{2} b^{2} - 7 \, b^{4} - \frac{1}{6} \, {\left(a^{5} + 16 \, a^{3} b^{2} + 10 \, a b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f + \frac{1}{4} \, \sqrt{\frac{1}{3}} {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f^{2} - 2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)} \sqrt{\frac{288 \, a^{4} b^{4} + 720 \, a^{2} b^{6} - 36 \, b^{8} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)}^{2} f^{2} - 12 \, {\left(2 \, a^{7} b^{2} - 3 \, a^{5} b^{4} + a b^{8}\right)} {\left(\frac{{\left(\frac{b^{4}}{a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{2}}{{\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}}} - 9 \, {\left(-\frac{b^{4}}{54 \, {\left(a^{7} f^{3} - 2 \, a^{5} b^{2} f^{3} + a^{3} b^{4} f^{3}\right)}} + \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)} b^{4}}{18 \, {\left(a^{6} f^{2} - 2 \, a^{4} b^{2} f^{2} + a^{2} b^{4} f^{2}\right)} {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}} - \frac{{\left(2 \, a^{2} b^{2} + b^{4}\right)}^{3}}{27 \, {\left(a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f\right)}^{3}} + \frac{{\left(a^{2} + 8 \, b^{2}\right)} b^{4}}{54 \, {\left(a^{2} - b^{2}\right)}^{4} a f^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} - \frac{6 \, {\left(2 \, a^{2} b^{2} + b^{4}\right)}}{a^{5} f - 2 \, a^{3} b^{2} f + a b^{4} f}\right)} f}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} f^{2}}} - 2 \, {\left(a^{3} b + 8 \, a b^{3}\right)} \cos\left(f x + e\right)\right) + 9 \, {\left(2 \, a^{4} - a^{3} b - 8 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(2 \, a^{4} - a^{3} b - 8 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 9 \, {\left(2 \, a^{4} + a^{3} b - 8 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(2 \, a^{4} + a^{3} b - 8 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{36 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)}}"," ",0,"1/36*(18*a^4 - 18*a^2*b^2 + 2*((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*log(1/12*(a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 + 2*a^2*b^2 + 7*b^4 + 1/6*(a^5 + 16*a^3*b^2 + 10*a*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f - (a^3*b + 8*a*b^3)*cos(f*x + e)) - 18*(a^3*b - a*b^3)*cos(f*x + e) + (36*a^2*b^2 + 18*b^4 - 18*(2*a^2*b^2 + b^4)*cos(f*x + e)^2 - ((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f)) + 3*sqrt(1/3)*((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^5*b^4 + a*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*f^2)))*log(1/12*(a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 + 2*a^2*b^2 + 7*b^4 + 1/6*(a^5 + 16*a^3*b^2 + 10*a*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f + 1/4*sqrt(1/3)*((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f^2 - 2*(a^5 - 2*a^3*b^2 + a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^5*b^4 + a*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*f^2)) + 2*(a^3*b + 8*a*b^3)*cos(f*x + e)) + (36*a^2*b^2 + 18*b^4 - 18*(2*a^2*b^2 + b^4)*cos(f*x + e)^2 - ((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 3*sqrt(1/3)*((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^5*b^4 + a*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*f^2)))*log(-1/12*(a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 2*a^2*b^2 - 7*b^4 - 1/6*(a^5 + 16*a^3*b^2 + 10*a*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f + 1/4*sqrt(1/3)*((a^6 - 2*a^4*b^2 + a^2*b^4)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f^2 - 2*(a^5 - 2*a^3*b^2 + a*b^4)*f)*sqrt((288*a^4*b^4 + 720*a^2*b^6 - 36*b^8 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))^2*f^2 - 12*(2*a^7*b^2 - 3*a^5*b^4 + a*b^8)*((b^4/(a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2) - (2*a^2*b^2 + b^4)^2/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^2)*(-I*sqrt(3) + 1)/(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3) - 9*(-1/54*b^4/(a^7*f^3 - 2*a^5*b^2*f^3 + a^3*b^4*f^3) + 1/18*(2*a^2*b^2 + b^4)*b^4/((a^6*f^2 - 2*a^4*b^2*f^2 + a^2*b^4*f^2)*(a^5*f - 2*a^3*b^2*f + a*b^4*f)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^5*f - 2*a^3*b^2*f + a*b^4*f)^3 + 1/54*(a^2 + 8*b^2)*b^4/((a^2 - b^2)^4*a*f^3))^(1/3)*(I*sqrt(3) + 1) - 6*(2*a^2*b^2 + b^4)/(a^5*f - 2*a^3*b^2*f + a*b^4*f))*f)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*f^2)) - 2*(a^3*b + 8*a*b^3)*cos(f*x + e)) + 9*(2*a^4 - a^3*b - 8*a^2*b^2 - 5*a*b^3 - (2*a^4 - a^3*b - 8*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 9*(2*a^4 + a^3*b - 8*a^2*b^2 + 5*a*b^3 - (2*a^4 + a^3*b - 8*a^2*b^2 + 5*a*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^5 - 2*a^3*b^2 + a*b^4)*f*cos(f*x + e)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*f)","C",0
462,0,0,0,0.579641," ","integrate((a+b*(c*sec(f*x+e))^n)^p*(d*tan(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*(d*tan(f*x + e))^m, x)","F",0
463,0,0,0,0.503902," ","integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^5,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \tan\left(f x + e\right)^{5}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e)^5, x)","F",0
464,0,0,0,0.571096," ","integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \tan\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e)^3, x)","F",0
465,0,0,0,0.595495," ","integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e),x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \tan\left(f x + e\right), x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e), x)","F",0
466,0,0,0,0.508592," ","integrate(cot(f*x+e)*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \cot\left(f x + e\right), x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e), x)","F",0
467,0,0,0,0.514766," ","integrate(cot(f*x+e)^3*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \cot\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^3, x)","F",0
468,0,0,0,0.513908," ","integrate((a+b*(c*sec(f*x+e))^n)^p*tan(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*tan(f*x + e)^2, x)","F",0
469,0,0,0,0.537064," ","integrate((a+b*(c*sec(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p, x)","F",0
470,0,0,0,0.593510," ","integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \sec\left(f x + e\right)\right)^{n} b + a\right)}^{p} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)","F",0
