1,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^5,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
2,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^4,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
3,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
4,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
5,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)*tan(e*x+d),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
6,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
7,-1,0,0,0.000000," ","integrate(cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
8,-1,0,0,0.000000," ","integrate(cot(e*x+d)^2*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
9,-1,0,0,0.000000," ","integrate(cot(e*x+d)^3*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
10,-1,0,0,0.000000," ","integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
11,-1,0,0,0.000000," ","integrate(tan(e*x+d)^4/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
12,-1,0,0,0.000000," ","integrate(tan(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
13,-1,0,0,0.000000," ","integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
14,-1,0,0,0.000000," ","integrate(tan(e*x+d)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
15,-1,0,0,0.000000," ","integrate(1/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
16,-1,0,0,0.000000," ","integrate(cot(e*x+d)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
17,-1,0,0,0.000000," ","integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
18,-1,0,0,0.000000," ","integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
19,-1,0,0,0.000000," ","integrate(tan(e*x+d)^7/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
20,-1,0,0,0.000000," ","integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
21,-1,0,0,0.000000," ","integrate(tan(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
22,-1,0,0,0.000000," ","integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
23,-1,0,0,0.000000," ","integrate(tan(e*x+d)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
24,-1,0,0,0.000000," ","integrate(cot(e*x+d)/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
25,-1,0,0,0.000000," ","integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
26,-1,0,0,0.000000," ","integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
27,1,1405,0,8.732277," ","integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^5,x, algorithm=""fricas"")","\left[\frac{48 \, \sqrt{a - b + c} c^{3} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 3 \, {\left(b^{3} - 8 \, {\left(a - b\right)} c^{2} - 16 \, c^{3} - 2 \, {\left(2 \, a b - b^{2}\right)} c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + 4 \, {\left(8 \, c^{3} \tan\left(e x + d\right)^{4} - 3 \, b^{2} c + 2 \, {\left(4 \, a - 3 \, b\right)} c^{2} + 24 \, c^{3} + 2 \, {\left(b c^{2} - 6 \, c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{192 \, c^{3} e}, \frac{24 \, \sqrt{a - b + c} c^{3} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 3 \, {\left(b^{3} - 8 \, {\left(a - b\right)} c^{2} - 16 \, c^{3} - 2 \, {\left(2 \, a b - b^{2}\right)} c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) + 2 \, {\left(8 \, c^{3} \tan\left(e x + d\right)^{4} - 3 \, b^{2} c + 2 \, {\left(4 \, a - 3 \, b\right)} c^{2} + 24 \, c^{3} + 2 \, {\left(b c^{2} - 6 \, c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{96 \, c^{3} e}, -\frac{96 \, \sqrt{-a + b - c} c^{3} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + 3 \, {\left(b^{3} - 8 \, {\left(a - b\right)} c^{2} - 16 \, c^{3} - 2 \, {\left(2 \, a b - b^{2}\right)} c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) - 4 \, {\left(8 \, c^{3} \tan\left(e x + d\right)^{4} - 3 \, b^{2} c + 2 \, {\left(4 \, a - 3 \, b\right)} c^{2} + 24 \, c^{3} + 2 \, {\left(b c^{2} - 6 \, c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{192 \, c^{3} e}, -\frac{48 \, \sqrt{-a + b - c} c^{3} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + 3 \, {\left(b^{3} - 8 \, {\left(a - b\right)} c^{2} - 16 \, c^{3} - 2 \, {\left(2 \, a b - b^{2}\right)} c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - 2 \, {\left(8 \, c^{3} \tan\left(e x + d\right)^{4} - 3 \, b^{2} c + 2 \, {\left(4 \, a - 3 \, b\right)} c^{2} + 24 \, c^{3} + 2 \, {\left(b c^{2} - 6 \, c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{96 \, c^{3} e}\right]"," ",0,"[1/192*(48*sqrt(a - b + c)*c^3*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + 4*(8*c^3*tan(e*x + d)^4 - 3*b^2*c + 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(c^3*e), 1/96*(24*sqrt(a - b + c)*c^3*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) + 2*(8*c^3*tan(e*x + d)^4 - 3*b^2*c + 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(c^3*e), -1/192*(96*sqrt(-a + b - c)*c^3*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) - 4*(8*c^3*tan(e*x + d)^4 - 3*b^2*c + 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(c^3*e), -1/96*(48*sqrt(-a + b - c)*c^3*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + 3*(b^3 - 8*(a - b)*c^2 - 16*c^3 - 2*(2*a*b - b^2)*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - 2*(8*c^3*tan(e*x + d)^4 - 3*b^2*c + 2*(4*a - 3*b)*c^2 + 24*c^3 + 2*(b*c^2 - 6*c^3)*tan(e*x + d)^2)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/(c^3*e)]","A",0
28,1,1199,0,6.366403," ","integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorithm=""fricas"")","\left[\frac{8 \, \sqrt{a - b + c} c^{2} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - {\left(b^{2} - 4 \, {\left(a - b\right)} c - 8 \, c^{2}\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c^{2} \tan\left(e x + d\right)^{2} + b c - 4 \, c^{2}\right)}}{32 \, c^{2} e}, \frac{4 \, \sqrt{a - b + c} c^{2} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + {\left(b^{2} - 4 \, {\left(a - b\right)} c - 8 \, c^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) + 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c^{2} \tan\left(e x + d\right)^{2} + b c - 4 \, c^{2}\right)}}{16 \, c^{2} e}, \frac{16 \, \sqrt{-a + b - c} c^{2} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - {\left(b^{2} - 4 \, {\left(a - b\right)} c - 8 \, c^{2}\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c^{2} \tan\left(e x + d\right)^{2} + b c - 4 \, c^{2}\right)}}{32 \, c^{2} e}, \frac{8 \, \sqrt{-a + b - c} c^{2} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(b^{2} - 4 \, {\left(a - b\right)} c - 8 \, c^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) + 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c^{2} \tan\left(e x + d\right)^{2} + b c - 4 \, c^{2}\right)}}{16 \, c^{2} e}\right]"," ",0,"[1/32*(8*sqrt(a - b + c)*c^2*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - (b^2 - 4*(a - b)*c - 8*c^2)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c^2*tan(e*x + d)^2 + b*c - 4*c^2))/(c^2*e), 1/16*(4*sqrt(a - b + c)*c^2*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + (b^2 - 4*(a - b)*c - 8*c^2)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) + 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c^2*tan(e*x + d)^2 + b*c - 4*c^2))/(c^2*e), 1/32*(16*sqrt(-a + b - c)*c^2*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - (b^2 - 4*(a - b)*c - 8*c^2)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c^2*tan(e*x + d)^2 + b*c - 4*c^2))/(c^2*e), 1/16*(8*sqrt(-a + b - c)*c^2*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (b^2 - 4*(a - b)*c - 8*c^2)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) + 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c^2*tan(e*x + d)^2 + b*c - 4*c^2))/(c^2*e)]","A",0
29,1,1057,0,3.292196," ","integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d),x, algorithm=""fricas"")","\left[-\frac{{\left(b - 2 \, c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) - 2 \, \sqrt{a - b + c} c \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} c}{8 \, c e}, -\frac{{\left(b - 2 \, c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - \sqrt{a - b + c} c \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} c}{4 \, c e}, -\frac{4 \, \sqrt{-a + b - c} c \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(b - 2 \, c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} c}{8 \, c e}, -\frac{2 \, \sqrt{-a + b - c} c \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(b - 2 \, c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} c}{4 \, c e}\right]"," ",0,"[-1/8*((b - 2*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) - 2*sqrt(a - b + c)*c*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*c)/(c*e), -1/4*((b - 2*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - sqrt(a - b + c)*c*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*c)/(c*e), -1/8*(4*sqrt(-a + b - c)*c*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (b - 2*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*c)/(c*e), -1/4*(2*sqrt(-a + b - c)*c*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (b - 2*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*c)/(c*e)]","A",0
30,1,2097,0,1.953427," ","integrate(cot(e*x+d)*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right)}{4 \, e}, -\frac{2 \, \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-c}}{2 \, c \tan\left(e x + d\right)^{2} + b}\right) - \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right)}{4 \, e}, \frac{2 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a}}{b \tan\left(e x + d\right)^{2} + 2 \, a}\right) + \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right)}{4 \, e}, \frac{2 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a}}{b \tan\left(e x + d\right)^{2} + 2 \, a}\right) - 2 \, \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-c}}{2 \, c \tan\left(e x + d\right)^{2} + b}\right) + \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right)}{4 \, e}, \frac{2 \, \sqrt{-a + b - c} \arctan\left(-\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a + b - c}}{{\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b}\right) + \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right)}{4 \, e}, \frac{2 \, \sqrt{-a + b - c} \arctan\left(-\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a + b - c}}{{\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b}\right) - 2 \, \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-c}}{2 \, c \tan\left(e x + d\right)^{2} + b}\right) + \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right)}{4 \, e}, \frac{2 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a}}{b \tan\left(e x + d\right)^{2} + 2 \, a}\right) + 2 \, \sqrt{-a + b - c} \arctan\left(-\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a + b - c}}{{\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b}\right) + \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right)}{4 \, e}, \frac{\sqrt{-a} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a}}{b \tan\left(e x + d\right)^{2} + 2 \, a}\right) + \sqrt{-a + b - c} \arctan\left(-\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-a + b - c}}{{\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b}\right) - \sqrt{-c} \arctan\left(\frac{2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \sqrt{-c}}{2 \, c \tan\left(e x + d\right)^{2} + b}\right)}{2 \, e}\right]"," ",0,"[1/4*(sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4))/e, -1/4*(2*sqrt(-c)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-c)/(2*c*tan(e*x + d)^2 + b)) - sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4))/e, 1/4*(2*sqrt(-a)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a)/(b*tan(e*x + d)^2 + 2*a)) + sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/e, 1/4*(2*sqrt(-a)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a)/(b*tan(e*x + d)^2 + 2*a)) - 2*sqrt(-c)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-c)/(2*c*tan(e*x + d)^2 + b)) + sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/e, 1/4*(2*sqrt(-a + b - c)*arctan(-2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a + b - c)/((b - 2*c)*tan(e*x + d)^2 + 2*a - b)) + sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4))/e, 1/4*(2*sqrt(-a + b - c)*arctan(-2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a + b - c)/((b - 2*c)*tan(e*x + d)^2 + 2*a - b)) - 2*sqrt(-c)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-c)/(2*c*tan(e*x + d)^2 + b)) + sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4))/e, 1/4*(2*sqrt(-a)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a)/(b*tan(e*x + d)^2 + 2*a)) + 2*sqrt(-a + b - c)*arctan(-2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a + b - c)/((b - 2*c)*tan(e*x + d)^2 + 2*a - b)) + sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c))/e, 1/2*(sqrt(-a)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a)/(b*tan(e*x + d)^2 + 2*a)) + sqrt(-a + b - c)*arctan(-2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-a + b - c)/((b - 2*c)*tan(e*x + d)^2 + 2*a - b)) - sqrt(-c)*arctan(2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*sqrt(-c)/(2*c*tan(e*x + d)^2 + b)))/e]","A",0
31,1,1186,0,5.566291," ","integrate(cot(e*x+d)^3*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a - b + c} a \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) \tan\left(e x + d\right)^{2} - {\left(2 \, a - b\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} a}{8 \, a e \tan\left(e x + d\right)^{2}}, -\frac{4 \, a \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) \tan\left(e x + d\right)^{2} + {\left(2 \, a - b\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} a}{8 \, a e \tan\left(e x + d\right)^{2}}, -\frac{\sqrt{-a} {\left(2 \, a - b\right)} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) \tan\left(e x + d\right)^{2} - \sqrt{a - b + c} a \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) \tan\left(e x + d\right)^{2} + 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} a}{4 \, a e \tan\left(e x + d\right)^{2}}, -\frac{\sqrt{-a} {\left(2 \, a - b\right)} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) \tan\left(e x + d\right)^{2} + 2 \, a \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) \tan\left(e x + d\right)^{2} + 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} a}{4 \, a e \tan\left(e x + d\right)^{2}}\right]"," ",0,"[1/8*(2*sqrt(a - b + c)*a*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1))*tan(e*x + d)^2 - (2*a - b)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*a)/(a*e*tan(e*x + d)^2), -1/8*(4*a*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c))*tan(e*x + d)^2 + (2*a - b)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*a)/(a*e*tan(e*x + d)^2), -1/4*(sqrt(-a)*(2*a - b)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2))*tan(e*x + d)^2 - sqrt(a - b + c)*a*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1))*tan(e*x + d)^2 + 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*a)/(a*e*tan(e*x + d)^2), -1/4*(sqrt(-a)*(2*a - b)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2))*tan(e*x + d)^2 + 2*a*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c))*tan(e*x + d)^2 + 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*a)/(a*e*tan(e*x + d)^2)]","A",0
32,0,0,0,1.671579," ","integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2)*tan(e*x+d)^2,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} \tan\left(e x + d\right)^{2}, x\right)"," ",0,"integral(sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*tan(e*x + d)^2, x)","F",0
33,-1,0,0,0.000000," ","integrate((a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
34,-1,0,0,0.000000," ","integrate(cot(e*x+d)^2*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
35,-1,0,0,0.000000," ","integrate(cot(e*x+d)^4*(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
36,1,1226,0,2.923468," ","integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a - b + c} c^{2} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + {\left(a b - b^{2} + {\left(2 \, a - b\right)} c + 2 \, c^{2}\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(a - b\right)} c + c^{2}\right)}}{8 \, {\left({\left(a - b\right)} c^{2} + c^{3}\right)} e}, \frac{\sqrt{a - b + c} c^{2} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + {\left(a b - b^{2} + {\left(2 \, a - b\right)} c + 2 \, c^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) + 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(a - b\right)} c + c^{2}\right)}}{4 \, {\left({\left(a - b\right)} c^{2} + c^{3}\right)} e}, -\frac{4 \, \sqrt{-a + b - c} c^{2} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - {\left(a b - b^{2} + {\left(2 \, a - b\right)} c + 2 \, c^{2}\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(a - b\right)} c + c^{2}\right)}}{8 \, {\left({\left(a - b\right)} c^{2} + c^{3}\right)} e}, -\frac{2 \, \sqrt{-a + b - c} c^{2} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - {\left(a b - b^{2} + {\left(2 \, a - b\right)} c + 2 \, c^{2}\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(a - b\right)} c + c^{2}\right)}}{4 \, {\left({\left(a - b\right)} c^{2} + c^{3}\right)} e}\right]"," ",0,"[1/8*(2*sqrt(a - b + c)*c^2*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((a - b)*c + c^2))/(((a - b)*c^2 + c^3)*e), 1/4*(sqrt(a - b + c)*c^2*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) + 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((a - b)*c + c^2))/(((a - b)*c^2 + c^3)*e), -1/8*(4*sqrt(-a + b - c)*c^2*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((a - b)*c + c^2))/(((a - b)*c^2 + c^3)*e), -1/4*(2*sqrt(-a + b - c)*c^2*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - (a*b - b^2 + (2*a - b)*c + 2*c^2)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((a - b)*c + c^2))/(((a - b)*c^2 + c^3)*e)]","A",0
37,1,993,0,2.200747," ","integrate(tan(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a - b + c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + \sqrt{a - b + c} c \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right)}{4 \, {\left({\left(a - b\right)} c + c^{2}\right)} e}, -\frac{2 \, {\left(a - b + c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - \sqrt{a - b + c} c \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right)}{4 \, {\left({\left(a - b\right)} c + c^{2}\right)} e}, \frac{2 \, \sqrt{-a + b - c} c \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(a - b + c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right)}{4 \, {\left({\left(a - b\right)} c + c^{2}\right)} e}, \frac{\sqrt{-a + b - c} c \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - {\left(a - b + c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right)}{2 \, {\left({\left(a - b\right)} c + c^{2}\right)} e}\right]"," ",0,"[1/4*((a - b + c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + sqrt(a - b + c)*c*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/(((a - b)*c + c^2)*e), -1/4*(2*(a - b + c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - sqrt(a - b + c)*c*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/(((a - b)*c + c^2)*e), 1/4*(2*sqrt(-a + b - c)*c*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (a - b + c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c))/(((a - b)*c + c^2)*e), 1/2*(sqrt(-a + b - c)*c*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - (a - b + c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)))/(((a - b)*c + c^2)*e)]","A",0
38,1,299,0,0.819956," ","integrate(tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right)}{4 \, \sqrt{a - b + c} e}, -\frac{\sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right)}{2 \, {\left(a - b + c\right)} e}\right]"," ",0,"[1/4*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1))/(sqrt(a - b + c)*e), -1/2*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c))/((a - b + c)*e)]","A",0
39,1,1015,0,1.924011," ","integrate(cot(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a - b + c} a \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + {\left(a - b + c\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right)}{4 \, {\left(a^{2} - a b + a c\right)} e}, \frac{2 \, \sqrt{-a} {\left(a - b + c\right)} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) + \sqrt{a - b + c} a \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right)}{4 \, {\left(a^{2} - a b + a c\right)} e}, \frac{2 \, a \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(a - b + c\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right)}{4 \, {\left(a^{2} - a b + a c\right)} e}, \frac{\sqrt{-a} {\left(a - b + c\right)} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) + a \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right)}{2 \, {\left(a^{2} - a b + a c\right)} e}\right]"," ",0,"[1/4*(sqrt(a - b + c)*a*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + (a - b + c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4))/((a^2 - a*b + a*c)*e), 1/4*(2*sqrt(-a)*(a - b + c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2)) + sqrt(a - b + c)*a*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/((a^2 - a*b + a*c)*e), 1/4*(2*a*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (a - b + c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4))/((a^2 - a*b + a*c)*e), 1/2*(sqrt(-a)*(a - b + c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2)) + a*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)))/((a^2 - a*b + a*c)*e)]","A",0
40,1,1350,0,3.024582," ","integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a - b + c} a^{2} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) \tan\left(e x + d\right)^{2} + {\left(2 \, a^{2} - a b - b^{2} + {\left(2 \, a + b\right)} c\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a^{2} - a b + a c\right)}}{8 \, {\left(a^{3} - a^{2} b + a^{2} c\right)} e \tan\left(e x + d\right)^{2}}, \frac{\sqrt{a - b + c} a^{2} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) \tan\left(e x + d\right)^{2} - {\left(2 \, a^{2} - a b - b^{2} + {\left(2 \, a + b\right)} c\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) \tan\left(e x + d\right)^{2} - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a^{2} - a b + a c\right)}}{4 \, {\left(a^{3} - a^{2} b + a^{2} c\right)} e \tan\left(e x + d\right)^{2}}, -\frac{4 \, a^{2} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) \tan\left(e x + d\right)^{2} - {\left(2 \, a^{2} - a b - b^{2} + {\left(2 \, a + b\right)} c\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a^{2} - a b + a c\right)}}{8 \, {\left(a^{3} - a^{2} b + a^{2} c\right)} e \tan\left(e x + d\right)^{2}}, -\frac{2 \, a^{2} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) \tan\left(e x + d\right)^{2} + {\left(2 \, a^{2} - a b - b^{2} + {\left(2 \, a + b\right)} c\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) \tan\left(e x + d\right)^{2} + 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a^{2} - a b + a c\right)}}{4 \, {\left(a^{3} - a^{2} b + a^{2} c\right)} e \tan\left(e x + d\right)^{2}}\right]"," ",0,"[1/8*(2*sqrt(a - b + c)*a^2*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1))*tan(e*x + d)^2 + (2*a^2 - a*b - b^2 + (2*a + b)*c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a^2 - a*b + a*c))/((a^3 - a^2*b + a^2*c)*e*tan(e*x + d)^2), 1/4*(sqrt(a - b + c)*a^2*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1))*tan(e*x + d)^2 - (2*a^2 - a*b - b^2 + (2*a + b)*c)*sqrt(-a)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2))*tan(e*x + d)^2 - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a^2 - a*b + a*c))/((a^3 - a^2*b + a^2*c)*e*tan(e*x + d)^2), -1/8*(4*a^2*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c))*tan(e*x + d)^2 - (2*a^2 - a*b - b^2 + (2*a + b)*c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a^2 - a*b + a*c))/((a^3 - a^2*b + a^2*c)*e*tan(e*x + d)^2), -1/4*(2*a^2*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c))*tan(e*x + d)^2 + (2*a^2 - a*b - b^2 + (2*a + b)*c)*sqrt(-a)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2))*tan(e*x + d)^2 + 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a^2 - a*b + a*c))/((a^3 - a^2*b + a^2*c)*e*tan(e*x + d)^2)]","A",0
41,-1,0,0,0.000000," ","integrate(tan(e*x+d)^4/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
42,-1,0,0,0.000000," ","integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
43,-1,0,0,0.000000," ","integrate(1/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
44,-1,0,0,0.000000," ","integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
45,1,3773,0,6.310108," ","integrate(tan(e*x+d)^7/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) + {\left(a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left(b^{2} c^{3} - 4 \, a c^{4}\right)} \tan\left(e x + d\right)^{4} + {\left(b^{3} c^{2} - 4 \, a b c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 4 \, {\left(2 \, a^{2} c^{3} + {\left(2 \, a^{3} - a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} - 3 \, a b\right)} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a c^{6} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{5} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{4} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c^{3}\right)} e \tan\left(e x + d\right)^{4} + {\left(4 \, a b c^{5} + {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{4} + 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c^{3} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} c^{2}\right)} e \tan\left(e x + d\right)^{2} + {\left(4 \, a^{2} c^{5} + {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{4} + 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c^{3} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} c^{2}\right)} e\right)}}, \frac{2 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - {\left(a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left(b^{2} c^{3} - 4 \, a c^{4}\right)} \tan\left(e x + d\right)^{4} + {\left(b^{3} c^{2} - 4 \, a b c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + 4 \, {\left(2 \, a^{2} c^{3} + {\left(2 \, a^{3} - a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} - 3 \, a b\right)} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a c^{6} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{5} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{4} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c^{3}\right)} e \tan\left(e x + d\right)^{4} + {\left(4 \, a b c^{5} + {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{4} + 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c^{3} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} c^{2}\right)} e \tan\left(e x + d\right)^{2} + {\left(4 \, a^{2} c^{5} + {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{4} + 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c^{3} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} c^{2}\right)} e\right)}}, -\frac{2 \, {\left(a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left(b^{2} c^{3} - 4 \, a c^{4}\right)} \tan\left(e x + d\right)^{4} + {\left(b^{3} c^{2} - 4 \, a b c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{c} \log\left(8 \, c^{2} \tan\left(e x + d\right)^{4} + 8 \, b c \tan\left(e x + d\right)^{2} + b^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{c} + 4 \, a c\right) - 4 \, {\left(2 \, a^{2} c^{3} + {\left(2 \, a^{3} - a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} - 3 \, a b\right)} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a c^{6} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{5} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{4} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c^{3}\right)} e \tan\left(e x + d\right)^{4} + {\left(4 \, a b c^{5} + {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{4} + 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c^{3} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} c^{2}\right)} e \tan\left(e x + d\right)^{2} + {\left(4 \, a^{2} c^{5} + {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{4} + 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c^{3} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} c^{2}\right)} e\right)}}, -\frac{{\left(a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left(b^{2} c^{3} - 4 \, a c^{4}\right)} \tan\left(e x + d\right)^{4} + {\left(b^{3} c^{2} - 4 \, a b c^{3}\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{-c} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, c \tan\left(e x + d\right)^{2} + b\right)} \sqrt{-c}}{2 \, {\left(c^{2} \tan\left(e x + d\right)^{4} + b c \tan\left(e x + d\right)^{2} + a c\right)}}\right) - 2 \, {\left(2 \, a^{2} c^{3} + {\left(2 \, a^{3} - a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} - 3 \, a b\right)} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{2 \, {\left({\left(4 \, a c^{6} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{5} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{4} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c^{3}\right)} e \tan\left(e x + d\right)^{4} + {\left(4 \, a b c^{5} + {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{4} + 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c^{3} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} c^{2}\right)} e \tan\left(e x + d\right)^{2} + {\left(4 \, a^{2} c^{5} + {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{4} + 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c^{3} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} c^{2}\right)} e\right)}}\right]"," ",0,"[-1/4*((a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) + (a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*tan(e*x + d)^4 + (b^3*c^2 - 4*a*b*c^3)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 4*(2*a^2*c^3 + (2*a^3 - a^2*b - a*b^2)*c^2 - ((2*a^2 - 3*a*b)*c^3 + (2*a^3 - 5*a^2*b + 2*a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a*c^6 + (8*a^2 - 8*a*b - b^2)*c^5 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^4 - (a^2*b^2 - 2*a*b^3 + b^4)*c^3)*e*tan(e*x + d)^4 + (4*a*b*c^5 + (8*a^2*b - 8*a*b^2 - b^3)*c^4 + 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c^3 - (a^2*b^3 - 2*a*b^4 + b^5)*c^2)*e*tan(e*x + d)^2 + (4*a^2*c^5 + (8*a^3 - 8*a^2*b - a*b^2)*c^4 + 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c^3 - (a^3*b^2 - 2*a^2*b^3 + a*b^4)*c^2)*e), 1/4*(2*(a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - (a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*tan(e*x + d)^4 + (b^3*c^2 - 4*a*b*c^3)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + 4*(2*a^2*c^3 + (2*a^3 - a^2*b - a*b^2)*c^2 - ((2*a^2 - 3*a*b)*c^3 + (2*a^3 - 5*a^2*b + 2*a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a*c^6 + (8*a^2 - 8*a*b - b^2)*c^5 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^4 - (a^2*b^2 - 2*a*b^3 + b^4)*c^3)*e*tan(e*x + d)^4 + (4*a*b*c^5 + (8*a^2*b - 8*a*b^2 - b^3)*c^4 + 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c^3 - (a^2*b^3 - 2*a*b^4 + b^5)*c^2)*e*tan(e*x + d)^2 + (4*a^2*c^5 + (8*a^3 - 8*a^2*b - a*b^2)*c^4 + 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c^3 - (a^3*b^2 - 2*a^2*b^3 + a*b^4)*c^2)*e), -1/4*(2*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*tan(e*x + d)^4 + (b^3*c^2 - 4*a*b*c^3)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(c)*log(8*c^2*tan(e*x + d)^4 + 8*b*c*tan(e*x + d)^2 + b^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(c) + 4*a*c) - 4*(2*a^2*c^3 + (2*a^3 - a^2*b - a*b^2)*c^2 - ((2*a^2 - 3*a*b)*c^3 + (2*a^3 - 5*a^2*b + 2*a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a*c^6 + (8*a^2 - 8*a*b - b^2)*c^5 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^4 - (a^2*b^2 - 2*a*b^3 + b^4)*c^3)*e*tan(e*x + d)^4 + (4*a*b*c^5 + (8*a^2*b - 8*a*b^2 - b^3)*c^4 + 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c^3 - (a^2*b^3 - 2*a*b^4 + b^5)*c^2)*e*tan(e*x + d)^2 + (4*a^2*c^5 + (8*a^3 - 8*a^2*b - a*b^2)*c^4 + 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c^3 - (a^3*b^2 - 2*a^2*b^3 + a*b^4)*c^2)*e), -1/2*((a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*tan(e*x + d)^4 + (b^3*c^2 - 4*a*b*c^3)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(-c)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*c*tan(e*x + d)^2 + b)*sqrt(-c)/(c^2*tan(e*x + d)^4 + b*c*tan(e*x + d)^2 + a*c)) - 2*(2*a^2*c^3 + (2*a^3 - a^2*b - a*b^2)*c^2 - ((2*a^2 - 3*a*b)*c^3 + (2*a^3 - 5*a^2*b + 2*a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a*c^6 + (8*a^2 - 8*a*b - b^2)*c^5 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^4 - (a^2*b^2 - 2*a*b^3 + b^4)*c^3)*e*tan(e*x + d)^4 + (4*a*b*c^5 + (8*a^2*b - 8*a*b^2 - b^3)*c^4 + 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c^3 - (a^2*b^3 - 2*a*b^4 + b^5)*c^2)*e*tan(e*x + d)^2 + (4*a^2*c^5 + (8*a^3 - 8*a^2*b - a*b^2)*c^4 + 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c^3 - (a^3*b^2 - 2*a^2*b^3 + a*b^4)*c^2)*e)]","B",0
46,1,1095,0,1.277361," ","integrate(tan(e*x+d)^5/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(b^{2} c - 4 \, a c^{2}\right)} \tan\left(e x + d\right)^{4} + a b^{2} - 4 \, a^{2} c + {\left(b^{3} - 4 \, a b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, a^{3} - 3 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3} + 2 \, a c^{2} + {\left(2 \, a^{2} - a b - b^{2}\right)} c\right)} \tan\left(e x + d\right)^{2} + {\left(2 \, a^{2} - a b\right)} c\right)}}{4 \, {\left({\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} e\right)}}, \frac{{\left({\left(b^{2} c - 4 \, a c^{2}\right)} \tan\left(e x + d\right)^{4} + a b^{2} - 4 \, a^{2} c + {\left(b^{3} - 4 \, a b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(2 \, a^{3} - 3 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3} + 2 \, a c^{2} + {\left(2 \, a^{2} - a b - b^{2}\right)} c\right)} \tan\left(e x + d\right)^{2} + {\left(2 \, a^{2} - a b\right)} c\right)}}{2 \, {\left({\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} e\right)}}\right]"," ",0,"[-1/4*(((b^2*c - 4*a*c^2)*tan(e*x + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*a^3 - 3*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3 + 2*a*c^2 + (2*a^2 - a*b - b^2)*c)*tan(e*x + d)^2 + (2*a^2 - a*b)*c))/((4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(e*x + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(e*x + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e), 1/2*(((b^2*c - 4*a*c^2)*tan(e*x + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(2*a^3 - 3*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3 + 2*a*c^2 + (2*a^2 - a*b - b^2)*c)*tan(e*x + d)^2 + (2*a^2 - a*b)*c))/((4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(e*x + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(e*x + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e)]","B",0
47,1,1077,0,1.151063," ","integrate(tan(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(b^{2} c - 4 \, a c^{2}\right)} \tan\left(e x + d\right)^{4} + a b^{2} - 4 \, a^{2} c + {\left(b^{3} - 4 \, a b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a^{2} b - a b^{2} - 2 \, a c^{2} + {\left({\left(2 \, a - b\right)} c^{2} + {\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(2 \, a^{2} - 3 \, a b\right)} c\right)}}{4 \, {\left({\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} e\right)}}, -\frac{{\left({\left(b^{2} c - 4 \, a c^{2}\right)} \tan\left(e x + d\right)^{4} + a b^{2} - 4 \, a^{2} c + {\left(b^{3} - 4 \, a b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a^{2} b - a b^{2} - 2 \, a c^{2} + {\left({\left(2 \, a - b\right)} c^{2} + {\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(2 \, a^{2} - 3 \, a b\right)} c\right)}}{2 \, {\left({\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} e\right)}}\right]"," ",0,"[-1/4*(((b^2*c - 4*a*c^2)*tan(e*x + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a^2*b - a*b^2 - 2*a*c^2 + ((2*a - b)*c^2 + (2*a^2 - 3*a*b + b^2)*c)*tan(e*x + d)^2 - (2*a^2 - 3*a*b)*c))/((4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(e*x + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(e*x + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e), -1/2*(((b^2*c - 4*a*c^2)*tan(e*x + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a^2*b - a*b^2 - 2*a*c^2 + ((2*a - b)*c^2 + (2*a^2 - 3*a*b + b^2)*c)*tan(e*x + d)^2 - (2*a^2 - 3*a*b)*c))/((4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(e*x + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(e*x + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e)]","B",0
48,1,1099,0,1.063219," ","integrate(tan(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(b^{2} c - 4 \, a c^{2}\right)} \tan\left(e x + d\right)^{4} + a b^{2} - 4 \, a^{2} c + {\left(b^{3} - 4 \, a b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a b^{2} - b^{3} - {\left(2 \, a + b\right)} c^{2} - {\left({\left(2 \, a - 3 \, b\right)} c^{2} + 2 \, c^{3} - {\left(a b - b^{2}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(2 \, a^{2} - a b - 2 \, b^{2}\right)} c\right)}}{4 \, {\left({\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} e\right)}}, \frac{{\left({\left(b^{2} c - 4 \, a c^{2}\right)} \tan\left(e x + d\right)^{4} + a b^{2} - 4 \, a^{2} c + {\left(b^{3} - 4 \, a b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - 2 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(a b^{2} - b^{3} - {\left(2 \, a + b\right)} c^{2} - {\left({\left(2 \, a - 3 \, b\right)} c^{2} + 2 \, c^{3} - {\left(a b - b^{2}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(2 \, a^{2} - a b - 2 \, b^{2}\right)} c\right)}}{2 \, {\left({\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} e\right)}}\right]"," ",0,"[-1/4*(((b^2*c - 4*a*c^2)*tan(e*x + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a*b^2 - b^3 - (2*a + b)*c^2 - ((2*a - 3*b)*c^2 + 2*c^3 - (a*b - b^2)*c)*tan(e*x + d)^2 - (2*a^2 - a*b - 2*b^2)*c))/((4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(e*x + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(e*x + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e), 1/2*(((b^2*c - 4*a*c^2)*tan(e*x + d)^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - 2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(a*b^2 - b^3 - (2*a + b)*c^2 - ((2*a - 3*b)*c^2 + 2*c^3 - (a*b - b^2)*c)*tan(e*x + d)^2 - (2*a^2 - a*b - 2*b^2)*c))/((4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*e*tan(e*x + d)^4 - (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*e*tan(e*x + d)^2 - (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*e)]","B",0
49,1,3951,0,7.466909," ","integrate(cot(e*x+d)/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{3} b^{2} - 4 \, a^{4} c + {\left(a^{2} b^{2} c - 4 \, a^{3} c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a^{2} b^{3} - 4 \, a^{3} b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) - 4 \, {\left(a^{2} b^{3} - a b^{4} + 2 \, a^{2} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} + a b\right)} c^{3} + {\left(2 \, a^{3} - a^{2} b - 2 \, a b^{2}\right)} c^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a^{3} c^{4} + {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{4} b^{3} - 2 \, a^{3} b^{4} + a^{2} b^{5} - 4 \, a^{3} b c^{3} - {\left(8 \, a^{4} b - 8 \, a^{3} b^{2} - a^{2} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{5} b - 4 \, a^{4} b^{2} + a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4} - 4 \, a^{4} c^{3} - {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c\right)} e\right)}}, -\frac{2 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) + {\left(a^{3} b^{2} - 4 \, a^{4} c + {\left(a^{2} b^{2} c - 4 \, a^{3} c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a^{2} b^{3} - 4 \, a^{3} b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 4 \, {\left(a^{2} b^{3} - a b^{4} + 2 \, a^{2} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} + a b\right)} c^{3} + {\left(2 \, a^{3} - a^{2} b - 2 \, a b^{2}\right)} c^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a^{3} c^{4} + {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{4} b^{3} - 2 \, a^{3} b^{4} + a^{2} b^{5} - 4 \, a^{3} b c^{3} - {\left(8 \, a^{4} b - 8 \, a^{3} b^{2} - a^{2} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{5} b - 4 \, a^{4} b^{2} + a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4} - 4 \, a^{4} c^{3} - {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c\right)} e\right)}}, -\frac{2 \, {\left(a^{3} b^{2} - 4 \, a^{4} c + {\left(a^{2} b^{2} c - 4 \, a^{3} c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a^{2} b^{3} - 4 \, a^{3} b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) - 4 \, {\left(a^{2} b^{3} - a b^{4} + 2 \, a^{2} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} + a b\right)} c^{3} + {\left(2 \, a^{3} - a^{2} b - 2 \, a b^{2}\right)} c^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a^{3} c^{4} + {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{4} b^{3} - 2 \, a^{3} b^{4} + a^{2} b^{5} - 4 \, a^{3} b c^{3} - {\left(8 \, a^{4} b - 8 \, a^{3} b^{2} - a^{2} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{5} b - 4 \, a^{4} b^{2} + a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4} - 4 \, a^{4} c^{3} - {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c\right)} e\right)}}, -\frac{{\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - 4 \, a^{2} c^{3} - {\left(4 \, a c^{4} + {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{3} - 4 \, a^{2} b + a b^{2} + b^{3}\right)} c^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{3} - 8 \, a^{2} b - a b^{2}\right)} c^{2} + {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5} - 4 \, a b c^{3} - {\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3} + b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{4} - 4 \, a^{3} b + a^{2} b^{2} + a b^{3}\right)} c\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) + {\left(a^{3} b^{2} - 4 \, a^{4} c + {\left(a^{2} b^{2} c - 4 \, a^{3} c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a^{2} b^{3} - 4 \, a^{3} b c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - 2 \, {\left(a^{2} b^{3} - a b^{4} + 2 \, a^{2} c^{3} + {\left(2 \, a^{3} - 5 \, a^{2} b - a b^{2}\right)} c^{2} - {\left({\left(2 \, a^{2} + a b\right)} c^{3} + {\left(2 \, a^{3} - a^{2} b - 2 \, a b^{2}\right)} c^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} c\right)} \tan\left(e x + d\right)^{2} - {\left(3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{2 \, {\left({\left(4 \, a^{3} c^{4} + {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{4} b^{3} - 2 \, a^{3} b^{4} + a^{2} b^{5} - 4 \, a^{3} b c^{3} - {\left(8 \, a^{4} b - 8 \, a^{3} b^{2} - a^{2} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{5} b - 4 \, a^{4} b^{2} + a^{3} b^{3} + a^{2} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4} - 4 \, a^{4} c^{3} - {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c\right)} e\right)}}\right]"," ",0,"[-1/4*((a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*tan(e*x + d)^4 + (a^2*b^3 - 4*a^3*b*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) + (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4) - 4*(a^2*b^3 - a*b^4 + 2*a^2*c^3 + (2*a^3 - 5*a^2*b - a*b^2)*c^2 - ((2*a^2 + a*b)*c^3 + (2*a^3 - a^2*b - 2*a*b^2)*c^2 - (a^2*b^2 - a*b^3)*c)*tan(e*x + d)^2 - (3*a^3*b - 2*a^2*b^2 - 2*a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^3*c^4 + (8*a^4 - 8*a^3*b - a^2*b^2)*c^3 + 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^4 - (a^4*b^3 - 2*a^3*b^4 + a^2*b^5 - 4*a^3*b*c^3 - (8*a^4*b - 8*a^3*b^2 - a^2*b^3)*c^2 - 2*(2*a^5*b - 4*a^4*b^2 + a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 4*a^4*c^3 - (8*a^5 - 8*a^4*b - a^3*b^2)*c^2 - 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c)*e), -1/4*(2*(a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(-a)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2)) + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*tan(e*x + d)^4 + (a^2*b^3 - 4*a^3*b*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 4*(a^2*b^3 - a*b^4 + 2*a^2*c^3 + (2*a^3 - 5*a^2*b - a*b^2)*c^2 - ((2*a^2 + a*b)*c^3 + (2*a^3 - a^2*b - 2*a*b^2)*c^2 - (a^2*b^2 - a*b^3)*c)*tan(e*x + d)^2 - (3*a^3*b - 2*a^2*b^2 - 2*a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^3*c^4 + (8*a^4 - 8*a^3*b - a^2*b^2)*c^3 + 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^4 - (a^4*b^3 - 2*a^3*b^4 + a^2*b^5 - 4*a^3*b*c^3 - (8*a^4*b - 8*a^3*b^2 - a^2*b^3)*c^2 - 2*(2*a^5*b - 4*a^4*b^2 + a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 4*a^4*c^3 - (8*a^5 - 8*a^4*b - a^3*b^2)*c^2 - 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c)*e), -1/4*(2*(a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*tan(e*x + d)^4 + (a^2*b^3 - 4*a^3*b*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + (a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4) - 4*(a^2*b^3 - a*b^4 + 2*a^2*c^3 + (2*a^3 - 5*a^2*b - a*b^2)*c^2 - ((2*a^2 + a*b)*c^3 + (2*a^3 - a^2*b - 2*a*b^2)*c^2 - (a^2*b^2 - a*b^3)*c)*tan(e*x + d)^2 - (3*a^3*b - 2*a^2*b^2 - 2*a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^3*c^4 + (8*a^4 - 8*a^3*b - a^2*b^2)*c^3 + 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^4 - (a^4*b^3 - 2*a^3*b^4 + a^2*b^5 - 4*a^3*b*c^3 - (8*a^4*b - 8*a^3*b^2 - a^2*b^3)*c^2 - 2*(2*a^5*b - 4*a^4*b^2 + a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 4*a^4*c^3 - (8*a^5 - 8*a^4*b - a^3*b^2)*c^2 - 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c)*e), -1/2*((a^3*b^2 - 2*a^2*b^3 + a*b^4 - 4*a^2*c^3 - (4*a*c^4 + (8*a^2 - 8*a*b - b^2)*c^3 + 2*(2*a^3 - 4*a^2*b + a*b^2 + b^3)*c^2 - (a^2*b^2 - 2*a*b^3 + b^4)*c)*tan(e*x + d)^4 - (8*a^3 - 8*a^2*b - a*b^2)*c^2 + (a^2*b^3 - 2*a*b^4 + b^5 - 4*a*b*c^3 - (8*a^2*b - 8*a*b^2 - b^3)*c^2 - 2*(2*a^3*b - 4*a^2*b^2 + a*b^3 + b^4)*c)*tan(e*x + d)^2 - 2*(2*a^4 - 4*a^3*b + a^2*b^2 + a*b^3)*c)*sqrt(-a)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2)) + (a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*tan(e*x + d)^4 + (a^2*b^3 - 4*a^3*b*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - 2*(a^2*b^3 - a*b^4 + 2*a^2*c^3 + (2*a^3 - 5*a^2*b - a*b^2)*c^2 - ((2*a^2 + a*b)*c^3 + (2*a^3 - a^2*b - 2*a*b^2)*c^2 - (a^2*b^2 - a*b^3)*c)*tan(e*x + d)^2 - (3*a^3*b - 2*a^2*b^2 - 2*a*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^3*c^4 + (8*a^4 - 8*a^3*b - a^2*b^2)*c^3 + 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^4 - (a^4*b^3 - 2*a^3*b^4 + a^2*b^5 - 4*a^3*b*c^3 - (8*a^4*b - 8*a^3*b^2 - a^2*b^3)*c^2 - 2*(2*a^5*b - 4*a^4*b^2 + a^3*b^3 + a^2*b^4)*c)*e*tan(e*x + d)^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4 - 4*a^4*c^3 - (8*a^5 - 8*a^4*b - a^3*b^2)*c^2 - 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c)*e)]","B",0
50,1,5189,0,9.152412," ","integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left({\left(a^{3} b^{2} c - 4 \, a^{4} c^{2}\right)} \tan\left(e x + d\right)^{6} + {\left(a^{3} b^{3} - 4 \, a^{4} b c\right)} \tan\left(e x + d\right)^{4} + {\left(a^{4} b^{2} - 4 \, a^{5} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - {\left({\left(4 \, {\left(2 \, a^{2} + 3 \, a b\right)} c^{4} + {\left(16 \, a^{3} + 8 \, a^{2} b - 26 \, a b^{2} - 3 \, b^{3}\right)} c^{3} + 2 \, {\left(4 \, a^{4} - 2 \, a^{3} b - 10 \, a^{2} b^{2} + 5 \, a b^{3} + 3 \, b^{4}\right)} c^{2} - {\left(2 \, a^{3} b^{2} - a^{2} b^{3} - 4 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{6} - {\left(2 \, a^{3} b^{3} - a^{2} b^{4} - 4 \, a b^{5} + 3 \, b^{6} - 4 \, {\left(2 \, a^{2} b + 3 \, a b^{2}\right)} c^{3} - {\left(16 \, a^{3} b + 8 \, a^{2} b^{2} - 26 \, a b^{3} - 3 \, b^{4}\right)} c^{2} - 2 \, {\left(4 \, a^{4} b - 2 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 4 \, {\left(2 \, a^{3} + 3 \, a^{2} b\right)} c^{3} - {\left(16 \, a^{4} + 8 \, a^{3} b - 26 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(4 \, a^{5} - 2 \, a^{4} b - 10 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4} - 4 \, a^{3} c^{3} - {\left(8 \, a^{2} c^{4} + 3 \, {\left(4 \, a^{3} - 6 \, a^{2} b - a b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{4} - 7 \, a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3}\right)} c^{2} - {\left(a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{2} + {\left(a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 2 \, {\left(2 \, a^{3} + 5 \, a^{2} b\right)} c^{3} - {\left(4 \, a^{4} + 10 \, a^{3} b - 22 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{4} b - 8 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{8 \, {\left({\left(4 \, a^{4} c^{4} + {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{6} - {\left(a^{5} b^{3} - 2 \, a^{4} b^{4} + a^{3} b^{5} - 4 \, a^{4} b c^{3} - {\left(8 \, a^{5} b - 8 \, a^{4} b^{2} - a^{3} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{6} b - 4 \, a^{5} b^{2} + a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{6} b^{2} - 2 \, a^{5} b^{3} + a^{4} b^{4} - 4 \, a^{5} c^{3} - {\left(8 \, a^{6} - 8 \, a^{5} b - a^{4} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{7} - 4 \, a^{6} b + a^{5} b^{2} + a^{4} b^{3}\right)} c\right)} e \tan\left(e x + d\right)^{2}\right)}}, -\frac{{\left({\left(4 \, {\left(2 \, a^{2} + 3 \, a b\right)} c^{4} + {\left(16 \, a^{3} + 8 \, a^{2} b - 26 \, a b^{2} - 3 \, b^{3}\right)} c^{3} + 2 \, {\left(4 \, a^{4} - 2 \, a^{3} b - 10 \, a^{2} b^{2} + 5 \, a b^{3} + 3 \, b^{4}\right)} c^{2} - {\left(2 \, a^{3} b^{2} - a^{2} b^{3} - 4 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{6} - {\left(2 \, a^{3} b^{3} - a^{2} b^{4} - 4 \, a b^{5} + 3 \, b^{6} - 4 \, {\left(2 \, a^{2} b + 3 \, a b^{2}\right)} c^{3} - {\left(16 \, a^{3} b + 8 \, a^{2} b^{2} - 26 \, a b^{3} - 3 \, b^{4}\right)} c^{2} - 2 \, {\left(4 \, a^{4} b - 2 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 4 \, {\left(2 \, a^{3} + 3 \, a^{2} b\right)} c^{3} - {\left(16 \, a^{4} + 8 \, a^{3} b - 26 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(4 \, a^{5} - 2 \, a^{4} b - 10 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) + {\left({\left(a^{3} b^{2} c - 4 \, a^{4} c^{2}\right)} \tan\left(e x + d\right)^{6} + {\left(a^{3} b^{3} - 4 \, a^{4} b c\right)} \tan\left(e x + d\right)^{4} + {\left(a^{4} b^{2} - 4 \, a^{5} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a - b + c} \log\left(\frac{{\left(b^{2} + 4 \, {\left(a - 2 \, b\right)} c + 8 \, c^{2}\right)} \tan\left(e x + d\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2} - 4 \, {\left(a - b\right)} c\right)} \tan\left(e x + d\right)^{2} - 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan\left(e x + d\right)^{4} + 2 \, \tan\left(e x + d\right)^{2} + 1}\right) - 2 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4} - 4 \, a^{3} c^{3} - {\left(8 \, a^{2} c^{4} + 3 \, {\left(4 \, a^{3} - 6 \, a^{2} b - a b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{4} - 7 \, a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3}\right)} c^{2} - {\left(a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{2} + {\left(a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 2 \, {\left(2 \, a^{3} + 5 \, a^{2} b\right)} c^{3} - {\left(4 \, a^{4} + 10 \, a^{3} b - 22 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{4} b - 8 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a^{4} c^{4} + {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{6} - {\left(a^{5} b^{3} - 2 \, a^{4} b^{4} + a^{3} b^{5} - 4 \, a^{4} b c^{3} - {\left(8 \, a^{5} b - 8 \, a^{4} b^{2} - a^{3} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{6} b - 4 \, a^{5} b^{2} + a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{6} b^{2} - 2 \, a^{5} b^{3} + a^{4} b^{4} - 4 \, a^{5} c^{3} - {\left(8 \, a^{6} - 8 \, a^{5} b - a^{4} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{7} - 4 \, a^{6} b + a^{5} b^{2} + a^{4} b^{3}\right)} c\right)} e \tan\left(e x + d\right)^{2}\right)}}, \frac{4 \, {\left({\left(a^{3} b^{2} c - 4 \, a^{4} c^{2}\right)} \tan\left(e x + d\right)^{6} + {\left(a^{3} b^{3} - 4 \, a^{4} b c\right)} \tan\left(e x + d\right)^{4} + {\left(a^{4} b^{2} - 4 \, a^{5} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) + {\left({\left(4 \, {\left(2 \, a^{2} + 3 \, a b\right)} c^{4} + {\left(16 \, a^{3} + 8 \, a^{2} b - 26 \, a b^{2} - 3 \, b^{3}\right)} c^{3} + 2 \, {\left(4 \, a^{4} - 2 \, a^{3} b - 10 \, a^{2} b^{2} + 5 \, a b^{3} + 3 \, b^{4}\right)} c^{2} - {\left(2 \, a^{3} b^{2} - a^{2} b^{3} - 4 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{6} - {\left(2 \, a^{3} b^{3} - a^{2} b^{4} - 4 \, a b^{5} + 3 \, b^{6} - 4 \, {\left(2 \, a^{2} b + 3 \, a b^{2}\right)} c^{3} - {\left(16 \, a^{3} b + 8 \, a^{2} b^{2} - 26 \, a b^{3} - 3 \, b^{4}\right)} c^{2} - 2 \, {\left(4 \, a^{4} b - 2 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 4 \, {\left(2 \, a^{3} + 3 \, a^{2} b\right)} c^{3} - {\left(16 \, a^{4} + 8 \, a^{3} b - 26 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(4 \, a^{5} - 2 \, a^{4} b - 10 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{a} \log\left(\frac{{\left(b^{2} + 4 \, a c\right)} \tan\left(e x + d\right)^{4} + 8 \, a b \tan\left(e x + d\right)^{2} + 4 \, \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{a} + 8 \, a^{2}}{\tan\left(e x + d\right)^{4}}\right) + 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4} - 4 \, a^{3} c^{3} - {\left(8 \, a^{2} c^{4} + 3 \, {\left(4 \, a^{3} - 6 \, a^{2} b - a b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{4} - 7 \, a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3}\right)} c^{2} - {\left(a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{2} + {\left(a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 2 \, {\left(2 \, a^{3} + 5 \, a^{2} b\right)} c^{3} - {\left(4 \, a^{4} + 10 \, a^{3} b - 22 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{4} b - 8 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{8 \, {\left({\left(4 \, a^{4} c^{4} + {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{6} - {\left(a^{5} b^{3} - 2 \, a^{4} b^{4} + a^{3} b^{5} - 4 \, a^{4} b c^{3} - {\left(8 \, a^{5} b - 8 \, a^{4} b^{2} - a^{3} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{6} b - 4 \, a^{5} b^{2} + a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{6} b^{2} - 2 \, a^{5} b^{3} + a^{4} b^{4} - 4 \, a^{5} c^{3} - {\left(8 \, a^{6} - 8 \, a^{5} b - a^{4} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{7} - 4 \, a^{6} b + a^{5} b^{2} + a^{4} b^{3}\right)} c\right)} e \tan\left(e x + d\right)^{2}\right)}}, -\frac{{\left({\left(4 \, {\left(2 \, a^{2} + 3 \, a b\right)} c^{4} + {\left(16 \, a^{3} + 8 \, a^{2} b - 26 \, a b^{2} - 3 \, b^{3}\right)} c^{3} + 2 \, {\left(4 \, a^{4} - 2 \, a^{3} b - 10 \, a^{2} b^{2} + 5 \, a b^{3} + 3 \, b^{4}\right)} c^{2} - {\left(2 \, a^{3} b^{2} - a^{2} b^{3} - 4 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{6} - {\left(2 \, a^{3} b^{3} - a^{2} b^{4} - 4 \, a b^{5} + 3 \, b^{6} - 4 \, {\left(2 \, a^{2} b + 3 \, a b^{2}\right)} c^{3} - {\left(16 \, a^{3} b + 8 \, a^{2} b^{2} - 26 \, a b^{3} - 3 \, b^{4}\right)} c^{2} - 2 \, {\left(4 \, a^{4} b - 2 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} + 3 \, b^{5}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 4 \, {\left(2 \, a^{3} + 3 \, a^{2} b\right)} c^{3} - {\left(16 \, a^{4} + 8 \, a^{3} b - 26 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(4 \, a^{5} - 2 \, a^{4} b - 10 \, a^{3} b^{2} + 5 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left(b \tan\left(e x + d\right)^{2} + 2 \, a\right)} \sqrt{-a}}{2 \, {\left(a c \tan\left(e x + d\right)^{4} + a b \tan\left(e x + d\right)^{2} + a^{2}\right)}}\right) - 2 \, {\left({\left(a^{3} b^{2} c - 4 \, a^{4} c^{2}\right)} \tan\left(e x + d\right)^{6} + {\left(a^{3} b^{3} - 4 \, a^{4} b c\right)} \tan\left(e x + d\right)^{4} + {\left(a^{4} b^{2} - 4 \, a^{5} c\right)} \tan\left(e x + d\right)^{2}\right)} \sqrt{-a + b - c} \arctan\left(-\frac{\sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a} {\left({\left(b - 2 \, c\right)} \tan\left(e x + d\right)^{2} + 2 \, a - b\right)} \sqrt{-a + b - c}}{2 \, {\left({\left({\left(a - b\right)} c + c^{2}\right)} \tan\left(e x + d\right)^{4} + {\left(a b - b^{2} + b c\right)} \tan\left(e x + d\right)^{2} + a^{2} - a b + a c\right)}}\right) - 2 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4} - 4 \, a^{3} c^{3} - {\left(8 \, a^{2} c^{4} + 3 \, {\left(4 \, a^{3} - 6 \, a^{2} b - a b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{4} - 7 \, a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3}\right)} c^{2} - {\left(a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{4} - {\left(8 \, a^{4} - 8 \, a^{3} b - a^{2} b^{2}\right)} c^{2} + {\left(a^{3} b^{3} - 4 \, a^{2} b^{4} + 3 \, a b^{5} - 2 \, {\left(2 \, a^{3} + 5 \, a^{2} b\right)} c^{3} - {\left(4 \, a^{4} + 10 \, a^{3} b - 22 \, a^{2} b^{2} - 3 \, a b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{4} b - 8 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} c\right)} \tan\left(e x + d\right)^{2} - 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + a^{3} b^{2} + a^{2} b^{3}\right)} c\right)} \sqrt{c \tan\left(e x + d\right)^{4} + b \tan\left(e x + d\right)^{2} + a}}{4 \, {\left({\left(4 \, a^{4} c^{4} + {\left(8 \, a^{5} - 8 \, a^{4} b - a^{3} b^{2}\right)} c^{3} + 2 \, {\left(2 \, a^{6} - 4 \, a^{5} b + a^{4} b^{2} + a^{3} b^{3}\right)} c^{2} - {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{6} - {\left(a^{5} b^{3} - 2 \, a^{4} b^{4} + a^{3} b^{5} - 4 \, a^{4} b c^{3} - {\left(8 \, a^{5} b - 8 \, a^{4} b^{2} - a^{3} b^{3}\right)} c^{2} - 2 \, {\left(2 \, a^{6} b - 4 \, a^{5} b^{2} + a^{4} b^{3} + a^{3} b^{4}\right)} c\right)} e \tan\left(e x + d\right)^{4} - {\left(a^{6} b^{2} - 2 \, a^{5} b^{3} + a^{4} b^{4} - 4 \, a^{5} c^{3} - {\left(8 \, a^{6} - 8 \, a^{5} b - a^{4} b^{2}\right)} c^{2} - 2 \, {\left(2 \, a^{7} - 4 \, a^{6} b + a^{5} b^{2} + a^{4} b^{3}\right)} c\right)} e \tan\left(e x + d\right)^{2}\right)}}\right]"," ",0,"[-1/8*(2*((a^3*b^2*c - 4*a^4*c^2)*tan(e*x + d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(e*x + d)^4 + (a^4*b^2 - 4*a^5*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - ((4*(2*a^2 + 3*a*b)*c^4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*b^2 + 5*a*b^3 + 3*b^4)*c^2 - (2*a^3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(e*x + d)^6 - (2*a^3*b^3 - a^2*b^4 - 4*a*b^5 + 3*b^6 - 4*(2*a^2*b + 3*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 + 3*b^5)*c)*tan(e*x + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2*a^3 + 3*a^2*b)*c^3 - (16*a^4 + 8*a^3*b - 26*a^2*b^2 - 3*a*b^3)*c^2 - 2*(4*a^5 - 2*a^4*b - 10*a^3*b^2 + 5*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4) - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 4*a^3*c^3 - (8*a^2*c^4 + 3*(4*a^3 - 6*a^2*b - a*b^2)*c^3 + 2*(2*a^4 - 7*a^3*b + 3*a^2*b^2 + 3*a*b^3)*c^2 - (a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^4 - (8*a^4 - 8*a^3*b - a^2*b^2)*c^2 + (a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 2*(2*a^3 + 5*a^2*b)*c^3 - (4*a^4 + 10*a^3*b - 22*a^2*b^2 - 3*a*b^3)*c^2 - 2*(2*a^4*b - 8*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2 - 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^4*c^4 + (8*a^5 - 8*a^4*b - a^3*b^2)*c^3 + 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^6 - (a^5*b^3 - 2*a^4*b^4 + a^3*b^5 - 4*a^4*b*c^3 - (8*a^5*b - 8*a^4*b^2 - a^3*b^3)*c^2 - 2*(2*a^6*b - 4*a^5*b^2 + a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^4 - (a^6*b^2 - 2*a^5*b^3 + a^4*b^4 - 4*a^5*c^3 - (8*a^6 - 8*a^5*b - a^4*b^2)*c^2 - 2*(2*a^7 - 4*a^6*b + a^5*b^2 + a^4*b^3)*c)*e*tan(e*x + d)^2), -1/4*(((4*(2*a^2 + 3*a*b)*c^4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*b^2 + 5*a*b^3 + 3*b^4)*c^2 - (2*a^3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(e*x + d)^6 - (2*a^3*b^3 - a^2*b^4 - 4*a*b^5 + 3*b^6 - 4*(2*a^2*b + 3*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 + 3*b^5)*c)*tan(e*x + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2*a^3 + 3*a^2*b)*c^3 - (16*a^4 + 8*a^3*b - 26*a^2*b^2 - 3*a*b^3)*c^2 - 2*(4*a^5 - 2*a^4*b - 10*a^3*b^2 + 5*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2)*sqrt(-a)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2)) + ((a^3*b^2*c - 4*a^4*c^2)*tan(e*x + d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(e*x + d)^4 + (a^4*b^2 - 4*a^5*c)*tan(e*x + d)^2)*sqrt(a - b + c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 - 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(a - b + c) + 8*a^2 - 8*a*b + b^2 + 4*a*c)/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)) - 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 4*a^3*c^3 - (8*a^2*c^4 + 3*(4*a^3 - 6*a^2*b - a*b^2)*c^3 + 2*(2*a^4 - 7*a^3*b + 3*a^2*b^2 + 3*a*b^3)*c^2 - (a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^4 - (8*a^4 - 8*a^3*b - a^2*b^2)*c^2 + (a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 2*(2*a^3 + 5*a^2*b)*c^3 - (4*a^4 + 10*a^3*b - 22*a^2*b^2 - 3*a*b^3)*c^2 - 2*(2*a^4*b - 8*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2 - 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^4*c^4 + (8*a^5 - 8*a^4*b - a^3*b^2)*c^3 + 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^6 - (a^5*b^3 - 2*a^4*b^4 + a^3*b^5 - 4*a^4*b*c^3 - (8*a^5*b - 8*a^4*b^2 - a^3*b^3)*c^2 - 2*(2*a^6*b - 4*a^5*b^2 + a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^4 - (a^6*b^2 - 2*a^5*b^3 + a^4*b^4 - 4*a^5*c^3 - (8*a^6 - 8*a^5*b - a^4*b^2)*c^2 - 2*(2*a^7 - 4*a^6*b + a^5*b^2 + a^4*b^3)*c)*e*tan(e*x + d)^2), 1/8*(4*((a^3*b^2*c - 4*a^4*c^2)*tan(e*x + d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(e*x + d)^4 + (a^4*b^2 - 4*a^5*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) + ((4*(2*a^2 + 3*a*b)*c^4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*b^2 + 5*a*b^3 + 3*b^4)*c^2 - (2*a^3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(e*x + d)^6 - (2*a^3*b^3 - a^2*b^4 - 4*a*b^5 + 3*b^6 - 4*(2*a^2*b + 3*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 + 3*b^5)*c)*tan(e*x + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2*a^3 + 3*a^2*b)*c^3 - (16*a^4 + 8*a^3*b - 26*a^2*b^2 - 3*a*b^3)*c^2 - 2*(4*a^5 - 2*a^4*b - 10*a^3*b^2 + 5*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2)*sqrt(a)*log(((b^2 + 4*a*c)*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + 4*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(a) + 8*a^2)/tan(e*x + d)^4) + 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 4*a^3*c^3 - (8*a^2*c^4 + 3*(4*a^3 - 6*a^2*b - a*b^2)*c^3 + 2*(2*a^4 - 7*a^3*b + 3*a^2*b^2 + 3*a*b^3)*c^2 - (a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^4 - (8*a^4 - 8*a^3*b - a^2*b^2)*c^2 + (a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 2*(2*a^3 + 5*a^2*b)*c^3 - (4*a^4 + 10*a^3*b - 22*a^2*b^2 - 3*a*b^3)*c^2 - 2*(2*a^4*b - 8*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2 - 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^4*c^4 + (8*a^5 - 8*a^4*b - a^3*b^2)*c^3 + 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^6 - (a^5*b^3 - 2*a^4*b^4 + a^3*b^5 - 4*a^4*b*c^3 - (8*a^5*b - 8*a^4*b^2 - a^3*b^3)*c^2 - 2*(2*a^6*b - 4*a^5*b^2 + a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^4 - (a^6*b^2 - 2*a^5*b^3 + a^4*b^4 - 4*a^5*c^3 - (8*a^6 - 8*a^5*b - a^4*b^2)*c^2 - 2*(2*a^7 - 4*a^6*b + a^5*b^2 + a^4*b^3)*c)*e*tan(e*x + d)^2), -1/4*(((4*(2*a^2 + 3*a*b)*c^4 + (16*a^3 + 8*a^2*b - 26*a*b^2 - 3*b^3)*c^3 + 2*(4*a^4 - 2*a^3*b - 10*a^2*b^2 + 5*a*b^3 + 3*b^4)*c^2 - (2*a^3*b^2 - a^2*b^3 - 4*a*b^4 + 3*b^5)*c)*tan(e*x + d)^6 - (2*a^3*b^3 - a^2*b^4 - 4*a*b^5 + 3*b^6 - 4*(2*a^2*b + 3*a*b^2)*c^3 - (16*a^3*b + 8*a^2*b^2 - 26*a*b^3 - 3*b^4)*c^2 - 2*(4*a^4*b - 2*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 + 3*b^5)*c)*tan(e*x + d)^4 - (2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 4*(2*a^3 + 3*a^2*b)*c^3 - (16*a^4 + 8*a^3*b - 26*a^2*b^2 - 3*a*b^3)*c^2 - 2*(4*a^5 - 2*a^4*b - 10*a^3*b^2 + 5*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2)*sqrt(-a)*arctan(1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*(b*tan(e*x + d)^2 + 2*a)*sqrt(-a)/(a*c*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a^2)) - 2*((a^3*b^2*c - 4*a^4*c^2)*tan(e*x + d)^6 + (a^3*b^3 - 4*a^4*b*c)*tan(e*x + d)^4 + (a^4*b^2 - 4*a^5*c)*tan(e*x + d)^2)*sqrt(-a + b - c)*arctan(-1/2*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a)*((b - 2*c)*tan(e*x + d)^2 + 2*a - b)*sqrt(-a + b - c)/(((a - b)*c + c^2)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + a^2 - a*b + a*c)) - 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 4*a^3*c^3 - (8*a^2*c^4 + 3*(4*a^3 - 6*a^2*b - a*b^2)*c^3 + 2*(2*a^4 - 7*a^3*b + 3*a^2*b^2 + 3*a*b^3)*c^2 - (a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^4 - (8*a^4 - 8*a^3*b - a^2*b^2)*c^2 + (a^3*b^3 - 4*a^2*b^4 + 3*a*b^5 - 2*(2*a^3 + 5*a^2*b)*c^3 - (4*a^4 + 10*a^3*b - 22*a^2*b^2 - 3*a*b^3)*c^2 - 2*(2*a^4*b - 8*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4)*c)*tan(e*x + d)^2 - 2*(2*a^5 - 4*a^4*b + a^3*b^2 + a^2*b^3)*c)*sqrt(c*tan(e*x + d)^4 + b*tan(e*x + d)^2 + a))/((4*a^4*c^4 + (8*a^5 - 8*a^4*b - a^3*b^2)*c^3 + 2*(2*a^6 - 4*a^5*b + a^4*b^2 + a^3*b^3)*c^2 - (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^6 - (a^5*b^3 - 2*a^4*b^4 + a^3*b^5 - 4*a^4*b*c^3 - (8*a^5*b - 8*a^4*b^2 - a^3*b^3)*c^2 - 2*(2*a^6*b - 4*a^5*b^2 + a^4*b^3 + a^3*b^4)*c)*e*tan(e*x + d)^4 - (a^6*b^2 - 2*a^5*b^3 + a^4*b^4 - 4*a^5*c^3 - (8*a^6 - 8*a^5*b - a^4*b^2)*c^2 - 2*(2*a^7 - 4*a^6*b + a^5*b^2 + a^4*b^3)*c)*e*tan(e*x + d)^2)]","B",0
51,-1,0,0,0.000000," ","integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)^2+c*tan(e*x+d)^4)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
