1,1,74,0,0.741998," ","integrate((b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(b^{2} \tan\left(f x + e\right)^{4} - 2 \, b^{2} \tan\left(f x + e\right)^{2} - 2 \, b^{2} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right) - 3 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2}}}{4 \, f \tan\left(f x + e\right)}"," ",0,"1/4*(b^2*tan(f*x + e)^4 - 2*b^2*tan(f*x + e)^2 - 2*b^2*log(1/(tan(f*x + e)^2 + 1)) - 3*b^2)*sqrt(b*tan(f*x + e)^2)/(f*tan(f*x + e))","A",0
2,1,52,0,0.401706," ","integrate((b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(b \tan\left(f x + e\right)^{2} + b \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right) + b\right)} \sqrt{b \tan\left(f x + e\right)^{2}}}{2 \, f \tan\left(f x + e\right)}"," ",0,"1/2*(b*tan(f*x + e)^2 + b*log(1/(tan(f*x + e)^2 + 1)) + b)*sqrt(b*tan(f*x + e)^2)/(f*tan(f*x + e))","A",0
3,1,38,0,0.414019," ","integrate((b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{b \tan\left(f x + e\right)^{2}} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f \tan\left(f x + e\right)}"," ",0,"-1/2*sqrt(b*tan(f*x + e)^2)*log(1/(tan(f*x + e)^2 + 1))/(f*tan(f*x + e))","A",0
4,1,50,0,0.412559," ","integrate(1/(b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{b \tan\left(f x + e\right)^{2}} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, b f \tan\left(f x + e\right)}"," ",0,"1/2*sqrt(b*tan(f*x + e)^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))/(b*f*tan(f*x + e))","A",0
5,1,69,0,0.410729," ","integrate(1/(b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{b \tan\left(f x + e\right)^{2}} {\left(\log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + \tan\left(f x + e\right)^{2} + 1\right)}}{2 \, b^{2} f \tan\left(f x + e\right)^{3}}"," ",0,"-1/2*sqrt(b*tan(f*x + e)^2)*(log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + tan(f*x + e)^2 + 1)/(b^2*f*tan(f*x + e)^3)","A",0
6,1,82,0,0.860491," ","integrate(1/(b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(2 \, \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + 3 \, \tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} - 1\right)} \sqrt{b \tan\left(f x + e\right)^{2}}}{4 \, b^{3} f \tan\left(f x + e\right)^{5}}"," ",0,"1/4*(2*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + 3*tan(f*x + e)^4 + 2*tan(f*x + e)^2 - 1)*sqrt(b*tan(f*x + e)^2)/(b^3*f*tan(f*x + e)^5)","A",0
7,-1,0,0,0.000000," ","integrate((b*tan(f*x+e)^3)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
8,-1,0,0,0.000000," ","integrate((b*tan(f*x+e)^3)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
9,-1,0,0,0.000000," ","integrate((b*tan(f*x+e)^3)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
10,-1,0,0,0.000000," ","integrate(1/(b*tan(f*x+e)^3)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
11,-1,0,0,0.000000," ","integrate(1/(b*tan(f*x+e)^3)^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
12,-1,0,0,0.000000," ","integrate(1/(b*tan(f*x+e)^3)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
13,1,96,0,0.981500," ","integrate((b*tan(f*x+e)^4)^(5/2),x, algorithm=""fricas"")","\frac{{\left(35 \, b^{2} \tan\left(f x + e\right)^{9} - 45 \, b^{2} \tan\left(f x + e\right)^{7} + 63 \, b^{2} \tan\left(f x + e\right)^{5} - 105 \, b^{2} \tan\left(f x + e\right)^{3} - 315 \, b^{2} f x + 315 \, b^{2} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{4}}}{315 \, f \tan\left(f x + e\right)^{2}}"," ",0,"1/315*(35*b^2*tan(f*x + e)^9 - 45*b^2*tan(f*x + e)^7 + 63*b^2*tan(f*x + e)^5 - 105*b^2*tan(f*x + e)^3 - 315*b^2*f*x + 315*b^2*tan(f*x + e))*sqrt(b*tan(f*x + e)^4)/(f*tan(f*x + e)^2)","A",0
14,1,62,0,0.761155," ","integrate((b*tan(f*x+e)^4)^(3/2),x, algorithm=""fricas"")","\frac{{\left(3 \, b \tan\left(f x + e\right)^{5} - 5 \, b \tan\left(f x + e\right)^{3} - 15 \, b f x + 15 \, b \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{4}}}{15 \, f \tan\left(f x + e\right)^{2}}"," ",0,"1/15*(3*b*tan(f*x + e)^5 - 5*b*tan(f*x + e)^3 - 15*b*f*x + 15*b*tan(f*x + e))*sqrt(b*tan(f*x + e)^4)/(f*tan(f*x + e)^2)","A",0
15,1,37,0,0.850236," ","integrate((b*tan(f*x+e)^4)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{b \tan\left(f x + e\right)^{4}} {\left(f x - \tan\left(f x + e\right)\right)}}{f \tan\left(f x + e\right)^{2}}"," ",0,"-sqrt(b*tan(f*x + e)^4)*(f*x - tan(f*x + e))/(f*tan(f*x + e)^2)","A",0
16,1,39,0,0.845217," ","integrate(1/(b*tan(f*x+e)^4)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{b \tan\left(f x + e\right)^{4}} {\left(f x \tan\left(f x + e\right) + 1\right)}}{b f \tan\left(f x + e\right)^{3}}"," ",0,"-sqrt(b*tan(f*x + e)^4)*(f*x*tan(f*x + e) + 1)/(b*f*tan(f*x + e)^3)","A",0
17,1,62,0,0.743554," ","integrate(1/(b*tan(f*x+e)^4)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(15 \, f x \tan\left(f x + e\right)^{5} + 15 \, \tan\left(f x + e\right)^{4} - 5 \, \tan\left(f x + e\right)^{2} + 3\right)} \sqrt{b \tan\left(f x + e\right)^{4}}}{15 \, b^{2} f \tan\left(f x + e\right)^{7}}"," ",0,"-1/15*(15*f*x*tan(f*x + e)^5 + 15*tan(f*x + e)^4 - 5*tan(f*x + e)^2 + 3)*sqrt(b*tan(f*x + e)^4)/(b^2*f*tan(f*x + e)^7)","A",0
18,1,82,0,0.966150," ","integrate(1/(b*tan(f*x+e)^4)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(315 \, f x \tan\left(f x + e\right)^{9} + 315 \, \tan\left(f x + e\right)^{8} - 105 \, \tan\left(f x + e\right)^{6} + 63 \, \tan\left(f x + e\right)^{4} - 45 \, \tan\left(f x + e\right)^{2} + 35\right)} \sqrt{b \tan\left(f x + e\right)^{4}}}{315 \, b^{3} f \tan\left(f x + e\right)^{11}}"," ",0,"-1/315*(315*f*x*tan(f*x + e)^9 + 315*tan(f*x + e)^8 - 105*tan(f*x + e)^6 + 63*tan(f*x + e)^4 - 45*tan(f*x + e)^2 + 35)*sqrt(b*tan(f*x + e)^4)/(b^3*f*tan(f*x + e)^11)","A",0
19,-2,0,0,0.000000," ","integrate((b*tan(f*x+e)^n)^(5/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
20,-2,0,0,0.000000," ","integrate((b*tan(f*x+e)^n)^(3/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
21,-2,0,0,0.000000," ","integrate((b*tan(f*x+e)^n)^(1/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
22,-2,0,0,0.000000," ","integrate(1/(b*tan(f*x+e)^n)^(1/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
23,-2,0,0,0.000000," ","integrate(1/(b*tan(f*x+e)^n)^(3/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
24,-2,0,0,0.000000," ","integrate(1/(b*tan(f*x+e)^n)^(5/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
25,0,0,0,0.450484," ","integrate((b*tan(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{n}\right)^{p}, x\right)"," ",0,"integral((b*tan(f*x + e)^n)^p, x)","F",0
26,0,0,0,0.694113," ","integrate((b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p, x)","F",0
27,0,0,0,0.440368," ","integrate((b*tan(f*x+e)^3)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{3}\right)^{p}, x\right)"," ",0,"integral((b*tan(f*x + e)^3)^p, x)","F",0
28,0,0,0,0.431066," ","integrate((b*tan(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{4}\right)^{p}, x\right)"," ",0,"integral((b*tan(f*x + e)^4)^p, x)","F",0
29,1,23,0,0.472855," ","integrate((b*tan(f*x+e)^n)^(1/n),x, algorithm=""fricas"")","-\frac{b^{\left(\frac{1}{n}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}"," ",0,"-1/2*b^(1/n)*log(1/(tan(f*x + e)^2 + 1))/f","A",0
30,1,64,0,0.412863," ","integrate(sin(f*x+e)^5*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{3 \, {\left(a - b\right)} \cos\left(f x + e\right)^{6} - 5 \, {\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{4} + 15 \, {\left(a - 3 \, b\right)} \cos\left(f x + e\right)^{2} - 15 \, b}{15 \, f \cos\left(f x + e\right)}"," ",0,"-1/15*(3*(a - b)*cos(f*x + e)^6 - 5*(2*a - 3*b)*cos(f*x + e)^4 + 15*(a - 3*b)*cos(f*x + e)^2 - 15*b)/(f*cos(f*x + e))","A",0
31,1,46,0,0.485109," ","integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b}{3 \, f \cos\left(f x + e\right)}"," ",0,"1/3*((a - b)*cos(f*x + e)^4 - 3*(a - 2*b)*cos(f*x + e)^2 + 3*b)/(f*cos(f*x + e))","A",0
32,1,31,0,0.443395," ","integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - b}{f \cos\left(f x + e\right)}"," ",0,"-((a - b)*cos(f*x + e)^2 - b)/(f*cos(f*x + e))","A",0
33,1,56,0,0.442629," ","integrate(csc(f*x+e)*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{a \cos\left(f x + e\right) \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - a \cos\left(f x + e\right) \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 2 \, b}{2 \, f \cos\left(f x + e\right)}"," ",0,"-1/2*(a*cos(f*x + e)*log(1/2*cos(f*x + e) + 1/2) - a*cos(f*x + e)*log(-1/2*cos(f*x + e) + 1/2) - 2*b)/(f*cos(f*x + e))","B",0
34,1,124,0,0.645409," ","integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{2 \, {\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 4 \, b}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}"," ",0,"1/4*(2*(a + 2*b)*cos(f*x + e)^2 - ((a + 2*b)*cos(f*x + e)^3 - (a + 2*b)*cos(f*x + e))*log(1/2*cos(f*x + e) + 1/2) + ((a + 2*b)*cos(f*x + e)^3 - (a + 2*b)*cos(f*x + e))*log(-1/2*cos(f*x + e) + 1/2) - 4*b)/(f*cos(f*x + e)^3 - f*cos(f*x + e))","B",0
35,1,178,0,0.699928," ","integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{6 \, {\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{4} - 10 \, {\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left({\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{3} + {\left(a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a + 4 \, b\right)} \cos\left(f x + e\right)^{3} + {\left(a + 4 \, b\right)} \cos\left(f x + e\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 16 \, b}{16 \, {\left(f \cos\left(f x + e\right)^{5} - 2 \, f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)\right)}}"," ",0,"1/16*(6*(a + 4*b)*cos(f*x + e)^4 - 10*(a + 4*b)*cos(f*x + e)^2 - 3*((a + 4*b)*cos(f*x + e)^5 - 2*(a + 4*b)*cos(f*x + e)^3 + (a + 4*b)*cos(f*x + e))*log(1/2*cos(f*x + e) + 1/2) + 3*((a + 4*b)*cos(f*x + e)^5 - 2*(a + 4*b)*cos(f*x + e)^3 + (a + 4*b)*cos(f*x + e))*log(-1/2*cos(f*x + e) + 1/2) + 16*b)/(f*cos(f*x + e)^5 - 2*f*cos(f*x + e)^3 + f*cos(f*x + e))","B",0
36,1,90,0,0.595017," ","integrate(sin(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{15 \, {\left(a - 7 \, b\right)} f x \cos\left(f x + e\right) - {\left(8 \, {\left(a - b\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(13 \, a - 19 \, b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(11 \, a - 29 \, b\right)} \cos\left(f x + e\right)^{2} - 48 \, b\right)} \sin\left(f x + e\right)}{48 \, f \cos\left(f x + e\right)}"," ",0,"1/48*(15*(a - 7*b)*f*x*cos(f*x + e) - (8*(a - b)*cos(f*x + e)^6 - 2*(13*a - 19*b)*cos(f*x + e)^4 + 3*(11*a - 29*b)*cos(f*x + e)^2 - 48*b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
37,1,72,0,0.578300," ","integrate(sin(f*x+e)^4*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(a - 5 \, b\right)} f x \cos\left(f x + e\right) + {\left(2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a - 9 \, b\right)} \cos\left(f x + e\right)^{2} + 8 \, b\right)} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}"," ",0,"1/8*(3*(a - 5*b)*f*x*cos(f*x + e) + (2*(a - b)*cos(f*x + e)^4 - (5*a - 9*b)*cos(f*x + e)^2 + 8*b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
38,1,54,0,0.420997," ","integrate(sin(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(a - 3 \, b\right)} f x \cos\left(f x + e\right) - {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, b\right)} \sin\left(f x + e\right)}{2 \, f \cos\left(f x + e\right)}"," ",0,"1/2*((a - 3*b)*f*x*cos(f*x + e) - ((a - b)*cos(f*x + e)^2 - 2*b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
39,1,21,0,0.423733," ","integrate(a+b*tan(f*x+e)^2,x, algorithm=""fricas"")","\frac{{\left(a - b\right)} f x + b \tan\left(f x + e\right)}{f}"," ",0,"((a - b)*f*x + b*tan(f*x + e))/f","A",0
40,1,37,0,0.437095," ","integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a + b\right)} \cos\left(f x + e\right)^{2} - b}{f \cos\left(f x + e\right) \sin\left(f x + e\right)}"," ",0,"-((a + b)*cos(f*x + e)^2 - b)/(f*cos(f*x + e)*sin(f*x + e))","A",0
41,1,66,0,0.408254," ","integrate(csc(f*x+e)^4*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b}{3 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a + 3*b)*cos(f*x + e)^4 - 3*(a + 3*b)*cos(f*x + e)^2 + 3*b)/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e))","A",0
42,1,91,0,0.436362," ","integrate(csc(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{8 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{6} - 20 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{4} + 15 \, {\left(a + 5 \, b\right)} \cos\left(f x + e\right)^{2} - 15 \, b}{15 \, {\left(f \cos\left(f x + e\right)^{5} - 2 \, f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a + 5*b)*cos(f*x + e)^6 - 20*(a + 5*b)*cos(f*x + e)^4 + 15*(a + 5*b)*cos(f*x + e)^2 - 15*b)/((f*cos(f*x + e)^5 - 2*f*cos(f*x + e)^3 + f*cos(f*x + e))*sin(f*x + e))","A",0
43,1,105,0,0.441722," ","integrate(sin(f*x+e)^5*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{8} - 10 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + 15 \, {\left(a^{2} - 6 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 30 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 5 \, b^{2}}{15 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/15*(3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^8 - 10*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^6 + 15*(a^2 - 6*a*b + 6*b^2)*cos(f*x + e)^4 - 30*(a*b - 2*b^2)*cos(f*x + e)^2 - 5*b^2)/(f*cos(f*x + e)^3)","A",0
44,1,80,0,0.483321," ","integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 3 \, {\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(2 \, a b - 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*((a^2 - 2*a*b + b^2)*cos(f*x + e)^6 - 3*(a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^4 + 3*(2*a*b - 3*b^2)*cos(f*x + e)^2 + b^2)/(f*cos(f*x + e)^3)","A",0
45,1,59,0,0.451503," ","integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 6 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - b^{2}}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/3*(3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 6*(a*b - b^2)*cos(f*x + e)^2 - b^2)/(f*cos(f*x + e)^3)","A",0
46,1,87,0,0.564201," ","integrate(csc(f*x+e)*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, a^{2} \cos\left(f x + e\right)^{3} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 3 \, a^{2} \cos\left(f x + e\right)^{3} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 6 \, {\left(2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, b^{2}}{6 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/6*(3*a^2*cos(f*x + e)^3*log(1/2*cos(f*x + e) + 1/2) - 3*a^2*cos(f*x + e)^3*log(-1/2*cos(f*x + e) + 1/2) - 6*(2*a*b - b^2)*cos(f*x + e)^2 - 2*b^2)/(f*cos(f*x + e)^3)","A",0
47,1,168,0,0.438384," ","integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{6 \, {\left(a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{4} - 4 \, {\left(6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, b^{2} - 3 \, {\left({\left(a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{5} - {\left(a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{5} - {\left(a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{12 \, {\left(f \cos\left(f x + e\right)^{5} - f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/12*(6*(a^2 + 4*a*b)*cos(f*x + e)^4 - 4*(6*a*b - b^2)*cos(f*x + e)^2 - 4*b^2 - 3*((a^2 + 4*a*b)*cos(f*x + e)^5 - (a^2 + 4*a*b)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^2 + 4*a*b)*cos(f*x + e)^5 - (a^2 + 4*a*b)*cos(f*x + e)^3)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^5 - f*cos(f*x + e)^3)","B",0
48,1,284,0,0.544620," ","integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{6 \, {\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - 10 \, {\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 16 \, {\left(6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 16 \, b^{2} - 3 \, {\left({\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + {\left(3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{48 \, {\left(f \cos\left(f x + e\right)^{7} - 2 \, f \cos\left(f x + e\right)^{5} + f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/48*(6*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^6 - 10*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^4 + 16*(6*a*b + b^2)*cos(f*x + e)^2 + 16*b^2 - 3*((3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^5 + (3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^5 + (3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^3)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^7 - 2*f*cos(f*x + e)^5 + f*cos(f*x + e)^3)","B",0
49,1,120,0,0.440720," ","integrate(sin(f*x+e)^4*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(3 \, a^{2} - 30 \, a b + 35 \, b^{2}\right)} f x \cos\left(f x + e\right)^{3} + {\left(6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 3 \, {\left(5 \, a^{2} - 18 \, a b + 13 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 16 \, {\left(3 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, b^{2}\right)} \sin\left(f x + e\right)}{24 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/24*(3*(3*a^2 - 30*a*b + 35*b^2)*f*x*cos(f*x + e)^3 + (6*(a^2 - 2*a*b + b^2)*cos(f*x + e)^6 - 3*(5*a^2 - 18*a*b + 13*b^2)*cos(f*x + e)^4 + 16*(3*a*b - 5*b^2)*cos(f*x + e)^2 + 8*b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
50,1,94,0,0.559078," ","integrate(sin(f*x+e)^2*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(a^{2} - 6 \, a b + 5 \, b^{2}\right)} f x \cos\left(f x + e\right)^{3} - {\left(3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(6 \, a b - 7 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, b^{2}\right)} \sin\left(f x + e\right)}{6 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/6*(3*(a^2 - 6*a*b + 5*b^2)*f*x*cos(f*x + e)^3 - (3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 2*(6*a*b - 7*b^2)*cos(f*x + e)^2 - 2*b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
51,1,51,0,0.459097," ","integrate((a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{b^{2} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x + 3 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(b^2*tan(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*x + 3*(2*a*b - b^2)*tan(f*x + e))/f","A",0
52,1,71,0,0.443607," ","integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - b^{2}}{3 \, f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right)}"," ",0,"-1/3*((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a*b - b^2)*cos(f*x + e)^2 - b^2)/(f*cos(f*x + e)^3*sin(f*x + e))","A",0
53,1,92,0,0.514191," ","integrate(csc(f*x+e)^4*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 3 \, {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 6 \, a b \cos\left(f x + e\right)^{2} + b^{2}}{3 \, {\left(f \cos\left(f x + e\right)^{5} - f \cos\left(f x + e\right)^{3}\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a^2 + 6*a*b + b^2)*cos(f*x + e)^6 - 3*(a^2 + 6*a*b + b^2)*cos(f*x + e)^4 + 6*a*b*cos(f*x + e)^2 + b^2)/((f*cos(f*x + e)^5 - f*cos(f*x + e)^3)*sin(f*x + e))","A",0
54,1,137,0,0.526499," ","integrate(csc(f*x+e)^6*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{8 \, {\left(a^{2} + 10 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{8} - 20 \, {\left(a^{2} + 10 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + 15 \, {\left(a^{2} + 10 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 10 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 5 \, b^{2}}{15 \, {\left(f \cos\left(f x + e\right)^{7} - 2 \, f \cos\left(f x + e\right)^{5} + f \cos\left(f x + e\right)^{3}\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a^2 + 10*a*b + 5*b^2)*cos(f*x + e)^8 - 20*(a^2 + 10*a*b + 5*b^2)*cos(f*x + e)^6 + 15*(a^2 + 10*a*b + 5*b^2)*cos(f*x + e)^4 - 10*(3*a*b + b^2)*cos(f*x + e)^2 - 5*b^2)/((f*cos(f*x + e)^7 - 2*f*cos(f*x + e)^5 + f*cos(f*x + e)^3)*sin(f*x + e))","A",0
55,1,294,0,0.475652," ","integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, a^{2} \sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 30 \, a^{2} \cos\left(f x + e\right)}{30 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}, -\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 5 \, {\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, a^{2} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + 15 \, a^{2} \cos\left(f x + e\right)}{15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}\right]"," ",0,"[-1/30*(6*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 10*(2*a^2 - 3*a*b + b^2)*cos(f*x + e)^3 + 15*a^2*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 30*a^2*cos(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f), -1/15*(3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 5*(2*a^2 - 3*a*b + b^2)*cos(f*x + e)^3 + 15*a^2*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + 15*a^2*cos(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f)]","A",0
56,1,206,0,0.547313," ","integrate(sin(f*x+e)^3/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{3} + 3 \, a \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - 6 \, a \cos\left(f x + e\right)}{6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, \frac{{\left(a - b\right)} \cos\left(f x + e\right)^{3} - 3 \, a \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) - 3 \, a \cos\left(f x + e\right)}{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}\right]"," ",0,"[1/6*(2*(a - b)*cos(f*x + e)^3 + 3*a*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 6*a*cos(f*x + e))/((a^2 - 2*a*b + b^2)*f), 1/3*((a - b)*cos(f*x + e)^3 - 3*a*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) - 3*a*cos(f*x + e))/((a^2 - 2*a*b + b^2)*f)]","A",0
57,1,158,0,0.503585," ","integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 2 \, \cos\left(f x + e\right)}{2 \, {\left(a - b\right)} f}, -\frac{\sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + \cos\left(f x + e\right)}{{\left(a - b\right)} f}\right]"," ",0,"[-1/2*(sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 2*cos(f*x + e))/((a - b)*f), -(sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + cos(f*x + e))/((a - b)*f)]","A",0
58,1,184,0,0.535692," ","integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, a f}, -\frac{2 \, \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, a f}\right]"," ",0,"[1/2*(sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - log(1/2*cos(f*x + e) + 1/2) + log(-1/2*cos(f*x + e) + 1/2))/(a*f), -1/2*(2*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + log(1/2*cos(f*x + e) + 1/2) - log(-1/2*cos(f*x + e) + 1/2))/(a*f)]","A",0
59,1,327,0,0.576659," ","integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{-a b + b^{2}} {\left(\cos\left(f x + e\right)^{2} - 1\right)} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-a b + b^{2}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 2 \, a \cos\left(f x + e\right) - {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} - a + 2 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} - a + 2 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}, \frac{4 \, \sqrt{a b - b^{2}} {\left(\cos\left(f x + e\right)^{2} - 1\right)} \arctan\left(\frac{\sqrt{a b - b^{2}} \cos\left(f x + e\right)}{b}\right) + 2 \, a \cos\left(f x + e\right) - {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} - a + 2 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} - a + 2 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}\right]"," ",0,"[1/4*(2*sqrt(-a*b + b^2)*(cos(f*x + e)^2 - 1)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(-a*b + b^2)*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 2*a*cos(f*x + e) - ((a - 2*b)*cos(f*x + e)^2 - a + 2*b)*log(1/2*cos(f*x + e) + 1/2) + ((a - 2*b)*cos(f*x + e)^2 - a + 2*b)*log(-1/2*cos(f*x + e) + 1/2))/(a^2*f*cos(f*x + e)^2 - a^2*f), 1/4*(4*sqrt(a*b - b^2)*(cos(f*x + e)^2 - 1)*arctan(sqrt(a*b - b^2)*cos(f*x + e)/b) + 2*a*cos(f*x + e) - ((a - 2*b)*cos(f*x + e)^2 - a + 2*b)*log(1/2*cos(f*x + e) + 1/2) + ((a - 2*b)*cos(f*x + e)^2 - a + 2*b)*log(-1/2*cos(f*x + e) + 1/2))/(a^2*f*cos(f*x + e)^2 - a^2*f)]","A",0
60,1,630,0,0.573301," ","integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \cos\left(f x + e\right)^{3} - 8 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{2} + a - b\right)} \sqrt{-a b + b^{2}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a b + b^{2}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - 2 \, {\left(5 \, a^{2} - 4 \, a b\right)} \cos\left(f x + e\right) - {\left({\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)}}, \frac{2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \cos\left(f x + e\right)^{3} + 16 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{2} + a - b\right)} \sqrt{a b - b^{2}} \arctan\left(\frac{\sqrt{a b - b^{2}} \cos\left(f x + e\right)}{b}\right) - 2 \, {\left(5 \, a^{2} - 4 \, a b\right)} \cos\left(f x + e\right) - {\left({\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)}}\right]"," ",0,"[1/16*(2*(3*a^2 - 4*a*b)*cos(f*x + e)^3 - 8*((a - b)*cos(f*x + e)^4 - 2*(a - b)*cos(f*x + e)^2 + a - b)*sqrt(-a*b + b^2)*log(((a - b)*cos(f*x + e)^2 - 2*sqrt(-a*b + b^2)*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 2*(5*a^2 - 4*a*b)*cos(f*x + e) - ((3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 - 12*a*b + 8*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 - 12*a*b + 8*b^2)*log(-1/2*cos(f*x + e) + 1/2))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f), 1/16*(2*(3*a^2 - 4*a*b)*cos(f*x + e)^3 + 16*((a - b)*cos(f*x + e)^4 - 2*(a - b)*cos(f*x + e)^2 + a - b)*sqrt(a*b - b^2)*arctan(sqrt(a*b - b^2)*cos(f*x + e)/b) - 2*(5*a^2 - 4*a*b)*cos(f*x + e) - ((3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 - 12*a*b + 8*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^2 + 3*a^2 - 12*a*b + 8*b^2)*log(-1/2*cos(f*x + e) + 1/2))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f)]","B",0
61,1,521,0,0.559624," ","integrate(sin(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{12 \, \sqrt{-a b} a^{2} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) + 3 \, {\left(5 \, a^{3} + 15 \, a^{2} b - 5 \, a b^{2} + b^{3}\right)} f x - {\left(8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} - 33 \, a^{2} b + 27 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(11 \, a^{3} - 15 \, a^{2} b + 5 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f}, \frac{24 \, \sqrt{a b} a^{2} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 3 \, {\left(5 \, a^{3} + 15 \, a^{2} b - 5 \, a b^{2} + b^{3}\right)} f x - {\left(8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(13 \, a^{3} - 33 \, a^{2} b + 27 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(11 \, a^{3} - 15 \, a^{2} b + 5 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f}\right]"," ",0,"[1/48*(12*sqrt(-a*b)*a^2*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a + b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) + 3*(5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*f*x - (8*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - 2*(13*a^3 - 33*a^2*b + 27*a*b^2 - 7*b^3)*cos(f*x + e)^3 + 3*(11*a^3 - 15*a^2*b + 5*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f), 1/48*(24*sqrt(a*b)*a^2*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(a*b)/(a*b*cos(f*x + e)*sin(f*x + e))) + 3*(5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*f*x - (8*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - 2*(13*a^3 - 33*a^2*b + 27*a*b^2 - 7*b^3)*cos(f*x + e)^3 + 3*(11*a^3 - 15*a^2*b + 5*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f)]","A",0
62,1,383,0,0.504540," ","integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} f x - 2 \, \sqrt{-a b} a \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}, \frac{{\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} f x + 4 \, \sqrt{a b} a \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}\right]"," ",0,"[1/8*((3*a^2 + 6*a*b - b^2)*f*x - 2*sqrt(-a*b)*a*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a + b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) + (2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 6*a*b + b^2)*cos(f*x + e))*sin(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f), 1/8*((3*a^2 + 6*a*b - b^2)*f*x + 4*sqrt(a*b)*a*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(a*b)/(a*b*cos(f*x + e)*sin(f*x + e))) + (2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 6*a*b + b^2)*cos(f*x + e))*sin(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f)]","A",0
63,1,274,0,0.475005," ","integrate(sin(f*x+e)^2/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a + b\right)} f x - 2 \, {\left(a - b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right)}{4 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, \frac{{\left(a + b\right)} f x - {\left(a - b\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}\right]"," ",0,"[1/4*(2*(a + b)*f*x - 2*(a - b)*cos(f*x + e)*sin(f*x + e) + sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a + b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)))/((a^2 - 2*a*b + b^2)*f), 1/2*((a + b)*f*x - (a - b)*cos(f*x + e)*sin(f*x + e) + sqrt(a*b)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(a*b)/(a*b*cos(f*x + e)*sin(f*x + e))))/((a^2 - 2*a*b + b^2)*f)]","A",0
64,1,182,0,0.539199," ","integrate(1/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, f x - \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{4 \, {\left(a - b\right)} f}, \frac{2 \, f x - \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{2 \, {\left(a - b\right)} f}\right]"," ",0,"[1/4*(4*f*x - sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a - b)*f), 1/2*(2*f*x - sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a - b)*f)]","A",0
65,1,257,0,0.470855," ","integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 4 \, \cos\left(f x + e\right)}{4 \, a f \sin\left(f x + e\right)}, \frac{\sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)}{2 \, a f \sin\left(f x + e\right)}\right]"," ",0,"[1/4*(sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 4*cos(f*x + e))/(a*f*sin(f*x + e)), 1/2*(sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 2*cos(f*x + e))/(a*f*sin(f*x + e))]","B",0
66,1,373,0,0.473299," ","integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, {\left(a - b\right)} \cos\left(f x + e\right)}{12 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, {\left(a - b\right)} \cos\left(f x + e\right)}{6 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/12*(4*(2*a - 3*b)*cos(f*x + e)^3 + 3*((a - b)*cos(f*x + e)^2 - a + b)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(a - b)*cos(f*x + e))/((a^2*f*cos(f*x + e)^2 - a^2*f)*sin(f*x + e)), -1/6*(2*(2*a - 3*b)*cos(f*x + e)^3 - 3*((a - b)*cos(f*x + e)^2 - a + b)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*(a - b)*cos(f*x + e))/((a^2*f*cos(f*x + e)^2 - a^2*f)*sin(f*x + e))]","B",0
67,1,543,0,0.464294," ","integrate(csc(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{2} - 25 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 20 \, {\left(4 \, a^{2} - 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)}{60 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(8 \, a^{2} - 25 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(4 \, a^{2} - 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} - 2 \, a b + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)}{30 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/60*(4*(8*a^2 - 25*a*b + 15*b^2)*cos(f*x + e)^5 - 20*(4*a^2 - 11*a*b + 6*b^2)*cos(f*x + e)^3 - 15*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a^2 - 2*a*b + b^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(a^2 - 2*a*b + b^2)*cos(f*x + e))/((a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f)*sin(f*x + e)), -1/30*(2*(8*a^2 - 25*a*b + 15*b^2)*cos(f*x + e)^5 - 10*(4*a^2 - 11*a*b + 6*b^2)*cos(f*x + e)^3 - 15*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a^2 - 2*a*b + b^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(a^2 - 2*a*b + b^2)*cos(f*x + e))/((a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f)*sin(f*x + e))]","B",0
68,1,593,0,0.559559," ","integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{12 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(10 \, a^{3} - 23 \, a^{2} b + 16 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(3 \, a^{2} b + 4 \, a b^{2} + {\left(3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 30 \, {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)}{60 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}, -\frac{6 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(10 \, a^{3} - 23 \, a^{2} b + 16 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left(3 \, a^{2} b + 4 \, a b^{2} + {\left(3 \, a^{3} + a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + 15 \, {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)}{30 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[-1/60*(12*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 4*(10*a^3 - 23*a^2*b + 16*a*b^2 - 3*b^3)*cos(f*x + e)^5 + 20*(3*a^3 + a^2*b - 4*a*b^2)*cos(f*x + e)^3 - 15*(3*a^2*b + 4*a*b^2 + (3*a^3 + a^2*b - 4*a*b^2)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 30*(3*a^2*b + 4*a*b^2)*cos(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f), -1/30*(6*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 2*(10*a^3 - 23*a^2*b + 16*a*b^2 - 3*b^3)*cos(f*x + e)^5 + 10*(3*a^3 + a^2*b - 4*a*b^2)*cos(f*x + e)^3 + 15*(3*a^2*b + 4*a*b^2 + (3*a^3 + a^2*b - 4*a*b^2)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + 15*(3*a^2*b + 4*a*b^2)*cos(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f)]","A",0
69,1,456,0,0.567043," ","integrate(sin(f*x+e)^3/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(3 \, a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(3 \, a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - 6 \, {\left(3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)}{12 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}, \frac{2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(3 \, a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(3 \, a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + 2 \, b^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) - 3 \, {\left(3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)}{6 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}\right]"," ",0,"[1/12*(4*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 4*(3*a^2 - a*b - 2*b^2)*cos(f*x + e)^3 - 3*((3*a^2 - a*b - 2*b^2)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 6*(3*a*b + 2*b^2)*cos(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f), 1/6*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 2*(3*a^2 - a*b - 2*b^2)*cos(f*x + e)^3 - 3*((3*a^2 - a*b - 2*b^2)*cos(f*x + e)^2 + 3*a*b + 2*b^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) - 3*(3*a*b + 2*b^2)*cos(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f)]","A",0
70,1,307,0,0.478019," ","integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a - b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 6 \, b \cos\left(f x + e\right)}{4 \, {\left({\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f\right)}}, -\frac{2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + 3 \, b \cos\left(f x + e\right)}{2 \, {\left({\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f\right)}}\right]"," ",0,"[-1/4*(4*(a - b)*cos(f*x + e)^3 - 3*((a - b)*cos(f*x + e)^2 + b)*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 6*b*cos(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cos(f*x + e)^2 + (a^2*b - 2*a*b^2 + b^3)*f), -1/2*(2*(a - b)*cos(f*x + e)^3 + 3*((a - b)*cos(f*x + e)^2 + b)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + 3*b*cos(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cos(f*x + e)^2 + (a^2*b - 2*a*b^2 + b^3)*f)]","A",0
71,1,470,0,0.619453," ","integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{2 \, a b \cos\left(f x + e\right) - {\left({\left(3 \, a^{2} - 5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a b - 2 \, b^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 2 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - a^{2} b^{2}\right)} f\right)}}, -\frac{a b \cos\left(f x + e\right) + {\left({\left(3 \, a^{2} - 5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, {\left({\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/4*(2*a*b*cos(f*x + e) - ((3*a^2 - 5*a*b + 2*b^2)*cos(f*x + e)^2 + 3*a*b - 2*b^2)*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 2*((a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a*b - b^2)*log(1/2*cos(f*x + e) + 1/2) - 2*((a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a*b - b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 - 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b - a^2*b^2)*f), -1/2*(a*b*cos(f*x + e) + ((3*a^2 - 5*a*b + 2*b^2)*cos(f*x + e)^2 + 3*a*b - 2*b^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + ((a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a*b - b^2)*log(1/2*cos(f*x + e) + 1/2) - ((a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a*b - b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 - 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b - a^2*b^2)*f)]","B",0
72,1,672,0,0.560509," ","integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{3} + 4 \, a b \cos\left(f x + e\right) - {\left({\left(3 \, a^{2} - 7 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 10 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + 4 \, b^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - {\left({\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 6 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 4 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 6 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 4 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{2 \, {\left(a^{2} - 2 \, a b\right)} \cos\left(f x + e\right)^{3} + 4 \, a b \cos\left(f x + e\right) - 2 \, {\left({\left(3 \, a^{2} - 7 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 10 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + 4 \, b^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) - {\left({\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 6 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 4 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 6 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 4 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/4*(2*(a^2 - 2*a*b)*cos(f*x + e)^3 + 4*a*b*cos(f*x + e) - ((3*a^2 - 7*a*b + 4*b^2)*cos(f*x + e)^4 - (3*a^2 - 10*a*b + 8*b^2)*cos(f*x + e)^2 - 3*a*b + 4*b^2)*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - ((a^2 - 5*a*b + 4*b^2)*cos(f*x + e)^4 - (a^2 - 6*a*b + 8*b^2)*cos(f*x + e)^2 - a*b + 4*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((a^2 - 5*a*b + 4*b^2)*cos(f*x + e)^4 - (a^2 - 6*a*b + 8*b^2)*cos(f*x + e)^2 - a*b + 4*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2), 1/4*(2*(a^2 - 2*a*b)*cos(f*x + e)^3 + 4*a*b*cos(f*x + e) - 2*((3*a^2 - 7*a*b + 4*b^2)*cos(f*x + e)^4 - (3*a^2 - 10*a*b + 8*b^2)*cos(f*x + e)^2 - 3*a*b + 4*b^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) - ((a^2 - 5*a*b + 4*b^2)*cos(f*x + e)^4 - (a^2 - 6*a*b + 8*b^2)*cos(f*x + e)^2 - a*b + 4*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((a^2 - 5*a*b + 4*b^2)*cos(f*x + e)^4 - (a^2 - 6*a*b + 8*b^2)*cos(f*x + e)^2 - a*b + 4*b^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2)]","B",0
73,1,1052,0,0.592912," ","integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{3} - 5 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(5 \, a^{3} - 24 \, a^{2} b + 24 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, {\left({\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{2} - 7 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{2} - 5 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - 2 \, b^{2}\right)} \sqrt{-a b + b^{2}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a b + b^{2}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - 6 \, {\left(3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{3} - 9 \, a^{2} b + 16 \, a b^{2} - 8 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 19 \, a^{2} b + 40 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 8 \, a b^{2} + 8 \, b^{3} + {\left(a^{3} - 11 \, a^{2} b + 32 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{3} - 9 \, a^{2} b + 16 \, a b^{2} - 8 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 19 \, a^{2} b + 40 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 8 \, a b^{2} + 8 \, b^{3} + {\left(a^{3} - 11 \, a^{2} b + 32 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{6 \, {\left(a^{3} - 5 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(5 \, a^{3} - 24 \, a^{2} b + 24 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 24 \, {\left({\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{2} - 7 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{2} - 5 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - 2 \, b^{2}\right)} \sqrt{a b - b^{2}} \arctan\left(\frac{\sqrt{a b - b^{2}} \cos\left(f x + e\right)}{b}\right) - 6 \, {\left(3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{3} - 9 \, a^{2} b + 16 \, a b^{2} - 8 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 19 \, a^{2} b + 40 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 8 \, a b^{2} + 8 \, b^{3} + {\left(a^{3} - 11 \, a^{2} b + 32 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{3} - 9 \, a^{2} b + 16 \, a b^{2} - 8 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 19 \, a^{2} b + 40 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 8 \, a b^{2} + 8 \, b^{3} + {\left(a^{3} - 11 \, a^{2} b + 32 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/16*(6*(a^3 - 5*a^2*b + 4*a*b^2)*cos(f*x + e)^5 - 2*(5*a^3 - 24*a^2*b + 24*a*b^2)*cos(f*x + e)^3 - 12*((a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^6 - (2*a^2 - 7*a*b + 6*b^2)*cos(f*x + e)^4 + (a^2 - 5*a*b + 6*b^2)*cos(f*x + e)^2 + a*b - 2*b^2)*sqrt(-a*b + b^2)*log(((a - b)*cos(f*x + e)^2 - 2*sqrt(-a*b + b^2)*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 6*(3*a^2*b - 4*a*b^2)*cos(f*x + e) - 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - (2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8*b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - (2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8*b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2), 1/16*(6*(a^3 - 5*a^2*b + 4*a*b^2)*cos(f*x + e)^5 - 2*(5*a^3 - 24*a^2*b + 24*a*b^2)*cos(f*x + e)^3 + 24*((a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^6 - (2*a^2 - 7*a*b + 6*b^2)*cos(f*x + e)^4 + (a^2 - 5*a*b + 6*b^2)*cos(f*x + e)^2 + a*b - 2*b^2)*sqrt(a*b - b^2)*arctan(sqrt(a*b - b^2)*cos(f*x + e)/b) - 6*(3*a^2*b - 4*a*b^2)*cos(f*x + e) - 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - (2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8*b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 - 9*a^2*b + 16*a*b^2 - 8*b^3)*cos(f*x + e)^6 - (2*a^3 - 19*a^2*b + 40*a*b^2 - 24*b^3)*cos(f*x + e)^4 + a^2*b - 8*a*b^2 + 8*b^3 + (a^3 - 11*a^2*b + 32*a*b^2 - 24*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2)]","B",0
74,1,705,0,0.639482," ","integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} + 5 \, a^{2} b - 5 \, a b^{2} - b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(a^{2} b + 6 \, a b^{2} + b^{3}\right)} f x + 3 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{3} - b \cos\left(f x + e\right)\right)} \sqrt{-a b} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) + {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} - 9 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}, \frac{3 \, {\left(a^{3} + 5 \, a^{2} b - 5 \, a b^{2} - b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 3 \, {\left(a^{2} b + 6 \, a b^{2} + b^{3}\right)} f x + 6 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} + a b + b^{2}\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} - 9 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[1/8*(3*(a^3 + 5*a^2*b - 5*a*b^2 - b^3)*f*x*cos(f*x + e)^2 + 3*(a^2*b + 6*a*b^2 + b^3)*f*x + 3*((a^2 - b^2)*cos(f*x + e)^2 + a*b + b^2)*sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a + b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) + (2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - (5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^3 - 3*(3*a^2*b - 2*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f), 1/8*(3*(a^3 + 5*a^2*b - 5*a*b^2 - b^3)*f*x*cos(f*x + e)^2 + 3*(a^2*b + 6*a*b^2 + b^3)*f*x + 6*((a^2 - b^2)*cos(f*x + e)^2 + a*b + b^2)*sqrt(a*b)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(a*b)/(a*b*cos(f*x + e)*sin(f*x + e))) + (2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - (5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e)^3 - 3*(3*a^2*b - 2*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f)]","A",0
75,1,568,0,0.595795," ","integrate(sin(f*x+e)^2/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} + 2 \, a b - 3 \, b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 4 \, {\left(a b + 3 \, b^{2}\right)} f x - {\left({\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}, \frac{2 \, {\left(a^{2} + 2 \, a b - 3 \, b^{2}\right)} f x \cos\left(f x + e\right)^{2} + 2 \, {\left(a b + 3 \, b^{2}\right)} f x + {\left({\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a b + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}\right]"," ",0,"[1/8*(4*(a^2 + 2*a*b - 3*b^2)*f*x*cos(f*x + e)^2 + 4*(a*b + 3*b^2)*f*x - ((3*a^2 - 2*a*b - b^2)*cos(f*x + e)^2 + 3*a*b + b^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) - 4*((a^2 - 2*a*b + b^2)*cos(f*x + e)^3 + 2*(a*b - b^2)*cos(f*x + e))*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f), 1/4*(2*(a^2 + 2*a*b - 3*b^2)*f*x*cos(f*x + e)^2 + 2*(a*b + 3*b^2)*f*x + ((3*a^2 - 2*a*b - b^2)*cos(f*x + e)^2 + 3*a*b + b^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e))) - 2*((a^2 - 2*a*b + b^2)*cos(f*x + e)^3 + 2*(a*b - b^2)*cos(f*x + e))*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f)]","A",0
76,1,390,0,0.497633," ","integrate(1/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, a b f x \tan\left(f x + e\right)^{2} + 8 \, a^{2} f x - {\left({\left(3 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2} - a b\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{8 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{4 \, a b f x \tan\left(f x + e\right)^{2} + 4 \, a^{2} f x - {\left({\left(3 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2} - a b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) - 2 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{4 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(8*a*b*f*x*tan(f*x + e)^2 + 8*a^2*f*x - ((3*a*b - b^2)*tan(f*x + e)^2 + 3*a^2 - a*b)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f), 1/4*(4*a*b*f*x*tan(f*x + e)^2 + 4*a^2*f*x - ((3*a*b - b^2)*tan(f*x + e)^2 + 3*a^2 - a*b)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))) - 2*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f)]","A",0
77,1,373,0,0.517579," ","integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 12 \, b \cos\left(f x + e\right)}{8 \, {\left(a^{2} b f + {\left(a^{3} - a^{2} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 6 \, b \cos\left(f x + e\right)}{4 \, {\left(a^{2} b f + {\left(a^{3} - a^{2} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/8*(4*(2*a - 3*b)*cos(f*x + e)^3 - 3*((a - b)*cos(f*x + e)^2 + b)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 12*b*cos(f*x + e))/((a^2*b*f + (a^3 - a^2*b)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/4*(2*(2*a - 3*b)*cos(f*x + e)^3 - 3*((a - b)*cos(f*x + e)^2 + b)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 6*b*cos(f*x + e))/((a^2*b*f + (a^3 - a^2*b)*f*cos(f*x + e)^2)*sin(f*x + e))]","B",0
78,1,587,0,0.653963," ","integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(4 \, a^{2} - 19 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(3 \, a^{2} - 14 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(3 \, a^{2} - 8 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 11 \, a b + 10 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + 5 \, b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 12 \, {\left(3 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)}{24 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(4 \, a^{2} - 19 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(3 \, a^{2} - 14 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left({\left(3 \, a^{2} - 8 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{2} - 11 \, a b + 10 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, a b + 5 \, b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 6 \, {\left(3 \, a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)}{12 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/24*(4*(4*a^2 - 19*a*b + 15*b^2)*cos(f*x + e)^5 - 8*(3*a^2 - 14*a*b + 15*b^2)*cos(f*x + e)^3 + 3*((3*a^2 - 8*a*b + 5*b^2)*cos(f*x + e)^4 - (3*a^2 - 11*a*b + 10*b^2)*cos(f*x + e)^2 - 3*a*b + 5*b^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 12*(3*a*b - 5*b^2)*cos(f*x + e))/(((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/12*(2*(4*a^2 - 19*a*b + 15*b^2)*cos(f*x + e)^5 - 4*(3*a^2 - 14*a*b + 15*b^2)*cos(f*x + e)^3 - 3*((3*a^2 - 8*a*b + 5*b^2)*cos(f*x + e)^4 - (3*a^2 - 11*a*b + 10*b^2)*cos(f*x + e)^2 - 3*a*b + 5*b^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 6*(3*a*b - 5*b^2)*cos(f*x + e))/(((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2)*sin(f*x + e))]","B",0
79,1,855,0,0.578356," ","integrate(csc(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(40 \, a^{3} - 321 \, a^{2} b + 590 \, a b^{2} - 315 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(6 \, a^{3} - 47 \, a^{2} b + 104 \, a b^{2} - 63 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(6 \, a^{3} - 29 \, a^{2} b + 44 \, a b^{2} - 21 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3} + {\left(3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right)} \cos\left(f x + e\right)}{120 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(40 \, a^{3} - 321 \, a^{2} b + 590 \, a b^{2} - 315 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(6 \, a^{3} - 47 \, a^{2} b + 104 \, a b^{2} - 63 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(6 \, a^{3} - 29 \, a^{2} b + 44 \, a b^{2} - 21 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3} + {\left(3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(3 \, a^{2} b - 10 \, a b^{2} + 7 \, b^{3}\right)} \cos\left(f x + e\right)}{60 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/120*(4*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 4*(40*a^3 - 321*a^2*b + 590*a*b^2 - 315*b^3)*cos(f*x + e)^5 + 20*(6*a^3 - 47*a^2*b + 104*a*b^2 - 63*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 13*a^2*b + 17*a*b^2 - 7*b^3)*cos(f*x + e)^6 - (6*a^3 - 29*a^2*b + 44*a*b^2 - 21*b^3)*cos(f*x + e)^4 + 3*a^2*b - 10*a*b^2 + 7*b^3 + (3*a^3 - 19*a^2*b + 37*a*b^2 - 21*b^3)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(3*a^2*b - 10*a*b^2 + 7*b^3)*cos(f*x + e))/(((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/60*(2*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 2*(40*a^3 - 321*a^2*b + 590*a*b^2 - 315*b^3)*cos(f*x + e)^5 + 10*(6*a^3 - 47*a^2*b + 104*a*b^2 - 63*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 13*a^2*b + 17*a*b^2 - 7*b^3)*cos(f*x + e)^6 - (6*a^3 - 29*a^2*b + 44*a*b^2 - 21*b^3)*cos(f*x + e)^4 + 3*a^2*b - 10*a*b^2 + 7*b^3 + (3*a^3 - 19*a^2*b + 37*a*b^2 - 21*b^3)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(3*a^2*b - 10*a*b^2 + 7*b^3)*cos(f*x + e))/(((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2)*sin(f*x + e))]","B",0
80,1,1018,0,0.775861," ","integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{48 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{9} - 16 \, {\left(10 \, a^{4} - 31 \, a^{3} b + 33 \, a^{2} b^{2} - 13 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{7} + 16 \, {\left(15 \, a^{4} + 10 \, a^{3} b - 57 \, a^{2} b^{2} + 24 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 50 \, {\left(15 \, a^{3} b + 25 \, a^{2} b^{2} - 32 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(15 \, a^{4} + 10 \, a^{3} b - 57 \, a^{2} b^{2} + 24 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 25 \, a^{2} b^{2} - 32 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 30 \, {\left(15 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)}{240 \, {\left({\left(a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right)} f\right)}}, -\frac{24 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{9} - 8 \, {\left(10 \, a^{4} - 31 \, a^{3} b + 33 \, a^{2} b^{2} - 13 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{7} + 8 \, {\left(15 \, a^{4} + 10 \, a^{3} b - 57 \, a^{2} b^{2} + 24 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 25 \, {\left(15 \, a^{3} b + 25 \, a^{2} b^{2} - 32 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(15 \, a^{4} + 10 \, a^{3} b - 57 \, a^{2} b^{2} + 24 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b + 25 \, a^{2} b^{2} - 32 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + 15 \, {\left(15 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)}{120 \, {\left({\left(a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right)} f\right)}}\right]"," ",0,"[-1/240*(48*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^9 - 16*(10*a^4 - 31*a^3*b + 33*a^2*b^2 - 13*a*b^3 + b^4)*cos(f*x + e)^7 + 16*(15*a^4 + 10*a^3*b - 57*a^2*b^2 + 24*a*b^3 + 8*b^4)*cos(f*x + e)^5 + 50*(15*a^3*b + 25*a^2*b^2 - 32*a*b^3 - 8*b^4)*cos(f*x + e)^3 + 15*((15*a^4 + 10*a^3*b - 57*a^2*b^2 + 24*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 + 40*a*b^3 + 8*b^4 + 2*(15*a^3*b + 25*a^2*b^2 - 32*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 30*(15*a^2*b^2 + 40*a*b^3 + 8*b^4)*cos(f*x + e))/((a^7 - 7*a^6*b + 21*a^5*b^2 - 35*a^4*b^3 + 35*a^3*b^4 - 21*a^2*b^5 + 7*a*b^6 - b^7)*f*cos(f*x + e)^4 + 2*(a^6*b - 6*a^5*b^2 + 15*a^4*b^3 - 20*a^3*b^4 + 15*a^2*b^5 - 6*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^5*b^2 - 5*a^4*b^3 + 10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 - b^7)*f), -1/120*(24*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^9 - 8*(10*a^4 - 31*a^3*b + 33*a^2*b^2 - 13*a*b^3 + b^4)*cos(f*x + e)^7 + 8*(15*a^4 + 10*a^3*b - 57*a^2*b^2 + 24*a*b^3 + 8*b^4)*cos(f*x + e)^5 + 25*(15*a^3*b + 25*a^2*b^2 - 32*a*b^3 - 8*b^4)*cos(f*x + e)^3 + 15*((15*a^4 + 10*a^3*b - 57*a^2*b^2 + 24*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 + 40*a*b^3 + 8*b^4 + 2*(15*a^3*b + 25*a^2*b^2 - 32*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + 15*(15*a^2*b^2 + 40*a*b^3 + 8*b^4)*cos(f*x + e))/((a^7 - 7*a^6*b + 21*a^5*b^2 - 35*a^4*b^3 + 35*a^3*b^4 - 21*a^2*b^5 + 7*a*b^6 - b^7)*f*cos(f*x + e)^4 + 2*(a^6*b - 6*a^5*b^2 + 15*a^4*b^3 - 20*a^3*b^4 + 15*a^2*b^5 - 6*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^5*b^2 - 5*a^4*b^3 + 10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 - b^7)*f)]","B",0
81,1,775,0,0.669318," ","integrate(sin(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 16 \, {\left(3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{5} - 50 \, {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, a b^{2} + 4 \, b^{3} + 2 \, {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - 30 \, {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)}{48 \, {\left({\left(a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} f\right)}}, \frac{8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 8 \, {\left(3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{5} - 25 \, {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(3 \, a^{3} - 2 \, a^{2} b - 5 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, a b^{2} + 4 \, b^{3} + 2 \, {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) - 15 \, {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)}{24 \, {\left({\left(a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} f\right)}}\right]"," ",0,"[1/48*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 16*(3*a^3 - 2*a^2*b - 5*a*b^2 + 4*b^3)*cos(f*x + e)^5 - 50*(3*a^2*b + a*b^2 - 4*b^3)*cos(f*x + e)^3 + 15*((3*a^3 - 2*a^2*b - 5*a*b^2 + 4*b^3)*cos(f*x + e)^4 + 3*a*b^2 + 4*b^3 + 2*(3*a^2*b + a*b^2 - 4*b^3)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 30*(3*a*b^2 + 4*b^3)*cos(f*x + e))/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*f*cos(f*x + e)^4 + 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*f*cos(f*x + e)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*f), 1/24*(8*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 8*(3*a^3 - 2*a^2*b - 5*a*b^2 + 4*b^3)*cos(f*x + e)^5 - 25*(3*a^2*b + a*b^2 - 4*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 2*a^2*b - 5*a*b^2 + 4*b^3)*cos(f*x + e)^4 + 3*a*b^2 + 4*b^3 + 2*(3*a^2*b + a*b^2 - 4*b^3)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) - 15*(3*a*b^2 + 4*b^3)*cos(f*x + e))/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*f*cos(f*x + e)^4 + 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*f*cos(f*x + e)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*f)]","B",0
82,1,556,0,0.608409," ","integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{16 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} + 50 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{3} + 30 \, b^{2} \cos\left(f x + e\right) + 15 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right)}{16 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f\right)}}, -\frac{8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} + 25 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{3} + 15 \, b^{2} \cos\left(f x + e\right) + 15 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right)}{8 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f\right)}}\right]"," ",0,"[-1/16*(16*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 + 50*(a*b - b^2)*cos(f*x + e)^3 + 30*b^2*cos(f*x + e) + 15*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^4 + 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f*cos(f*x + e)^2 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f), -1/8*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 + 25*(a*b - b^2)*cos(f*x + e)^3 + 15*b^2*cos(f*x + e) + 15*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^4 + 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f*cos(f*x + e)^2 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f)]","B",0
83,1,1050,0,0.788134," ","integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(9 \, a^{3} b - 13 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(15 \, a^{4} - 50 \, a^{3} b + 63 \, a^{2} b^{2} - 36 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} - 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b - 35 \, a^{2} b^{2} + 28 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(7 \, a^{2} b^{2} - 4 \, a b^{3}\right)} \cos\left(f x + e\right) + 8 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 8 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, -\frac{{\left(9 \, a^{3} b - 13 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left({\left(15 \, a^{4} - 50 \, a^{3} b + 63 \, a^{2} b^{2} - 36 \, a b^{3} + 8 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} - 20 \, a b^{3} + 8 \, b^{4} + 2 \, {\left(15 \, a^{3} b - 35 \, a^{2} b^{2} + 28 \, a b^{3} - 8 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + {\left(7 \, a^{2} b^{2} - 4 \, a b^{3}\right)} \cos\left(f x + e\right) + 4 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - 4 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{8 \, {\left({\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[-1/16*(2*(9*a^3*b - 13*a^2*b^2 + 4*a*b^3)*cos(f*x + e)^3 - ((15*a^4 - 50*a^3*b + 63*a^2*b^2 - 36*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 - 20*a*b^3 + 8*b^4 + 2*(15*a^3*b - 35*a^2*b^2 + 28*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 2*(7*a^2*b^2 - 4*a*b^3)*cos(f*x + e) + 8*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 2*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) - 8*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 2*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f), -1/8*((9*a^3*b - 13*a^2*b^2 + 4*a*b^3)*cos(f*x + e)^3 + ((15*a^4 - 50*a^3*b + 63*a^2*b^2 - 36*a*b^3 + 8*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 - 20*a*b^3 + 8*b^4 + 2*(15*a^3*b - 35*a^2*b^2 + 28*a*b^3 - 8*b^4)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + (7*a^2*b^2 - 4*a*b^3)*cos(f*x + e) + 4*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 2*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) - 4*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 2*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f)]","B",0
84,1,1419,0,0.959968," ","integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(4 \, a^{4} - 21 \, a^{3} b + 29 \, a^{2} b^{2} - 12 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(17 \, a^{3} b - 40 \, a^{2} b^{2} + 24 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(15 \, a^{4} - 70 \, a^{3} b + 119 \, a^{2} b^{2} - 88 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(15 \, a^{4} - 100 \, a^{3} b + 229 \, a^{2} b^{2} - 216 \, a b^{3} + 72 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 15 \, a^{2} b^{2} + 40 \, a b^{3} - 24 \, b^{4} - {\left(30 \, a^{3} b - 125 \, a^{2} b^{2} + 168 \, a b^{3} - 72 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) + 2 \, {\left(11 \, a^{2} b^{2} - 12 \, a b^{3}\right)} \cos\left(f x + e\right) - 4 \, {\left({\left(a^{4} - 9 \, a^{3} b + 21 \, a^{2} b^{2} - 19 \, a b^{3} + 6 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 11 \, a^{3} b + 37 \, a^{2} b^{2} - 45 \, a b^{3} + 18 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 7 \, a b^{3} - 6 \, b^{4} - {\left(2 \, a^{3} b - 17 \, a^{2} b^{2} + 33 \, a b^{3} - 18 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 4 \, {\left({\left(a^{4} - 9 \, a^{3} b + 21 \, a^{2} b^{2} - 19 \, a b^{3} + 6 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 11 \, a^{3} b + 37 \, a^{2} b^{2} - 45 \, a b^{3} + 18 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 7 \, a b^{3} - 6 \, b^{4} - {\left(2 \, a^{3} b - 17 \, a^{2} b^{2} + 33 \, a b^{3} - 18 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} - 5 \, a^{6} b + 7 \, a^{5} b^{2} - 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b - 5 \, a^{5} b^{2} + 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}, \frac{{\left(4 \, a^{4} - 21 \, a^{3} b + 29 \, a^{2} b^{2} - 12 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(17 \, a^{3} b - 40 \, a^{2} b^{2} + 24 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left({\left(15 \, a^{4} - 70 \, a^{3} b + 119 \, a^{2} b^{2} - 88 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(15 \, a^{4} - 100 \, a^{3} b + 229 \, a^{2} b^{2} - 216 \, a b^{3} + 72 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - 15 \, a^{2} b^{2} + 40 \, a b^{3} - 24 \, b^{4} - {\left(30 \, a^{3} b - 125 \, a^{2} b^{2} + 168 \, a b^{3} - 72 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) + {\left(11 \, a^{2} b^{2} - 12 \, a b^{3}\right)} \cos\left(f x + e\right) - 2 \, {\left({\left(a^{4} - 9 \, a^{3} b + 21 \, a^{2} b^{2} - 19 \, a b^{3} + 6 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 11 \, a^{3} b + 37 \, a^{2} b^{2} - 45 \, a b^{3} + 18 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 7 \, a b^{3} - 6 \, b^{4} - {\left(2 \, a^{3} b - 17 \, a^{2} b^{2} + 33 \, a b^{3} - 18 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 2 \, {\left({\left(a^{4} - 9 \, a^{3} b + 21 \, a^{2} b^{2} - 19 \, a b^{3} + 6 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 11 \, a^{3} b + 37 \, a^{2} b^{2} - 45 \, a b^{3} + 18 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 7 \, a b^{3} - 6 \, b^{4} - {\left(2 \, a^{3} b - 17 \, a^{2} b^{2} + 33 \, a b^{3} - 18 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{8 \, {\left({\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} - 5 \, a^{6} b + 7 \, a^{5} b^{2} - 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b - 5 \, a^{5} b^{2} + 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[1/16*(2*(4*a^4 - 21*a^3*b + 29*a^2*b^2 - 12*a*b^3)*cos(f*x + e)^5 + 2*(17*a^3*b - 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^3 - ((15*a^4 - 70*a^3*b + 119*a^2*b^2 - 88*a*b^3 + 24*b^4)*cos(f*x + e)^6 - (15*a^4 - 100*a^3*b + 229*a^2*b^2 - 216*a*b^3 + 72*b^4)*cos(f*x + e)^4 - 15*a^2*b^2 + 40*a*b^3 - 24*b^4 - (30*a^3*b - 125*a^2*b^2 + 168*a*b^3 - 72*b^4)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(-((a - b)*cos(f*x + e)^2 - 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) + 2*(11*a^2*b^2 - 12*a*b^3)*cos(f*x + e) - 4*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 4*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*a^6*b + 7*a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 - (2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)*f*cos(f*x + e)^2 - (a^5*b^2 - a^4*b^3)*f), 1/8*((4*a^4 - 21*a^3*b + 29*a^2*b^2 - 12*a*b^3)*cos(f*x + e)^5 + (17*a^3*b - 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^3 - ((15*a^4 - 70*a^3*b + 119*a^2*b^2 - 88*a*b^3 + 24*b^4)*cos(f*x + e)^6 - (15*a^4 - 100*a^3*b + 229*a^2*b^2 - 216*a*b^3 + 72*b^4)*cos(f*x + e)^4 - 15*a^2*b^2 + 40*a*b^3 - 24*b^4 - (30*a^3*b - 125*a^2*b^2 + 168*a*b^3 - 72*b^4)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) + (11*a^2*b^2 - 12*a*b^3)*cos(f*x + e) - 2*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 2*((a^4 - 9*a^3*b + 21*a^2*b^2 - 19*a*b^3 + 6*b^4)*cos(f*x + e)^6 - (a^4 - 11*a^3*b + 37*a^2*b^2 - 45*a*b^3 + 18*b^4)*cos(f*x + e)^4 - a^2*b^2 + 7*a*b^3 - 6*b^4 - (2*a^3*b - 17*a^2*b^2 + 33*a*b^3 - 18*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*a^6*b + 7*a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 - (2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)*f*cos(f*x + e)^2 - (a^5*b^2 - a^4*b^3)*f)]","B",0
85,1,1693,0,0.887396," ","integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{4} - 9 \, a^{3} b + 16 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(5 \, a^{4} - 46 \, a^{3} b + 108 \, a^{2} b^{2} - 72 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(19 \, a^{3} b - 72 \, a^{2} b^{2} + 72 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left({\left(5 \, a^{4} - 30 \, a^{3} b + 61 \, a^{2} b^{2} - 52 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(5 \, a^{4} - 35 \, a^{3} b + 86 \, a^{2} b^{2} - 88 \, a b^{3} + 32 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(5 \, a^{4} - 50 \, a^{3} b + 166 \, a^{2} b^{2} - 216 \, a b^{3} + 96 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} - 20 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(5 \, a^{3} b - 30 \, a^{2} b^{2} + 56 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a - b}} \log\left(\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-\frac{b}{a - b}} \cos\left(f x + e\right) - b}{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}\right) - 24 \, {\left(a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{4} - 14 \, a^{3} b + 41 \, a^{2} b^{2} - 44 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 15 \, a^{3} b + 54 \, a^{2} b^{2} - 72 \, a b^{3} + 32 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 18 \, a^{3} b + 94 \, a^{2} b^{2} - 168 \, a b^{3} + 96 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 12 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(a^{3} b - 14 \, a^{2} b^{2} + 40 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{4} - 14 \, a^{3} b + 41 \, a^{2} b^{2} - 44 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 15 \, a^{3} b + 54 \, a^{2} b^{2} - 72 \, a b^{3} + 32 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 18 \, a^{3} b + 94 \, a^{2} b^{2} - 168 \, a b^{3} + 96 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 12 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(a^{3} b - 14 \, a^{2} b^{2} + 40 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{8} + a^{5} b^{2} f - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} - 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{6 \, {\left(a^{4} - 9 \, a^{3} b + 16 \, a^{2} b^{2} - 8 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(5 \, a^{4} - 46 \, a^{3} b + 108 \, a^{2} b^{2} - 72 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(19 \, a^{3} b - 72 \, a^{2} b^{2} + 72 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left({\left(5 \, a^{4} - 30 \, a^{3} b + 61 \, a^{2} b^{2} - 52 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(5 \, a^{4} - 35 \, a^{3} b + 86 \, a^{2} b^{2} - 88 \, a b^{3} + 32 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(5 \, a^{4} - 50 \, a^{3} b + 166 \, a^{2} b^{2} - 216 \, a b^{3} + 96 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} - 20 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(5 \, a^{3} b - 30 \, a^{2} b^{2} + 56 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a - b}} \arctan\left(-\frac{{\left(a - b\right)} \sqrt{\frac{b}{a - b}} \cos\left(f x + e\right)}{b}\right) - 24 \, {\left(a^{2} b^{2} - 2 \, a b^{3}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{4} - 14 \, a^{3} b + 41 \, a^{2} b^{2} - 44 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 15 \, a^{3} b + 54 \, a^{2} b^{2} - 72 \, a b^{3} + 32 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 18 \, a^{3} b + 94 \, a^{2} b^{2} - 168 \, a b^{3} + 96 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 12 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(a^{3} b - 14 \, a^{2} b^{2} + 40 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{4} - 14 \, a^{3} b + 41 \, a^{2} b^{2} - 44 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{4} - 15 \, a^{3} b + 54 \, a^{2} b^{2} - 72 \, a b^{3} + 32 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(a^{4} - 18 \, a^{3} b + 94 \, a^{2} b^{2} - 168 \, a b^{3} + 96 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 12 \, a b^{3} + 16 \, b^{4} + 2 \, {\left(a^{3} b - 14 \, a^{2} b^{2} + 40 \, a b^{3} - 32 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{8} + a^{5} b^{2} f - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} - 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/16*(6*(a^4 - 9*a^3*b + 16*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 46*a^3*b + 108*a^2*b^2 - 72*a*b^3)*cos(f*x + e)^5 - 2*(19*a^3*b - 72*a^2*b^2 + 72*a*b^3)*cos(f*x + e)^3 + 3*((5*a^4 - 30*a^3*b + 61*a^2*b^2 - 52*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(5*a^4 - 35*a^3*b + 86*a^2*b^2 - 88*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (5*a^4 - 50*a^3*b + 166*a^2*b^2 - 216*a*b^3 + 96*b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 20*a*b^3 + 16*b^4 + 2*(5*a^3*b - 30*a^2*b^2 + 56*a*b^3 - 32*b^4)*cos(f*x + e)^2)*sqrt(-b/(a - b))*log(((a - b)*cos(f*x + e)^2 + 2*(a - b)*sqrt(-b/(a - b))*cos(f*x + e) - b)/((a - b)*cos(f*x + e)^2 + b)) - 24*(a^2*b^2 - 2*a*b^3)*cos(f*x + e) - 3*((a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2), 1/16*(6*(a^4 - 9*a^3*b + 16*a^2*b^2 - 8*a*b^3)*cos(f*x + e)^7 - 2*(5*a^4 - 46*a^3*b + 108*a^2*b^2 - 72*a*b^3)*cos(f*x + e)^5 - 2*(19*a^3*b - 72*a^2*b^2 + 72*a*b^3)*cos(f*x + e)^3 - 6*((5*a^4 - 30*a^3*b + 61*a^2*b^2 - 52*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(5*a^4 - 35*a^3*b + 86*a^2*b^2 - 88*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (5*a^4 - 50*a^3*b + 166*a^2*b^2 - 216*a*b^3 + 96*b^4)*cos(f*x + e)^4 + 5*a^2*b^2 - 20*a*b^3 + 16*b^4 + 2*(5*a^3*b - 30*a^2*b^2 + 56*a*b^3 - 32*b^4)*cos(f*x + e)^2)*sqrt(b/(a - b))*arctan(-(a - b)*sqrt(b/(a - b))*cos(f*x + e)/b) - 24*(a^2*b^2 - 2*a*b^3)*cos(f*x + e) - 3*((a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^4 - 14*a^3*b + 41*a^2*b^2 - 44*a*b^3 + 16*b^4)*cos(f*x + e)^8 - 2*(a^4 - 15*a^3*b + 54*a^2*b^2 - 72*a*b^3 + 32*b^4)*cos(f*x + e)^6 + (a^4 - 18*a^3*b + 94*a^2*b^2 - 168*a*b^3 + 96*b^4)*cos(f*x + e)^4 + a^2*b^2 - 12*a*b^3 + 16*b^4 + 2*(a^3*b - 14*a^2*b^2 + 40*a*b^3 - 32*b^4)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/((a^7 - 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2)]","B",0
86,1,1191,0,0.973790," ","integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{4} + 8 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, b^{4}\right)} f x \cos\left(f x + e\right)^{4} + 24 \, {\left(a^{3} b + 9 \, a^{2} b^{2} - 5 \, a b^{3} - 5 \, b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 12 \, {\left(a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4}\right)} f x - 3 \, {\left({\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4} + 2 \, {\left(5 \, a^{3} b + 5 \, a^{2} b^{2} - 9 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) + 4 \, {\left(2 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{7} - {\left(5 \, a^{4} - 12 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} - 3 \, b^{4}\right)} \cos\left(f x + e\right)^{5} - {\left(19 \, a^{3} b - 21 \, a^{2} b^{2} - 15 \, a b^{3} + 17 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 12 \, {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right)} f\right)}}, \frac{6 \, {\left(a^{4} + 8 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, b^{4}\right)} f x \cos\left(f x + e\right)^{4} + 12 \, {\left(a^{3} b + 9 \, a^{2} b^{2} - 5 \, a b^{3} - 5 \, b^{4}\right)} f x \cos\left(f x + e\right)^{2} + 6 \, {\left(a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4}\right)} f x + 3 \, {\left({\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + 5 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4} + 2 \, {\left(5 \, a^{3} b + 5 \, a^{2} b^{2} - 9 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + 2 \, {\left(2 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{7} - {\left(5 \, a^{4} - 12 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} - 3 \, b^{4}\right)} \cos\left(f x + e\right)^{5} - {\left(19 \, a^{3} b - 21 \, a^{2} b^{2} - 15 \, a b^{3} + 17 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 12 \, {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right)} f\right)}}\right]"," ",0,"[1/32*(12*(a^4 + 8*a^3*b - 14*a^2*b^2 + 5*b^4)*f*x*cos(f*x + e)^4 + 24*(a^3*b + 9*a^2*b^2 - 5*a*b^3 - 5*b^4)*f*x*cos(f*x + e)^2 + 12*(a^2*b^2 + 10*a*b^3 + 5*b^4)*f*x - 3*((5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(f*x + e)^4 + 5*a^2*b^2 + 10*a*b^3 + b^4 + 2*(5*a^3*b + 5*a^2*b^2 - 9*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) + 4*(2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^7 - (5*a^4 - 12*a^3*b + 6*a^2*b^2 + 4*a*b^3 - 3*b^4)*cos(f*x + e)^5 - (19*a^3*b - 21*a^2*b^2 - 15*a*b^3 + 17*b^4)*cos(f*x + e)^3 - 12*(a^2*b^2 - b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 - 7*a^6*b + 21*a^5*b^2 - 35*a^4*b^3 + 35*a^3*b^4 - 21*a^2*b^5 + 7*a*b^6 - b^7)*f*cos(f*x + e)^4 + 2*(a^6*b - 6*a^5*b^2 + 15*a^4*b^3 - 20*a^3*b^4 + 15*a^2*b^5 - 6*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^5*b^2 - 5*a^4*b^3 + 10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 - b^7)*f), 1/16*(6*(a^4 + 8*a^3*b - 14*a^2*b^2 + 5*b^4)*f*x*cos(f*x + e)^4 + 12*(a^3*b + 9*a^2*b^2 - 5*a*b^3 - 5*b^4)*f*x*cos(f*x + e)^2 + 6*(a^2*b^2 + 10*a*b^3 + 5*b^4)*f*x + 3*((5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(f*x + e)^4 + 5*a^2*b^2 + 10*a*b^3 + b^4 + 2*(5*a^3*b + 5*a^2*b^2 - 9*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e))) + 2*(2*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^7 - (5*a^4 - 12*a^3*b + 6*a^2*b^2 + 4*a*b^3 - 3*b^4)*cos(f*x + e)^5 - (19*a^3*b - 21*a^2*b^2 - 15*a*b^3 + 17*b^4)*cos(f*x + e)^3 - 12*(a^2*b^2 - b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 - 7*a^6*b + 21*a^5*b^2 - 35*a^4*b^3 + 35*a^3*b^4 - 21*a^2*b^5 + 7*a*b^6 - b^7)*f*cos(f*x + e)^4 + 2*(a^6*b - 6*a^5*b^2 + 15*a^4*b^3 - 20*a^3*b^4 + 15*a^2*b^5 - 6*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^5*b^2 - 5*a^4*b^3 + 10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 - b^7)*f)]","B",0
87,1,1076,0,0.882174," ","integrate(sin(f*x+e)^2/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{16 \, {\left(a^{4} + 3 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 32 \, {\left(a^{3} b + 4 \, a^{2} b^{2} - 5 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 16 \, {\left(a^{2} b^{2} + 5 \, a b^{3}\right)} f x - {\left({\left(15 \, a^{4} - 20 \, a^{3} b - 6 \, a^{2} b^{2} + 12 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4} + 2 \, {\left(15 \, a^{3} b - 5 \, a^{2} b^{2} - 11 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) - 4 \, {\left(4 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(17 \, a^{3} b - 33 \, a^{2} b^{2} + 15 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(11 \, a^{2} b^{2} - 10 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 10 \, a^{4} b^{3} - 10 \, a^{3} b^{4} + 5 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} - 4 \, a^{2} b^{5} + a b^{6}\right)} f\right)}}, \frac{8 \, {\left(a^{4} + 3 \, a^{3} b - 9 \, a^{2} b^{2} + 5 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{4} + 16 \, {\left(a^{3} b + 4 \, a^{2} b^{2} - 5 \, a b^{3}\right)} f x \cos\left(f x + e\right)^{2} + 8 \, {\left(a^{2} b^{2} + 5 \, a b^{3}\right)} f x + {\left({\left(15 \, a^{4} - 20 \, a^{3} b - 6 \, a^{2} b^{2} + 12 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4} + 2 \, {\left(15 \, a^{3} b - 5 \, a^{2} b^{2} - 11 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - 2 \, {\left(4 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(17 \, a^{3} b - 33 \, a^{2} b^{2} + 15 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(11 \, a^{2} b^{2} - 10 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 10 \, a^{4} b^{3} - 10 \, a^{3} b^{4} + 5 \, a^{2} b^{5} - a b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} - 4 \, a^{2} b^{5} + a b^{6}\right)} f\right)}}\right]"," ",0,"[1/32*(16*(a^4 + 3*a^3*b - 9*a^2*b^2 + 5*a*b^3)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 4*a^2*b^2 - 5*a*b^3)*f*x*cos(f*x + e)^2 + 16*(a^2*b^2 + 5*a*b^3)*f*x - ((15*a^4 - 20*a^3*b - 6*a^2*b^2 + 12*a*b^3 - b^4)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 - b^4 + 2*(15*a^3*b - 5*a^2*b^2 - 11*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) - 4*(4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(f*x + e)^5 + (17*a^3*b - 33*a^2*b^2 + 15*a*b^3 + b^4)*cos(f*x + e)^3 + (11*a^2*b^2 - 10*a*b^3 - b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*f*cos(f*x + e)^4 + 2*(a^6*b - 5*a^5*b^2 + 10*a^4*b^3 - 10*a^3*b^4 + 5*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 + (a^5*b^2 - 4*a^4*b^3 + 6*a^3*b^4 - 4*a^2*b^5 + a*b^6)*f), 1/16*(8*(a^4 + 3*a^3*b - 9*a^2*b^2 + 5*a*b^3)*f*x*cos(f*x + e)^4 + 16*(a^3*b + 4*a^2*b^2 - 5*a*b^3)*f*x*cos(f*x + e)^2 + 8*(a^2*b^2 + 5*a*b^3)*f*x + ((15*a^4 - 20*a^3*b - 6*a^2*b^2 + 12*a*b^3 - b^4)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 - b^4 + 2*(15*a^3*b - 5*a^2*b^2 - 11*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e))) - 2*(4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(f*x + e)^5 + (17*a^3*b - 33*a^2*b^2 + 15*a*b^3 + b^4)*cos(f*x + e)^3 + (11*a^2*b^2 - 10*a*b^3 - b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*f*cos(f*x + e)^4 + 2*(a^6*b - 5*a^5*b^2 + 10*a^4*b^3 - 10*a^3*b^4 + 5*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 + (a^5*b^2 - 4*a^4*b^3 + 6*a^3*b^4 - 4*a^2*b^5 + a*b^6)*f)]","B",0
88,1,742,0,0.697484," ","integrate(1/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, a^{2} b^{2} f x \tan\left(f x + e\right)^{4} + 64 \, a^{3} b f x \tan\left(f x + e\right)^{2} + 32 \, a^{4} f x - 4 \, {\left(7 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(15 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + 15 \, a^{4} - 10 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, {\left(15 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(9 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(f x + e\right)}{32 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}, \frac{16 \, a^{2} b^{2} f x \tan\left(f x + e\right)^{4} + 32 \, a^{3} b f x \tan\left(f x + e\right)^{2} + 16 \, a^{4} f x - 2 \, {\left(7 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(15 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + 15 \, a^{4} - 10 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, {\left(15 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) - 2 \, {\left(9 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(f x + e\right)}{16 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[1/32*(32*a^2*b^2*f*x*tan(f*x + e)^4 + 64*a^3*b*f*x*tan(f*x + e)^2 + 32*a^4*f*x - 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^3 - ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^4 + 15*a^4 - 10*a^3*b + 3*a^2*b^2 + 2*(15*a^3*b - 10*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(9*a^3*b - 14*a^2*b^2 + 5*a*b^3)*tan(f*x + e))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f), 1/16*(16*a^2*b^2*f*x*tan(f*x + e)^4 + 32*a^3*b*f*x*tan(f*x + e)^2 + 16*a^4*f*x - 2*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^3 - ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^4 + 15*a^4 - 10*a^3*b + 3*a^2*b^2 + 2*(15*a^3*b - 10*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))) - 2*(9*a^3*b - 14*a^2*b^2 + 5*a*b^3)*tan(f*x + e))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f)]","B",0
89,1,555,0,0.854536," ","integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{2} - 25 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(5 \, a b - 6 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, b^{2} \cos\left(f x + e\right)}{32 \, {\left(a^{3} b^{2} f + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(8 \, a^{2} - 25 \, a b + 15 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(5 \, a b - 6 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, b^{2} \cos\left(f x + e\right)}{16 \, {\left(a^{3} b^{2} f + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/32*(4*(8*a^2 - 25*a*b + 15*b^2)*cos(f*x + e)^5 + 20*(5*a*b - 6*b^2)*cos(f*x + e)^3 - 15*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*b^2*cos(f*x + e))/((a^3*b^2*f + (a^5 - 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^4 + 2*(a^4*b - a^3*b^2)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/16*(2*(8*a^2 - 25*a*b + 15*b^2)*cos(f*x + e)^5 + 10*(5*a*b - 6*b^2)*cos(f*x + e)^3 - 15*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*b^2*cos(f*x + e))/((a^3*b^2*f + (a^5 - 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^4 + 2*(a^4*b - a^3*b^2)*f*cos(f*x + e)^2)*sin(f*x + e))]","B",0
90,1,857,0,0.762503," ","integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(24 \, a^{3} - 206 \, a^{2} b + 485 \, a b^{2} - 315 \, b^{3}\right)} \cos\left(f x + e\right)^{5} - 20 \, {\left(15 \, a^{2} b - 62 \, a b^{2} + 63 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left({\left(3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 3 \, a b^{2} + 7 \, b^{3} - {\left(6 \, a^{2} b - 23 \, a b^{2} + 21 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) - 60 \, {\left(3 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)}{96 \, {\left({\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{6} - a^{4} b^{2} f - {\left(a^{6} - 4 \, a^{5} b + 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b - 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(16 \, a^{3} - 131 \, a^{2} b + 220 \, a b^{2} - 105 \, b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(24 \, a^{3} - 206 \, a^{2} b + 485 \, a b^{2} - 315 \, b^{3}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(15 \, a^{2} b - 62 \, a b^{2} + 63 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(3 \, a^{3} - 13 \, a^{2} b + 17 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(3 \, a^{3} - 19 \, a^{2} b + 37 \, a b^{2} - 21 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 3 \, a b^{2} + 7 \, b^{3} - {\left(6 \, a^{2} b - 23 \, a b^{2} + 21 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) - 30 \, {\left(3 \, a b^{2} - 7 \, b^{3}\right)} \cos\left(f x + e\right)}{48 \, {\left({\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{6} - a^{4} b^{2} f - {\left(a^{6} - 4 \, a^{5} b + 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{5} b - 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/96*(4*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 4*(24*a^3 - 206*a^2*b + 485*a*b^2 - 315*b^3)*cos(f*x + e)^5 - 20*(15*a^2*b - 62*a*b^2 + 63*b^3)*cos(f*x + e)^3 + 15*((3*a^3 - 13*a^2*b + 17*a*b^2 - 7*b^3)*cos(f*x + e)^6 - (3*a^3 - 19*a^2*b + 37*a*b^2 - 21*b^3)*cos(f*x + e)^4 - 3*a*b^2 + 7*b^3 - (6*a^2*b - 23*a*b^2 + 21*b^3)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) - 60*(3*a*b^2 - 7*b^3)*cos(f*x + e))/(((a^6 - 2*a^5*b + a^4*b^2)*f*cos(f*x + e)^6 - a^4*b^2*f - (a^6 - 4*a^5*b + 3*a^4*b^2)*f*cos(f*x + e)^4 - (2*a^5*b - 3*a^4*b^2)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/48*(2*(16*a^3 - 131*a^2*b + 220*a*b^2 - 105*b^3)*cos(f*x + e)^7 - 2*(24*a^3 - 206*a^2*b + 485*a*b^2 - 315*b^3)*cos(f*x + e)^5 - 10*(15*a^2*b - 62*a*b^2 + 63*b^3)*cos(f*x + e)^3 - 15*((3*a^3 - 13*a^2*b + 17*a*b^2 - 7*b^3)*cos(f*x + e)^6 - (3*a^3 - 19*a^2*b + 37*a*b^2 - 21*b^3)*cos(f*x + e)^4 - 3*a*b^2 + 7*b^3 - (6*a^2*b - 23*a*b^2 + 21*b^3)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) - 30*(3*a*b^2 - 7*b^3)*cos(f*x + e))/(((a^6 - 2*a^5*b + a^4*b^2)*f*cos(f*x + e)^6 - a^4*b^2*f - (a^6 - 4*a^5*b + 3*a^4*b^2)*f*cos(f*x + e)^4 - (2*a^5*b - 3*a^4*b^2)*f*cos(f*x + e)^2)*sin(f*x + e))]","B",0
91,1,1199,0,0.576092," ","integrate(csc(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(64 \, a^{4} - 863 \, a^{3} b + 2479 \, a^{2} b^{2} - 2625 \, a b^{3} + 945 \, b^{4}\right)} \cos\left(f x + e\right)^{9} - 4 \, {\left(160 \, a^{4} - 2173 \, a^{3} b + 7158 \, a^{2} b^{2} - 8925 \, a b^{3} + 3780 \, b^{4}\right)} \cos\left(f x + e\right)^{7} + 4 \, {\left(120 \, a^{4} - 1685 \, a^{3} b + 7104 \, a^{2} b^{2} - 11025 \, a b^{3} + 5670 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 20 \, {\left(75 \, a^{3} b - 530 \, a^{2} b^{2} + 1155 \, a b^{3} - 756 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(15 \, a^{4} - 100 \, a^{3} b + 218 \, a^{2} b^{2} - 196 \, a b^{3} + 63 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(15 \, a^{4} - 115 \, a^{3} b + 303 \, a^{2} b^{2} - 329 \, a b^{3} + 126 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(15 \, a^{4} - 160 \, a^{3} b + 573 \, a^{2} b^{2} - 798 \, a b^{3} + 378 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} - 70 \, a b^{3} + 63 \, b^{4} + 2 \, {\left(15 \, a^{3} b - 100 \, a^{2} b^{2} + 203 \, a b^{3} - 126 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - a b \cos\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(f x + e\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}}\right) \sin\left(f x + e\right) + 60 \, {\left(15 \, a^{2} b^{2} - 70 \, a b^{3} + 63 \, b^{4}\right)} \cos\left(f x + e\right)}{480 \, {\left({\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{8} + a^{5} b^{2} f - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} - 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, -\frac{2 \, {\left(64 \, a^{4} - 863 \, a^{3} b + 2479 \, a^{2} b^{2} - 2625 \, a b^{3} + 945 \, b^{4}\right)} \cos\left(f x + e\right)^{9} - 2 \, {\left(160 \, a^{4} - 2173 \, a^{3} b + 7158 \, a^{2} b^{2} - 8925 \, a b^{3} + 3780 \, b^{4}\right)} \cos\left(f x + e\right)^{7} + 2 \, {\left(120 \, a^{4} - 1685 \, a^{3} b + 7104 \, a^{2} b^{2} - 11025 \, a b^{3} + 5670 \, b^{4}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(75 \, a^{3} b - 530 \, a^{2} b^{2} + 1155 \, a b^{3} - 756 \, b^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left({\left(15 \, a^{4} - 100 \, a^{3} b + 218 \, a^{2} b^{2} - 196 \, a b^{3} + 63 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(15 \, a^{4} - 115 \, a^{3} b + 303 \, a^{2} b^{2} - 329 \, a b^{3} + 126 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(15 \, a^{4} - 160 \, a^{3} b + 573 \, a^{2} b^{2} - 798 \, a b^{3} + 378 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 15 \, a^{2} b^{2} - 70 \, a b^{3} + 63 \, b^{4} + 2 \, {\left(15 \, a^{3} b - 100 \, a^{2} b^{2} + 203 \, a b^{3} - 126 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 30 \, {\left(15 \, a^{2} b^{2} - 70 \, a b^{3} + 63 \, b^{4}\right)} \cos\left(f x + e\right)}{240 \, {\left({\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{8} + a^{5} b^{2} f - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} - 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right]"," ",0,"[-1/480*(4*(64*a^4 - 863*a^3*b + 2479*a^2*b^2 - 2625*a*b^3 + 945*b^4)*cos(f*x + e)^9 - 4*(160*a^4 - 2173*a^3*b + 7158*a^2*b^2 - 8925*a*b^3 + 3780*b^4)*cos(f*x + e)^7 + 4*(120*a^4 - 1685*a^3*b + 7104*a^2*b^2 - 11025*a*b^3 + 5670*b^4)*cos(f*x + e)^5 + 20*(75*a^3*b - 530*a^2*b^2 + 1155*a*b^3 - 756*b^4)*cos(f*x + e)^3 - 15*((15*a^4 - 100*a^3*b + 218*a^2*b^2 - 196*a*b^3 + 63*b^4)*cos(f*x + e)^8 - 2*(15*a^4 - 115*a^3*b + 303*a^2*b^2 - 329*a*b^3 + 126*b^4)*cos(f*x + e)^6 + (15*a^4 - 160*a^3*b + 573*a^2*b^2 - 798*a*b^3 + 378*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 - 70*a*b^3 + 63*b^4 + 2*(15*a^3*b - 100*a^2*b^2 + 203*a*b^3 - 126*b^4)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 + 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2))*sin(f*x + e) + 60*(15*a^2*b^2 - 70*a*b^3 + 63*b^4)*cos(f*x + e))/(((a^7 - 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2)*sin(f*x + e)), -1/240*(2*(64*a^4 - 863*a^3*b + 2479*a^2*b^2 - 2625*a*b^3 + 945*b^4)*cos(f*x + e)^9 - 2*(160*a^4 - 2173*a^3*b + 7158*a^2*b^2 - 8925*a*b^3 + 3780*b^4)*cos(f*x + e)^7 + 2*(120*a^4 - 1685*a^3*b + 7104*a^2*b^2 - 11025*a*b^3 + 5670*b^4)*cos(f*x + e)^5 + 10*(75*a^3*b - 530*a^2*b^2 + 1155*a*b^3 - 756*b^4)*cos(f*x + e)^3 - 15*((15*a^4 - 100*a^3*b + 218*a^2*b^2 - 196*a*b^3 + 63*b^4)*cos(f*x + e)^8 - 2*(15*a^4 - 115*a^3*b + 303*a^2*b^2 - 329*a*b^3 + 126*b^4)*cos(f*x + e)^6 + (15*a^4 - 160*a^3*b + 573*a^2*b^2 - 798*a*b^3 + 378*b^4)*cos(f*x + e)^4 + 15*a^2*b^2 - 70*a*b^3 + 63*b^4 + 2*(15*a^3*b - 100*a^2*b^2 + 203*a*b^3 - 126*b^4)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(b/a)/(b*cos(f*x + e)*sin(f*x + e)))*sin(f*x + e) + 30*(15*a^2*b^2 - 70*a*b^3 + 63*b^4)*cos(f*x + e))/(((a^7 - 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2)*sin(f*x + e))]","B",0
92,1,378,0,0.657757," ","integrate(sin(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(10 \, a^{2} - 21 \, a b + 11 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} - 40 \, a b + 23 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{30 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, -\frac{15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left(3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(10 \, a^{2} - 21 \, a b + 11 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} - 40 \, a b + 23 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}\right]"," ",0,"[1/30*(15*(a^2 - 2*a*b + b^2)*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - 2*(3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - (10*a^2 - 21*a*b + 11*b^2)*cos(f*x + e)^3 + (15*a^2 - 40*a*b + 23*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 - 2*a*b + b^2)*f), -1/15*(15*(a^2 - 2*a*b + b^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + (3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - (10*a^2 - 21*a*b + 11*b^2)*cos(f*x + e)^3 + (15*a^2 - 40*a*b + 23*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2 - 2*a*b + b^2)*f)]","A",0
93,1,277,0,0.610937," ","integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a - b\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a - 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left(a - b\right)} f}, -\frac{3 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a - 4 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a - b\right)} f}\right]"," ",0,"[1/6*(3*(a - b)*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((a - b)*cos(f*x + e)^3 - (3*a - 4*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a - b)*f), -1/3*(3*(a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - ((a - b)*cos(f*x + e)^3 - (3*a - 4*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a - b)*f)]","A",0
94,1,203,0,0.703628," ","integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, f}, -\frac{\sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{f}\right]"," ",0,"[-1/2*(2*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/f, -(sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/f]","A",0
95,1,514,0,0.679814," ","integrate(csc(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, f}, -\frac{2 \, \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, f}, \frac{2 \, \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, f}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) - \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right)}{f}\right]"," ",0,"[1/2*(sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/f, -1/2*(2*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)))/f, 1/2*(2*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/f, (sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b))/f]","A",0
96,1,849,0,1.384934," ","integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, a \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{4 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}, \frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + a \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}, -\frac{4 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - 2 \, a \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{4 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}, \frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) - 2 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + a \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{2 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}\right]"," ",0,"[1/4*(2*a*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 2*(a*cos(f*x + e)^2 - a)*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^2 - a*f), 1/2*(((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + a*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + (a*cos(f*x + e)^2 - a)*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^2 - a*f), -1/4*(4*(a*cos(f*x + e)^2 - a)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - 2*a*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)))/(a*f*cos(f*x + e)^2 - a*f), 1/2*(((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - 2*(a*cos(f*x + e)^2 - a)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + a*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(a*f*cos(f*x + e)^2 - a*f)]","A",0
97,1,1273,0,2.015119," ","integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 6 \, a b - b^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 8 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}, \frac{{\left({\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 6 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + 4 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + {\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}, -\frac{16 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left({\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 6 \, a b - b^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}, \frac{{\left({\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 6 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 6 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) - 8 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}\right]"," ",0,"[-1/16*(((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 8*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - 2*((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/8*(((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + 4*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + ((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), -1/16*(16*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + ((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/8*(((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - 8*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + ((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)]","A",0
98,1,2068,0,13.416914," ","integrate(sin(f*x+e)^4*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 16 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 8 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, -\frac{32 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, \frac{{\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, \frac{{\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 16 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 11 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}\right]"," ",0,"[1/64*((3*a^2 - 12*a*b + 8*b^2)*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 16*(a^2 - 2*a*b + b^2)*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 8*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 11*a*b + 6*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^2 - 2*a*b + b^2)*f), -1/64*(32*(a^2 - 2*a*b + b^2)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - (3*a^2 - 12*a*b + 8*b^2)*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 11*a*b + 6*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^2 - 2*a*b + b^2)*f), 1/32*((3*a^2 - 12*a*b + 8*b^2)*sqrt(a - b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 8*(a^2 - 2*a*b + b^2)*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 11*a*b + 6*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^2 - 2*a*b + b^2)*f), 1/32*((3*a^2 - 12*a*b + 8*b^2)*sqrt(a - b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 16*(a^2 - 2*a*b + b^2)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + 4*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 11*a*b + 6*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^2 - 2*a*b + b^2)*f)]","B",0
99,1,1847,0,1.366982," ","integrate(sin(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a - 2 \, b\right)} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 4 \, {\left(a - b\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{16 \, {\left(a - b\right)} f}, -\frac{8 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) + 8 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(a - 2 \, b\right)} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, {\left(a - b\right)} f}, -\frac{4 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - \sqrt{a - b} {\left(a - 2 \, b\right)} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, {\left(a - b\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, {\left(a - b\right)} f}, -\frac{4 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - \sqrt{a - b} {\left(a - 2 \, b\right)} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, {\left(a - b\right)} f}\right]"," ",0,"[-1/16*(8*(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - (a - 2*b)*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 4*(a - b)*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/((a - b)*f), -1/16*(8*(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) + 8*(a - b)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))) - (a - 2*b)*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/((a - b)*f), -1/8*(4*(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - sqrt(a - b)*(a - 2*b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 2*(a - b)*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4))/((a - b)*f), -1/8*(4*(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - sqrt(a - b)*(a - 2*b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(a - b)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/((a - b)*f)]","B",0
100,1,410,0,0.547881," ","integrate((a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}, \frac{2 \, \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right)}{2 \, f}, -\frac{2 \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}, \frac{\sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right)}{f}\right]"," ",0,"[1/2*(sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)))/f, 1/2*(2*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a))/f, -1/2*(2*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)))/f, (sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))))/f]","A",0
101,1,331,0,0.558920," ","integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{4 \, f \sin\left(f x + e\right)}, -\frac{\sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{2 \, f \sin\left(f x + e\right)}\right]"," ",0,"[1/4*(sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e)), -1/2*(sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) + 2*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(f*sin(f*x + e))]","B",0
102,1,435,0,0.684794," ","integrate(csc(f*x+e)^4*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left(a \cos\left(f x + e\right)^{2} - a\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/12*(3*(a*cos(f*x + e)^2 - a)*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((2*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a*f*cos(f*x + e)^2 - a*f)*sin(f*x + e)), -1/6*(3*(a*cos(f*x + e)^2 - a)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) + 2*((2*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a*f*cos(f*x + e)^2 - a*f)*sin(f*x + e))]","B",0
103,1,587,0,1.367766," ","integrate(csc(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(8 \, a^{2} + 9 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(20 \, a^{2} + 19 \, a b - 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)} \sin\left(f x + e\right)}, -\frac{15 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, {\left({\left(8 \, a^{2} + 9 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(20 \, a^{2} + 19 \, a b - 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} + 10 \, a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{30 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/60*(15*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((8*a^2 + 9*a*b - 2*b^2)*cos(f*x + e)^5 - (20*a^2 + 19*a*b - 4*b^2)*cos(f*x + e)^3 + (15*a^2 + 10*a*b - 2*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)*sin(f*x + e)), -1/30*(15*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) + 2*((8*a^2 + 9*a*b - 2*b^2)*cos(f*x + e)^5 - (20*a^2 + 19*a*b - 4*b^2)*cos(f*x + e)^3 + (15*a^2 + 10*a*b - 2*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)*sin(f*x + e))]","B",0
104,1,426,0,0.907219," ","integrate(sin(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(3 \, a^{2} - 10 \, a b + 7 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 4 \, {\left(5 \, a^{2} - 13 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(15 \, a^{2} - 70 \, a b + 58 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b + 15 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left(a - b\right)} f \cos\left(f x + e\right)}, -\frac{15 \, {\left(3 \, a^{2} - 10 \, a b + 7 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) + {\left(6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 4 \, {\left(5 \, a^{2} - 13 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(15 \, a^{2} - 70 \, a b + 58 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b + 15 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{30 \, {\left(a - b\right)} f \cos\left(f x + e\right)}\right]"," ",0,"[-1/60*(15*(3*a^2 - 10*a*b + 7*b^2)*sqrt(b)*cos(f*x + e)*log(-((a - b)*cos(f*x + e)^2 - 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(6*(a^2 - 2*a*b + b^2)*cos(f*x + e)^6 - 4*(5*a^2 - 13*a*b + 8*b^2)*cos(f*x + e)^4 + 2*(15*a^2 - 70*a*b + 58*b^2)*cos(f*x + e)^2 - 15*a*b + 15*b^2)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a - b)*f*cos(f*x + e)), -1/30*(15*(3*a^2 - 10*a*b + 7*b^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) + (6*(a^2 - 2*a*b + b^2)*cos(f*x + e)^6 - 4*(5*a^2 - 13*a*b + 8*b^2)*cos(f*x + e)^4 + 2*(15*a^2 - 70*a*b + 58*b^2)*cos(f*x + e)^2 - 15*a*b + 15*b^2)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a - b)*f*cos(f*x + e))]","A",0
105,1,307,0,0.822454," ","integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(3 \, a - 5 \, b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a - 7 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, f \cos\left(f x + e\right)}, -\frac{3 \, {\left(3 \, a - 5 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) - {\left(2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a - 7 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, f \cos\left(f x + e\right)}\right]"," ",0,"[-1/12*(3*(3*a - 5*b)*sqrt(b)*cos(f*x + e)*log(-((a - b)*cos(f*x + e)^2 - 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) - 2*(2*(a - b)*cos(f*x + e)^4 - 2*(3*a - 7*b)*cos(f*x + e)^2 + 3*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), -1/6*(3*(3*a - 5*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) - (2*(a - b)*cos(f*x + e)^4 - 2*(3*a - 7*b)*cos(f*x + e)^2 + 3*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e))]","A",0
106,1,268,0,1.003267," ","integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right)}, -\frac{3 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) + {\left(2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, f \cos\left(f x + e\right)}\right]"," ",0,"[-1/4*(3*(a - b)*sqrt(b)*cos(f*x + e)*log(-((a - b)*cos(f*x + e)^2 - 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(2*(a - b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), -1/2*(3*(a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) + (2*(a - b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e))]","A",0
107,1,747,0,1.520423," ","integrate(csc(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{2 \, a^{\frac{3}{2}} \cos\left(f x + e\right) \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - {\left(3 \, a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, b \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right)}, -\frac{{\left(3 \, a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) - a^{\frac{3}{2}} \cos\left(f x + e\right) \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - b \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, f \cos\left(f x + e\right)}, \frac{4 \, \sqrt{-a} a \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) \cos\left(f x + e\right) - {\left(3 \, a - b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, b \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right)}, \frac{2 \, \sqrt{-a} a \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) \cos\left(f x + e\right) - {\left(3 \, a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) \cos\left(f x + e\right) + b \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, f \cos\left(f x + e\right)}\right]"," ",0,"[1/4*(2*a^(3/2)*cos(f*x + e)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - (3*a - b)*sqrt(b)*cos(f*x + e)*log(-((a - b)*cos(f*x + e)^2 - 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*b*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), -1/2*((3*a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) - a^(3/2)*cos(f*x + e)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - b*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), 1/4*(4*sqrt(-a)*a*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a)*cos(f*x + e) - (3*a - b)*sqrt(b)*cos(f*x + e)*log(-((a - b)*cos(f*x + e)^2 - 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*b*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)), 1/2*(2*sqrt(-a)*a*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a)*cos(f*x + e) - (3*a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b)*cos(f*x + e) + b*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e))]","A",0
108,1,994,0,1.971626," ","integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + {\left({\left(3 \, a + b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}, -\frac{2 \, {\left({\left(3 \, a + b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}, \frac{2 \, {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + {\left({\left(3 \, a + b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}, \frac{{\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) - {\left({\left(3 \, a + b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*(((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + ((3*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), -1/4*(2*((3*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - ((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*((a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), 1/4*(2*((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + ((3*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*((a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e)), 1/2*(((a + 3*b)*cos(f*x + e)^3 - (a + 3*b)*cos(f*x + e))*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - ((3*a + b)*cos(f*x + e)^3 - (3*a + b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + ((a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)^3 - f*cos(f*x + e))]","A",0
109,1,1365,0,2.055681," ","integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 12 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(3 \, {\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 13 \, a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)}}, \frac{3 \, {\left({\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + 6 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(-\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + {\left(3 \, {\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 13 \, a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)}}, -\frac{24 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) - 3 \, {\left({\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 13 \, a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)}}, \frac{3 \, {\left({\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) - 12 \, {\left({\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{b}\right) + {\left(3 \, {\left(a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{4} - {\left(5 \, a^{2} + 13 \, a b\right)} \cos\left(f x + e\right)^{2} + 4 \, a b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/16*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 12*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + 2*(3*(a^2 + 3*a*b)*cos(f*x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)), 1/8*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + 6*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(b)*log(-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)^2) + (3*(a^2 + 3*a*b)*cos(f*x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)), -1/16*(24*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) - 3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*(3*(a^2 + 3*a*b)*cos(f*x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)), 1/8*(3*((a^2 + 6*a*b + b^2)*cos(f*x + e)^5 - 2*(a^2 + 6*a*b + b^2)*cos(f*x + e)^3 + (a^2 + 6*a*b + b^2)*cos(f*x + e))*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - 12*((a^2 + a*b)*cos(f*x + e)^5 - 2*(a^2 + a*b)*cos(f*x + e)^3 + (a^2 + a*b)*cos(f*x + e))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) + (3*(a^2 + 3*a*b)*cos(f*x + e)^4 - (5*a^2 + 13*a*b)*cos(f*x + e)^2 + 4*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e))]","A",0
110,1,2163,0,120.240841," ","integrate(sin(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \sqrt{-a + b} \cos\left(f x + e\right) \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 24 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 8 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b - 4 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left(a - b\right)} f \cos\left(f x + e\right)}, -\frac{48 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 3 \, {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \sqrt{-a + b} \cos\left(f x + e\right) \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b - 4 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left(a - b\right)} f \cos\left(f x + e\right)}, \frac{3 \, {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 12 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b - 4 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left(a - b\right)} f \cos\left(f x + e\right)}, \frac{3 \, {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 24 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + 4 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a b - 4 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left(a - b\right)} f \cos\left(f x + e\right)}\right]"," ",0,"[-1/64*(3*(a^2 - 8*a*b + 8*b^2)*sqrt(-a + b)*cos(f*x + e)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 24*(a^2 - 3*a*b + 2*b^2)*sqrt(b)*cos(f*x + e)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 - 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 8*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 5*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 + 4*a*b - 4*b^2)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a - b)*f*cos(f*x + e)), -1/64*(48*(a^2 - 3*a*b + 2*b^2)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + 3*(a^2 - 8*a*b + 8*b^2)*sqrt(-a + b)*cos(f*x + e)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 5*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 + 4*a*b - 4*b^2)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a - b)*f*cos(f*x + e)), 1/32*(3*(a^2 - 8*a*b + 8*b^2)*sqrt(a - b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - 12*(a^2 - 3*a*b + 2*b^2)*sqrt(b)*cos(f*x + e)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 - 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 4*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 5*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 + 4*a*b - 4*b^2)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a - b)*f*cos(f*x + e)), 1/32*(3*(a^2 - 8*a*b + 8*b^2)*sqrt(a - b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - 24*(a^2 - 3*a*b + 2*b^2)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + 4*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 5*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^2 + 4*a*b - 4*b^2)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a - b)*f*cos(f*x + e))]","B",0
111,1,1931,0,8.000534," ","integrate(sin(f*x+e)^2*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a - 4 \, b\right)} \sqrt{-a + b} \cos\left(f x + e\right) \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 2 \, {\left(3 \, a - 4 \, b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 8 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, f \cos\left(f x + e\right)}, -\frac{4 \, {\left(3 \, a - 4 \, b\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + {\left(a - 4 \, b\right)} \sqrt{-a + b} \cos\left(f x + e\right) \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, f \cos\left(f x + e\right)}, \frac{\sqrt{a - b} {\left(a - 4 \, b\right)} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - {\left(3 \, a - 4 \, b\right)} \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}, \frac{\sqrt{a - b} {\left(a - 4 \, b\right)} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 2 \, {\left(3 \, a - 4 \, b\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)}\right]"," ",0,"[-1/16*((a - 4*b)*sqrt(-a + b)*cos(f*x + e)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 2*(3*a - 4*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 - 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) + 8*((a - b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), -1/16*(4*(3*a - 4*b)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) + (a - 4*b)*sqrt(-a + b)*cos(f*x + e)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*((a - b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), 1/8*(sqrt(a - b)*(a - 4*b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - (3*a - 4*b)*sqrt(b)*cos(f*x + e)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 - 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4) - 4*((a - b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e)), 1/8*(sqrt(a - b)*(a - 4*b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e) - 2*(3*a - 4*b)*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e) - 4*((a - b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(f*cos(f*x + e))]","B",0
112,1,537,0,0.878731," ","integrate((a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(3 \, a - 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(a - b\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{4 \, f}, -\frac{{\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(-a + b\right)}^{\frac{3}{2}} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{2 \, f}, \frac{4 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a - 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{4 \, f}, \frac{2 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{2 \, f}\right]"," ",0,"[-1/4*((3*a - 2*b)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(a - b)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f, -1/2*((3*a - 2*b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (-a + b)^(3/2)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f, 1/4*(4*(a - b)^(3/2)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a - 2*b)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f, 1/2*(2*(a - b)^(3/2)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a - 2*b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f]","A",0
113,1,387,0,0.764814," ","integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{b} \cos\left(f x + e\right) \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, f \cos\left(f x + e\right) \sin\left(f x + e\right)}, -\frac{3 \, a \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) \sin\left(f x + e\right) + 2 \, {\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, f \cos\left(f x + e\right) \sin\left(f x + e\right)}\right]"," ",0,"[1/8*(3*a*sqrt(b)*cos(f*x + e)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((2*a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)*sin(f*x + e)), -1/4*(3*a*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)*sin(f*x + e) + 2*((2*a + b)*cos(f*x + e)^2 - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(f*cos(f*x + e)*sin(f*x + e))]","B",0
114,1,497,0,1.585768," ","integrate(csc(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(4 \, a + 11 \, b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a + 7 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}, -\frac{3 \, {\left({\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a + 2 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, {\left({\left(4 \, a + 11 \, b\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a + 7 \, b\right)} \cos\left(f x + e\right)^{2} + 3 \, b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/24*(3*((3*a + 2*b)*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((4*a + 11*b)*cos(f*x + e)^4 - 2*(3*a + 7*b)*cos(f*x + e)^2 + 3*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e)), -1/12*(3*((3*a + 2*b)*cos(f*x + e)^3 - (3*a + 2*b)*cos(f*x + e))*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) + 2*((4*a + 11*b)*cos(f*x + e)^4 - 2*(3*a + 7*b)*cos(f*x + e)^2 + 3*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e))]","A",0
115,1,655,0,5.219900," ","integrate(csc(f*x+e)^6*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left({\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{b} \log\left(\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 8 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right) + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) \sin\left(f x + e\right) - 4 \, {\left({\left(16 \, a^{2} + 83 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(40 \, a^{2} + 193 \, a b + 12 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(30 \, a^{2} + 125 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{120 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}, -\frac{15 \, {\left({\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, a^{2} + 4 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b} \arctan\left(\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2}\right)} \sin\left(f x + e\right)}\right) \sin\left(f x + e\right) + 2 \, {\left({\left(16 \, a^{2} + 83 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{6} - {\left(40 \, a^{2} + 193 \, a b + 12 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(30 \, a^{2} + 125 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, a b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{60 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/120*(15*((3*a^2 + 4*a*b)*cos(f*x + e)^5 - 2*(3*a^2 + 4*a*b)*cos(f*x + e)^3 + (3*a^2 + 4*a*b)*cos(f*x + e))*sqrt(b)*log(((a^2 - 8*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(a*b - 2*b^2)*cos(f*x + e)^2 + 4*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)*sin(f*x + e) - 4*((16*a^2 + 83*a*b + 6*b^2)*cos(f*x + e)^6 - (40*a^2 + 193*a*b + 12*b^2)*cos(f*x + e)^4 + (30*a^2 + 125*a*b + 6*b^2)*cos(f*x + e)^2 - 15*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e))*sin(f*x + e)), -1/60*(15*((3*a^2 + 4*a*b)*cos(f*x + e)^5 - 2*(3*a^2 + 4*a*b)*cos(f*x + e)^3 + (3*a^2 + 4*a*b)*cos(f*x + e))*sqrt(-b)*arctan(1/2*((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a*b - b^2)*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*sin(f*x + e) + 2*((16*a^2 + 83*a*b + 6*b^2)*cos(f*x + e)^6 - (40*a^2 + 193*a*b + 12*b^2)*cos(f*x + e)^4 + (30*a^2 + 125*a*b + 6*b^2)*cos(f*x + e)^2 - 15*a*b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e))*sin(f*x + e))]","A",0
116,1,124,0,0.851111," ","integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(5 \, a^{2} - 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} - 10 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}"," ",0,"-1/15*(3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 2*(5*a^2 - 8*a*b + 3*b^2)*cos(f*x + e)^3 + (15*a^2 - 10*a*b + 3*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f)","A",0
117,1,75,0,0.585319," ","integrate(sin(f*x+e)^3/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a - b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}"," ",0,"1/3*((a - b)*cos(f*x + e)^3 - (3*a - b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2 - 2*a*b + b^2)*f)","A",0
118,1,45,0,0.618783," ","integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{{\left(a - b\right)} f}"," ",0,"-sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/((a - b)*f)","A",0
119,1,134,0,0.759121," ","integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, \sqrt{a} f}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right)}{a f}\right]"," ",0,"[1/2*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1))/(sqrt(a)*f), sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a)/(a*f)]","A",0
120,1,284,0,0.631208," ","integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, a \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}, \frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + a \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{2 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}\right]"," ",0,"[1/4*(2*a*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - ((a - b)*cos(f*x + e)^2 - a + b)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)))/(a^2*f*cos(f*x + e)^2 - a^2*f), 1/2*(((a - b)*cos(f*x + e)^2 - a + b)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + a*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f)]","A",0
121,1,437,0,0.653441," ","integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} - 2 \, a b + b^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(3 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)}}, \frac{3 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + {\left(3 \, {\left(a^{2} - a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)}}\right]"," ",0,"[1/16*(3*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a^2 - 2*a*b + b^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 2*(3*(a^2 - a*b)*cos(f*x + e)^3 - (5*a^2 - 3*a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f), 1/8*(3*((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a^2 - 2*a*b + b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + (3*(a^2 - a*b)*cos(f*x + e)^3 - (5*a^2 - 3*a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f)]","A",0
122,1,788,0,4.690905," ","integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, a^{2} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 7 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}, \frac{3 \, \sqrt{a - b} a^{2} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} - 7 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f}\right]"," ",0,"[-1/64*(3*a^2*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 8*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 7*a*b + 2*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f), 1/32*(3*sqrt(a - b)*a^2*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*(a^2 - 2*a*b + b^2)*cos(f*x + e)^3 - (5*a^2 - 7*a*b + 2*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f)]","B",0
123,1,696,0,0.702802," ","integrate(sin(f*x+e)^2/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - a \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right)}{16 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}, -\frac{4 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) \sin\left(f x + e\right) - \sqrt{a - b} a \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f}\right]"," ",0,"[-1/16*(8*(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - a*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/((a^2 - 2*a*b + b^2)*f), -1/8*(4*(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)*sin(f*x + e) - sqrt(a - b)*a*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))))/((a^2 - 2*a*b + b^2)*f)]","B",0
124,1,125,0,0.462969," ","integrate(1/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a - b\right)} f}, \frac{\arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right)}{\sqrt{a - b} f}\right]"," ",0,"[-1/2*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1))/((a - b)*f), arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e)))/(sqrt(a - b)*f)]","A",0
125,1,49,0,0.447024," ","integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a f \sin\left(f x + e\right)}"," ",0,"-sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/(a*f*sin(f*x + e))","A",0
126,1,90,0,0.637267," ","integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, {\left(a - b\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a - 2 \, b\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a - b)*cos(f*x + e)^3 - (3*a - 2*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2*f*cos(f*x + e)^2 - a^2*f)*sin(f*x + e))","A",0
127,1,141,0,1.108407," ","integrate(csc(f*x+e)^6/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(5 \, a^{2} - 9 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 4*(5*a^2 - 9*a*b + 4*b^2)*cos(f*x + e)^3 + (15*a^2 - 20*a*b + 8*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f)*sin(f*x + e))","A",0
128,1,233,0,0.652959," ","integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 2 \, {\left(5 \, a^{3} - 12 \, a^{2} b + 9 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{3} - 5 \, a^{2} b - 11 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(15 \, a^{2} b + 10 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}"," ",0,"-1/15*(3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 2*(5*a^3 - 12*a^2*b + 9*a*b^2 - 2*b^3)*cos(f*x + e)^5 + (15*a^3 - 5*a^2*b - 11*a*b^2 + b^3)*cos(f*x + e)^3 + 2*(15*a^2*b + 10*a*b^2 - b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f)","A",0
129,1,158,0,0.654107," ","integrate(sin(f*x+e)^3/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}"," ",0,"1/3*((a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - (3*a^2 - 2*a*b - b^2)*cos(f*x + e)^3 - 2*(3*a*b + b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f)","A",0
130,1,104,0,0.569109," ","integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{{\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f}"," ",0,"-((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*f*cos(f*x + e)^2 + (a^2*b - 2*a*b^2 + b^3)*f)","A",0
131,1,355,0,0.747788," ","integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, a b \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, {\left({\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - a^{2} b^{2}\right)} f\right)}}, -\frac{a b \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) - {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right)}{{\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - a^{2} b^{2}\right)} f}\right]"," ",0,"[-1/2*(2*a*b*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - ((a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a*b - b^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)))/((a^4 - 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b - a^2*b^2)*f), -(a*b*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) - ((a^2 - 2*a*b + b^2)*cos(f*x + e)^2 + a*b - b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a))/((a^4 - 2*a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^3*b - a^2*b^2)*f)]","B",0
132,1,455,0,0.774213," ","integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 5 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 3 \, b^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, a b \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{{\left({\left(a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} - 5 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a b + 3 \, b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + {\left({\left(a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{3} + 3 \, a b \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{2 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[-1/4*(((a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^4 - (a^2 - 5*a*b + 6*b^2)*cos(f*x + e)^2 - a*b + 3*b^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*((a^2 - 3*a*b)*cos(f*x + e)^3 + 3*a*b*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2), 1/2*(((a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^4 - (a^2 - 5*a*b + 6*b^2)*cos(f*x + e)^2 - a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + ((a^2 - 3*a*b)*cos(f*x + e)^3 + 3*a*b*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2)]","A",0
133,1,705,0,1.120548," ","integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{3} - 7 \, a^{2} b + 11 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 15 \, a^{2} b + 28 \, a b^{2} - 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 6 \, a b^{2} + 5 \, b^{3} + {\left(a^{3} - 9 \, a^{2} b + 23 \, a b^{2} - 15 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(3 \, {\left(a^{3} - 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} - 31 \, a^{2} b + 30 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b - 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{16 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{3 \, {\left({\left(a^{3} - 7 \, a^{2} b + 11 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(2 \, a^{3} - 15 \, a^{2} b + 28 \, a b^{2} - 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{2} b - 6 \, a b^{2} + 5 \, b^{3} + {\left(a^{3} - 9 \, a^{2} b + 23 \, a b^{2} - 15 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + {\left(3 \, {\left(a^{3} - 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(5 \, a^{3} - 31 \, a^{2} b + 30 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b - 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{8 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/16*(3*((a^3 - 7*a^2*b + 11*a*b^2 - 5*b^3)*cos(f*x + e)^6 - (2*a^3 - 15*a^2*b + 28*a*b^2 - 15*b^3)*cos(f*x + e)^4 + a^2*b - 6*a*b^2 + 5*b^3 + (a^3 - 9*a^2*b + 23*a*b^2 - 15*b^3)*cos(f*x + e)^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 2*(3*(a^3 - 6*a^2*b + 5*a*b^2)*cos(f*x + e)^5 - (5*a^3 - 31*a^2*b + 30*a*b^2)*cos(f*x + e)^3 - (13*a^2*b - 15*a*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2), 1/8*(3*((a^3 - 7*a^2*b + 11*a*b^2 - 5*b^3)*cos(f*x + e)^6 - (2*a^3 - 15*a^2*b + 28*a*b^2 - 15*b^3)*cos(f*x + e)^4 + a^2*b - 6*a*b^2 + 5*b^3 + (a^3 - 9*a^2*b + 23*a*b^2 - 15*b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + (3*(a^3 - 6*a^2*b + 5*a*b^2)*cos(f*x + e)^5 - (5*a^3 - 31*a^2*b + 30*a*b^2)*cos(f*x + e)^3 - (13*a^2*b - 15*a*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2)]","B",0
134,1,1046,0,128.622997," ","integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b + 4 \, a b^{2} + {\left(a^{3} + 3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - 5 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b - 11 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{64 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}, \frac{3 \, {\left(a^{2} b + 4 \, a b^{2} + {\left(a^{3} + 3 \, a^{2} b - 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{5} - 5 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, a^{2} b - 11 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{32 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[1/64*(3*(a^2*b + 4*a*b^2 + (a^3 + 3*a^2*b - 4*a*b^2)*cos(f*x + e)^2)*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*(2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - 5*(a^3 - 2*a^2*b + a*b^2)*cos(f*x + e)^3 - (13*a^2*b - 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f), 1/32*(3*(a^2*b + 4*a*b^2 + (a^3 + 3*a^2*b - 4*a*b^2)*cos(f*x + e)^2)*sqrt(a - b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - 5*(a^3 - 2*a^2*b + a*b^2)*cos(f*x + e)^3 - (13*a^2*b - 11*a*b^2 - 2*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f)]","B",0
135,1,908,0,6.274164," ","integrate(sin(f*x+e)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a^{2} + a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{-a + b} \log\left(128 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{8} - 256 \, {\left(a^{4} - 5 \, a^{3} b + 9 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{4} - 34 \, a^{3} b + 77 \, a^{2} b^{2} - 72 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} - 32 \, a^{3} b + 160 \, a^{2} b^{2} - 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{4} - 11 \, a^{3} b + 34 \, a^{2} b^{2} - 40 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 24 \, {\left(a^{3} - 4 \, a^{2} b + 5 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(5 \, a^{3} - 29 \, a^{2} b + 48 \, a b^{2} - 24 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)\right) + 8 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{16 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}, \frac{{\left({\left(a^{2} + a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a b + 2 \, b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{{\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{4 \, {\left(2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{2} b + 3 \, a b^{2} - 2 \, b^{3} - {\left(a^{3} - 6 \, a^{2} b + 9 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} f\right)}}\right]"," ",0,"[-1/16*(((a^2 + a*b - 2*b^2)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(-a + b)*log(128*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^8 - 256*(a^4 - 5*a^3*b + 9*a^2*b^2 - 7*a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^4 - 34*a^3*b + 77*a^2*b^2 - 72*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 - 32*a^3*b + 160*a^2*b^2 - 256*a*b^3 + 128*b^4 - 32*(a^4 - 11*a^3*b + 34*a^2*b^2 - 40*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 24*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cos(f*x + e)^5 + 2*(5*a^3 - 29*a^2*b + 48*a*b^2 - 24*b^3)*cos(f*x + e)^3 - (a^3 - 10*a^2*b + 24*a*b^2 - 16*b^3)*cos(f*x + e))*sqrt(-a + b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 8*((a^2 - 2*a*b + b^2)*cos(f*x + e)^3 + 3*(a*b - b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f), 1/8*(((a^2 + a*b - 2*b^2)*cos(f*x + e)^2 + a*b + 2*b^2)*sqrt(a - b)*arctan(-1/4*(8*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 - 8*(a^2 - 3*a*b + 2*b^2)*cos(f*x + e)^3 + (a^2 - 8*a*b + 8*b^2)*cos(f*x + e))*sqrt(a - b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - a^2*b + 3*a*b^2 - 2*b^3 - (a^3 - 6*a^2*b + 9*a*b^2 - 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 4*((a^2 - 2*a*b + b^2)*cos(f*x + e)^3 + 3*(a*b - b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*f)]","B",0
136,1,310,0,0.481567," ","integrate(1/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{2 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{{\left(a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f}\right]"," ",0,"[1/2*((a*b*tan(f*x + e)^2 + a^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f), ((a*b*tan(f*x + e)^2 + a^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f)]","A",0
137,1,90,0,0.754442," ","integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{3} + 2 \, b \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{{\left(a^{2} b f + {\left(a^{3} - a^{2} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}"," ",0,"-((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2*b*f + (a^3 - a^2*b)*f*cos(f*x + e)^2)*sin(f*x + e))","A",0
138,1,155,0,3.410076," ","integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, {\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{5} - {\left(3 \, a^{2} - 16 \, a b + 16 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, a b - 4 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{4} - a^{3} b\right)} f \cos\left(f x + e\right)^{4} - a^{3} b f - {\left(a^{4} - 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(2*(a^2 - 5*a*b + 4*b^2)*cos(f*x + e)^5 - (3*a^2 - 16*a*b + 16*b^2)*cos(f*x + e)^3 - 2*(3*a*b - 4*b^2)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^4 - 2*a^3*b)*f*cos(f*x + e)^2)*sin(f*x + e))","A",0
139,1,232,0,31.862462," ","integrate(csc(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, {\left(a^{3} - 8 \, a^{2} b + 13 \, a b^{2} - 6 \, b^{3}\right)} \cos\left(f x + e\right)^{7} - 4 \, {\left(5 \, a^{3} - 41 \, a^{2} b + 72 \, a b^{2} - 36 \, b^{3}\right)} \cos\left(f x + e\right)^{5} + {\left(15 \, a^{3} - 130 \, a^{2} b + 264 \, a b^{2} - 144 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(15 \, a^{2} b - 40 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left({\left(a^{5} - a^{4} b\right)} f \cos\left(f x + e\right)^{6} + a^{4} b f - {\left(2 \, a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{5} - 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(8*(a^3 - 8*a^2*b + 13*a*b^2 - 6*b^3)*cos(f*x + e)^7 - 4*(5*a^3 - 41*a^2*b + 72*a*b^2 - 36*b^3)*cos(f*x + e)^5 + (15*a^3 - 130*a^2*b + 264*a*b^2 - 144*b^3)*cos(f*x + e)^3 + 2*(15*a^2*b - 40*a*b^2 + 24*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^5 - a^4*b)*f*cos(f*x + e)^6 + a^4*b*f - (2*a^5 - 3*a^4*b)*f*cos(f*x + e)^4 + (a^5 - 3*a^4*b)*f*cos(f*x + e)^2)*sin(f*x + e))","A",0
140,1,370,0,0.878278," ","integrate(sin(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{9} - 2 \, {\left(5 \, a^{4} - 16 \, a^{3} b + 18 \, a^{2} b^{2} - 8 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{7} + 3 \, {\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{5} + 12 \, {\left(5 \, a^{3} b + 5 \, a^{2} b^{2} - 9 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{3} + 8 \, {\left(5 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{15 \, {\left({\left(a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right)} f\right)}}"," ",0,"-1/15*(3*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^9 - 2*(5*a^4 - 16*a^3*b + 18*a^2*b^2 - 8*a*b^3 + b^4)*cos(f*x + e)^7 + 3*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(f*x + e)^5 + 12*(5*a^3*b + 5*a^2*b^2 - 9*a*b^3 - b^4)*cos(f*x + e)^3 + 8*(5*a^2*b^2 + 10*a*b^3 + b^4)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^7 - 7*a^6*b + 21*a^5*b^2 - 35*a^4*b^3 + 35*a^3*b^4 - 21*a^2*b^5 + 7*a*b^6 - b^7)*f*cos(f*x + e)^4 + 2*(a^6*b - 6*a^5*b^2 + 15*a^4*b^3 - 20*a^3*b^4 + 15*a^2*b^5 - 6*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^5*b^2 - 5*a^4*b^3 + 10*a^3*b^4 - 10*a^2*b^5 + 5*a*b^6 - b^7)*f)","A",0
141,1,270,0,0.725987," ","integrate(sin(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left({\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{7} - 3 \, {\left(a^{3} - a^{2} b - a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} - 12 \, {\left(a^{2} b - b^{3}\right)} \cos\left(f x + e\right)^{3} - 8 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} f\right)}}"," ",0,"1/3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^7 - 3*(a^3 - a^2*b - a*b^2 + b^3)*cos(f*x + e)^5 - 12*(a^2*b - b^3)*cos(f*x + e)^3 - 8*(a*b^2 + b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*f*cos(f*x + e)^4 + 2*(a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*f*cos(f*x + e)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*f)","A",0
142,1,202,0,0.535415," ","integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{5} + 12 \, {\left(a b - b^{2}\right)} \cos\left(f x + e\right)^{3} + 8 \, b^{2} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f\right)}}"," ",0,"-1/3*(3*(a^2 - 2*a*b + b^2)*cos(f*x + e)^5 + 12*(a*b - b^2)*cos(f*x + e)^3 + 8*b^2*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^4 + 2*(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f*cos(f*x + e)^2 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f)","A",0
143,1,696,0,0.624802," ","integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(2 \, a^{3} b - 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left({\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{2} b^{2} - 2 \, a b^{3} + b^{4} + 2 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) - {\left(3 \, {\left(2 \, a^{3} b - 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(5 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left({\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[1/6*(3*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 2*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*(3*(2*a^3*b - 3*a^2*b^2 + a*b^3)*cos(f*x + e)^3 + (5*a^2*b^2 - 3*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f), 1/3*(3*((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*cos(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 2*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - (3*(2*a^3*b - 3*a^2*b^2 + a*b^3)*cos(f*x + e)^3 + (5*a^2*b^2 - 3*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*cos(f*x + e)^2 + (a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*f)]","B",0
144,1,889,0,0.702233," ","integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} - 8 \, a^{3} b + 18 \, a^{2} b^{2} - 16 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 10 \, a^{3} b + 32 \, a^{2} b^{2} - 38 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 6 \, a b^{3} - 5 \, b^{4} - {\left(2 \, a^{3} b - 15 \, a^{2} b^{2} + 28 \, a b^{3} - 15 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(a^{4} - 7 \, a^{3} b + 11 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{3} b - 23 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(13 \, a^{2} b^{2} - 15 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{12 \, {\left({\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} - 5 \, a^{6} b + 7 \, a^{5} b^{2} - 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b - 5 \, a^{5} b^{2} + 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{4} - 8 \, a^{3} b + 18 \, a^{2} b^{2} - 16 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - {\left(a^{4} - 10 \, a^{3} b + 32 \, a^{2} b^{2} - 38 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(f x + e\right)^{4} - a^{2} b^{2} + 6 \, a b^{3} - 5 \, b^{4} - {\left(2 \, a^{3} b - 15 \, a^{2} b^{2} + 28 \, a b^{3} - 15 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + {\left(3 \, {\left(a^{4} - 7 \, a^{3} b + 11 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(9 \, a^{3} b - 23 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(13 \, a^{2} b^{2} - 15 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{6 \, {\left({\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{7} - 5 \, a^{6} b + 7 \, a^{5} b^{2} - 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{6} b - 5 \, a^{5} b^{2} + 3 \, a^{4} b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/12*(3*((a^4 - 8*a^3*b + 18*a^2*b^2 - 16*a*b^3 + 5*b^4)*cos(f*x + e)^6 - (a^4 - 10*a^3*b + 32*a^2*b^2 - 38*a*b^3 + 15*b^4)*cos(f*x + e)^4 - a^2*b^2 + 6*a*b^3 - 5*b^4 - (2*a^3*b - 15*a^2*b^2 + 28*a*b^3 - 15*b^4)*cos(f*x + e)^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) - 2*(3*(a^4 - 7*a^3*b + 11*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^5 + 2*(9*a^3*b - 23*a^2*b^2 + 15*a*b^3)*cos(f*x + e)^3 + (13*a^2*b^2 - 15*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*a^6*b + 7*a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 - (2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)*f*cos(f*x + e)^2 - (a^5*b^2 - a^4*b^3)*f), 1/6*(3*((a^4 - 8*a^3*b + 18*a^2*b^2 - 16*a*b^3 + 5*b^4)*cos(f*x + e)^6 - (a^4 - 10*a^3*b + 32*a^2*b^2 - 38*a*b^3 + 15*b^4)*cos(f*x + e)^4 - a^2*b^2 + 6*a*b^3 - 5*b^4 - (2*a^3*b - 15*a^2*b^2 + 28*a*b^3 - 15*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + (3*(a^4 - 7*a^3*b + 11*a^2*b^2 - 5*a*b^3)*cos(f*x + e)^5 + 2*(9*a^3*b - 23*a^2*b^2 + 15*a*b^3)*cos(f*x + e)^3 + (13*a^2*b^2 - 15*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f*cos(f*x + e)^6 - (a^7 - 5*a^6*b + 7*a^5*b^2 - 3*a^4*b^3)*f*cos(f*x + e)^4 - (2*a^6*b - 5*a^5*b^2 + 3*a^4*b^3)*f*cos(f*x + e)^2 - (a^5*b^2 - a^4*b^3)*f)]","B",0
145,1,1037,0,0.798712," ","integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(3 \, a^{4} - 36 \, a^{3} b + 98 \, a^{2} b^{2} - 100 \, a b^{3} + 35 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(3 \, a^{4} - 39 \, a^{3} b + 131 \, a^{2} b^{2} - 165 \, a b^{3} + 70 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(3 \, a^{4} - 48 \, a^{3} b + 233 \, a^{2} b^{2} - 390 \, a b^{3} + 210 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} - 30 \, a b^{3} + 35 \, b^{4} + 2 \, {\left(3 \, a^{3} b - 36 \, a^{2} b^{2} + 95 \, a b^{3} - 70 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right) + a + b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(3 \, {\left(3 \, a^{4} - 33 \, a^{3} b + 65 \, a^{2} b^{2} - 35 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} - {\left(15 \, a^{4} - 177 \, a^{3} b + 445 \, a^{2} b^{2} - 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - {\left(78 \, a^{3} b - 305 \, a^{2} b^{2} + 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(11 \, a^{2} b^{2} - 21 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{48 \, {\left({\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{8} + a^{5} b^{2} f - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} - 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{3 \, {\left({\left(3 \, a^{4} - 36 \, a^{3} b + 98 \, a^{2} b^{2} - 100 \, a b^{3} + 35 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(3 \, a^{4} - 39 \, a^{3} b + 131 \, a^{2} b^{2} - 165 \, a b^{3} + 70 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(3 \, a^{4} - 48 \, a^{3} b + 233 \, a^{2} b^{2} - 390 \, a b^{3} + 210 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b^{2} - 30 \, a b^{3} + 35 \, b^{4} + 2 \, {\left(3 \, a^{3} b - 36 \, a^{2} b^{2} + 95 \, a b^{3} - 70 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}} \cos\left(f x + e\right)}{a}\right) + {\left(3 \, {\left(3 \, a^{4} - 33 \, a^{3} b + 65 \, a^{2} b^{2} - 35 \, a b^{3}\right)} \cos\left(f x + e\right)^{7} - {\left(15 \, a^{4} - 177 \, a^{3} b + 445 \, a^{2} b^{2} - 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{5} - {\left(78 \, a^{3} b - 305 \, a^{2} b^{2} + 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(11 \, a^{2} b^{2} - 21 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{24 \, {\left({\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{8} + a^{5} b^{2} f - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} - 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/48*(3*((3*a^4 - 36*a^3*b + 98*a^2*b^2 - 100*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(3*a^4 - 39*a^3*b + 131*a^2*b^2 - 165*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (3*a^4 - 48*a^3*b + 233*a^2*b^2 - 390*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 30*a*b^3 + 35*b^4 + 2*(3*a^3*b - 36*a^2*b^2 + 95*a*b^3 - 70*b^4)*cos(f*x + e)^2)*sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1)) + 2*(3*(3*a^4 - 33*a^3*b + 65*a^2*b^2 - 35*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 177*a^3*b + 445*a^2*b^2 - 315*a*b^3)*cos(f*x + e)^5 - (78*a^3*b - 305*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^3 - 5*(11*a^2*b^2 - 21*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2), 1/24*(3*((3*a^4 - 36*a^3*b + 98*a^2*b^2 - 100*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(3*a^4 - 39*a^3*b + 131*a^2*b^2 - 165*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (3*a^4 - 48*a^3*b + 233*a^2*b^2 - 390*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 3*a^2*b^2 - 30*a*b^3 + 35*b^4 + 2*(3*a^3*b - 36*a^2*b^2 + 95*a*b^3 - 70*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + (3*(3*a^4 - 33*a^3*b + 65*a^2*b^2 - 35*a*b^3)*cos(f*x + e)^7 - (15*a^4 - 177*a^3*b + 445*a^2*b^2 - 315*a*b^3)*cos(f*x + e)^5 - (78*a^3*b - 305*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^3 - 5*(11*a^2*b^2 - 21*a*b^3)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/((a^7 - 2*a^6*b + a^5*b^2)*f*cos(f*x + e)^8 + a^5*b^2*f - 2*(a^7 - 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 - 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 + 2*(a^6*b - 2*a^5*b^2)*f*cos(f*x + e)^2)]","B",0
146,-1,0,0,0.000000," ","integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
147,-1,0,0,0.000000," ","integrate(sin(f*x+e)^2/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
148,1,561,0,1.941217," ","integrate(1/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{3} b \tan\left(f x + e\right)^{2} + a^{4}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(5 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(2 \, a^{3} b - 3 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}, \frac{3 \, {\left(a^{2} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{3} b \tan\left(f x + e\right)^{2} + a^{4}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left({\left(5 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(2 \, a^{3} b - 3 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{3 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/6*(3*(a^2*b^2*tan(f*x + e)^4 + 2*a^3*b*tan(f*x + e)^2 + a^4)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*tan(f*x + e)^3 + 3*(2*a^3*b - 3*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f), 1/3*(3*(a^2*b^2*tan(f*x + e)^4 + 2*a^3*b*tan(f*x + e)^2 + a^4)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - ((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*tan(f*x + e)^3 + 3*(2*a^3*b - 3*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f)]","B",0
149,1,156,0,10.473545," ","integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left({\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + 4 \, {\left(3 \, a b - 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 8 \, b^{2} \cos\left(f x + e\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(f x + e\right)^{2} + b}{\cos\left(f x + e\right)^{2}}}}{3 \, {\left(a^{3} b^{2} f + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*((3*a^2 - 12*a*b + 8*b^2)*cos(f*x + e)^5 + 4*(3*a*b - 4*b^2)*cos(f*x + e)^3 + 8*b^2*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^3*b^2*f + (a^5 - 2*a^4*b + a^3*b^2)*f*cos(f*x + e)^4 + 2*(a^4*b - a^3*b^2)*f*cos(f*x + e)^2)*sin(f*x + e))","A",0
150,-1,0,0,0.000000," ","integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
151,-1,0,0,0.000000," ","integrate(csc(f*x+e)^6/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
152,0,0,0,0.718131," ","integrate((d*sin(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p*(d*sin(f*x + e))^m, x)","F",0
153,0,0,0,1.375681," ","integrate((d*sin(f*x+e))^m*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*(d*sin(f*x + e))^m, x)","F",0
154,0,0,0,0.564533," ","integrate(sin(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} {\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(b*tan(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
155,0,0,0,0.796170," ","integrate(sin(f*x+e)^3*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(b*tan(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
156,0,0,0,0.767959," ","integrate(sin(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
157,0,0,0,0.487277," ","integrate(csc(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right), x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*csc(f*x + e), x)","F",0
158,0,0,0,0.920662," ","integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^3, x)","F",0
159,0,0,0,0.423853," ","integrate(sin(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(b*tan(f*x + e)^2 + a)^p, x)","F",0
160,0,0,0,0.408956," ","integrate((a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p, x)","F",0
161,0,0,0,0.452099," ","integrate(csc(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^2, x)","F",0
162,0,0,0,0.498302," ","integrate(csc(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^4, x)","F",0
163,0,0,0,0.550778," ","integrate(csc(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{6}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*csc(f*x + e)^6, x)","F",0
164,0,0,0,0.490309," ","integrate((d*sin(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*(d*sin(f*x + e))^m, x)","F",0
165,0,0,0,0.418426," ","integrate(sin(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} \left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p}, x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*((c*tan(f*x + e))^n*b)^p, x)","F",0
166,0,0,0,0.433062," ","integrate((b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p, x)","F",0
167,1,51,0,0.450214," ","integrate(csc(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","\frac{\cos\left(f x + e\right) e^{\left(n p \log\left(\frac{c \sin\left(f x + e\right)}{\cos\left(f x + e\right)}\right) + p \log\left(b\right)\right)}}{{\left(f n p - f\right)} \sin\left(f x + e\right)}"," ",0,"cos(f*x + e)*e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))/((f*n*p - f)*sin(f*x + e))","A",0
168,1,104,0,0.425626," ","integrate(csc(f*x+e)^4*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","\frac{{\left(2 \, \cos\left(f x + e\right)^{3} + {\left(n p - 3\right)} \cos\left(f x + e\right)\right)} e^{\left(n p \log\left(\frac{c \sin\left(f x + e\right)}{\cos\left(f x + e\right)}\right) + p \log\left(b\right)\right)}}{{\left(f n^{2} p^{2} - 4 \, f n p - {\left(f n^{2} p^{2} - 4 \, f n p + 3 \, f\right)} \cos\left(f x + e\right)^{2} + 3 \, f\right)} \sin\left(f x + e\right)}"," ",0,"(2*cos(f*x + e)^3 + (n*p - 3)*cos(f*x + e))*e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))/((f*n^2*p^2 - 4*f*n*p - (f*n^2*p^2 - 4*f*n*p + 3*f)*cos(f*x + e)^2 + 3*f)*sin(f*x + e))","A",0
169,1,180,0,0.438257," ","integrate(csc(f*x+e)^6*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","\frac{{\left(8 \, \cos\left(f x + e\right)^{5} + 4 \, {\left(n p - 5\right)} \cos\left(f x + e\right)^{3} + {\left(n^{2} p^{2} - 8 \, n p + 15\right)} \cos\left(f x + e\right)\right)} e^{\left(n p \log\left(\frac{c \sin\left(f x + e\right)}{\cos\left(f x + e\right)}\right) + p \log\left(b\right)\right)}}{{\left(f n^{3} p^{3} - 9 \, f n^{2} p^{2} + {\left(f n^{3} p^{3} - 9 \, f n^{2} p^{2} + 23 \, f n p - 15 \, f\right)} \cos\left(f x + e\right)^{4} + 23 \, f n p - 2 \, {\left(f n^{3} p^{3} - 9 \, f n^{2} p^{2} + 23 \, f n p - 15 \, f\right)} \cos\left(f x + e\right)^{2} - 15 \, f\right)} \sin\left(f x + e\right)}"," ",0,"(8*cos(f*x + e)^5 + 4*(n*p - 5)*cos(f*x + e)^3 + (n^2*p^2 - 8*n*p + 15)*cos(f*x + e))*e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))/((f*n^3*p^3 - 9*f*n^2*p^2 + (f*n^3*p^3 - 9*f*n^2*p^2 + 23*f*n*p - 15*f)*cos(f*x + e)^4 + 23*f*n*p - 2*(f*n^3*p^3 - 9*f*n^2*p^2 + 23*f*n*p - 15*f)*cos(f*x + e)^2 - 15*f)*sin(f*x + e))","A",0
170,0,0,0,0.440981," ","integrate(sin(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} \left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*((c*tan(f*x + e))^n*b)^p*sin(f*x + e), x)","F",0
171,0,0,0,0.437689," ","integrate(sin(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*sin(f*x + e), x)","F",0
172,0,0,0,0.461626," ","integrate(csc(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \csc\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*csc(f*x + e), x)","F",0
173,0,0,0,0.443959," ","integrate(csc(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \csc\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*csc(f*x + e)^3, x)","F",0
174,0,0,0,0.491504," ","integrate((d*sin(f*x+e))^m*(a+b*tan(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{n} + a\right)}^{p} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^n + a)^p*(d*sin(f*x + e))^m, x)","F",0
175,0,0,0,0.464096," ","integrate((d*cos(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p} \left(d \cos\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p*(d*cos(f*x + e))^m, x)","F",0
176,0,0,0,0.491941," ","integrate((d*cos(f*x+e))^m*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \left(d \cos\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*(d*cos(f*x + e))^m, x)","F",0
177,0,0,0,0.444707," ","integrate((d*cos(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \left(d \cos\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*(d*cos(f*x + e))^m, x)","F",0
178,0,0,0,0.455492," ","integrate((d*cos(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \left(d \cos\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*(d*cos(f*x + e))^m, x)","F",0
179,1,56,0,0.398018," ","integrate((a+a*tan(d*x+c)^2)^4,x, algorithm=""fricas"")","\frac{5 \, a^{4} \tan\left(d x + c\right)^{7} + 21 \, a^{4} \tan\left(d x + c\right)^{5} + 35 \, a^{4} \tan\left(d x + c\right)^{3} + 35 \, a^{4} \tan\left(d x + c\right)}{35 \, d}"," ",0,"1/35*(5*a^4*tan(d*x + c)^7 + 21*a^4*tan(d*x + c)^5 + 35*a^4*tan(d*x + c)^3 + 35*a^4*tan(d*x + c))/d","A",0
180,1,43,0,0.394593," ","integrate((a+a*tan(d*x+c)^2)^3,x, algorithm=""fricas"")","\frac{3 \, a^{3} \tan\left(d x + c\right)^{5} + 10 \, a^{3} \tan\left(d x + c\right)^{3} + 15 \, a^{3} \tan\left(d x + c\right)}{15 \, d}"," ",0,"1/15*(3*a^3*tan(d*x + c)^5 + 10*a^3*tan(d*x + c)^3 + 15*a^3*tan(d*x + c))/d","A",0
181,1,29,0,0.400437," ","integrate((a+a*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{a^{2} \tan\left(d x + c\right)^{3} + 3 \, a^{2} \tan\left(d x + c\right)}{3 \, d}"," ",0,"1/3*(a^2*tan(d*x + c)^3 + 3*a^2*tan(d*x + c))/d","A",0
182,1,40,0,0.423814," ","integrate(1/(a+a*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{d x \tan\left(d x + c\right)^{2} + d x + \tan\left(d x + c\right)}{2 \, {\left(a d \tan\left(d x + c\right)^{2} + a d\right)}}"," ",0,"1/2*(d*x*tan(d*x + c)^2 + d*x + tan(d*x + c))/(a*d*tan(d*x + c)^2 + a*d)","A",0
183,1,84,0,0.404732," ","integrate(1/(a+a*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{3 \, d x \tan\left(d x + c\right)^{4} + 6 \, d x \tan\left(d x + c\right)^{2} + 3 \, \tan\left(d x + c\right)^{3} + 3 \, d x + 5 \, \tan\left(d x + c\right)}{8 \, {\left(a^{2} d \tan\left(d x + c\right)^{4} + 2 \, a^{2} d \tan\left(d x + c\right)^{2} + a^{2} d\right)}}"," ",0,"1/8*(3*d*x*tan(d*x + c)^4 + 6*d*x*tan(d*x + c)^2 + 3*tan(d*x + c)^3 + 3*d*x + 5*tan(d*x + c))/(a^2*d*tan(d*x + c)^4 + 2*a^2*d*tan(d*x + c)^2 + a^2*d)","A",0
184,1,120,0,0.394016," ","integrate(1/(a+a*tan(d*x+c)^2)^3,x, algorithm=""fricas"")","\frac{15 \, d x \tan\left(d x + c\right)^{6} + 45 \, d x \tan\left(d x + c\right)^{4} + 15 \, \tan\left(d x + c\right)^{5} + 45 \, d x \tan\left(d x + c\right)^{2} + 40 \, \tan\left(d x + c\right)^{3} + 15 \, d x + 33 \, \tan\left(d x + c\right)}{48 \, {\left(a^{3} d \tan\left(d x + c\right)^{6} + 3 \, a^{3} d \tan\left(d x + c\right)^{4} + 3 \, a^{3} d \tan\left(d x + c\right)^{2} + a^{3} d\right)}}"," ",0,"1/48*(15*d*x*tan(d*x + c)^6 + 45*d*x*tan(d*x + c)^4 + 15*tan(d*x + c)^5 + 45*d*x*tan(d*x + c)^2 + 40*tan(d*x + c)^3 + 15*d*x + 33*tan(d*x + c))/(a^3*d*tan(d*x + c)^6 + 3*a^3*d*tan(d*x + c)^4 + 3*a^3*d*tan(d*x + c)^2 + a^3*d)","A",0
185,1,67,0,0.448689," ","integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{2 \, b \tan\left(f x + e\right)^{6} + 3 \, {\left(a - b\right)} \tan\left(f x + e\right)^{4} - 6 \, {\left(a - b\right)} \tan\left(f x + e\right)^{2} - 6 \, {\left(a - b\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{12 \, f}"," ",0,"1/12*(2*b*tan(f*x + e)^6 + 3*(a - b)*tan(f*x + e)^4 - 6*(a - b)*tan(f*x + e)^2 - 6*(a - b)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
186,1,51,0,0.415645," ","integrate(tan(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{b \tan\left(f x + e\right)^{4} + 2 \, {\left(a - b\right)} \tan\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, f}"," ",0,"1/4*(b*tan(f*x + e)^4 + 2*(a - b)*tan(f*x + e)^2 + 2*(a - b)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
187,1,36,0,0.419858," ","integrate(tan(f*x+e)*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{b \tan\left(f x + e\right)^{2} - {\left(a - b\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}"," ",0,"1/2*(b*tan(f*x + e)^2 - (a - b)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
188,1,46,0,0.459066," ","integrate(cot(f*x+e)*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{a \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) - b \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}"," ",0,"1/2*(a*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) - b*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
189,1,61,0,0.419707," ","integrate(cot(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a - b\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + a \tan\left(f x + e\right)^{2} + a}{2 \, f \tan\left(f x + e\right)^{2}}"," ",0,"-1/2*((a - b)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + a*tan(f*x + e)^2 + a)/(f*tan(f*x + e)^2)","A",0
190,1,85,0,0.414494," ","integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{2 \, {\left(a - b\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left(3 \, a - 2 \, b\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a - b\right)} \tan\left(f x + e\right)^{2} - a}{4 \, f \tan\left(f x + e\right)^{4}}"," ",0,"1/4*(2*(a - b)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (3*a - 2*b)*tan(f*x + e)^4 + 2*(a - b)*tan(f*x + e)^2 - a)/(f*tan(f*x + e)^4)","A",0
191,1,69,0,0.410793," ","integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{15 \, b \tan\left(f x + e\right)^{7} + 21 \, {\left(a - b\right)} \tan\left(f x + e\right)^{5} - 35 \, {\left(a - b\right)} \tan\left(f x + e\right)^{3} - 105 \, {\left(a - b\right)} f x + 105 \, {\left(a - b\right)} \tan\left(f x + e\right)}{105 \, f}"," ",0,"1/105*(15*b*tan(f*x + e)^7 + 21*(a - b)*tan(f*x + e)^5 - 35*(a - b)*tan(f*x + e)^3 - 105*(a - b)*f*x + 105*(a - b)*tan(f*x + e))/f","A",0
192,1,54,0,0.418817," ","integrate(tan(f*x+e)^4*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, b \tan\left(f x + e\right)^{5} + 5 \, {\left(a - b\right)} \tan\left(f x + e\right)^{3} + 15 \, {\left(a - b\right)} f x - 15 \, {\left(a - b\right)} \tan\left(f x + e\right)}{15 \, f}"," ",0,"1/15*(3*b*tan(f*x + e)^5 + 5*(a - b)*tan(f*x + e)^3 + 15*(a - b)*f*x - 15*(a - b)*tan(f*x + e))/f","A",0
193,1,38,0,0.431695," ","integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{b \tan\left(f x + e\right)^{3} - 3 \, {\left(a - b\right)} f x + 3 \, {\left(a - b\right)} \tan\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(b*tan(f*x + e)^3 - 3*(a - b)*f*x + 3*(a - b)*tan(f*x + e))/f","A",0
194,1,21,0,0.399644," ","integrate(a+b*tan(f*x+e)^2,x, algorithm=""fricas"")","\frac{{\left(a - b\right)} f x + b \tan\left(f x + e\right)}{f}"," ",0,"((a - b)*f*x + b*tan(f*x + e))/f","A",0
195,1,29,0,0.407615," ","integrate(cot(f*x+e)^2*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{{\left(a - b\right)} f x \tan\left(f x + e\right) + a}{f \tan\left(f x + e\right)}"," ",0,"-((a - b)*f*x*tan(f*x + e) + a)/(f*tan(f*x + e))","A",0
196,1,49,0,0.421711," ","integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(a - b\right)} f x \tan\left(f x + e\right)^{3} + 3 \, {\left(a - b\right)} \tan\left(f x + e\right)^{2} - a}{3 \, f \tan\left(f x + e\right)^{3}}"," ",0,"1/3*(3*(a - b)*f*x*tan(f*x + e)^3 + 3*(a - b)*tan(f*x + e)^2 - a)/(f*tan(f*x + e)^3)","A",0
197,1,64,0,0.396698," ","integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{15 \, {\left(a - b\right)} f x \tan\left(f x + e\right)^{5} + 15 \, {\left(a - b\right)} \tan\left(f x + e\right)^{4} - 5 \, {\left(a - b\right)} \tan\left(f x + e\right)^{2} + 3 \, a}{15 \, f \tan\left(f x + e\right)^{5}}"," ",0,"-1/15*(15*(a - b)*f*x*tan(f*x + e)^5 + 15*(a - b)*tan(f*x + e)^4 - 5*(a - b)*tan(f*x + e)^2 + 3*a)/(f*tan(f*x + e)^5)","A",0
198,1,107,0,0.459152," ","integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} \tan\left(f x + e\right)^{8} + 4 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{6} + 6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{4} - 12 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{2} - 12 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{24 \, f}"," ",0,"1/24*(3*b^2*tan(f*x + e)^8 + 4*(2*a*b - b^2)*tan(f*x + e)^6 + 6*(a^2 - 2*a*b + b^2)*tan(f*x + e)^4 - 12*(a^2 - 2*a*b + b^2)*tan(f*x + e)^2 - 12*(a^2 - 2*a*b + b^2)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
199,1,86,0,0.471064," ","integrate(tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{2 \, b^{2} \tan\left(f x + e\right)^{6} + 3 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{4} + 6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{2} + 6 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{12 \, f}"," ",0,"1/12*(2*b^2*tan(f*x + e)^6 + 3*(2*a*b - b^2)*tan(f*x + e)^4 + 6*(a^2 - 2*a*b + b^2)*tan(f*x + e)^2 + 6*(a^2 - 2*a*b + b^2)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
200,1,64,0,0.424270," ","integrate(tan(f*x+e)*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{2} - 2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, f}"," ",0,"1/4*(b^2*tan(f*x + e)^4 + 2*(2*a*b - b^2)*tan(f*x + e)^2 - 2*(a^2 - 2*a*b + b^2)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
201,1,69,0,0.424176," ","integrate(cot(f*x+e)*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{b^{2} \tan\left(f x + e\right)^{2} + a^{2} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(2 \, a b - b^{2}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}"," ",0,"1/2*(b^2*tan(f*x + e)^2 + a^2*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) - (2*a*b - b^2)*log(1/(tan(f*x + e)^2 + 1)))/f","A",0
202,1,93,0,0.444833," ","integrate(cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{b^{2} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + a^{2} \tan\left(f x + e\right)^{2} + {\left(a^{2} - 2 \, a b\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + a^{2}}{2 \, f \tan\left(f x + e\right)^{2}}"," ",0,"-1/2*(b^2*log(1/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + a^2*tan(f*x + e)^2 + (a^2 - 2*a*b)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + a^2)/(f*tan(f*x + e)^2)","A",0
203,1,99,0,0.439065," ","integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{2} - 2 \, a b\right)} \tan\left(f x + e\right)^{2} - a^{2}}{4 \, f \tan\left(f x + e\right)^{4}}"," ",0,"1/4*(2*(a^2 - 2*a*b + b^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (3*a^2 - 4*a*b)*tan(f*x + e)^4 + 2*(a^2 - 2*a*b)*tan(f*x + e)^2 - a^2)/(f*tan(f*x + e)^4)","A",0
204,1,115,0,0.441307," ","integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{35 \, b^{2} \tan\left(f x + e\right)^{9} + 45 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{7} + 63 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{5} - 105 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{3} - 315 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x + 315 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)}{315 \, f}"," ",0,"1/315*(35*b^2*tan(f*x + e)^9 + 45*(2*a*b - b^2)*tan(f*x + e)^7 + 63*(a^2 - 2*a*b + b^2)*tan(f*x + e)^5 - 105*(a^2 - 2*a*b + b^2)*tan(f*x + e)^3 - 315*(a^2 - 2*a*b + b^2)*f*x + 315*(a^2 - 2*a*b + b^2)*tan(f*x + e))/f","A",0
205,1,94,0,0.414087," ","integrate(tan(f*x+e)^4*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{15 \, b^{2} \tan\left(f x + e\right)^{7} + 21 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{5} + 35 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{3} + 105 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x - 105 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)}{105 \, f}"," ",0,"1/105*(15*b^2*tan(f*x + e)^7 + 21*(2*a*b - b^2)*tan(f*x + e)^5 + 35*(a^2 - 2*a*b + b^2)*tan(f*x + e)^3 + 105*(a^2 - 2*a*b + b^2)*f*x - 105*(a^2 - 2*a*b + b^2)*tan(f*x + e))/f","A",0
206,1,73,0,0.414468," ","integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} \tan\left(f x + e\right)^{5} + 5 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{3} - 15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x + 15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)}{15 \, f}"," ",0,"1/15*(3*b^2*tan(f*x + e)^5 + 5*(2*a*b - b^2)*tan(f*x + e)^3 - 15*(a^2 - 2*a*b + b^2)*f*x + 15*(a^2 - 2*a*b + b^2)*tan(f*x + e))/f","A",0
207,1,51,0,0.422338," ","integrate((a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{b^{2} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x + 3 \, {\left(2 \, a b - b^{2}\right)} \tan\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(b^2*tan(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*x + 3*(2*a*b - b^2)*tan(f*x + e))/f","A",0
208,1,50,0,0.410227," ","integrate(cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(a^{2} - 2 \, a b + b^{2}\right)} f x \tan\left(f x + e\right) - b^{2} \tan\left(f x + e\right)^{2} + a^{2}}{f \tan\left(f x + e\right)}"," ",0,"-((a^2 - 2*a*b + b^2)*f*x*tan(f*x + e) - b^2*tan(f*x + e)^2 + a^2)/(f*tan(f*x + e))","A",0
209,1,60,0,0.423024," ","integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} - 2 \, a b\right)} \tan\left(f x + e\right)^{2} - a^{2}}{3 \, f \tan\left(f x + e\right)^{3}}"," ",0,"1/3*(3*(a^2 - 2*a*b + b^2)*f*x*tan(f*x + e)^3 + 3*(a^2 - 2*a*b)*tan(f*x + e)^2 - a^2)/(f*tan(f*x + e)^3)","A",0
210,1,81,0,0.408316," ","integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} f x \tan\left(f x + e\right)^{5} + 15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{4} - 5 \, {\left(a^{2} - 2 \, a b\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2}}{15 \, f \tan\left(f x + e\right)^{5}}"," ",0,"-1/15*(15*(a^2 - 2*a*b + b^2)*f*x*tan(f*x + e)^5 + 15*(a^2 - 2*a*b + b^2)*tan(f*x + e)^4 - 5*(a^2 - 2*a*b)*tan(f*x + e)^2 + 3*a^2)/(f*tan(f*x + e)^5)","A",0
211,1,92,0,0.476146," ","integrate(tan(f*x+e)^5/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{a^{2} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{2} - {\left(a^{2} - b^{2}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a b^{2} - b^{3}\right)} f}"," ",0,"-1/2*(a^2*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)) - (a*b - b^2)*tan(f*x + e)^2 - (a^2 - b^2)*log(1/(tan(f*x + e)^2 + 1)))/((a*b^2 - b^3)*f)","A",0
212,1,65,0,0.449533," ","integrate(tan(f*x+e)^3/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{a \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(a - b\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a b - b^{2}\right)} f}"," ",0,"1/2*(a*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)) - (a - b)*log(1/(tan(f*x + e)^2 + 1)))/((a*b - b^2)*f)","A",0
213,1,38,0,0.427619," ","integrate(tan(f*x+e)/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{\log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a - b\right)} f}"," ",0,"-1/2*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1))/((a - b)*f)","A",0
214,1,72,0,0.451551," ","integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(a - b\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + b \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a^{2} - a b\right)} f}"," ",0,"1/2*((a - b)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + b*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^2 - a*b)*f)","A",0
215,1,128,0,0.461076," ","integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","-\frac{b^{2} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + {\left(a^{2} - b^{2}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + {\left(a^{2} - a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - a b}{2 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{2}}"," ",0,"-1/2*(b^2*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + (a^2 - b^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + (a^2 - a*b)*tan(f*x + e)^2 + a^2 - a*b)/((a^3 - a^2*b)*f*tan(f*x + e)^2)","A",0
216,1,163,0,0.454640," ","integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\frac{2 \, b^{3} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{3} - b^{3}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left(3 \, a^{3} - a^{2} b - 2 \, a b^{2}\right)} \tan\left(f x + e\right)^{4} - a^{3} + a^{2} b + 2 \, {\left(a^{3} - a b^{2}\right)} \tan\left(f x + e\right)^{2}}{4 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{4}}"," ",0,"1/4*(2*b^3*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + 2*(a^3 - b^3)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (3*a^3 - a^2*b - 2*a*b^2)*tan(f*x + e)^4 - a^3 + a^2*b + 2*(a^3 - a*b^2)*tan(f*x + e)^2)/((a^4 - a^3*b)*f*tan(f*x + e)^4)","A",0
217,1,278,0,0.488541," ","integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{12 \, b^{2} f x - 4 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{3} + 3 \, a^{2} \sqrt{-\frac{a}{b}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b^{2} \tan\left(f x + e\right)^{3} - a b \tan\left(f x + e\right)\right)} \sqrt{-\frac{a}{b}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) + 12 \, {\left(a^{2} - b^{2}\right)} \tan\left(f x + e\right)}{12 \, {\left(a b^{2} - b^{3}\right)} f}, -\frac{6 \, b^{2} f x - 2 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{3} - 3 \, a^{2} \sqrt{\frac{a}{b}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{a}{b}}}{2 \, a \tan\left(f x + e\right)}\right) + 6 \, {\left(a^{2} - b^{2}\right)} \tan\left(f x + e\right)}{6 \, {\left(a b^{2} - b^{3}\right)} f}\right]"," ",0,"[-1/12*(12*b^2*f*x - 4*(a*b - b^2)*tan(f*x + e)^3 + 3*a^2*sqrt(-a/b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) + 12*(a^2 - b^2)*tan(f*x + e))/((a*b^2 - b^3)*f), -1/6*(6*b^2*f*x - 2*(a*b - b^2)*tan(f*x + e)^3 - 3*a^2*sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))) + 6*(a^2 - b^2)*tan(f*x + e))/((a*b^2 - b^3)*f)]","A",0
218,1,220,0,0.451751," ","integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, b f x - a \sqrt{-\frac{a}{b}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(b^{2} \tan\left(f x + e\right)^{3} - a b \tan\left(f x + e\right)\right)} \sqrt{-\frac{a}{b}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) + 4 \, {\left(a - b\right)} \tan\left(f x + e\right)}{4 \, {\left(a b - b^{2}\right)} f}, \frac{2 \, b f x - a \sqrt{\frac{a}{b}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{a}{b}}}{2 \, a \tan\left(f x + e\right)}\right) + 2 \, {\left(a - b\right)} \tan\left(f x + e\right)}{2 \, {\left(a b - b^{2}\right)} f}\right]"," ",0,"[1/4*(4*b*f*x - a*sqrt(-a/b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) + 4*(a - b)*tan(f*x + e))/((a*b - b^2)*f), 1/2*(2*b*f*x - a*sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))) + 2*(a - b)*tan(f*x + e))/((a*b - b^2)*f)]","A",0
219,1,181,0,0.448345," ","integrate(tan(f*x+e)^2/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, f x + \sqrt{-\frac{a}{b}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b^{2} \tan\left(f x + e\right)^{3} - a b \tan\left(f x + e\right)\right)} \sqrt{-\frac{a}{b}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{4 \, {\left(a - b\right)} f}, -\frac{2 \, f x - \sqrt{\frac{a}{b}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{a}{b}}}{2 \, a \tan\left(f x + e\right)}\right)}{2 \, {\left(a - b\right)} f}\right]"," ",0,"[-1/4*(4*f*x + sqrt(-a/b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a - b)*f), -1/2*(2*f*x - sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))))/((a - b)*f)]","A",0
220,1,182,0,0.461703," ","integrate(1/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{4 \, f x - \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{4 \, {\left(a - b\right)} f}, \frac{2 \, f x - \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{2 \, {\left(a - b\right)} f}\right]"," ",0,"[1/4*(4*f*x - sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a - b)*f), 1/2*(2*f*x - sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a - b)*f)]","A",0
221,1,243,0,0.472901," ","integrate(cot(f*x+e)^2/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, a f x \tan\left(f x + e\right) + b \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) \tan\left(f x + e\right) + 4 \, a - 4 \, b}{4 \, {\left(a^{2} - a b\right)} f \tan\left(f x + e\right)}, -\frac{2 \, a f x \tan\left(f x + e\right) - b \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) \tan\left(f x + e\right) + 2 \, a - 2 \, b}{2 \, {\left(a^{2} - a b\right)} f \tan\left(f x + e\right)}\right]"," ",0,"[-1/4*(4*a*f*x*tan(f*x + e) + b*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2))*tan(f*x + e) + 4*a - 4*b)/((a^2 - a*b)*f*tan(f*x + e)), -1/2*(2*a*f*x*tan(f*x + e) - b*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e)))*tan(f*x + e) + 2*a - 2*b)/((a^2 - a*b)*f*tan(f*x + e))]","A",0
222,1,308,0,0.506820," ","integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[\frac{12 \, a^{2} f x \tan\left(f x + e\right)^{3} - 3 \, b^{2} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) \tan\left(f x + e\right)^{3} + 12 \, {\left(a^{2} - b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, a^{2} + 4 \, a b}{12 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{3}}, \frac{6 \, a^{2} f x \tan\left(f x + e\right)^{3} - 3 \, b^{2} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) \tan\left(f x + e\right)^{3} + 6 \, {\left(a^{2} - b^{2}\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2} + 2 \, a b}{6 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{3}}\right]"," ",0,"[1/12*(12*a^2*f*x*tan(f*x + e)^3 - 3*b^2*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2))*tan(f*x + e)^3 + 12*(a^2 - b^2)*tan(f*x + e)^2 - 4*a^2 + 4*a*b)/((a^3 - a^2*b)*f*tan(f*x + e)^3), 1/6*(6*a^2*f*x*tan(f*x + e)^3 - 3*b^2*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e)))*tan(f*x + e)^3 + 6*(a^2 - b^2)*tan(f*x + e)^2 - 2*a^2 + 2*a*b)/((a^3 - a^2*b)*f*tan(f*x + e)^3)]","A",0
223,1,352,0,0.459374," ","integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm=""fricas"")","\left[-\frac{60 \, a^{3} f x \tan\left(f x + e\right)^{5} + 15 \, b^{3} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) \tan\left(f x + e\right)^{5} + 60 \, {\left(a^{3} - b^{3}\right)} \tan\left(f x + e\right)^{4} + 12 \, a^{3} - 12 \, a^{2} b - 20 \, {\left(a^{3} - a b^{2}\right)} \tan\left(f x + e\right)^{2}}{60 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{5}}, -\frac{30 \, a^{3} f x \tan\left(f x + e\right)^{5} - 15 \, b^{3} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) \tan\left(f x + e\right)^{5} + 30 \, {\left(a^{3} - b^{3}\right)} \tan\left(f x + e\right)^{4} + 6 \, a^{3} - 6 \, a^{2} b - 10 \, {\left(a^{3} - a b^{2}\right)} \tan\left(f x + e\right)^{2}}{30 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{5}}\right]"," ",0,"[-1/60*(60*a^3*f*x*tan(f*x + e)^5 + 15*b^3*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2))*tan(f*x + e)^5 + 60*(a^3 - b^3)*tan(f*x + e)^4 + 12*a^3 - 12*a^2*b - 20*(a^3 - a*b^2)*tan(f*x + e)^2)/((a^4 - a^3*b)*f*tan(f*x + e)^5), -1/30*(30*a^3*f*x*tan(f*x + e)^5 - 15*b^3*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e)))*tan(f*x + e)^5 + 30*(a^3 - b^3)*tan(f*x + e)^4 + 6*a^3 - 6*a^2*b - 10*(a^3 - a*b^2)*tan(f*x + e)^2)/((a^4 - a^3*b)*f*tan(f*x + e)^5)]","A",0
224,1,186,0,0.446184," ","integrate(tan(f*x+e)^5/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{a^{2} b \tan\left(f x + e\right)^{2} + a^{2} b - {\left(a^{3} - 2 \, a^{2} b + {\left(a^{2} b - 2 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}"," ",0,"-1/2*(a^2*b*tan(f*x + e)^2 + a^2*b - (a^3 - 2*a^2*b + (a^2*b - 2*a*b^2)*tan(f*x + e)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)) + (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*log(1/(tan(f*x + e)^2 + 1)))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f)","B",0
225,1,98,0,0.427440," ","integrate(tan(f*x+e)^3/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{a \tan\left(f x + e\right)^{2} + {\left(b \tan\left(f x + e\right)^{2} + a\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) + a}{2 \, {\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} f\right)}}"," ",0,"1/2*(a*tan(f*x + e)^2 + (b*tan(f*x + e)^2 + a)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)) + a)/((a^2*b - 2*a*b^2 + b^3)*f*tan(f*x + e)^2 + (a^3 - 2*a^2*b + a*b^2)*f)","A",0
226,1,98,0,0.445820," ","integrate(tan(f*x+e)/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{b \tan\left(f x + e\right)^{2} + {\left(b \tan\left(f x + e\right)^{2} + a\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right) + b}{2 \, {\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} f\right)}}"," ",0,"-1/2*(b*tan(f*x + e)^2 + (b*tan(f*x + e)^2 + a)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)) + b)/((a^2*b - 2*a*b^2 + b^3)*f*tan(f*x + e)^2 + (a^3 - 2*a^2*b + a*b^2)*f)","A",0
227,1,197,0,0.475015," ","integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{a b^{2} \tan\left(f x + e\right)^{2} + a b^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(2 \, a^{2} b - a b^{2} + {\left(2 \, a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}"," ",0,"1/2*(a*b^2*tan(f*x + e)^2 + a*b^2 + (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + (2*a^2*b - a*b^2 + (2*a*b^2 - b^3)*tan(f*x + e)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f)","A",0
228,1,292,0,0.492676," ","integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(a^{3} b - 2 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \tan\left(f x + e\right)^{4} + a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{4} - a^{3} b - a^{2} b^{2} + 2 \, a b^{3}\right)} \tan\left(f x + e\right)^{2} + {\left({\left(a^{3} b - 3 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(a^{4} - 3 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left({\left(3 \, a b^{3} - 2 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(3 \, a^{2} b^{2} - 2 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{2}\right)}}"," ",0,"-1/2*((a^3*b - 2*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^4 + a^4 - 2*a^3*b + a^2*b^2 + (a^4 - a^3*b - a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2 + ((a^3*b - 3*a*b^3 + 2*b^4)*tan(f*x + e)^4 + (a^4 - 3*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + ((3*a*b^3 - 2*b^4)*tan(f*x + e)^4 + (3*a^2*b^2 - 2*a*b^3)*tan(f*x + e)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^2)","B",0
229,1,347,0,0.521474," ","integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(3 \, a^{4} b - 2 \, a^{3} b^{2} - 5 \, a^{2} b^{3} + 6 \, a b^{4}\right)} \tan\left(f x + e\right)^{6} - a^{5} + 2 \, a^{4} b - a^{3} b^{2} + {\left(3 \, a^{5} - 5 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + 6 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{5} - a^{4} b - 4 \, a^{3} b^{2} + 3 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2} + 2 \, {\left({\left(a^{4} b - 4 \, a b^{4} + 3 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + {\left(a^{5} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right)} \tan\left(f x + e\right)^{4}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(4 \, a b^{4} - 3 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + {\left(4 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \tan\left(f x + e\right)^{4}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{4}\right)}}"," ",0,"1/4*((3*a^4*b - 2*a^3*b^2 - 5*a^2*b^3 + 6*a*b^4)*tan(f*x + e)^6 - a^5 + 2*a^4*b - a^3*b^2 + (3*a^5 - 5*a^3*b^2 - 2*a^2*b^3 + 6*a*b^4)*tan(f*x + e)^4 + (2*a^5 - a^4*b - 4*a^3*b^2 + 3*a^2*b^3)*tan(f*x + e)^2 + 2*((a^4*b - 4*a*b^4 + 3*b^5)*tan(f*x + e)^6 + (a^5 - 4*a^2*b^3 + 3*a*b^4)*tan(f*x + e)^4)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + 2*((4*a*b^4 - 3*b^5)*tan(f*x + e)^6 + (4*a^2*b^3 - 3*a*b^4)*tan(f*x + e)^4)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4)","B",0
230,1,474,0,0.474121," ","integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{8 \, b^{3} f x \tan\left(f x + e\right)^{2} + 8 \, a b^{2} f x - 8 \, {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{3} + {\left(3 \, a^{3} - 5 \, a^{2} b + {\left(3 \, a^{2} b - 5 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-\frac{a}{b}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(b^{2} \tan\left(f x + e\right)^{3} - a b \tan\left(f x + e\right)\right)} \sqrt{-\frac{a}{b}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(3 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2}\right)} \tan\left(f x + e\right)}{8 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}, -\frac{4 \, b^{3} f x \tan\left(f x + e\right)^{2} + 4 \, a b^{2} f x - 4 \, {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{3} + {\left(3 \, a^{3} - 5 \, a^{2} b + {\left(3 \, a^{2} b - 5 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{\frac{a}{b}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{a}{b}}}{2 \, a \tan\left(f x + e\right)}\right) - 2 \, {\left(3 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2}\right)} \tan\left(f x + e\right)}{4 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*b^3*f*x*tan(f*x + e)^2 + 8*a*b^2*f*x - 8*(a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^3 + (3*a^3 - 5*a^2*b + (3*a^2*b - 5*a*b^2)*tan(f*x + e)^2)*sqrt(-a/b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(3*a^3 - 5*a^2*b + 2*a*b^2)*tan(f*x + e))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f), -1/4*(4*b^3*f*x*tan(f*x + e)^2 + 4*a*b^2*f*x - 4*(a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^3 + (3*a^3 - 5*a^2*b + (3*a^2*b - 5*a*b^2)*tan(f*x + e)^2)*sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))) - 2*(3*a^3 - 5*a^2*b + 2*a*b^2)*tan(f*x + e))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f)]","A",0
231,1,381,0,0.486753," ","integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, b^{2} f x \tan\left(f x + e\right)^{2} + 8 \, a b f x - {\left({\left(a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + a^{2} - 3 \, a b\right)} \sqrt{-\frac{a}{b}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b^{2} \tan\left(f x + e\right)^{3} - a b \tan\left(f x + e\right)\right)} \sqrt{-\frac{a}{b}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(a^{2} - a b\right)} \tan\left(f x + e\right)}{8 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, \frac{4 \, b^{2} f x \tan\left(f x + e\right)^{2} + 4 \, a b f x + {\left({\left(a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + a^{2} - 3 \, a b\right)} \sqrt{\frac{a}{b}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{a}{b}}}{2 \, a \tan\left(f x + e\right)}\right) - 2 \, {\left(a^{2} - a b\right)} \tan\left(f x + e\right)}{4 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[1/8*(8*b^2*f*x*tan(f*x + e)^2 + 8*a*b*f*x - ((a*b - 3*b^2)*tan(f*x + e)^2 + a^2 - 3*a*b)*sqrt(-a/b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a^2 - a*b)*tan(f*x + e))/((a^2*b^2 - 2*a*b^3 + b^4)*f*tan(f*x + e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f), 1/4*(4*b^2*f*x*tan(f*x + e)^2 + 4*a*b*f*x + ((a*b - 3*b^2)*tan(f*x + e)^2 + a^2 - 3*a*b)*sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))) - 2*(a^2 - a*b)*tan(f*x + e))/((a^2*b^2 - 2*a*b^3 + b^4)*f*tan(f*x + e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f)]","A",0
232,1,393,0,0.472400," ","integrate(tan(f*x+e)^2/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{8 \, a b^{2} f x \tan\left(f x + e\right)^{2} + 8 \, a^{2} b f x + {\left({\left(a b + b^{2}\right)} \tan\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{-a b} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{-a b}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(a^{2} b - a b^{2}\right)} \tan\left(f x + e\right)}{8 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}, -\frac{4 \, a b^{2} f x \tan\left(f x + e\right)^{2} + 4 \, a^{2} b f x - {\left({\left(a b + b^{2}\right)} \tan\left(f x + e\right)^{2} + a^{2} + a b\right)} \sqrt{a b} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{a b}}{2 \, a b \tan\left(f x + e\right)}\right) - 2 \, {\left(a^{2} b - a b^{2}\right)} \tan\left(f x + e\right)}{4 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*a*b^2*f*x*tan(f*x + e)^2 + 8*a^2*b*f*x + ((a*b + b^2)*tan(f*x + e)^2 + a^2 + a*b)*sqrt(-a*b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(-a*b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a^2*b - a*b^2)*tan(f*x + e))/((a^3*b^2 - 2*a^2*b^3 + a*b^4)*f*tan(f*x + e)^2 + (a^4*b - 2*a^3*b^2 + a^2*b^3)*f), -1/4*(4*a*b^2*f*x*tan(f*x + e)^2 + 4*a^2*b*f*x - ((a*b + b^2)*tan(f*x + e)^2 + a^2 + a*b)*sqrt(a*b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a*b)/(a*b*tan(f*x + e))) - 2*(a^2*b - a*b^2)*tan(f*x + e))/((a^3*b^2 - 2*a^2*b^3 + a*b^4)*f*tan(f*x + e)^2 + (a^4*b - 2*a^3*b^2 + a^2*b^3)*f)]","B",0
233,1,390,0,0.464494," ","integrate(1/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, a b f x \tan\left(f x + e\right)^{2} + 8 \, a^{2} f x - {\left({\left(3 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2} - a b\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{8 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{4 \, a b f x \tan\left(f x + e\right)^{2} + 4 \, a^{2} f x - {\left({\left(3 \, a b - b^{2}\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2} - a b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) - 2 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{4 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[1/8*(8*a*b*f*x*tan(f*x + e)^2 + 8*a^2*f*x - ((3*a*b - b^2)*tan(f*x + e)^2 + 3*a^2 - a*b)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f), 1/4*(4*a*b*f*x*tan(f*x + e)^2 + 4*a^2*f*x - ((3*a*b - b^2)*tan(f*x + e)^2 + 3*a^2 - a*b)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))) - 2*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f)]","A",0
234,1,503,0,0.477307," ","integrate(cot(f*x+e)^2/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{8 \, a^{2} b f x \tan\left(f x + e\right)^{3} + 8 \, a^{3} f x \tan\left(f x + e\right) + 8 \, a^{3} - 16 \, a^{2} b + 8 \, a b^{2} + 4 \, {\left(2 \, a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right)} \tan\left(f x + e\right)^{2} + {\left({\left(5 \, a b^{2} - 3 \, b^{3}\right)} \tan\left(f x + e\right)^{3} + {\left(5 \, a^{2} b - 3 \, a b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{8 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \tan\left(f x + e\right)\right)}}, -\frac{4 \, a^{2} b f x \tan\left(f x + e\right)^{3} + 4 \, a^{3} f x \tan\left(f x + e\right) + 4 \, a^{3} - 8 \, a^{2} b + 4 \, a b^{2} + 2 \, {\left(2 \, a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right)} \tan\left(f x + e\right)^{2} - {\left({\left(5 \, a b^{2} - 3 \, b^{3}\right)} \tan\left(f x + e\right)^{3} + {\left(5 \, a^{2} b - 3 \, a b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{4 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \tan\left(f x + e\right)\right)}}\right]"," ",0,"[-1/8*(8*a^2*b*f*x*tan(f*x + e)^3 + 8*a^3*f*x*tan(f*x + e) + 8*a^3 - 16*a^2*b + 8*a*b^2 + 4*(2*a^2*b - 5*a*b^2 + 3*b^3)*tan(f*x + e)^2 + ((5*a*b^2 - 3*b^3)*tan(f*x + e)^3 + (5*a^2*b - 3*a*b^2)*tan(f*x + e))*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^3 + (a^5 - 2*a^4*b + a^3*b^2)*f*tan(f*x + e)), -1/4*(4*a^2*b*f*x*tan(f*x + e)^3 + 4*a^3*f*x*tan(f*x + e) + 4*a^3 - 8*a^2*b + 4*a*b^2 + 2*(2*a^2*b - 5*a*b^2 + 3*b^3)*tan(f*x + e)^2 - ((5*a*b^2 - 3*b^3)*tan(f*x + e)^3 + (5*a^2*b - 3*a*b^2)*tan(f*x + e))*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^3 + (a^5 - 2*a^4*b + a^3*b^2)*f*tan(f*x + e))]","A",0
235,1,596,0,0.517230," ","integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[\frac{24 \, a^{3} b f x \tan\left(f x + e\right)^{5} + 24 \, a^{4} f x \tan\left(f x + e\right)^{3} + 12 \, {\left(2 \, a^{3} b - 7 \, a b^{3} + 5 \, b^{4}\right)} \tan\left(f x + e\right)^{4} - 8 \, a^{4} + 16 \, a^{3} b - 8 \, a^{2} b^{2} + 8 \, {\left(3 \, a^{4} - a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(f x + e\right)^{2} - 3 \, {\left({\left(7 \, a b^{3} - 5 \, b^{4}\right)} \tan\left(f x + e\right)^{5} + {\left(7 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \tan\left(f x + e\right)^{3}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{24 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{3}\right)}}, \frac{12 \, a^{3} b f x \tan\left(f x + e\right)^{5} + 12 \, a^{4} f x \tan\left(f x + e\right)^{3} + 6 \, {\left(2 \, a^{3} b - 7 \, a b^{3} + 5 \, b^{4}\right)} \tan\left(f x + e\right)^{4} - 4 \, a^{4} + 8 \, a^{3} b - 4 \, a^{2} b^{2} + 4 \, {\left(3 \, a^{4} - a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(f x + e\right)^{2} - 3 \, {\left({\left(7 \, a b^{3} - 5 \, b^{4}\right)} \tan\left(f x + e\right)^{5} + {\left(7 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \tan\left(f x + e\right)^{3}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{12 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{3}\right)}}\right]"," ",0,"[1/24*(24*a^3*b*f*x*tan(f*x + e)^5 + 24*a^4*f*x*tan(f*x + e)^3 + 12*(2*a^3*b - 7*a*b^3 + 5*b^4)*tan(f*x + e)^4 - 8*a^4 + 16*a^3*b - 8*a^2*b^2 + 8*(3*a^4 - a^3*b - 7*a^2*b^2 + 5*a*b^3)*tan(f*x + e)^2 - 3*((7*a*b^3 - 5*b^4)*tan(f*x + e)^5 + (7*a^2*b^2 - 5*a*b^3)*tan(f*x + e)^3)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^5 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^3), 1/12*(12*a^3*b*f*x*tan(f*x + e)^5 + 12*a^4*f*x*tan(f*x + e)^3 + 6*(2*a^3*b - 7*a*b^3 + 5*b^4)*tan(f*x + e)^4 - 4*a^4 + 8*a^3*b - 4*a^2*b^2 + 4*(3*a^4 - a^3*b - 7*a^2*b^2 + 5*a*b^3)*tan(f*x + e)^2 - 3*((7*a*b^3 - 5*b^4)*tan(f*x + e)^5 + (7*a^2*b^2 - 5*a*b^3)*tan(f*x + e)^3)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^5 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^3)]","A",0
236,1,672,0,0.536595," ","integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x, algorithm=""fricas"")","\left[-\frac{120 \, a^{4} b f x \tan\left(f x + e\right)^{7} + 120 \, a^{5} f x \tan\left(f x + e\right)^{5} + 60 \, {\left(2 \, a^{4} b - 9 \, a b^{4} + 7 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + 24 \, a^{5} - 48 \, a^{4} b + 24 \, a^{3} b^{2} + 40 \, {\left(3 \, a^{5} - a^{4} b - 9 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - 8 \, {\left(5 \, a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2} + 15 \, {\left({\left(9 \, a b^{4} - 7 \, b^{5}\right)} \tan\left(f x + e\right)^{7} + {\left(9 \, a^{2} b^{3} - 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{5}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{120 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{5}\right)}}, -\frac{60 \, a^{4} b f x \tan\left(f x + e\right)^{7} + 60 \, a^{5} f x \tan\left(f x + e\right)^{5} + 30 \, {\left(2 \, a^{4} b - 9 \, a b^{4} + 7 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + 12 \, a^{5} - 24 \, a^{4} b + 12 \, a^{3} b^{2} + 20 \, {\left(3 \, a^{5} - a^{4} b - 9 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - 4 \, {\left(5 \, a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2} - 15 \, {\left({\left(9 \, a b^{4} - 7 \, b^{5}\right)} \tan\left(f x + e\right)^{7} + {\left(9 \, a^{2} b^{3} - 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{5}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{60 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{5}\right)}}\right]"," ",0,"[-1/120*(120*a^4*b*f*x*tan(f*x + e)^7 + 120*a^5*f*x*tan(f*x + e)^5 + 60*(2*a^4*b - 9*a*b^4 + 7*b^5)*tan(f*x + e)^6 + 24*a^5 - 48*a^4*b + 24*a^3*b^2 + 40*(3*a^5 - a^4*b - 9*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^4 - 8*(5*a^5 - 3*a^4*b - 9*a^3*b^2 + 7*a^2*b^3)*tan(f*x + e)^2 + 15*((9*a*b^4 - 7*b^5)*tan(f*x + e)^7 + (9*a^2*b^3 - 7*a*b^4)*tan(f*x + e)^5)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5), -1/60*(60*a^4*b*f*x*tan(f*x + e)^7 + 60*a^5*f*x*tan(f*x + e)^5 + 30*(2*a^4*b - 9*a*b^4 + 7*b^5)*tan(f*x + e)^6 + 12*a^5 - 24*a^4*b + 12*a^3*b^2 + 20*(3*a^5 - a^4*b - 9*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^4 - 4*(5*a^5 - 3*a^4*b - 9*a^3*b^2 + 7*a^2*b^3)*tan(f*x + e)^2 - 15*((9*a*b^4 - 7*b^5)*tan(f*x + e)^7 + (9*a^2*b^3 - 7*a*b^4)*tan(f*x + e)^5)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5)]","A",0
237,1,206,0,0.450749," ","integrate(tan(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{{\left(a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b\right)} \tan\left(f x + e\right)^{2} - 3 \, a^{2} - 2 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}"," ",0,"1/4*((a^2 - 4*a*b)*tan(f*x + e)^4 - 2*(a^2 + 2*a*b)*tan(f*x + e)^2 - 3*a^2 - 2*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)","B",0
238,1,212,0,0.454901," ","integrate(tan(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{2} + a b + b^{2}\right)} \tan\left(f x + e\right)^{2} + 2 \, a^{2} + a b + 2 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}"," ",0,"1/4*((a*b + 2*b^2)*tan(f*x + e)^4 + 2*(a^2 + a*b + b^2)*tan(f*x + e)^2 + 2*a^2 + a*b + 2*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)","B",0
239,1,206,0,0.454926," ","integrate(tan(f*x+e)/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","-\frac{3 \, b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(2 \, a b + b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, a b - b^{2} + 2 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}"," ",0,"-1/4*(3*b^2*tan(f*x + e)^4 + 2*(2*a*b + b^2)*tan(f*x + e)^2 + 4*a*b - b^2 + 2*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)","B",0
240,1,422,0,0.504399," ","integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{6 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + {\left(5 \, a^{2} b^{3} - 2 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{3} b^{2} + a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2} + 2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left(3 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3} + {\left(3 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f\right)}}"," ",0,"1/4*(6*a^3*b^2 - 3*a^2*b^3 + (5*a^2*b^3 - 2*a*b^4)*tan(f*x + e)^4 + 2*(3*a^3*b^2 + a^2*b^3 - a*b^4)*tan(f*x + e)^2 + 2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + 2*(3*a^4*b - 3*a^3*b^2 + a^2*b^3 + (3*a^2*b^3 - 3*a*b^4 + b^5)*tan(f*x + e)^4 + 2*(3*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(f*x + e)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^2 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f)","B",0
241,1,545,0,0.579338," ","integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","-\frac{{\left(2 \, a^{4} b^{2} - 6 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 6 \, a b^{5}\right)} \tan\left(f x + e\right)^{6} + 2 \, a^{6} - 6 \, a^{5} b + 6 \, a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, {\left(2 \, a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{6} - 2 \, a^{5} b - 6 \, a^{4} b^{2} + 18 \, a^{3} b^{3} - 9 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2} + 2 \, {\left({\left(a^{4} b^{2} - 6 \, a^{2} b^{4} + 8 \, a b^{5} - 3 \, b^{6}\right)} \tan\left(f x + e\right)^{6} + 2 \, {\left(a^{5} b - 6 \, a^{3} b^{3} + 8 \, a^{2} b^{4} - 3 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(a^{6} - 6 \, a^{4} b^{2} + 8 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(6 \, a^{2} b^{4} - 8 \, a b^{5} + 3 \, b^{6}\right)} \tan\left(f x + e\right)^{6} + 2 \, {\left(6 \, a^{3} b^{3} - 8 \, a^{2} b^{4} + 3 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(6 \, a^{4} b^{2} - 8 \, a^{3} b^{3} + 3 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{6} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{2}\right)}}"," ",0,"-1/4*((2*a^4*b^2 - 6*a^3*b^3 + 13*a^2*b^4 - 6*a*b^5)*tan(f*x + e)^6 + 2*a^6 - 6*a^5*b + 6*a^4*b^2 - 2*a^3*b^3 + 2*(2*a^5*b - 5*a^4*b^2 + 7*a^3*b^3 + 2*a^2*b^4 - 3*a*b^5)*tan(f*x + e)^4 + (2*a^6 - 2*a^5*b - 6*a^4*b^2 + 18*a^3*b^3 - 9*a^2*b^4)*tan(f*x + e)^2 + 2*((a^4*b^2 - 6*a^2*b^4 + 8*a*b^5 - 3*b^6)*tan(f*x + e)^6 + 2*(a^5*b - 6*a^3*b^3 + 8*a^2*b^4 - 3*a*b^5)*tan(f*x + e)^4 + (a^6 - 6*a^4*b^2 + 8*a^3*b^3 - 3*a^2*b^4)*tan(f*x + e)^2)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + 2*((6*a^2*b^4 - 8*a*b^5 + 3*b^6)*tan(f*x + e)^6 + 2*(6*a^3*b^3 - 8*a^2*b^4 + 3*a*b^5)*tan(f*x + e)^4 + (6*a^4*b^2 - 8*a^3*b^3 + 3*a^2*b^4)*tan(f*x + e)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^6 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^4 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^2)","B",0
242,1,611,0,0.619381," ","integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\frac{3 \, {\left(a^{5} b^{2} - a^{4} b^{3} - 3 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - 4 \, a b^{6}\right)} \tan\left(f x + e\right)^{8} - a^{7} + 3 \, a^{6} b - 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left(3 \, a^{6} b - 2 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 14 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - 6 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(3 \, a^{7} + a^{6} b - 10 \, a^{5} b^{2} - 6 \, a^{4} b^{3} + 33 \, a^{3} b^{4} - 18 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2} + 2 \, {\left({\left(a^{5} b^{2} - 10 \, a^{2} b^{5} + 15 \, a b^{6} - 6 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 2 \, {\left(a^{6} b - 10 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(a^{7} - 10 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4}\right)} \log\left(\frac{\tan\left(f x + e\right)^{2}}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(10 \, a^{2} b^{5} - 15 \, a b^{6} + 6 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 2 \, {\left(10 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(10 \, a^{4} b^{3} - 15 \, a^{3} b^{4} + 6 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4}\right)} \log\left(\frac{b \tan\left(f x + e\right)^{2} + a}{\tan\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{8} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{4}\right)}}"," ",0,"1/4*(3*(a^5*b^2 - a^4*b^3 - 3*a^3*b^4 + 8*a^2*b^5 - 4*a*b^6)*tan(f*x + e)^8 - a^7 + 3*a^6*b - 3*a^5*b^2 + a^4*b^3 + 2*(3*a^6*b - 2*a^5*b^2 - 9*a^4*b^3 + 14*a^3*b^4 + 3*a^2*b^5 - 6*a*b^6)*tan(f*x + e)^6 + (3*a^7 + a^6*b - 10*a^5*b^2 - 6*a^4*b^3 + 33*a^3*b^4 - 18*a^2*b^5)*tan(f*x + e)^4 + 2*(a^7 - a^6*b - 3*a^5*b^2 + 5*a^4*b^3 - 2*a^3*b^4)*tan(f*x + e)^2 + 2*((a^5*b^2 - 10*a^2*b^5 + 15*a*b^6 - 6*b^7)*tan(f*x + e)^8 + 2*(a^6*b - 10*a^3*b^4 + 15*a^2*b^5 - 6*a*b^6)*tan(f*x + e)^6 + (a^7 - 10*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5)*tan(f*x + e)^4)*log(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + 2*((10*a^2*b^5 - 15*a*b^6 + 6*b^7)*tan(f*x + e)^8 + 2*(10*a^3*b^4 - 15*a^2*b^5 + 6*a*b^6)*tan(f*x + e)^6 + (10*a^4*b^3 - 15*a^3*b^4 + 6*a^2*b^5)*tan(f*x + e)^4)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4)","B",0
243,1,743,0,0.484759," ","integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{32 \, b^{4} f x \tan\left(f x + e\right)^{4} + 64 \, a b^{3} f x \tan\left(f x + e\right)^{2} + 32 \, a^{2} b^{2} f x + 4 \, {\left(5 \, a^{3} b - 14 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \tan\left(f x + e\right)^{3} + {\left({\left(3 \, a^{2} b^{2} - 10 \, a b^{3} + 15 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + 3 \, a^{4} - 10 \, a^{3} b + 15 \, a^{2} b^{2} + 2 \, {\left(3 \, a^{3} b - 10 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-\frac{a}{b}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b^{2} \tan\left(f x + e\right)^{3} - a b \tan\left(f x + e\right)\right)} \sqrt{-\frac{a}{b}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) + 4 \, {\left(3 \, a^{4} - 10 \, a^{3} b + 7 \, a^{2} b^{2}\right)} \tan\left(f x + e\right)}{32 \, {\left({\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{3} - 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f\right)}}, -\frac{16 \, b^{4} f x \tan\left(f x + e\right)^{4} + 32 \, a b^{3} f x \tan\left(f x + e\right)^{2} + 16 \, a^{2} b^{2} f x + 2 \, {\left(5 \, a^{3} b - 14 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(3 \, a^{2} b^{2} - 10 \, a b^{3} + 15 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + 3 \, a^{4} - 10 \, a^{3} b + 15 \, a^{2} b^{2} + 2 \, {\left(3 \, a^{3} b - 10 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{\frac{a}{b}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{a}{b}}}{2 \, a \tan\left(f x + e\right)}\right) + 2 \, {\left(3 \, a^{4} - 10 \, a^{3} b + 7 \, a^{2} b^{2}\right)} \tan\left(f x + e\right)}{16 \, {\left({\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{3} - 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[-1/32*(32*b^4*f*x*tan(f*x + e)^4 + 64*a*b^3*f*x*tan(f*x + e)^2 + 32*a^2*b^2*f*x + 4*(5*a^3*b - 14*a^2*b^2 + 9*a*b^3)*tan(f*x + e)^3 + ((3*a^2*b^2 - 10*a*b^3 + 15*b^4)*tan(f*x + e)^4 + 3*a^4 - 10*a^3*b + 15*a^2*b^2 + 2*(3*a^3*b - 10*a^2*b^2 + 15*a*b^3)*tan(f*x + e)^2)*sqrt(-a/b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) + 4*(3*a^4 - 10*a^3*b + 7*a^2*b^2)*tan(f*x + e))/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*f*tan(f*x + e)^4 + 2*(a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 - a*b^6)*f*tan(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f), -1/16*(16*b^4*f*x*tan(f*x + e)^4 + 32*a*b^3*f*x*tan(f*x + e)^2 + 16*a^2*b^2*f*x + 2*(5*a^3*b - 14*a^2*b^2 + 9*a*b^3)*tan(f*x + e)^3 - ((3*a^2*b^2 - 10*a*b^3 + 15*b^4)*tan(f*x + e)^4 + 3*a^4 - 10*a^3*b + 15*a^2*b^2 + 2*(3*a^3*b - 10*a^2*b^2 + 15*a*b^3)*tan(f*x + e)^2)*sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))) + 2*(3*a^4 - 10*a^3*b + 7*a^2*b^2)*tan(f*x + e))/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*f*tan(f*x + e)^4 + 2*(a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 - a*b^6)*f*tan(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f)]","B",0
244,1,749,0,0.497789," ","integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, a b^{4} f x \tan\left(f x + e\right)^{4} + 64 \, a^{2} b^{3} f x \tan\left(f x + e\right)^{2} + 32 \, a^{3} b^{2} f x + 4 \, {\left(a^{3} b^{2} - 6 \, a^{2} b^{3} + 5 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(a^{2} b^{2} - 6 \, a b^{3} - 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + a^{4} - 6 \, a^{3} b - 3 \, a^{2} b^{2} + 2 \, {\left(a^{3} b - 6 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a b} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{-a b}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(a^{4} b + 2 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)}{32 \, {\left({\left(a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f\right)}}, \frac{16 \, a b^{4} f x \tan\left(f x + e\right)^{4} + 32 \, a^{2} b^{3} f x \tan\left(f x + e\right)^{2} + 16 \, a^{3} b^{2} f x + 2 \, {\left(a^{3} b^{2} - 6 \, a^{2} b^{3} + 5 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left({\left(a^{2} b^{2} - 6 \, a b^{3} - 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + a^{4} - 6 \, a^{3} b - 3 \, a^{2} b^{2} + 2 \, {\left(a^{3} b - 6 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a b} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{a b}}{2 \, a b \tan\left(f x + e\right)}\right) - 2 \, {\left(a^{4} b + 2 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)}{16 \, {\left({\left(a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f\right)}}\right]"," ",0,"[1/32*(32*a*b^4*f*x*tan(f*x + e)^4 + 64*a^2*b^3*f*x*tan(f*x + e)^2 + 32*a^3*b^2*f*x + 4*(a^3*b^2 - 6*a^2*b^3 + 5*a*b^4)*tan(f*x + e)^3 - ((a^2*b^2 - 6*a*b^3 - 3*b^4)*tan(f*x + e)^4 + a^4 - 6*a^3*b - 3*a^2*b^2 + 2*(a^3*b - 6*a^2*b^2 - 3*a*b^3)*tan(f*x + e)^2)*sqrt(-a*b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(-a*b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a^4*b + 2*a^3*b^2 - 3*a^2*b^3)*tan(f*x + e))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^4 + 2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^2 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f), 1/16*(16*a*b^4*f*x*tan(f*x + e)^4 + 32*a^2*b^3*f*x*tan(f*x + e)^2 + 16*a^3*b^2*f*x + 2*(a^3*b^2 - 6*a^2*b^3 + 5*a*b^4)*tan(f*x + e)^3 + ((a^2*b^2 - 6*a*b^3 - 3*b^4)*tan(f*x + e)^4 + a^4 - 6*a^3*b - 3*a^2*b^2 + 2*(a^3*b - 6*a^2*b^2 - 3*a*b^3)*tan(f*x + e)^2)*sqrt(a*b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a*b)/(a*b*tan(f*x + e))) - 2*(a^4*b + 2*a^3*b^2 - 3*a^2*b^3)*tan(f*x + e))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^4 + 2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^2 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f)]","B",0
245,1,759,0,0.756557," ","integrate(tan(f*x+e)^2/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{32 \, a^{2} b^{3} f x \tan\left(f x + e\right)^{4} + 64 \, a^{3} b^{2} f x \tan\left(f x + e\right)^{2} + 32 \, a^{4} b f x - 4 \, {\left(3 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left({\left(3 \, a^{2} b^{2} + 6 \, a b^{3} - b^{4}\right)} \tan\left(f x + e\right)^{4} + 3 \, a^{4} + 6 \, a^{3} b - a^{2} b^{2} + 2 \, {\left(3 \, a^{3} b + 6 \, a^{2} b^{2} - a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a b} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{-a b}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(5 \, a^{4} b - 6 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(f x + e\right)}{32 \, {\left({\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f\right)}}, -\frac{16 \, a^{2} b^{3} f x \tan\left(f x + e\right)^{4} + 32 \, a^{3} b^{2} f x \tan\left(f x + e\right)^{2} + 16 \, a^{4} b f x - 2 \, {\left(3 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(3 \, a^{2} b^{2} + 6 \, a b^{3} - b^{4}\right)} \tan\left(f x + e\right)^{4} + 3 \, a^{4} + 6 \, a^{3} b - a^{2} b^{2} + 2 \, {\left(3 \, a^{3} b + 6 \, a^{2} b^{2} - a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a b} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{a b}}{2 \, a b \tan\left(f x + e\right)}\right) - 2 \, {\left(5 \, a^{4} b - 6 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(f x + e\right)}{16 \, {\left({\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f\right)}}\right]"," ",0,"[-1/32*(32*a^2*b^3*f*x*tan(f*x + e)^4 + 64*a^3*b^2*f*x*tan(f*x + e)^2 + 32*a^4*b*f*x - 4*(3*a^3*b^2 - 2*a^2*b^3 - a*b^4)*tan(f*x + e)^3 + ((3*a^2*b^2 + 6*a*b^3 - b^4)*tan(f*x + e)^4 + 3*a^4 + 6*a^3*b - a^2*b^2 + 2*(3*a^3*b + 6*a^2*b^2 - a*b^3)*tan(f*x + e)^2)*sqrt(-a*b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(-a*b))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(5*a^4*b - 6*a^3*b^2 + a^2*b^3)*tan(f*x + e))/((a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^4 + 2*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^2 + (a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f), -1/16*(16*a^2*b^3*f*x*tan(f*x + e)^4 + 32*a^3*b^2*f*x*tan(f*x + e)^2 + 16*a^4*b*f*x - 2*(3*a^3*b^2 - 2*a^2*b^3 - a*b^4)*tan(f*x + e)^3 - ((3*a^2*b^2 + 6*a*b^3 - b^4)*tan(f*x + e)^4 + 3*a^4 + 6*a^3*b - a^2*b^2 + 2*(3*a^3*b + 6*a^2*b^2 - a*b^3)*tan(f*x + e)^2)*sqrt(a*b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a*b)/(a*b*tan(f*x + e))) - 2*(5*a^4*b - 6*a^3*b^2 + a^2*b^3)*tan(f*x + e))/((a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^4 + 2*(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^2 + (a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f)]","B",0
246,1,742,0,0.526519," ","integrate(1/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, a^{2} b^{2} f x \tan\left(f x + e\right)^{4} + 64 \, a^{3} b f x \tan\left(f x + e\right)^{2} + 32 \, a^{4} f x - 4 \, {\left(7 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(15 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + 15 \, a^{4} - 10 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, {\left(15 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right) - 4 \, {\left(9 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(f x + e\right)}{32 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}, \frac{16 \, a^{2} b^{2} f x \tan\left(f x + e\right)^{4} + 32 \, a^{3} b f x \tan\left(f x + e\right)^{2} + 16 \, a^{4} f x - 2 \, {\left(7 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - {\left({\left(15 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + 15 \, a^{4} - 10 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, {\left(15 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right) - 2 \, {\left(9 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(f x + e\right)}{16 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[1/32*(32*a^2*b^2*f*x*tan(f*x + e)^4 + 64*a^3*b*f*x*tan(f*x + e)^2 + 32*a^4*f*x - 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^3 - ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^4 + 15*a^4 - 10*a^3*b + 3*a^2*b^2 + 2*(15*a^3*b - 10*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(9*a^3*b - 14*a^2*b^2 + 5*a*b^3)*tan(f*x + e))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f), 1/16*(16*a^2*b^2*f*x*tan(f*x + e)^4 + 32*a^3*b*f*x*tan(f*x + e)^2 + 16*a^4*f*x - 2*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^3 - ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(f*x + e)^4 + 15*a^4 - 10*a^3*b + 3*a^2*b^2 + 2*(15*a^3*b - 10*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))) - 2*(9*a^3*b - 14*a^2*b^2 + 5*a*b^3)*tan(f*x + e))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f)]","B",0
247,1,881,0,0.512922," ","integrate(cot(f*x+e)^2/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{32 \, a^{3} b^{2} f x \tan\left(f x + e\right)^{5} + 64 \, a^{4} b f x \tan\left(f x + e\right)^{3} + 32 \, a^{5} f x \tan\left(f x + e\right) + 32 \, a^{5} - 96 \, a^{4} b + 96 \, a^{3} b^{2} - 32 \, a^{2} b^{3} + 4 \, {\left(8 \, a^{3} b^{2} - 35 \, a^{2} b^{3} + 42 \, a b^{4} - 15 \, b^{5}\right)} \tan\left(f x + e\right)^{4} + 4 \, {\left(16 \, a^{4} b - 61 \, a^{3} b^{2} + 70 \, a^{2} b^{3} - 25 \, a b^{4}\right)} \tan\left(f x + e\right)^{2} + {\left({\left(35 \, a^{2} b^{3} - 42 \, a b^{4} + 15 \, b^{5}\right)} \tan\left(f x + e\right)^{5} + 2 \, {\left(35 \, a^{3} b^{2} - 42 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left(35 \, a^{4} b - 42 \, a^{3} b^{2} + 15 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{32 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{5} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f \tan\left(f x + e\right)\right)}}, -\frac{16 \, a^{3} b^{2} f x \tan\left(f x + e\right)^{5} + 32 \, a^{4} b f x \tan\left(f x + e\right)^{3} + 16 \, a^{5} f x \tan\left(f x + e\right) + 16 \, a^{5} - 48 \, a^{4} b + 48 \, a^{3} b^{2} - 16 \, a^{2} b^{3} + 2 \, {\left(8 \, a^{3} b^{2} - 35 \, a^{2} b^{3} + 42 \, a b^{4} - 15 \, b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(16 \, a^{4} b - 61 \, a^{3} b^{2} + 70 \, a^{2} b^{3} - 25 \, a b^{4}\right)} \tan\left(f x + e\right)^{2} - {\left({\left(35 \, a^{2} b^{3} - 42 \, a b^{4} + 15 \, b^{5}\right)} \tan\left(f x + e\right)^{5} + 2 \, {\left(35 \, a^{3} b^{2} - 42 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left(35 \, a^{4} b - 42 \, a^{3} b^{2} + 15 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{16 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{5} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f \tan\left(f x + e\right)\right)}}\right]"," ",0,"[-1/32*(32*a^3*b^2*f*x*tan(f*x + e)^5 + 64*a^4*b*f*x*tan(f*x + e)^3 + 32*a^5*f*x*tan(f*x + e) + 32*a^5 - 96*a^4*b + 96*a^3*b^2 - 32*a^2*b^3 + 4*(8*a^3*b^2 - 35*a^2*b^3 + 42*a*b^4 - 15*b^5)*tan(f*x + e)^4 + 4*(16*a^4*b - 61*a^3*b^2 + 70*a^2*b^3 - 25*a*b^4)*tan(f*x + e)^2 + ((35*a^2*b^3 - 42*a*b^4 + 15*b^5)*tan(f*x + e)^5 + 2*(35*a^3*b^2 - 42*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^3 + (35*a^4*b - 42*a^3*b^2 + 15*a^2*b^3)*tan(f*x + e))*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^5 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^3 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f*tan(f*x + e)), -1/16*(16*a^3*b^2*f*x*tan(f*x + e)^5 + 32*a^4*b*f*x*tan(f*x + e)^3 + 16*a^5*f*x*tan(f*x + e) + 16*a^5 - 48*a^4*b + 48*a^3*b^2 - 16*a^2*b^3 + 2*(8*a^3*b^2 - 35*a^2*b^3 + 42*a*b^4 - 15*b^5)*tan(f*x + e)^4 + 2*(16*a^4*b - 61*a^3*b^2 + 70*a^2*b^3 - 25*a*b^4)*tan(f*x + e)^2 - ((35*a^2*b^3 - 42*a*b^4 + 15*b^5)*tan(f*x + e)^5 + 2*(35*a^3*b^2 - 42*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^3 + (35*a^4*b - 42*a^3*b^2 + 15*a^2*b^3)*tan(f*x + e))*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^5 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^3 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f*tan(f*x + e))]","B",0
248,1,1006,0,0.554324," ","integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[\frac{96 \, a^{4} b^{2} f x \tan\left(f x + e\right)^{7} + 192 \, a^{5} b f x \tan\left(f x + e\right)^{5} + 96 \, a^{6} f x \tan\left(f x + e\right)^{3} + 12 \, {\left(8 \, a^{4} b^{2} - 63 \, a^{2} b^{4} + 90 \, a b^{5} - 35 \, b^{6}\right)} \tan\left(f x + e\right)^{6} - 32 \, a^{6} + 96 \, a^{5} b - 96 \, a^{4} b^{2} + 32 \, a^{3} b^{3} + 4 \, {\left(48 \, a^{5} b - 8 \, a^{4} b^{2} - 315 \, a^{3} b^{3} + 450 \, a^{2} b^{4} - 175 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 32 \, {\left(3 \, a^{6} - 2 \, a^{5} b - 12 \, a^{4} b^{2} + 18 \, a^{3} b^{3} - 7 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2} - 3 \, {\left({\left(63 \, a^{2} b^{4} - 90 \, a b^{5} + 35 \, b^{6}\right)} \tan\left(f x + e\right)^{7} + 2 \, {\left(63 \, a^{3} b^{3} - 90 \, a^{2} b^{4} + 35 \, a b^{5}\right)} \tan\left(f x + e\right)^{5} + {\left(63 \, a^{4} b^{2} - 90 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{3}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{96 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{7} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{3}\right)}}, \frac{48 \, a^{4} b^{2} f x \tan\left(f x + e\right)^{7} + 96 \, a^{5} b f x \tan\left(f x + e\right)^{5} + 48 \, a^{6} f x \tan\left(f x + e\right)^{3} + 6 \, {\left(8 \, a^{4} b^{2} - 63 \, a^{2} b^{4} + 90 \, a b^{5} - 35 \, b^{6}\right)} \tan\left(f x + e\right)^{6} - 16 \, a^{6} + 48 \, a^{5} b - 48 \, a^{4} b^{2} + 16 \, a^{3} b^{3} + 2 \, {\left(48 \, a^{5} b - 8 \, a^{4} b^{2} - 315 \, a^{3} b^{3} + 450 \, a^{2} b^{4} - 175 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 16 \, {\left(3 \, a^{6} - 2 \, a^{5} b - 12 \, a^{4} b^{2} + 18 \, a^{3} b^{3} - 7 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2} - 3 \, {\left({\left(63 \, a^{2} b^{4} - 90 \, a b^{5} + 35 \, b^{6}\right)} \tan\left(f x + e\right)^{7} + 2 \, {\left(63 \, a^{3} b^{3} - 90 \, a^{2} b^{4} + 35 \, a b^{5}\right)} \tan\left(f x + e\right)^{5} + {\left(63 \, a^{4} b^{2} - 90 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{3}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{48 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{7} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{3}\right)}}\right]"," ",0,"[1/96*(96*a^4*b^2*f*x*tan(f*x + e)^7 + 192*a^5*b*f*x*tan(f*x + e)^5 + 96*a^6*f*x*tan(f*x + e)^3 + 12*(8*a^4*b^2 - 63*a^2*b^4 + 90*a*b^5 - 35*b^6)*tan(f*x + e)^6 - 32*a^6 + 96*a^5*b - 96*a^4*b^2 + 32*a^3*b^3 + 4*(48*a^5*b - 8*a^4*b^2 - 315*a^3*b^3 + 450*a^2*b^4 - 175*a*b^5)*tan(f*x + e)^4 + 32*(3*a^6 - 2*a^5*b - 12*a^4*b^2 + 18*a^3*b^3 - 7*a^2*b^4)*tan(f*x + e)^2 - 3*((63*a^2*b^4 - 90*a*b^5 + 35*b^6)*tan(f*x + e)^7 + 2*(63*a^3*b^3 - 90*a^2*b^4 + 35*a*b^5)*tan(f*x + e)^5 + (63*a^4*b^2 - 90*a^3*b^3 + 35*a^2*b^4)*tan(f*x + e)^3)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 + 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^7 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^3), 1/48*(48*a^4*b^2*f*x*tan(f*x + e)^7 + 96*a^5*b*f*x*tan(f*x + e)^5 + 48*a^6*f*x*tan(f*x + e)^3 + 6*(8*a^4*b^2 - 63*a^2*b^4 + 90*a*b^5 - 35*b^6)*tan(f*x + e)^6 - 16*a^6 + 48*a^5*b - 48*a^4*b^2 + 16*a^3*b^3 + 2*(48*a^5*b - 8*a^4*b^2 - 315*a^3*b^3 + 450*a^2*b^4 - 175*a*b^5)*tan(f*x + e)^4 + 16*(3*a^6 - 2*a^5*b - 12*a^4*b^2 + 18*a^3*b^3 - 7*a^2*b^4)*tan(f*x + e)^2 - 3*((63*a^2*b^4 - 90*a*b^5 + 35*b^6)*tan(f*x + e)^7 + 2*(63*a^3*b^3 - 90*a^2*b^4 + 35*a*b^5)*tan(f*x + e)^5 + (63*a^4*b^2 - 90*a^3*b^3 + 35*a^2*b^4)*tan(f*x + e)^3)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^7 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^3)]","B",0
249,1,1114,0,0.609054," ","integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm=""fricas"")","\left[-\frac{480 \, a^{5} b^{2} f x \tan\left(f x + e\right)^{9} + 960 \, a^{6} b f x \tan\left(f x + e\right)^{7} + 480 \, a^{7} f x \tan\left(f x + e\right)^{5} + 60 \, {\left(8 \, a^{5} b^{2} - 99 \, a^{2} b^{5} + 154 \, a b^{6} - 63 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 96 \, a^{7} - 288 \, a^{6} b + 288 \, a^{5} b^{2} - 96 \, a^{4} b^{3} + 20 \, {\left(48 \, a^{6} b - 8 \, a^{5} b^{2} - 495 \, a^{3} b^{4} + 770 \, a^{2} b^{5} - 315 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + 32 \, {\left(15 \, a^{7} - 10 \, a^{6} b + 3 \, a^{5} b^{2} - 99 \, a^{4} b^{3} + 154 \, a^{3} b^{4} - 63 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - 32 \, {\left(5 \, a^{7} - 6 \, a^{6} b - 12 \, a^{5} b^{2} + 22 \, a^{4} b^{3} - 9 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2} + 15 \, {\left({\left(99 \, a^{2} b^{5} - 154 \, a b^{6} + 63 \, b^{7}\right)} \tan\left(f x + e\right)^{9} + 2 \, {\left(99 \, a^{3} b^{4} - 154 \, a^{2} b^{5} + 63 \, a b^{6}\right)} \tan\left(f x + e\right)^{7} + {\left(99 \, a^{4} b^{3} - 154 \, a^{3} b^{4} + 63 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{5}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(f x + e\right)^{4} - 6 \, a b \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left(a b \tan\left(f x + e\right)^{3} - a^{2} \tan\left(f x + e\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}}\right)}{480 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{9} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{5}\right)}}, -\frac{240 \, a^{5} b^{2} f x \tan\left(f x + e\right)^{9} + 480 \, a^{6} b f x \tan\left(f x + e\right)^{7} + 240 \, a^{7} f x \tan\left(f x + e\right)^{5} + 30 \, {\left(8 \, a^{5} b^{2} - 99 \, a^{2} b^{5} + 154 \, a b^{6} - 63 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 48 \, a^{7} - 144 \, a^{6} b + 144 \, a^{5} b^{2} - 48 \, a^{4} b^{3} + 10 \, {\left(48 \, a^{6} b - 8 \, a^{5} b^{2} - 495 \, a^{3} b^{4} + 770 \, a^{2} b^{5} - 315 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + 16 \, {\left(15 \, a^{7} - 10 \, a^{6} b + 3 \, a^{5} b^{2} - 99 \, a^{4} b^{3} + 154 \, a^{3} b^{4} - 63 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - 16 \, {\left(5 \, a^{7} - 6 \, a^{6} b - 12 \, a^{5} b^{2} + 22 \, a^{4} b^{3} - 9 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2} - 15 \, {\left({\left(99 \, a^{2} b^{5} - 154 \, a b^{6} + 63 \, b^{7}\right)} \tan\left(f x + e\right)^{9} + 2 \, {\left(99 \, a^{3} b^{4} - 154 \, a^{2} b^{5} + 63 \, a b^{6}\right)} \tan\left(f x + e\right)^{7} + {\left(99 \, a^{4} b^{3} - 154 \, a^{3} b^{4} + 63 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{5}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(f x + e\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(f x + e\right)}\right)}{240 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{9} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{5}\right)}}\right]"," ",0,"[-1/480*(480*a^5*b^2*f*x*tan(f*x + e)^9 + 960*a^6*b*f*x*tan(f*x + e)^7 + 480*a^7*f*x*tan(f*x + e)^5 + 60*(8*a^5*b^2 - 99*a^2*b^5 + 154*a*b^6 - 63*b^7)*tan(f*x + e)^8 + 96*a^7 - 288*a^6*b + 288*a^5*b^2 - 96*a^4*b^3 + 20*(48*a^6*b - 8*a^5*b^2 - 495*a^3*b^4 + 770*a^2*b^5 - 315*a*b^6)*tan(f*x + e)^6 + 32*(15*a^7 - 10*a^6*b + 3*a^5*b^2 - 99*a^4*b^3 + 154*a^3*b^4 - 63*a^2*b^5)*tan(f*x + e)^4 - 32*(5*a^7 - 6*a^6*b - 12*a^5*b^2 + 22*a^4*b^3 - 9*a^3*b^4)*tan(f*x + e)^2 + 15*((99*a^2*b^5 - 154*a*b^6 + 63*b^7)*tan(f*x + e)^9 + 2*(99*a^3*b^4 - 154*a^2*b^5 + 63*a*b^6)*tan(f*x + e)^7 + (99*a^4*b^3 - 154*a^3*b^4 + 63*a^2*b^5)*tan(f*x + e)^5)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^9 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^7 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^5), -1/240*(240*a^5*b^2*f*x*tan(f*x + e)^9 + 480*a^6*b*f*x*tan(f*x + e)^7 + 240*a^7*f*x*tan(f*x + e)^5 + 30*(8*a^5*b^2 - 99*a^2*b^5 + 154*a*b^6 - 63*b^7)*tan(f*x + e)^8 + 48*a^7 - 144*a^6*b + 144*a^5*b^2 - 48*a^4*b^3 + 10*(48*a^6*b - 8*a^5*b^2 - 495*a^3*b^4 + 770*a^2*b^5 - 315*a*b^6)*tan(f*x + e)^6 + 16*(15*a^7 - 10*a^6*b + 3*a^5*b^2 - 99*a^4*b^3 + 154*a^3*b^4 - 63*a^2*b^5)*tan(f*x + e)^4 - 16*(5*a^7 - 6*a^6*b - 12*a^5*b^2 + 22*a^4*b^3 - 9*a^3*b^4)*tan(f*x + e)^2 - 15*((99*a^2*b^5 - 154*a*b^6 + 63*b^7)*tan(f*x + e)^9 + 2*(99*a^3*b^4 - 154*a^2*b^5 + 63*a*b^6)*tan(f*x + e)^7 + (99*a^4*b^3 - 154*a^3*b^4 + 63*a^2*b^5)*tan(f*x + e)^5)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^9 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^7 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^5)]","A",0
250,1,134,0,0.406029," ","integrate((a+b*tan(d*x+c)^2)^4,x, algorithm=""fricas"")","\frac{15 \, b^{4} \tan\left(d x + c\right)^{7} + 21 \, {\left(4 \, a b^{3} - b^{4}\right)} \tan\left(d x + c\right)^{5} + 35 \, {\left(6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)^{3} + 105 \, {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)} d x + 105 \, {\left(4 \, a^{3} b - 6 \, a^{2} b^{2} + 4 \, a b^{3} - b^{4}\right)} \tan\left(d x + c\right)}{105 \, d}"," ",0,"1/105*(15*b^4*tan(d*x + c)^7 + 21*(4*a*b^3 - b^4)*tan(d*x + c)^5 + 35*(6*a^2*b^2 - 4*a*b^3 + b^4)*tan(d*x + c)^3 + 105*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*d*x + 105*(4*a^3*b - 6*a^2*b^2 + 4*a*b^3 - b^4)*tan(d*x + c))/d","A",0
251,1,90,0,0.417344," ","integrate((a+b*tan(d*x+c)^2)^3,x, algorithm=""fricas"")","\frac{3 \, b^{3} \tan\left(d x + c\right)^{5} + 5 \, {\left(3 \, a b^{2} - b^{3}\right)} \tan\left(d x + c\right)^{3} + 15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d x + 15 \, {\left(3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right)} \tan\left(d x + c\right)}{15 \, d}"," ",0,"1/15*(3*b^3*tan(d*x + c)^5 + 5*(3*a*b^2 - b^3)*tan(d*x + c)^3 + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*x + 15*(3*a^2*b - 3*a*b^2 + b^3)*tan(d*x + c))/d","A",0
252,1,51,0,0.441278," ","integrate((a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{b^{2} \tan\left(d x + c\right)^{3} + 3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d x + 3 \, {\left(2 \, a b - b^{2}\right)} \tan\left(d x + c\right)}{3 \, d}"," ",0,"1/3*(b^2*tan(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*d*x + 3*(2*a*b - b^2)*tan(d*x + c))/d","A",0
253,1,21,0,0.425566," ","integrate(a+b*tan(d*x+c)^2,x, algorithm=""fricas"")","\frac{{\left(a - b\right)} d x + b \tan\left(d x + c\right)}{d}"," ",0,"((a - b)*d*x + b*tan(d*x + c))/d","A",0
254,1,182,0,0.475691," ","integrate(1/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{4 \, d x - \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(d x + c\right)^{4} - 6 \, a b \tan\left(d x + c\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(d x + c\right)^{3} - a^{2} \tan\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(d x + c\right)^{4} + 2 \, a b \tan\left(d x + c\right)^{2} + a^{2}}\right)}{4 \, {\left(a - b\right)} d}, \frac{2 \, d x - \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(d x + c\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(d x + c\right)}\right)}{2 \, {\left(a - b\right)} d}\right]"," ",0,"[1/4*(4*d*x - sqrt(-b/a)*log((b^2*tan(d*x + c)^4 - 6*a*b*tan(d*x + c)^2 + a^2 + 4*(a*b*tan(d*x + c)^3 - a^2*tan(d*x + c))*sqrt(-b/a))/(b^2*tan(d*x + c)^4 + 2*a*b*tan(d*x + c)^2 + a^2)))/((a - b)*d), 1/2*(2*d*x - sqrt(b/a)*arctan(1/2*(b*tan(d*x + c)^2 - a)*sqrt(b/a)/(b*tan(d*x + c))))/((a - b)*d)]","A",0
255,1,390,0,0.448837," ","integrate(1/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, a b d x \tan\left(d x + c\right)^{2} + 8 \, a^{2} d x - {\left({\left(3 \, a b - b^{2}\right)} \tan\left(d x + c\right)^{2} + 3 \, a^{2} - a b\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(d x + c\right)^{4} - 6 \, a b \tan\left(d x + c\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(d x + c\right)^{3} - a^{2} \tan\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(d x + c\right)^{4} + 2 \, a b \tan\left(d x + c\right)^{2} + a^{2}}\right) - 4 \, {\left(a b - b^{2}\right)} \tan\left(d x + c\right)}{8 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}, \frac{4 \, a b d x \tan\left(d x + c\right)^{2} + 4 \, a^{2} d x - {\left({\left(3 \, a b - b^{2}\right)} \tan\left(d x + c\right)^{2} + 3 \, a^{2} - a b\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(d x + c\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(d x + c\right)}\right) - 2 \, {\left(a b - b^{2}\right)} \tan\left(d x + c\right)}{4 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}\right]"," ",0,"[1/8*(8*a*b*d*x*tan(d*x + c)^2 + 8*a^2*d*x - ((3*a*b - b^2)*tan(d*x + c)^2 + 3*a^2 - a*b)*sqrt(-b/a)*log((b^2*tan(d*x + c)^4 - 6*a*b*tan(d*x + c)^2 + a^2 + 4*(a*b*tan(d*x + c)^3 - a^2*tan(d*x + c))*sqrt(-b/a))/(b^2*tan(d*x + c)^4 + 2*a*b*tan(d*x + c)^2 + a^2)) - 4*(a*b - b^2)*tan(d*x + c))/((a^3*b - 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^2 + (a^4 - 2*a^3*b + a^2*b^2)*d), 1/4*(4*a*b*d*x*tan(d*x + c)^2 + 4*a^2*d*x - ((3*a*b - b^2)*tan(d*x + c)^2 + 3*a^2 - a*b)*sqrt(b/a)*arctan(1/2*(b*tan(d*x + c)^2 - a)*sqrt(b/a)/(b*tan(d*x + c))) - 2*(a*b - b^2)*tan(d*x + c))/((a^3*b - 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^2 + (a^4 - 2*a^3*b + a^2*b^2)*d)]","A",0
256,1,742,0,0.487571," ","integrate(1/(a+b*tan(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, a^{2} b^{2} d x \tan\left(d x + c\right)^{4} + 64 \, a^{3} b d x \tan\left(d x + c\right)^{2} + 32 \, a^{4} d x - 4 \, {\left(7 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(d x + c\right)^{3} - {\left({\left(15 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(d x + c\right)^{4} + 15 \, a^{4} - 10 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, {\left(15 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{b^{2} \tan\left(d x + c\right)^{4} - 6 \, a b \tan\left(d x + c\right)^{2} + a^{2} + 4 \, {\left(a b \tan\left(d x + c\right)^{3} - a^{2} \tan\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}}}{b^{2} \tan\left(d x + c\right)^{4} + 2 \, a b \tan\left(d x + c\right)^{2} + a^{2}}\right) - 4 \, {\left(9 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(d x + c\right)}{32 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \tan\left(d x + c\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \tan\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d\right)}}, \frac{16 \, a^{2} b^{2} d x \tan\left(d x + c\right)^{4} + 32 \, a^{3} b d x \tan\left(d x + c\right)^{2} + 16 \, a^{4} d x - 2 \, {\left(7 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(d x + c\right)^{3} - {\left({\left(15 \, a^{2} b^{2} - 10 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(d x + c\right)^{4} + 15 \, a^{4} - 10 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, {\left(15 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left(b \tan\left(d x + c\right)^{2} - a\right)} \sqrt{\frac{b}{a}}}{2 \, b \tan\left(d x + c\right)}\right) - 2 \, {\left(9 \, a^{3} b - 14 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \tan\left(d x + c\right)}{16 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \tan\left(d x + c\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \tan\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d\right)}}\right]"," ",0,"[1/32*(32*a^2*b^2*d*x*tan(d*x + c)^4 + 64*a^3*b*d*x*tan(d*x + c)^2 + 32*a^4*d*x - 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(d*x + c)^3 - ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(d*x + c)^4 + 15*a^4 - 10*a^3*b + 3*a^2*b^2 + 2*(15*a^3*b - 10*a^2*b^2 + 3*a*b^3)*tan(d*x + c)^2)*sqrt(-b/a)*log((b^2*tan(d*x + c)^4 - 6*a*b*tan(d*x + c)^2 + a^2 + 4*(a*b*tan(d*x + c)^3 - a^2*tan(d*x + c))*sqrt(-b/a))/(b^2*tan(d*x + c)^4 + 2*a*b*tan(d*x + c)^2 + a^2)) - 4*(9*a^3*b - 14*a^2*b^2 + 5*a*b^3)*tan(d*x + c))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*tan(d*x + c)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*tan(d*x + c)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d), 1/16*(16*a^2*b^2*d*x*tan(d*x + c)^4 + 32*a^3*b*d*x*tan(d*x + c)^2 + 16*a^4*d*x - 2*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(d*x + c)^3 - ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*tan(d*x + c)^4 + 15*a^4 - 10*a^3*b + 3*a^2*b^2 + 2*(15*a^3*b - 10*a^2*b^2 + 3*a*b^3)*tan(d*x + c)^2)*sqrt(b/a)*arctan(1/2*(b*tan(d*x + c)^2 - a)*sqrt(b/a)/(b*tan(d*x + c))) - 2*(9*a^3*b - 14*a^2*b^2 + 5*a*b^3)*tan(d*x + c))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*tan(d*x + c)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*tan(d*x + c)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d)]","B",0
257,1,56,0,0.421811," ","integrate((a+a*tan(x)^2)^(1/2)*tan(x)^4,x, algorithm=""fricas"")","\frac{1}{8} \, \sqrt{a \tan\left(x\right)^{2} + a} {\left(2 \, \tan\left(x\right)^{3} - 3 \, \tan\left(x\right)\right)} + \frac{3}{16} \, \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} \tan\left(x\right) + a\right)"," ",0,"1/8*sqrt(a*tan(x)^2 + a)*(2*tan(x)^3 - 3*tan(x)) + 3/16*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a*tan(x)^2 + a)*sqrt(a)*tan(x) + a)","A",0
258,1,18,0,0.379008," ","integrate((a+a*tan(x)^2)^(1/2)*tan(x)^3,x, algorithm=""fricas"")","\frac{1}{3} \, \sqrt{a \tan\left(x\right)^{2} + a} {\left(\tan\left(x\right)^{2} - 2\right)}"," ",0,"1/3*sqrt(a*tan(x)^2 + a)*(tan(x)^2 - 2)","A",0
259,1,47,0,0.420961," ","integrate((a+a*tan(x)^2)^(1/2)*tan(x)^2,x, algorithm=""fricas"")","\frac{1}{4} \, \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} \tan\left(x\right) + a\right) + \frac{1}{2} \, \sqrt{a \tan\left(x\right)^{2} + a} \tan\left(x\right)"," ",0,"1/4*sqrt(a)*log(2*a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a)*tan(x) + a) + 1/2*sqrt(a*tan(x)^2 + a)*tan(x)","A",0
260,1,10,0,0.415515," ","integrate((a+a*tan(x)^2)^(1/2)*tan(x),x, algorithm=""fricas"")","\sqrt{a \tan\left(x\right)^{2} + a}"," ",0,"sqrt(a*tan(x)^2 + a)","A",0
261,1,63,0,0.442679," ","integrate(cot(x)*(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{1}{2} \, \sqrt{a} \log\left(\frac{a \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{2}}\right), \sqrt{-a} \arctan\left(\frac{\sqrt{a \tan\left(x\right)^{2} + a} \sqrt{-a}}{a}\right)\right]"," ",0,"[1/2*sqrt(a)*log((a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2), sqrt(-a)*arctan(sqrt(a*tan(x)^2 + a)*sqrt(-a)/a)]","A",0
262,1,16,0,0.385819," ","integrate(cot(x)^2*(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \tan\left(x\right)^{2} + a}}{\tan\left(x\right)}"," ",0,"-sqrt(a*tan(x)^2 + a)/tan(x)","A",0
263,1,58,0,0.425781," ","integrate(cot(x)^3*(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a} \log\left(\frac{a \tan\left(x\right)^{2} + 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{2}}\right) \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a}}{4 \, \tan\left(x\right)^{2}}"," ",0,"1/4*(sqrt(a)*log((a*tan(x)^2 + 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2)*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a))/tan(x)^2","A",0
264,1,24,0,0.417488," ","integrate(cot(x)^4*(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \tan\left(x\right)^{2} + a} {\left(2 \, \tan\left(x\right)^{2} - 1\right)}}{3 \, \tan\left(x\right)^{3}}"," ",0,"1/3*sqrt(a*tan(x)^2 + a)*(2*tan(x)^2 - 1)/tan(x)^3","A",0
265,1,90,0,0.434692," ","integrate((a+a*tan(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(2 \, a \tan\left(d x + c\right)^{2} + 2 \, \sqrt{a \tan\left(d x + c\right)^{2} + a} \sqrt{a} \tan\left(d x + c\right) + a\right)}{2 \, d}, -\frac{\sqrt{-a} \arctan\left(\frac{\sqrt{a \tan\left(d x + c\right)^{2} + a} \sqrt{-a}}{a \tan\left(d x + c\right)}\right)}{d}\right]"," ",0,"[1/2*sqrt(a)*log(2*a*tan(d*x + c)^2 + 2*sqrt(a*tan(d*x + c)^2 + a)*sqrt(a)*tan(d*x + c) + a)/d, -sqrt(-a)*arctan(sqrt(a*tan(d*x + c)^2 + a)*sqrt(-a)/(a*tan(d*x + c)))/d]","A",0
266,1,29,0,0.402033," ","integrate(tan(x)^3*(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{1}{15} \, {\left(3 \, a \tan\left(x\right)^{4} + a \tan\left(x\right)^{2} - 2 \, a\right)} \sqrt{a \tan\left(x\right)^{2} + a}"," ",0,"1/15*(3*a*tan(x)^4 + a*tan(x)^2 - 2*a)*sqrt(a*tan(x)^2 + a)","A",0
267,1,57,0,0.426142," ","integrate(tan(x)^2*(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{1}{16} \, a^{\frac{3}{2}} \log\left(2 \, a \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} \tan\left(x\right) + a\right) + \frac{1}{8} \, {\left(2 \, a \tan\left(x\right)^{3} + a \tan\left(x\right)\right)} \sqrt{a \tan\left(x\right)^{2} + a}"," ",0,"1/16*a^(3/2)*log(2*a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a)*tan(x) + a) + 1/8*(2*a*tan(x)^3 + a*tan(x))*sqrt(a*tan(x)^2 + a)","A",0
268,1,12,0,0.454420," ","integrate(tan(x)*(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{1}{3} \, {\left(a \tan\left(x\right)^{2} + a\right)}^{\frac{3}{2}}"," ",0,"1/3*(a*tan(x)^2 + a)^(3/2)","A",0
269,1,49,0,0.442331," ","integrate(cot(x)*(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{1}{2} \, a^{\frac{3}{2}} \log\left(\frac{a \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{2}}\right) + \sqrt{a \tan\left(x\right)^{2} + a} a"," ",0,"1/2*a^(3/2)*log((a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2) + sqrt(a*tan(x)^2 + a)*a","A",0
270,1,53,0,0.409950," ","integrate(cot(x)^2*(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{a^{\frac{3}{2}} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} \tan\left(x\right) + a\right) \tan\left(x\right) - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} a}{2 \, \tan\left(x\right)}"," ",0,"1/2*(a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a*tan(x)^2 + a)*sqrt(a)*tan(x) + a)*tan(x) - 2*sqrt(a*tan(x)^2 + a)*a)/tan(x)","A",0
271,1,72,0,0.411924," ","integrate((a+a*tan(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","\frac{a^{\frac{3}{2}} \log\left(2 \, a \tan\left(d x + c\right)^{2} + 2 \, \sqrt{a \tan\left(d x + c\right)^{2} + a} \sqrt{a} \tan\left(d x + c\right) + a\right) + 2 \, \sqrt{a \tan\left(d x + c\right)^{2} + a} a \tan\left(d x + c\right)}{4 \, d}"," ",0,"1/4*(a^(3/2)*log(2*a*tan(d*x + c)^2 + 2*sqrt(a*tan(d*x + c)^2 + a)*sqrt(a)*tan(d*x + c) + a) + 2*sqrt(a*tan(d*x + c)^2 + a)*a*tan(d*x + c))/d","A",0
272,1,91,0,0.402529," ","integrate((a+a*tan(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\frac{3 \, a^{\frac{5}{2}} \log\left(2 \, a \tan\left(d x + c\right)^{2} + 2 \, \sqrt{a \tan\left(d x + c\right)^{2} + a} \sqrt{a} \tan\left(d x + c\right) + a\right) + 2 \, {\left(2 \, a^{2} \tan\left(d x + c\right)^{3} + 5 \, a^{2} \tan\left(d x + c\right)\right)} \sqrt{a \tan\left(d x + c\right)^{2} + a}}{16 \, d}"," ",0,"1/16*(3*a^(5/2)*log(2*a*tan(d*x + c)^2 + 2*sqrt(a*tan(d*x + c)^2 + a)*sqrt(a)*tan(d*x + c) + a) + 2*(2*a^2*tan(d*x + c)^3 + 5*a^2*tan(d*x + c))*sqrt(a*tan(d*x + c)^2 + a))/d","A",0
273,1,17,0,0.387256," ","integrate(tan(x)^3/(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\tan\left(x\right)^{2} + 2}{\sqrt{a \tan\left(x\right)^{2} + a}}"," ",0,"(tan(x)^2 + 2)/sqrt(a*tan(x)^2 + a)","A",0
274,1,64,0,0.421925," ","integrate(tan(x)^2/(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(\tan\left(x\right)^{2} + 1\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} \tan\left(x\right) + a\right) - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \tan\left(x\right)}{2 \, {\left(a \tan\left(x\right)^{2} + a\right)}}"," ",0,"1/2*((tan(x)^2 + 1)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a*tan(x)^2 + a)*sqrt(a)*tan(x) + a) - 2*sqrt(a*tan(x)^2 + a)*tan(x))/(a*tan(x)^2 + a)","B",0
275,1,12,0,0.408838," ","integrate(tan(x)/(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{1}{\sqrt{a \tan\left(x\right)^{2} + a}}"," ",0,"-1/sqrt(a*tan(x)^2 + a)","A",0
276,1,66,0,0.438009," ","integrate(cot(x)/(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(\tan\left(x\right)^{2} + 1\right)} \sqrt{a} \log\left(\frac{a \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{2}}\right) + 2 \, \sqrt{a \tan\left(x\right)^{2} + a}}{2 \, {\left(a \tan\left(x\right)^{2} + a\right)}}"," ",0,"1/2*((tan(x)^2 + 1)*sqrt(a)*log((a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2) + 2*sqrt(a*tan(x)^2 + a))/(a*tan(x)^2 + a)","B",0
277,1,33,0,0.394977," ","integrate(cot(x)^2/(a+a*tan(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \tan\left(x\right)^{2} + a} {\left(2 \, \tan\left(x\right)^{2} + 1\right)}}{a \tan\left(x\right)^{3} + a \tan\left(x\right)}"," ",0,"-sqrt(a*tan(x)^2 + a)*(2*tan(x)^2 + 1)/(a*tan(x)^3 + a*tan(x))","A",0
278,1,43,0,0.409643," ","integrate(tan(x)^3/(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \tan\left(x\right)^{2} + a} {\left(3 \, \tan\left(x\right)^{2} + 2\right)}}{3 \, {\left(a^{2} \tan\left(x\right)^{4} + 2 \, a^{2} \tan\left(x\right)^{2} + a^{2}\right)}}"," ",0,"-1/3*sqrt(a*tan(x)^2 + a)*(3*tan(x)^2 + 2)/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)","A",0
279,1,39,0,0.388865," ","integrate(tan(x)^2/(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{a \tan\left(x\right)^{2} + a} \tan\left(x\right)^{3}}{3 \, {\left(a^{2} \tan\left(x\right)^{4} + 2 \, a^{2} \tan\left(x\right)^{2} + a^{2}\right)}}"," ",0,"1/3*sqrt(a*tan(x)^2 + a)*tan(x)^3/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)","B",0
280,1,35,0,0.401899," ","integrate(tan(x)/(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \tan\left(x\right)^{2} + a}}{3 \, {\left(a^{2} \tan\left(x\right)^{4} + 2 \, a^{2} \tan\left(x\right)^{2} + a^{2}\right)}}"," ",0,"-1/3*sqrt(a*tan(x)^2 + a)/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)","B",0
281,1,94,0,0.416553," ","integrate(cot(x)/(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1\right)} \sqrt{a} \log\left(\frac{a \tan\left(x\right)^{2} - 2 \, \sqrt{a \tan\left(x\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{2}}\right) + 2 \, \sqrt{a \tan\left(x\right)^{2} + a} {\left(3 \, \tan\left(x\right)^{2} + 4\right)}}{6 \, {\left(a^{2} \tan\left(x\right)^{4} + 2 \, a^{2} \tan\left(x\right)^{2} + a^{2}\right)}}"," ",0,"1/6*(3*(tan(x)^4 + 2*tan(x)^2 + 1)*sqrt(a)*log((a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2) + 2*sqrt(a*tan(x)^2 + a)*(3*tan(x)^2 + 4))/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)","B",0
282,1,52,0,0.422284," ","integrate(cot(x)^2/(a+a*tan(x)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(8 \, \tan\left(x\right)^{4} + 12 \, \tan\left(x\right)^{2} + 3\right)} \sqrt{a \tan\left(x\right)^{2} + a}}{3 \, {\left(a^{2} \tan\left(x\right)^{5} + 2 \, a^{2} \tan\left(x\right)^{3} + a^{2} \tan\left(x\right)\right)}}"," ",0,"-1/3*(8*tan(x)^4 + 12*tan(x)^2 + 3)*sqrt(a*tan(x)^2 + a)/(a^2*tan(x)^5 + 2*a^2*tan(x)^3 + a^2*tan(x))","A",0
283,1,38,0,0.395550," ","integrate(1/(a+a*tan(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \tan\left(d x + c\right)^{2} + a} \tan\left(d x + c\right)}{a d \tan\left(d x + c\right)^{2} + a d}"," ",0,"sqrt(a*tan(d*x + c)^2 + a)*tan(d*x + c)/(a*d*tan(d*x + c)^2 + a*d)","A",0
284,1,70,0,0.429705," ","integrate(1/(a+a*tan(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{a \tan\left(d x + c\right)^{2} + a} {\left(2 \, \tan\left(d x + c\right)^{3} + 3 \, \tan\left(d x + c\right)\right)}}{3 \, {\left(a^{2} d \tan\left(d x + c\right)^{4} + 2 \, a^{2} d \tan\left(d x + c\right)^{2} + a^{2} d\right)}}"," ",0,"1/3*sqrt(a*tan(d*x + c)^2 + a)*(2*tan(d*x + c)^3 + 3*tan(d*x + c))/(a^2*d*tan(d*x + c)^4 + 2*a^2*d*tan(d*x + c)^2 + a^2*d)","A",0
285,1,94,0,0.419529," ","integrate(1/(a+a*tan(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(8 \, \tan\left(d x + c\right)^{5} + 20 \, \tan\left(d x + c\right)^{3} + 15 \, \tan\left(d x + c\right)\right)} \sqrt{a \tan\left(d x + c\right)^{2} + a}}{15 \, {\left(a^{3} d \tan\left(d x + c\right)^{6} + 3 \, a^{3} d \tan\left(d x + c\right)^{4} + 3 \, a^{3} d \tan\left(d x + c\right)^{2} + a^{3} d\right)}}"," ",0,"1/15*(8*tan(d*x + c)^5 + 20*tan(d*x + c)^3 + 15*tan(d*x + c))*sqrt(a*tan(d*x + c)^2 + a)/(a^3*d*tan(d*x + c)^6 + 3*a^3*d*tan(d*x + c)^4 + 3*a^3*d*tan(d*x + c)^2 + a^3*d)","A",0
286,1,118,0,0.393018," ","integrate(1/(a+a*tan(d*x+c)^2)^(7/2),x, algorithm=""fricas"")","\frac{{\left(16 \, \tan\left(d x + c\right)^{7} + 56 \, \tan\left(d x + c\right)^{5} + 70 \, \tan\left(d x + c\right)^{3} + 35 \, \tan\left(d x + c\right)\right)} \sqrt{a \tan\left(d x + c\right)^{2} + a}}{35 \, {\left(a^{4} d \tan\left(d x + c\right)^{8} + 4 \, a^{4} d \tan\left(d x + c\right)^{6} + 6 \, a^{4} d \tan\left(d x + c\right)^{4} + 4 \, a^{4} d \tan\left(d x + c\right)^{2} + a^{4} d\right)}}"," ",0,"1/35*(16*tan(d*x + c)^7 + 56*tan(d*x + c)^5 + 70*tan(d*x + c)^3 + 35*tan(d*x + c))*sqrt(a*tan(d*x + c)^2 + a)/(a^4*d*tan(d*x + c)^8 + 4*a^4*d*tan(d*x + c)^6 + 6*a^4*d*tan(d*x + c)^4 + 4*a^4*d*tan(d*x + c)^2 + a^4*d)","A",0
287,1,72,0,0.393044," ","integrate((1+tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{1}{2} \, \sqrt{\tan\left(x\right)^{2} + 1} \tan\left(x\right) + \frac{1}{4} \, \log\left(\frac{\tan\left(x\right)^{2} + \sqrt{\tan\left(x\right)^{2} + 1} \tan\left(x\right) + 1}{\tan\left(x\right)^{2} + 1}\right) - \frac{1}{4} \, \log\left(\frac{\tan\left(x\right)^{2} - \sqrt{\tan\left(x\right)^{2} + 1} \tan\left(x\right) + 1}{\tan\left(x\right)^{2} + 1}\right)"," ",0,"1/2*sqrt(tan(x)^2 + 1)*tan(x) + 1/4*log((tan(x)^2 + sqrt(tan(x)^2 + 1)*tan(x) + 1)/(tan(x)^2 + 1)) - 1/4*log((tan(x)^2 - sqrt(tan(x)^2 + 1)*tan(x) + 1)/(tan(x)^2 + 1))","B",0
288,1,60,0,0.405188," ","integrate((1+tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \log\left(\frac{\tan\left(x\right)^{2} + \sqrt{\tan\left(x\right)^{2} + 1} \tan\left(x\right) + 1}{\tan\left(x\right)^{2} + 1}\right) - \frac{1}{2} \, \log\left(\frac{\tan\left(x\right)^{2} - \sqrt{\tan\left(x\right)^{2} + 1} \tan\left(x\right) + 1}{\tan\left(x\right)^{2} + 1}\right)"," ",0,"1/2*log((tan(x)^2 + sqrt(tan(x)^2 + 1)*tan(x) + 1)/(tan(x)^2 + 1)) - 1/2*log((tan(x)^2 - sqrt(tan(x)^2 + 1)*tan(x) + 1)/(tan(x)^2 + 1))","B",0
289,1,11,0,0.449149," ","integrate(1/(1+tan(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\tan\left(x\right)}{\sqrt{\tan\left(x\right)^{2} + 1}}"," ",0,"tan(x)/sqrt(tan(x)^2 + 1)","A",0
290,1,73,0,0.446199," ","integrate((-1-tan(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(-i \, e^{\left(4 i \, x\right)} - 2 i \, e^{\left(2 i \, x\right)} - i\right)} \log\left(e^{\left(i \, x\right)} + i\right) + {\left(i \, e^{\left(4 i \, x\right)} + 2 i \, e^{\left(2 i \, x\right)} + i\right)} \log\left(e^{\left(i \, x\right)} - i\right) - 2 \, e^{\left(3 i \, x\right)} + 2 \, e^{\left(i \, x\right)}}{2 \, {\left(e^{\left(4 i \, x\right)} + 2 \, e^{\left(2 i \, x\right)} + 1\right)}}"," ",0,"1/2*((-I*e^(4*I*x) - 2*I*e^(2*I*x) - I)*log(e^(I*x) + I) + (I*e^(4*I*x) + 2*I*e^(2*I*x) + I)*log(e^(I*x) - I) - 2*e^(3*I*x) + 2*e^(I*x))/(e^(4*I*x) + 2*e^(2*I*x) + 1)","C",0
291,1,19,0,0.413863," ","integrate((-1-tan(x)^2)^(1/2),x, algorithm=""fricas"")","i \, \log\left(e^{\left(i \, x\right)} + i\right) - i \, \log\left(e^{\left(i \, x\right)} - i\right)"," ",0,"I*log(e^(I*x) + I) - I*log(e^(I*x) - I)","C",0
292,1,12,0,0.387810," ","integrate(1/(-1-tan(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{1}{2} \, {\left(e^{\left(2 i \, x\right)} - 1\right)} e^{\left(-i \, x\right)}"," ",0,"-1/2*(e^(2*I*x) - 1)*e^(-I*x)","C",0
293,1,320,0,0.585136," ","integrate((a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^5,x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{a - b} b^{2} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left(3 \, b^{2} \tan\left(f x + e\right)^{4} + {\left(a b - 5 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2} - 5 \, a b + 15 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, b^{2} f}, \frac{15 \, \sqrt{-a + b} b^{2} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left(3 \, b^{2} \tan\left(f x + e\right)^{4} + {\left(a b - 5 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2} - 5 \, a b + 15 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, b^{2} f}\right]"," ",0,"[1/60*(15*sqrt(a - b)*b^2*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*(3*b^2*tan(f*x + e)^4 + (a*b - 5*b^2)*tan(f*x + e)^2 - 2*a^2 - 5*a*b + 15*b^2)*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/30*(15*sqrt(-a + b)*b^2*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(3*b^2*tan(f*x + e)^4 + (a*b - 5*b^2)*tan(f*x + e)^2 - 2*a^2 - 5*a*b + 15*b^2)*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f)]","A",0
294,1,254,0,0.511441," ","integrate((a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a - b} b \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left(b \tan\left(f x + e\right)^{2} + a - 3 \, b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, b f}, -\frac{3 \, \sqrt{-a + b} b \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) - 2 \, {\left(b \tan\left(f x + e\right)^{2} + a - 3 \, b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, b f}\right]"," ",0,"[1/12*(3*sqrt(a - b)*b*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*(b*tan(f*x + e)^2 + a - 3*b)*sqrt(b*tan(f*x + e)^2 + a))/(b*f), -1/6*(3*sqrt(-a + b)*b*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) - 2*(b*tan(f*x + e)^2 + a - 3*b)*sqrt(b*tan(f*x + e)^2 + a))/(b*f)]","A",0
295,1,214,0,0.508867," ","integrate((a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e),x, algorithm=""fricas"")","\left[\frac{\sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, f}, \frac{\sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, f}\right]"," ",0,"[1/4*(sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*sqrt(b*tan(f*x + e)^2 + a))/f, 1/2*(sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*sqrt(b*tan(f*x + e)^2 + a))/f]","A",0
296,1,382,0,0.431630," ","integrate(cot(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right)}{2 \, f}, \frac{2 \, \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right)}{2 \, f}, \frac{2 \, \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right)}{f}\right]"," ",0,"[1/2*(sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2))/f, 1/2*(2*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2))/f, 1/2*(2*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)))/f, (sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)))/f]","A",0
297,1,592,0,0.447673," ","integrate(cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a - b} a \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} - {\left(2 \, a - b\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{4 \, a f \tan\left(f x + e\right)^{2}}, -\frac{4 \, a \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{2} + {\left(2 \, a - b\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{4 \, a f \tan\left(f x + e\right)^{2}}, -\frac{\sqrt{-a} {\left(2 \, a - b\right)} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{2} - \sqrt{a - b} a \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{2 \, a f \tan\left(f x + e\right)^{2}}, -\frac{\sqrt{-a} {\left(2 \, a - b\right)} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{2} + 2 \, a \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{2} + \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{2 \, a f \tan\left(f x + e\right)^{2}}\right]"," ",0,"[1/4*(2*sqrt(a - b)*a*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 - (2*a - b)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*a)/(a*f*tan(f*x + e)^2), -1/4*(4*a*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 + (2*a - b)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*a)/(a*f*tan(f*x + e)^2), -1/2*(sqrt(-a)*(2*a - b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 - sqrt(a - b)*a*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + sqrt(b*tan(f*x + e)^2 + a)*a)/(a*f*tan(f*x + e)^2), -1/2*(sqrt(-a)*(2*a - b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 + 2*a*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 + sqrt(b*tan(f*x + e)^2 + a)*a)/(a*f*tan(f*x + e)^2)]","A",0
298,1,729,0,0.446314," ","integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, \sqrt{a - b} a^{2} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} - {\left(8 \, a^{2} - 4 \, a b - b^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{4} + 2 \, {\left({\left(4 \, a^{2} - a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, a^{2} f \tan\left(f x + e\right)^{4}}, \frac{16 \, a^{2} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{4} - {\left(8 \, a^{2} - 4 \, a b - b^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{4} + 2 \, {\left({\left(4 \, a^{2} - a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, a^{2} f \tan\left(f x + e\right)^{4}}, \frac{4 \, \sqrt{a - b} a^{2} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{2} - 4 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{4} + {\left({\left(4 \, a^{2} - a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, a^{2} f \tan\left(f x + e\right)^{4}}, \frac{8 \, a^{2} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{2} - 4 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{4} + {\left({\left(4 \, a^{2} - a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, a^{2} f \tan\left(f x + e\right)^{4}}\right]"," ",0,"[1/16*(8*sqrt(a - b)*a^2*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 - (8*a^2 - 4*a*b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 + 2*((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4), 1/16*(16*a^2*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 - (8*a^2 - 4*a*b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 + 2*((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4), 1/8*(4*sqrt(a - b)*a^2*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (8*a^2 - 4*a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 + ((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4), 1/8*(8*a^2*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^2 - 4*a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 + ((4*a^2 - a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^4)]","A",0
299,1,826,0,2.099103," ","integrate((a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^6,x, algorithm=""fricas"")","\left[\frac{48 \, \sqrt{-a + b} b^{3} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 3 \, {\left(a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{96 \, b^{3} f}, -\frac{96 \, \sqrt{a - b} b^{3} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + 3 \, {\left(a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{96 \, b^{3} f}, \frac{24 \, \sqrt{-a + b} b^{3} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 3 \, {\left(a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{48 \, b^{3} f}, -\frac{48 \, \sqrt{a - b} b^{3} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + 3 \, {\left(a^{3} + 2 \, a^{2} b + 8 \, a b^{2} - 16 \, b^{3}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(a^{2} b + 2 \, a b^{2} - 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{48 \, b^{3} f}\right]"," ",0,"[1/96*(48*sqrt(-a + b)*b^3*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 3*(a^3 + 2*a^2*b + 8*a*b^2 - 16*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(8*b^3*tan(f*x + e)^5 + 2*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/96*(96*sqrt(a - b)*b^3*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(a^3 + 2*a^2*b + 8*a*b^2 - 16*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*(8*b^3*tan(f*x + e)^5 + 2*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), 1/48*(24*sqrt(-a + b)*b^3*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 3*(a^3 + 2*a^2*b + 8*a*b^2 - 16*b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + (8*b^3*tan(f*x + e)^5 + 2*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/48*(48*sqrt(a - b)*b^3*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(a^3 + 2*a^2*b + 8*a*b^2 - 16*b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (8*b^3*tan(f*x + e)^5 + 2*(a*b^2 - 6*b^3)*tan(f*x + e)^3 - 3*(a^2*b + 2*a*b^2 - 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f)]","A",0
300,1,671,0,1.201296," ","integrate((a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\left[\frac{8 \, \sqrt{-a + b} b^{2} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(a^{2} + 4 \, a b - 8 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, b^{2} f}, \frac{16 \, \sqrt{a - b} b^{2} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(a^{2} + 4 \, a b - 8 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, b^{2} f}, \frac{4 \, \sqrt{-a + b} b^{2} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(a^{2} + 4 \, a b - 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, b^{2} f}, \frac{8 \, \sqrt{a - b} b^{2} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a^{2} + 4 \, a b - 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, b^{2} f}\right]"," ",0,"[1/16*(8*sqrt(-a + b)*b^2*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - (a^2 + 4*a*b - 8*b^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(2*b^2*tan(f*x + e)^3 + (a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/16*(16*sqrt(a - b)*b^2*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (a^2 + 4*a*b - 8*b^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(2*b^2*tan(f*x + e)^3 + (a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/8*(4*sqrt(-a + b)*b^2*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + (a^2 + 4*a*b - 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + (2*b^2*tan(f*x + e)^3 + (a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/8*(8*sqrt(a - b)*b^2*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a^2 + 4*a*b - 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + (2*b^2*tan(f*x + e)^3 + (a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f)]","A",0
301,1,539,0,0.723053," ","integrate((a+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","\left[-\frac{{\left(a - 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, \sqrt{-a + b} b \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{4 \, b f}, -\frac{4 \, \sqrt{a - b} b \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a - 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{4 \, b f}, -\frac{{\left(a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - \sqrt{-a + b} b \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{2 \, b f}, -\frac{2 \, \sqrt{a - b} b \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{2 \, b f}\right]"," ",0,"[-1/4*((a - 2*b)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*sqrt(-a + b)*b*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/(b*f), -1/4*(4*sqrt(a - b)*b*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a - 2*b)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/(b*f), -1/2*((a - 2*b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - sqrt(-a + b)*b*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/(b*f), -1/2*(2*sqrt(a - b)*b*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a - 2*b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/(b*f)]","A",0
302,1,410,0,0.508517," ","integrate((a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}, \frac{2 \, \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right)}{2 \, f}, -\frac{2 \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, f}, \frac{\sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right)}{f}\right]"," ",0,"[1/2*(sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)))/f, 1/2*(2*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a))/f, -1/2*(2*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)))/f, (sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))))/f]","A",0
303,1,257,0,0.528702," ","integrate(cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right) - 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, f \tan\left(f x + e\right)}, -\frac{\sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, f \tan\left(f x + e\right)}\right]"," ",0,"[1/4*(sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e) - 4*sqrt(b*tan(f*x + e)^2 + a))/(f*tan(f*x + e)), -1/2*(sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e) + 2*sqrt(b*tan(f*x + e)^2 + a))/(f*tan(f*x + e))]","A",0
304,1,311,0,0.610192," ","integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{3} + 4 \, {\left({\left(3 \, a - b\right)} \tan\left(f x + e\right)^{2} - a\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, a f \tan\left(f x + e\right)^{3}}, \frac{3 \, \sqrt{a - b} a \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right)^{3} + 2 \, {\left({\left(3 \, a - b\right)} \tan\left(f x + e\right)^{2} - a\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, a f \tan\left(f x + e\right)^{3}}\right]"," ",0,"[1/12*(3*a*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^3 + 4*((3*a - b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^3), 1/6*(3*sqrt(a - b)*a*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e)^3 + 2*((3*a - b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^3)]","A",0
305,1,375,0,0.542818," ","integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{15 \, a^{2} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{5} - 4 \, {\left({\left(15 \, a^{2} - 5 \, a b - 2 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{2} - a b\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, a^{2} f \tan\left(f x + e\right)^{5}}, -\frac{15 \, \sqrt{a - b} a^{2} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right)^{5} + 2 \, {\left({\left(15 \, a^{2} - 5 \, a b - 2 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{2} - a b\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, a^{2} f \tan\left(f x + e\right)^{5}}\right]"," ",0,"[1/60*(15*a^2*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^5 - 4*((15*a^2 - 5*a*b - 2*b^2)*tan(f*x + e)^4 - (5*a^2 - a*b)*tan(f*x + e)^2 + 3*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^5), -1/30*(15*sqrt(a - b)*a^2*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e)^5 + 2*((15*a^2 - 5*a*b - 2*b^2)*tan(f*x + e)^4 - (5*a^2 - a*b)*tan(f*x + e)^2 + 3*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a^2*f*tan(f*x + e)^5)]","A",0
306,1,414,0,0.550861," ","integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{105 \, {\left(a b^{2} - b^{3}\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(15 \, b^{3} \tan\left(f x + e\right)^{6} + 3 \, {\left(8 \, a b^{2} - 7 \, b^{3}\right)} \tan\left(f x + e\right)^{4} - 6 \, a^{3} - 21 \, a^{2} b + 140 \, a b^{2} - 105 \, b^{3} + {\left(3 \, a^{2} b - 42 \, a b^{2} + 35 \, b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{420 \, b^{2} f}, \frac{105 \, {\left(a b^{2} - b^{3}\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left(15 \, b^{3} \tan\left(f x + e\right)^{6} + 3 \, {\left(8 \, a b^{2} - 7 \, b^{3}\right)} \tan\left(f x + e\right)^{4} - 6 \, a^{3} - 21 \, a^{2} b + 140 \, a b^{2} - 105 \, b^{3} + {\left(3 \, a^{2} b - 42 \, a b^{2} + 35 \, b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{210 \, b^{2} f}\right]"," ",0,"[-1/420*(105*(a*b^2 - b^3)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(15*b^3*tan(f*x + e)^6 + 3*(8*a*b^2 - 7*b^3)*tan(f*x + e)^4 - 6*a^3 - 21*a^2*b + 140*a*b^2 - 105*b^3 + (3*a^2*b - 42*a*b^2 + 35*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/210*(105*(a*b^2 - b^3)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(15*b^3*tan(f*x + e)^6 + 3*(8*a*b^2 - 7*b^3)*tan(f*x + e)^4 - 6*a^3 - 21*a^2*b + 140*a*b^2 - 105*b^3 + (3*a^2*b - 42*a*b^2 + 35*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f)]","A",0
307,1,334,0,0.533494," ","integrate(tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a b - b^{2}\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(3 \, b^{2} \tan\left(f x + e\right)^{4} + {\left(6 \, a b - 5 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2} - 20 \, a b + 15 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, b f}, -\frac{15 \, {\left(a b - b^{2}\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) - 2 \, {\left(3 \, b^{2} \tan\left(f x + e\right)^{4} + {\left(6 \, a b - 5 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2} - 20 \, a b + 15 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, b f}\right]"," ",0,"[-1/60*(15*(a*b - b^2)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(3*b^2*tan(f*x + e)^4 + (6*a*b - 5*b^2)*tan(f*x + e)^2 + 3*a^2 - 20*a*b + 15*b^2)*sqrt(b*tan(f*x + e)^2 + a))/(b*f), -1/30*(15*(a*b - b^2)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) - 2*(3*b^2*tan(f*x + e)^4 + (6*a*b - 5*b^2)*tan(f*x + e)^2 + 3*a^2 - 20*a*b + 15*b^2)*sqrt(b*tan(f*x + e)^2 + a))/(b*f)]","A",0
308,1,255,0,0.593792," ","integrate(tan(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a - b\right)}^{\frac{3}{2}} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 4 \, a - 3 \, b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, f}, \frac{3 \, {\left(a - b\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left(b \tan\left(f x + e\right)^{2} + 4 \, a - 3 \, b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, f}\right]"," ",0,"[-1/12*(3*(a - b)^(3/2)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(b*tan(f*x + e)^2 + 4*a - 3*b)*sqrt(b*tan(f*x + e)^2 + a))/f, 1/6*(3*(a - b)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(b*tan(f*x + e)^2 + 4*a - 3*b)*sqrt(b*tan(f*x + e)^2 + a))/f]","A",0
309,1,591,0,1.222924," ","integrate(cot(f*x+e)*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a - b\right)}^{\frac{3}{2}} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 2 \, a^{\frac{3}{2}} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b}{4 \, f}, \frac{4 \, \sqrt{-a} a \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) - {\left(a - b\right)}^{\frac{3}{2}} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b}{4 \, f}, \frac{{\left(-a + b\right)}^{\frac{3}{2}} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + a^{\frac{3}{2}} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b}{2 \, f}, \frac{2 \, \sqrt{-a} a \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + {\left(-a + b\right)}^{\frac{3}{2}} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b}{2 \, f}\right]"," ",0,"[-1/4*((a - b)^(3/2)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 2*a^(3/2)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 4*sqrt(b*tan(f*x + e)^2 + a)*b)/f, 1/4*(4*sqrt(-a)*a*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - (a - b)^(3/2)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*sqrt(b*tan(f*x + e)^2 + a)*b)/f, 1/2*((-a + b)^(3/2)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + a^(3/2)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*sqrt(b*tan(f*x + e)^2 + a)*b)/f, 1/2*(2*sqrt(-a)*a*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + (-a + b)^(3/2)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*sqrt(b*tan(f*x + e)^2 + a)*b)/f]","A",0
310,1,584,0,0.480321," ","integrate(cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a - b\right)}^{\frac{3}{2}} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + {\left(2 \, a - 3 \, b\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{4 \, f \tan\left(f x + e\right)^{2}}, -\frac{4 \, {\left(a - b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{2} + {\left(2 \, a - 3 \, b\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{4 \, f \tan\left(f x + e\right)^{2}}, -\frac{\sqrt{-a} {\left(2 \, a - 3 \, b\right)} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{2} + {\left(a - b\right)}^{\frac{3}{2}} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{2 \, f \tan\left(f x + e\right)^{2}}, -\frac{\sqrt{-a} {\left(2 \, a - 3 \, b\right)} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{2} + 2 \, {\left(a - b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{2} + \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{2 \, f \tan\left(f x + e\right)^{2}}\right]"," ",0,"[-1/4*(2*(a - b)^(3/2)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + (2*a - 3*b)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)^2), -1/4*(4*(a - b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 + (2*a - 3*b)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)^2), -1/2*(sqrt(-a)*(2*a - 3*b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 + (a - b)^(3/2)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)^2), -1/2*(sqrt(-a)*(2*a - 3*b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 + 2*(a - b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 + sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)^2)]","A",0
311,1,748,0,0.492081," ","integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{2} - a b\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} - {\left(8 \, a^{2} - 12 \, a b + 3 \, b^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{4} - 2 \, {\left({\left(4 \, a^{2} - 5 \, a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, a f \tan\left(f x + e\right)^{4}}, \frac{16 \, {\left(a^{2} - a b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{2} - 12 \, a b + 3 \, b^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{4} + 2 \, {\left({\left(4 \, a^{2} - 5 \, a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, a f \tan\left(f x + e\right)^{4}}, \frac{{\left(8 \, a^{2} - 12 \, a b + 3 \, b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{4} - 4 \, {\left(a^{2} - a b\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left({\left(4 \, a^{2} - 5 \, a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, a f \tan\left(f x + e\right)^{4}}, \frac{{\left(8 \, a^{2} - 12 \, a b + 3 \, b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{4} + 8 \, {\left(a^{2} - a b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{4} + {\left({\left(4 \, a^{2} - 5 \, a b\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, a f \tan\left(f x + e\right)^{4}}\right]"," ",0,"[-1/16*(8*(a^2 - a*b)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 - (8*a^2 - 12*a*b + 3*b^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 - 2*((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/16*(16*(a^2 - a*b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^2 - 12*a*b + 3*b^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 + 2*((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/8*((8*a^2 - 12*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 - 4*(a^2 - a*b)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + ((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4), 1/8*((8*a^2 - 12*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 + 8*(a^2 - a*b)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + ((4*a^2 - 5*a*b)*tan(f*x + e)^2 - 2*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^4)]","A",0
312,1,1059,0,4.944471," ","integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(3 \, a^{4} + 8 \, a^{3} b + 48 \, a^{2} b^{2} - 192 \, a b^{3} + 128 \, b^{4}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 384 \, {\left(a b^{3} - b^{4}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left(48 \, b^{4} \tan\left(f x + e\right)^{7} + 8 \, {\left(9 \, a b^{3} - 8 \, b^{4}\right)} \tan\left(f x + e\right)^{5} + 2 \, {\left(3 \, a^{2} b^{2} - 56 \, a b^{3} + 48 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{768 \, b^{3} f}, -\frac{3 \, {\left(3 \, a^{4} + 8 \, a^{3} b + 48 \, a^{2} b^{2} - 192 \, a b^{3} + 128 \, b^{4}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + 192 \, {\left(a b^{3} - b^{4}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(48 \, b^{4} \tan\left(f x + e\right)^{7} + 8 \, {\left(9 \, a b^{3} - 8 \, b^{4}\right)} \tan\left(f x + e\right)^{5} + 2 \, {\left(3 \, a^{2} b^{2} - 56 \, a b^{3} + 48 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{384 \, b^{3} f}, -\frac{768 \, {\left(a b^{3} - b^{4}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - 3 \, {\left(3 \, a^{4} + 8 \, a^{3} b + 48 \, a^{2} b^{2} - 192 \, a b^{3} + 128 \, b^{4}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left(48 \, b^{4} \tan\left(f x + e\right)^{7} + 8 \, {\left(9 \, a b^{3} - 8 \, b^{4}\right)} \tan\left(f x + e\right)^{5} + 2 \, {\left(3 \, a^{2} b^{2} - 56 \, a b^{3} + 48 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{768 \, b^{3} f}, -\frac{384 \, {\left(a b^{3} - b^{4}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + 3 \, {\left(3 \, a^{4} + 8 \, a^{3} b + 48 \, a^{2} b^{2} - 192 \, a b^{3} + 128 \, b^{4}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(48 \, b^{4} \tan\left(f x + e\right)^{7} + 8 \, {\left(9 \, a b^{3} - 8 \, b^{4}\right)} \tan\left(f x + e\right)^{5} + 2 \, {\left(3 \, a^{2} b^{2} - 56 \, a b^{3} + 48 \, b^{4}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{3} b + 8 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{384 \, b^{3} f}\right]"," ",0,"[1/768*(3*(3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 384*(a*b^3 - b^4)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*(48*b^4*tan(f*x + e)^7 + 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3*a^2*b^2 - 56*a*b^3 + 48*b^4)*tan(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80*a*b^3 + 64*b^4)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/384*(3*(3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + 192*(a*b^3 - b^4)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - (48*b^4*tan(f*x + e)^7 + 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3*a^2*b^2 - 56*a*b^3 + 48*b^4)*tan(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80*a*b^3 + 64*b^4)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/768*(768*(a*b^3 - b^4)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - 3*(3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*(48*b^4*tan(f*x + e)^7 + 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3*a^2*b^2 - 56*a*b^3 + 48*b^4)*tan(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80*a*b^3 + 64*b^4)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f), -1/384*(384*(a*b^3 - b^4)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (48*b^4*tan(f*x + e)^7 + 8*(9*a*b^3 - 8*b^4)*tan(f*x + e)^5 + 2*(3*a^2*b^2 - 56*a*b^3 + 48*b^4)*tan(f*x + e)^3 - 3*(3*a^3*b + 8*a^2*b^2 - 80*a*b^3 + 64*b^4)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^3*f)]","A",0
313,1,861,0,2.987069," ","integrate(tan(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 48 \, {\left(a b^{2} - b^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(7 \, a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{96 \, b^{2} f}, \frac{3 \, {\left(a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - 24 \, {\left(a b^{2} - b^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(7 \, a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{48 \, b^{2} f}, \frac{96 \, {\left(a b^{2} - b^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + 3 \, {\left(a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(7 \, a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{96 \, b^{2} f}, \frac{48 \, {\left(a b^{2} - b^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + 3 \, {\left(a^{3} + 6 \, a^{2} b - 24 \, a b^{2} + 16 \, b^{3}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + {\left(8 \, b^{3} \tan\left(f x + e\right)^{5} + 2 \, {\left(7 \, a b^{2} - 6 \, b^{3}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{2} b - 10 \, a b^{2} + 8 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{48 \, b^{2} f}\right]"," ",0,"[1/96*(3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 48*(a*b^2 - b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*(8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2*b - 10*a*b^2 + 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/48*(3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - 24*(a*b^2 - b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + (8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2*b - 10*a*b^2 + 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/96*(96*(a*b^2 - b^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2*b - 10*a*b^2 + 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f), 1/48*(48*(a*b^2 - b^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + (8*b^3*tan(f*x + e)^5 + 2*(7*a*b^2 - 6*b^3)*tan(f*x + e)^3 + 3*(a^2*b - 10*a*b^2 + 8*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b^2*f)]","A",0
314,1,708,0,1.244750," ","integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 8 \, {\left(a b - b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(5 \, a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, b f}, -\frac{{\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + 4 \, {\left(a b - b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(5 \, a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, b f}, -\frac{16 \, {\left(a b - b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(5 \, a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, b f}, -\frac{8 \, {\left(a b - b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(2 \, b^{2} \tan\left(f x + e\right)^{3} + {\left(5 \, a b - 4 \, b^{2}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, b f}\right]"," ",0,"[1/16*((3*a^2 - 12*a*b + 8*b^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 8*(a*b - b^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*(2*b^2*tan(f*x + e)^3 + (5*a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b*f), -1/8*((3*a^2 - 12*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + 4*(a*b - b^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - (2*b^2*tan(f*x + e)^3 + (5*a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b*f), -1/16*(16*(a*b - b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a^2 - 12*a*b + 8*b^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*(2*b^2*tan(f*x + e)^3 + (5*a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b*f), -1/8*(8*(a*b - b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (3*a^2 - 12*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (2*b^2*tan(f*x + e)^3 + (5*a*b - 4*b^2)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/(b*f)]","A",0
315,1,537,0,0.934635," ","integrate((a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(3 \, a - 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(a - b\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{4 \, f}, -\frac{{\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(-a + b\right)}^{\frac{3}{2}} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{2 \, f}, \frac{4 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a - 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{4 \, f}, \frac{2 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + \sqrt{b \tan\left(f x + e\right)^{2} + a} b \tan\left(f x + e\right)}{2 \, f}\right]"," ",0,"[-1/4*((3*a - 2*b)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(a - b)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f, -1/2*((3*a - 2*b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (-a + b)^(3/2)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f, 1/4*(4*(a - b)^(3/2)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a - 2*b)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f, 1/2*(2*(a - b)^(3/2)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a - 2*b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + sqrt(b*tan(f*x + e)^2 + a)*b*tan(f*x + e))/f]","A",0
316,1,710,0,1.275800," ","integrate(cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{2 \, b^{\frac{3}{2}} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) \tan\left(f x + e\right) - {\left(a - b\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right) - 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{4 \, f \tan\left(f x + e\right)}, -\frac{4 \, \sqrt{-b} b \arctan\left(\frac{\sqrt{-b} \tan\left(f x + e\right)}{\sqrt{b \tan\left(f x + e\right)^{2} + a}}\right) \tan\left(f x + e\right) + {\left(a - b\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right) + 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{4 \, f \tan\left(f x + e\right)}, -\frac{{\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right) - b^{\frac{3}{2}} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) \tan\left(f x + e\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{2 \, f \tan\left(f x + e\right)}, -\frac{{\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right) + 2 \, \sqrt{-b} b \arctan\left(\frac{\sqrt{-b} \tan\left(f x + e\right)}{\sqrt{b \tan\left(f x + e\right)^{2} + a}}\right) \tan\left(f x + e\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} a}{2 \, f \tan\left(f x + e\right)}\right]"," ",0,"[1/4*(2*b^(3/2)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a)*tan(f*x + e) - (a - b)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e) - 4*sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)), -1/4*(4*sqrt(-b)*b*arctan(sqrt(-b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a))*tan(f*x + e) + (a - b)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e) + 4*sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)), -1/2*((a - b)^(3/2)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e) - b^(3/2)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a)*tan(f*x + e) + 2*sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e)), -1/2*((a - b)^(3/2)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e) + 2*sqrt(-b)*b*arctan(sqrt(-b)*tan(f*x + e)/sqrt(b*tan(f*x + e)^2 + a))*tan(f*x + e) + 2*sqrt(b*tan(f*x + e)^2 + a)*a)/(f*tan(f*x + e))]","A",0
317,1,308,0,0.554378," ","integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a - b\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{3} - 4 \, {\left({\left(3 \, a - 4 \, b\right)} \tan\left(f x + e\right)^{2} - a\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, f \tan\left(f x + e\right)^{3}}, \frac{3 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right)^{3} + 2 \, {\left({\left(3 \, a - 4 \, b\right)} \tan\left(f x + e\right)^{2} - a\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, f \tan\left(f x + e\right)^{3}}\right]"," ",0,"[-1/12*(3*(a - b)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^3 - 4*((3*a - 4*b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(f*tan(f*x + e)^3), 1/6*(3*(a - b)^(3/2)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e)^3 + 2*((3*a - 4*b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(f*tan(f*x + e)^3)]","A",0
318,1,385,0,0.570059," ","integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{2} - a b\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{5} + 4 \, {\left({\left(15 \, a^{2} - 20 \, a b + 3 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{2} - 6 \, a b\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, a f \tan\left(f x + e\right)^{5}}, -\frac{15 \, {\left(a^{2} - a b\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right)^{5} + 2 \, {\left({\left(15 \, a^{2} - 20 \, a b + 3 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{2} - 6 \, a b\right)} \tan\left(f x + e\right)^{2} + 3 \, a^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, a f \tan\left(f x + e\right)^{5}}\right]"," ",0,"[-1/60*(15*(a^2 - a*b)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^5 + 4*((15*a^2 - 20*a*b + 3*b^2)*tan(f*x + e)^4 - (5*a^2 - 6*a*b)*tan(f*x + e)^2 + 3*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^5), -1/30*(15*(a^2 - a*b)*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e)^5 + 2*((15*a^2 - 20*a*b + 3*b^2)*tan(f*x + e)^4 - (5*a^2 - 6*a*b)*tan(f*x + e)^2 + 3*a^2)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^5)]","A",0
319,1,703,0,2.224355," ","integrate((a+b*tan(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{{\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(d x + c\right)^{2} + 2 \, \sqrt{b \tan\left(d x + c\right)^{2} + a} \sqrt{b} \tan\left(d x + c\right) + a\right) + 8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(d x + c\right)^{2} + 2 \, \sqrt{b \tan\left(d x + c\right)^{2} + a} \sqrt{-a + b} \tan\left(d x + c\right) - a}{\tan\left(d x + c\right)^{2} + 1}\right) + 2 \, {\left(2 \, b^{2} \tan\left(d x + c\right)^{3} + {\left(9 \, a b - 4 \, b^{2}\right)} \tan\left(d x + c\right)\right)} \sqrt{b \tan\left(d x + c\right)^{2} + a}}{16 \, d}, \frac{16 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(d x + c\right)^{2} + a}}{\sqrt{a - b} \tan\left(d x + c\right)}\right) + {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(d x + c\right)^{2} + 2 \, \sqrt{b \tan\left(d x + c\right)^{2} + a} \sqrt{b} \tan\left(d x + c\right) + a\right) + 2 \, {\left(2 \, b^{2} \tan\left(d x + c\right)^{3} + {\left(9 \, a b - 4 \, b^{2}\right)} \tan\left(d x + c\right)\right)} \sqrt{b \tan\left(d x + c\right)^{2} + a}}{16 \, d}, -\frac{{\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(d x + c\right)^{2} + a} \sqrt{-b}}{b \tan\left(d x + c\right)}\right) - 4 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(d x + c\right)^{2} + 2 \, \sqrt{b \tan\left(d x + c\right)^{2} + a} \sqrt{-a + b} \tan\left(d x + c\right) - a}{\tan\left(d x + c\right)^{2} + 1}\right) - {\left(2 \, b^{2} \tan\left(d x + c\right)^{3} + {\left(9 \, a b - 4 \, b^{2}\right)} \tan\left(d x + c\right)\right)} \sqrt{b \tan\left(d x + c\right)^{2} + a}}{8 \, d}, \frac{8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(d x + c\right)^{2} + a}}{\sqrt{a - b} \tan\left(d x + c\right)}\right) - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(d x + c\right)^{2} + a} \sqrt{-b}}{b \tan\left(d x + c\right)}\right) + {\left(2 \, b^{2} \tan\left(d x + c\right)^{3} + {\left(9 \, a b - 4 \, b^{2}\right)} \tan\left(d x + c\right)\right)} \sqrt{b \tan\left(d x + c\right)^{2} + a}}{8 \, d}\right]"," ",0,"[1/16*((15*a^2 - 20*a*b + 8*b^2)*sqrt(b)*log(2*b*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sqrt(b)*tan(d*x + c) + a) + 8*(a^2 - 2*a*b + b^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sqrt(-a + b)*tan(d*x + c) - a)/(tan(d*x + c)^2 + 1)) + 2*(2*b^2*tan(d*x + c)^3 + (9*a*b - 4*b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d, 1/16*(16*(a^2 - 2*a*b + b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(d*x + c)^2 + a)/(sqrt(a - b)*tan(d*x + c))) + (15*a^2 - 20*a*b + 8*b^2)*sqrt(b)*log(2*b*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sqrt(b)*tan(d*x + c) + a) + 2*(2*b^2*tan(d*x + c)^3 + (9*a*b - 4*b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d, -1/8*((15*a^2 - 20*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(d*x + c)^2 + a)*sqrt(-b)/(b*tan(d*x + c))) - 4*(a^2 - 2*a*b + b^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sqrt(-a + b)*tan(d*x + c) - a)/(tan(d*x + c)^2 + 1)) - (2*b^2*tan(d*x + c)^3 + (9*a*b - 4*b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d, 1/8*(8*(a^2 - 2*a*b + b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(d*x + c)^2 + a)/(sqrt(a - b)*tan(d*x + c))) - (15*a^2 - 20*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(d*x + c)^2 + a)*sqrt(-b)/(b*tan(d*x + c))) + (2*b^2*tan(d*x + c)^3 + (9*a*b - 4*b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d]","A",0
320,1,314,0,0.553681," ","integrate(tan(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a - b} b^{2} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left(a b^{2} - b^{3}\right)} f}, \frac{3 \, \sqrt{-a + b} b^{2} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left({\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{2} - 2 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left(a b^{2} - b^{3}\right)} f}\right]"," ",0,"[1/12*(3*sqrt(a - b)*b^2*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*((a*b - b^2)*tan(f*x + e)^2 - 2*a^2 - a*b + 3*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a*b^2 - b^3)*f), 1/6*(3*sqrt(-a + b)*b^2*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*((a*b - b^2)*tan(f*x + e)^2 - 2*a^2 - a*b + 3*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a*b^2 - b^3)*f)]","A",0
321,1,248,0,0.543397," ","integrate(tan(f*x+e)^3/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a - b} b \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)}}{4 \, {\left(a b - b^{2}\right)} f}, -\frac{\sqrt{-a + b} b \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)}}{2 \, {\left(a b - b^{2}\right)} f}\right]"," ",0,"[1/4*(sqrt(a - b)*b*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*sqrt(b*tan(f*x + e)^2 + a)*(a - b))/((a*b - b^2)*f), -1/2*(sqrt(-a + b)*b*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) - 2*sqrt(b*tan(f*x + e)^2 + a)*(a - b))/((a*b - b^2)*f)]","A",0
322,1,185,0,0.570051," ","integrate(tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right)}{4 \, \sqrt{a - b} f}, \frac{\sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right)}{2 \, {\left(a - b\right)} f}\right]"," ",0,"[1/4*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))/(sqrt(a - b)*f), 1/2*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b))/((a - b)*f)]","A",0
323,1,446,0,0.463121," ","integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a - b} a \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(a - b\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right)}{2 \, {\left(a^{2} - a b\right)} f}, \frac{2 \, a \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + {\left(a - b\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right)}{2 \, {\left(a^{2} - a b\right)} f}, \frac{2 \, \sqrt{-a} {\left(a - b\right)} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + \sqrt{a - b} a \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a^{2} - a b\right)} f}, \frac{\sqrt{-a} {\left(a - b\right)} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + a \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right)}{{\left(a^{2} - a b\right)} f}\right]"," ",0,"[1/2*(sqrt(a - b)*a*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + (a - b)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2))/((a^2 - a*b)*f), 1/2*(2*a*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + (a - b)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2))/((a^2 - a*b)*f), 1/2*(2*sqrt(-a)*(a - b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + sqrt(a - b)*a*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)))/((a^2 - a*b)*f), (sqrt(-a)*(a - b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + a*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)))/((a^2 - a*b)*f)]","A",0
324,1,697,0,0.460987," ","integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a - b} a^{2} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} + {\left(2 \, a^{2} - a b - b^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a^{2} - a b\right)}}{4 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{2}}, -\frac{4 \, a^{2} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{2} - {\left(2 \, a^{2} - a b - b^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a^{2} - a b\right)}}{4 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{2}}, \frac{\sqrt{a - b} a^{2} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{2} - {\left(2 \, a^{2} - a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{2} - \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a^{2} - a b\right)}}{2 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{2}}, -\frac{2 \, a^{2} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{2} + {\left(2 \, a^{2} - a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{2} + \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a^{2} - a b\right)}}{2 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{2}}\right]"," ",0,"[1/4*(2*sqrt(a - b)*a^2*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + (2*a^2 - a*b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), -1/4*(4*a^2*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 - (2*a^2 - a*b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), 1/2*(sqrt(a - b)*a^2*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 - (2*a^2 - a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 - sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), -1/2*(2*a^2*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^2 + (2*a^2 - a*b - b^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^2 + sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2)]","A",0
325,1,857,0,0.493282," ","integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, \sqrt{a - b} a^{3} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{4} - 2 \, {\left(2 \, a^{3} - 2 \, a^{2} b - {\left(4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{4}}, \frac{16 \, a^{3} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) \tan\left(f x + e\right)^{4} - 2 \, {\left(2 \, a^{3} - 2 \, a^{2} b - {\left(4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{4}}, \frac{4 \, \sqrt{a - b} a^{3} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{4} - {\left(2 \, a^{3} - 2 \, a^{2} b - {\left(4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{4}}, \frac{8 \, a^{3} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) \tan\left(f x + e\right)^{4} + {\left(8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) \tan\left(f x + e\right)^{4} - {\left(2 \, a^{3} - 2 \, a^{2} b - {\left(4 \, a^{3} - a^{2} b - 3 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{4}}\right]"," ",0,"[1/16*(8*sqrt(a - b)*a^3*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 - 2*(2*a^3 - 2*a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^4), 1/16*(16*a^3*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^4 - 2*(2*a^3 - 2*a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^4), 1/8*(4*sqrt(a - b)*a^3*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 - (2*a^3 - 2*a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^4), 1/8*(8*a^3*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b))*tan(f*x + e)^4 + (8*a^3 - 4*a^2*b - a*b^2 - 3*b^3)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a)*tan(f*x + e)^4 - (2*a^3 - 2*a^2*b - (4*a^3 - a^2*b - 3*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^4)]","A",0
326,1,817,0,2.079005," ","integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, \sqrt{-a + b} b^{3} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(3 \, a^{3} + a^{2} b + 4 \, a b^{2} - 8 \, b^{3}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left(2 \, {\left(a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{3} - {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, {\left(a b^{3} - b^{4}\right)} f}, -\frac{4 \, \sqrt{-a + b} b^{3} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(3 \, a^{3} + a^{2} b + 4 \, a b^{2} - 8 \, b^{3}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(2 \, {\left(a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{3} - {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, {\left(a b^{3} - b^{4}\right)} f}, -\frac{16 \, \sqrt{a - b} b^{3} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a^{3} + a^{2} b + 4 \, a b^{2} - 8 \, b^{3}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left(2 \, {\left(a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{3} - {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, {\left(a b^{3} - b^{4}\right)} f}, -\frac{8 \, \sqrt{a - b} b^{3} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(3 \, a^{3} + a^{2} b + 4 \, a b^{2} - 8 \, b^{3}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(2 \, {\left(a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{3} - {\left(3 \, a^{2} b + a b^{2} - 4 \, b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, {\left(a b^{3} - b^{4}\right)} f}\right]"," ",0,"[-1/16*(8*sqrt(-a + b)*b^3*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - (3*a^3 + a^2*b + 4*a*b^2 - 8*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*(2*(a*b^2 - b^3)*tan(f*x + e)^3 - (3*a^2*b + a*b^2 - 4*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a*b^3 - b^4)*f), -1/8*(4*sqrt(-a + b)*b^3*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + (3*a^3 + a^2*b + 4*a*b^2 - 8*b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (2*(a*b^2 - b^3)*tan(f*x + e)^3 - (3*a^2*b + a*b^2 - 4*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a*b^3 - b^4)*f), -1/16*(16*sqrt(a - b)*b^3*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a^3 + a^2*b + 4*a*b^2 - 8*b^3)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*(2*(a*b^2 - b^3)*tan(f*x + e)^3 - (3*a^2*b + a*b^2 - 4*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a*b^3 - b^4)*f), -1/8*(8*sqrt(a - b)*b^3*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (3*a^3 + a^2*b + 4*a*b^2 - 8*b^3)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (2*(a*b^2 - b^3)*tan(f*x + e)^3 - (3*a^2*b + a*b^2 - 4*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a*b^3 - b^4)*f)]","A",0
327,1,647,0,1.185901," ","integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{-a + b} b^{2} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(a^{2} + a b - 2 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{4 \, {\left(a b^{2} - b^{3}\right)} f}, -\frac{\sqrt{-a + b} b^{2} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(a^{2} + a b - 2 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{2 \, {\left(a b^{2} - b^{3}\right)} f}, \frac{4 \, \sqrt{a - b} b^{2} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a^{2} + a b - 2 \, b^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{4 \, {\left(a b^{2} - b^{3}\right)} f}, \frac{2 \, \sqrt{a - b} b^{2} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a^{2} + a b - 2 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{2 \, {\left(a b^{2} - b^{3}\right)} f}\right]"," ",0,"[-1/4*(2*sqrt(-a + b)*b^2*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - (a^2 + a*b - 2*b^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a*b^2 - b^3)*f), -1/2*(sqrt(-a + b)*b^2*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - (a^2 + a*b - 2*b^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a*b^2 - b^3)*f), 1/4*(4*sqrt(a - b)*b^2*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a^2 + a*b - 2*b^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a*b^2 - b^3)*f), 1/2*(2*sqrt(a - b)*b^2*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a^2 + a*b - 2*b^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a*b^2 - b^3)*f)]","A",0
328,1,479,0,0.547027," ","integrate(tan(f*x+e)^2/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a - b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - \sqrt{-a + b} b \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a b - b^{2}\right)} f}, -\frac{2 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + \sqrt{-a + b} b \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a b - b^{2}\right)} f}, -\frac{2 \, \sqrt{a - b} b \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(a - b\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right)}{2 \, {\left(a b - b^{2}\right)} f}, -\frac{\sqrt{a - b} b \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right)}{{\left(a b - b^{2}\right)} f}\right]"," ",0,"[1/2*((a - b)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - sqrt(-a + b)*b*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)))/((a*b - b^2)*f), -1/2*(2*(a - b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + sqrt(-a + b)*b*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)))/((a*b - b^2)*f), -1/2*(2*sqrt(a - b)*b*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (a - b)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a))/((a*b - b^2)*f), -(sqrt(a - b)*b*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a - b)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))))/((a*b - b^2)*f)]","A",0
329,1,125,0,0.444235," ","integrate(1/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right)}{2 \, {\left(a - b\right)} f}, \frac{\arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right)}{\sqrt{a - b} f}\right]"," ",0,"[-1/2*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1))/((a - b)*f), arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e)))/(sqrt(a - b)*f)]","A",0
330,1,289,0,0.565553," ","integrate(cot(f*x+e)^2/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{a \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right) + 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)}}{4 \, {\left(a^{2} - a b\right)} f \tan\left(f x + e\right)}, -\frac{\sqrt{a - b} a \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)}}{2 \, {\left(a^{2} - a b\right)} f \tan\left(f x + e\right)}\right]"," ",0,"[-1/4*(a*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e) + 4*sqrt(b*tan(f*x + e)^2 + a)*(a - b))/((a^2 - a*b)*f*tan(f*x + e)), -1/2*(sqrt(a - b)*a*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e) + 2*sqrt(b*tan(f*x + e)^2 + a)*(a - b))/((a^2 - a*b)*f*tan(f*x + e))]","A",0
331,1,359,0,0.563650," ","integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{3 \, a^{2} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{3} - 4 \, {\left({\left(3 \, a^{2} - a b - 2 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - a^{2} + a b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{3}}, \frac{3 \, \sqrt{a - b} a^{2} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right)^{3} + 2 \, {\left({\left(3 \, a^{2} - a b - 2 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - a^{2} + a b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left(a^{3} - a^{2} b\right)} f \tan\left(f x + e\right)^{3}}\right]"," ",0,"[-1/12*(3*a^2*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^3 - 4*((3*a^2 - a*b - 2*b^2)*tan(f*x + e)^2 - a^2 + a*b)*sqrt(b*tan(f*x + e)^2 + a))/((a^3 - a^2*b)*f*tan(f*x + e)^3), 1/6*(3*sqrt(a - b)*a^2*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e)^3 + 2*((3*a^2 - a*b - 2*b^2)*tan(f*x + e)^2 - a^2 + a*b)*sqrt(b*tan(f*x + e)^2 + a))/((a^3 - a^2*b)*f*tan(f*x + e)^3)]","A",0
332,1,437,0,0.579199," ","integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{15 \, a^{3} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) \tan\left(f x + e\right)^{5} + 4 \, {\left({\left(15 \, a^{3} - 5 \, a^{2} b - 2 \, a b^{2} - 8 \, b^{3}\right)} \tan\left(f x + e\right)^{4} + 3 \, a^{3} - 3 \, a^{2} b - {\left(5 \, a^{3} - a^{2} b - 4 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{5}}, -\frac{15 \, \sqrt{a - b} a^{3} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) \tan\left(f x + e\right)^{5} + 2 \, {\left({\left(15 \, a^{3} - 5 \, a^{2} b - 2 \, a b^{2} - 8 \, b^{3}\right)} \tan\left(f x + e\right)^{4} + 3 \, a^{3} - 3 \, a^{2} b - {\left(5 \, a^{3} - a^{2} b - 4 \, a b^{2}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, {\left(a^{4} - a^{3} b\right)} f \tan\left(f x + e\right)^{5}}\right]"," ",0,"[-1/60*(15*a^3*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1))*tan(f*x + e)^5 + 4*((15*a^3 - 5*a^2*b - 2*a*b^2 - 8*b^3)*tan(f*x + e)^4 + 3*a^3 - 3*a^2*b - (5*a^3 - a^2*b - 4*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^5), -1/30*(15*sqrt(a - b)*a^3*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a))*tan(f*x + e)^5 + 2*((15*a^3 - 5*a^2*b - 2*a*b^2 - 8*b^3)*tan(f*x + e)^4 + 3*a^3 - 3*a^2*b - (5*a^3 - a^2*b - 4*a*b^2)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4 - a^3*b)*f*tan(f*x + e)^5)]","A",0
333,1,432,0,0.555010," ","integrate(tan(f*x+e)^5/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(b^{3} \tan\left(f x + e\right)^{2} + a b^{2}\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(2 \, a^{3} - 3 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}, \frac{{\left(b^{3} \tan\left(f x + e\right)^{2} + a b^{2}\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left(2 \, a^{3} - 3 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}\right]"," ",0,"[-1/4*((b^3*tan(f*x + e)^2 + a*b^2)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(2*a^3 - 3*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f), 1/2*((b^3*tan(f*x + e)^2 + a*b^2)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(2*a^3 - 3*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f)]","B",0
334,1,358,0,0.549790," ","integrate(tan(f*x+e)^3/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(b^{2} \tan\left(f x + e\right)^{2} + a b\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a^{2} - a b\right)}}{4 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, -\frac{{\left(b^{2} \tan\left(f x + e\right)^{2} + a b\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a^{2} - a b\right)}}{2 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[-1/4*((b^2*tan(f*x + e)^2 + a*b)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^2*b^2 - 2*a*b^3 + b^4)*f*tan(f*x + e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f), -1/2*((b^2*tan(f*x + e)^2 + a*b)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^2*b^2 - 2*a*b^3 + b^4)*f*tan(f*x + e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f)]","B",0
335,1,332,0,0.575768," ","integrate(tan(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(b \tan\left(f x + e\right)^{2} + a\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)}}{4 \, {\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} f\right)}}, \frac{{\left(b \tan\left(f x + e\right)^{2} + a\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)}}{2 \, {\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} f\right)}}\right]"," ",0,"[-1/4*((b*tan(f*x + e)^2 + a)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*sqrt(b*tan(f*x + e)^2 + a)*(a - b))/((a^2*b - 2*a*b^2 + b^3)*f*tan(f*x + e)^2 + (a^3 - 2*a^2*b + a*b^2)*f), 1/2*((b*tan(f*x + e)^2 + a)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*sqrt(b*tan(f*x + e)^2 + a)*(a - b))/((a^2*b - 2*a*b^2 + b^3)*f*tan(f*x + e)^2 + (a^3 - 2*a^2*b + a*b^2)*f)]","B",0
336,1,920,0,0.490669," ","integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} b \tan\left(f x + e\right)^{2} + a^{3}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) + 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}, \frac{2 \, {\left(a^{2} b \tan\left(f x + e\right)^{2} + a^{3}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}, \frac{2 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) - {\left(a^{2} b \tan\left(f x + e\right)^{2} + a^{3}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}, \frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + {\left(a^{2} b \tan\left(f x + e\right)^{2} + a^{3}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) - {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{{\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f}\right]"," ",0,"[-1/2*((a^2*b*tan(f*x + e)^2 + a^3)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), 1/2*(2*(a^2*b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), 1/2*(2*(a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - (a^2*b*tan(f*x + e)^2 + a^3)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), ((a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + (a^2*b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f)]","B",0
337,1,1252,0,0.544459," ","integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{3} b \tan\left(f x + e\right)^{4} + a^{4} \tan\left(f x + e\right)^{2}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left({\left(2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) + 2 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{2}\right)}}, -\frac{4 \, {\left(a^{3} b \tan\left(f x + e\right)^{4} + a^{4} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) - {\left({\left(2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) + 2 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{2}\right)}}, -\frac{{\left({\left(2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + {\left(a^{3} b \tan\left(f x + e\right)^{4} + a^{4} \tan\left(f x + e\right)^{2}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{2}\right)}}, -\frac{{\left({\left(2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} + 3 \, b^{4}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 2 \, {\left(a^{3} b \tan\left(f x + e\right)^{4} + a^{4} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[-1/4*(2*(a^3*b*tan(f*x + e)^4 + a^4*tan(f*x + e)^2)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - ((2*a^3*b - a^2*b^2 - 4*a*b^3 + 3*b^4)*tan(f*x + e)^4 + (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(a^4 - 2*a^3*b + a^2*b^2 + (a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^2), -1/4*(4*(a^3*b*tan(f*x + e)^4 + a^4*tan(f*x + e)^2)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - ((2*a^3*b - a^2*b^2 - 4*a*b^3 + 3*b^4)*tan(f*x + e)^4 + (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(a^4 - 2*a^3*b + a^2*b^2 + (a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^2), -1/2*(((2*a^3*b - a^2*b^2 - 4*a*b^3 + 3*b^4)*tan(f*x + e)^4 + (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + (a^3*b*tan(f*x + e)^4 + a^4*tan(f*x + e)^2)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + (a^4 - 2*a^3*b + a^2*b^2 + (a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^2), -1/2*(((2*a^3*b - a^2*b^2 - 4*a*b^3 + 3*b^4)*tan(f*x + e)^4 + (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 2*(a^3*b*tan(f*x + e)^4 + a^4*tan(f*x + e)^2)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + (a^4 - 2*a^3*b + a^2*b^2 + (a^3*b - 4*a^2*b^2 + 3*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^2)]","B",0
338,1,1522,0,0.564374," ","integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{4} b \tan\left(f x + e\right)^{6} + a^{5} \tan\left(f x + e\right)^{4}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left({\left(8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) + 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left(4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - {\left(4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{4}\right)}}, \frac{16 \, {\left(a^{4} b \tan\left(f x + e\right)^{6} + a^{5} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + {\left({\left(8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left(4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - {\left(4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{16 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{4}\right)}}, \frac{{\left({\left(8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) - 4 \, {\left(a^{4} b \tan\left(f x + e\right)^{6} + a^{5} \tan\left(f x + e\right)^{4}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left(4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - {\left(4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{4}\right)}}, \frac{{\left({\left(8 \, a^{4} b - 4 \, a^{3} b^{2} - a^{2} b^{3} - 18 \, a b^{4} + 15 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{5} - 4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 8 \, {\left(a^{4} b \tan\left(f x + e\right)^{6} + a^{5} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) - {\left(2 \, a^{5} - 4 \, a^{4} b + 2 \, a^{3} b^{2} - {\left(4 \, a^{4} b - a^{3} b^{2} - 18 \, a^{2} b^{3} + 15 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - {\left(4 \, a^{5} - 3 \, a^{4} b - 6 \, a^{3} b^{2} + 5 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{8 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{4}\right)}}\right]"," ",0,"[-1/16*(8*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - ((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/16*(16*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + ((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/8*(((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - 4*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - (2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4), 1/8*(((8*a^4*b - 4*a^3*b^2 - a^2*b^3 - 18*a*b^4 + 15*b^5)*tan(f*x + e)^6 + (8*a^5 - 4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 8*(a^4*b*tan(f*x + e)^6 + a^5*tan(f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (2*a^5 - 4*a^4*b + 2*a^3*b^2 - (4*a^4*b - a^3*b^2 - 18*a^2*b^3 + 15*a*b^4)*tan(f*x + e)^4 - (4*a^5 - 3*a^4*b - 6*a^3*b^2 + 5*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^6 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^4)]","A",0
339,1,1207,0,2.336937," ","integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{4} - 4 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left(b^{4} \tan\left(f x + e\right)^{2} + a b^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} f\right)}}, \frac{{\left(3 \, a^{4} - 4 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + {\left(b^{4} \tan\left(f x + e\right)^{2} + a b^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} f\right)}}, -\frac{4 \, {\left(b^{4} \tan\left(f x + e\right)^{2} + a b^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a^{4} - 4 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} f\right)}}, -\frac{2 \, {\left(b^{4} \tan\left(f x + e\right)^{2} + a b^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(3 \, a^{4} - 4 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \tan\left(f x + e\right)^{3} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} f\right)}}\right]"," ",0,"[1/4*((3*a^4 - 4*a^3*b - a^2*b^2 + 2*a*b^3 + (3*a^3*b - 4*a^2*b^2 - a*b^3 + 2*b^4)*tan(f*x + e)^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*(b^4*tan(f*x + e)^2 + a*b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*((a^2*b^2 - 2*a*b^3 + b^4)*tan(f*x + e)^3 + (3*a^3*b - 4*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^2*b^4 - 2*a*b^5 + b^6)*f*tan(f*x + e)^2 + (a^3*b^3 - 2*a^2*b^4 + a*b^5)*f), 1/2*((3*a^4 - 4*a^3*b - a^2*b^2 + 2*a*b^3 + (3*a^3*b - 4*a^2*b^2 - a*b^3 + 2*b^4)*tan(f*x + e)^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + (b^4*tan(f*x + e)^2 + a*b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + ((a^2*b^2 - 2*a*b^3 + b^4)*tan(f*x + e)^3 + (3*a^3*b - 4*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^2*b^4 - 2*a*b^5 + b^6)*f*tan(f*x + e)^2 + (a^3*b^3 - 2*a^2*b^4 + a*b^5)*f), -1/4*(4*(b^4*tan(f*x + e)^2 + a*b^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a^4 - 4*a^3*b - a^2*b^2 + 2*a*b^3 + (3*a^3*b - 4*a^2*b^2 - a*b^3 + 2*b^4)*tan(f*x + e)^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*((a^2*b^2 - 2*a*b^3 + b^4)*tan(f*x + e)^3 + (3*a^3*b - 4*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^2*b^4 - 2*a*b^5 + b^6)*f*tan(f*x + e)^2 + (a^3*b^3 - 2*a^2*b^4 + a*b^5)*f), -1/2*(2*(b^4*tan(f*x + e)^2 + a*b^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (3*a^4 - 4*a^3*b - a^2*b^2 + 2*a*b^3 + (3*a^3*b - 4*a^2*b^2 - a*b^3 + 2*b^4)*tan(f*x + e)^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - ((a^2*b^2 - 2*a*b^3 + b^4)*tan(f*x + e)^3 + (3*a^3*b - 4*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^2*b^4 - 2*a*b^5 + b^6)*f*tan(f*x + e)^2 + (a^3*b^3 - 2*a^2*b^4 + a*b^5)*f)]","A",0
340,1,974,0,1.360148," ","integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + {\left(b^{3} \tan\left(f x + e\right)^{2} + a b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \tan\left(f x + e\right)}{2 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}, -\frac{2 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(b^{3} \tan\left(f x + e\right)^{2} + a b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \tan\left(f x + e\right)}{2 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}, \frac{2 \, {\left(b^{3} \tan\left(f x + e\right)^{2} + a b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \tan\left(f x + e\right)}{2 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f\right)}}, \frac{{\left(b^{3} \tan\left(f x + e\right)^{2} + a b^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) - {\left(a^{2} b - a b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \tan\left(f x + e\right)}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} f}\right]"," ",0,"[1/2*((a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + (b^3*tan(f*x + e)^2 + a*b^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a)*tan(f*x + e))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f), -1/2*(2*(a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (b^3*tan(f*x + e)^2 + a*b^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a)*tan(f*x + e))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f), 1/2*(2*(b^3*tan(f*x + e)^2 + a*b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a)*tan(f*x + e))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f), ((b^3*tan(f*x + e)^2 + a*b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) - (a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a)*tan(f*x + e))/((a^2*b^3 - 2*a*b^4 + b^5)*f*tan(f*x + e)^2 + (a^3*b^2 - 2*a^2*b^3 + a*b^4)*f)]","B",0
341,1,285,0,0.472110," ","integrate(tan(f*x+e)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(b \tan\left(f x + e\right)^{2} + a\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)} \tan\left(f x + e\right)}{2 \, {\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} f\right)}}, -\frac{{\left(b \tan\left(f x + e\right)^{2} + a\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a - b\right)} \tan\left(f x + e\right)}{{\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} f}\right]"," ",0,"[1/2*((b*tan(f*x + e)^2 + a)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*sqrt(b*tan(f*x + e)^2 + a)*(a - b)*tan(f*x + e))/((a^2*b - 2*a*b^2 + b^3)*f*tan(f*x + e)^2 + (a^3 - 2*a^2*b + a*b^2)*f), -((b*tan(f*x + e)^2 + a)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - sqrt(b*tan(f*x + e)^2 + a)*(a - b)*tan(f*x + e))/((a^2*b - 2*a*b^2 + b^3)*f*tan(f*x + e)^2 + (a^3 - 2*a^2*b + a*b^2)*f)]","A",0
342,1,310,0,0.453020," ","integrate(1/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{2 \, {\left({\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}, \frac{{\left(a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - \sqrt{b \tan\left(f x + e\right)^{2} + a} {\left(a b - b^{2}\right)} \tan\left(f x + e\right)}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} f}\right]"," ",0,"[1/2*((a*b*tan(f*x + e)^2 + a^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f), ((a*b*tan(f*x + e)^2 + a^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - sqrt(b*tan(f*x + e)^2 + a)*(a*b - b^2)*tan(f*x + e))/((a^3*b - 2*a^2*b^2 + a*b^3)*f*tan(f*x + e)^2 + (a^4 - 2*a^3*b + a^2*b^2)*f)]","A",0
343,1,471,0,1.024584," ","integrate(cot(f*x+e)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} b \tan\left(f x + e\right)^{3} + a^{3} \tan\left(f x + e\right)\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{4 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \tan\left(f x + e\right)\right)}}, -\frac{{\left(a^{2} b \tan\left(f x + e\right)^{3} + a^{3} \tan\left(f x + e\right)\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) + 2 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{2 \, {\left({\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} f \tan\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*((a^2*b*tan(f*x + e)^3 + a^3*tan(f*x + e))*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(a^3 - 2*a^2*b + a*b^2 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^3 + (a^5 - 2*a^4*b + a^3*b^2)*f*tan(f*x + e)), -1/2*((a^2*b*tan(f*x + e)^3 + a^3*tan(f*x + e))*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*(a^3 - 2*a^2*b + a*b^2 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^3 + (a^5 - 2*a^4*b + a^3*b^2)*f*tan(f*x + e))]","A",0
344,1,579,0,0.579407," ","integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} b \tan\left(f x + e\right)^{5} + a^{4} \tan\left(f x + e\right)^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(3 \, a^{3} b - a^{2} b^{2} - 10 \, a b^{3} + 8 \, b^{4}\right)} \tan\left(f x + e\right)^{4} - a^{4} + 2 \, a^{3} b - a^{2} b^{2} + {\left(3 \, a^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{3}\right)}}, \frac{3 \, {\left(a^{3} b \tan\left(f x + e\right)^{5} + a^{4} \tan\left(f x + e\right)^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) + 2 \, {\left({\left(3 \, a^{3} b - a^{2} b^{2} - 10 \, a b^{3} + 8 \, b^{4}\right)} \tan\left(f x + e\right)^{4} - a^{4} + 2 \, a^{3} b - a^{2} b^{2} + {\left(3 \, a^{4} - 2 \, a^{3} b - 5 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} f \tan\left(f x + e\right)^{3}\right)}}\right]"," ",0,"[1/12*(3*(a^3*b*tan(f*x + e)^5 + a^4*tan(f*x + e)^3)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*((3*a^3*b - a^2*b^2 - 10*a*b^3 + 8*b^4)*tan(f*x + e)^4 - a^4 + 2*a^3*b - a^2*b^2 + (3*a^4 - 2*a^3*b - 5*a^2*b^2 + 4*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^5 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^3), 1/6*(3*(a^3*b*tan(f*x + e)^5 + a^4*tan(f*x + e)^3)*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*((3*a^3*b - a^2*b^2 - 10*a*b^3 + 8*b^4)*tan(f*x + e)^4 - a^4 + 2*a^3*b - a^2*b^2 + (3*a^4 - 2*a^3*b - 5*a^2*b^2 + 4*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^5 + (a^6 - 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^3)]","A",0
345,1,687,0,0.562589," ","integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{4} b \tan\left(f x + e\right)^{7} + a^{5} \tan\left(f x + e\right)^{5}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(15 \, a^{4} b - 5 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 56 \, a b^{4} + 48 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + 3 \, a^{5} - 6 \, a^{4} b + 3 \, a^{3} b^{2} + {\left(15 \, a^{5} - 10 \, a^{4} b - a^{3} b^{2} - 28 \, a^{2} b^{3} + 24 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{5} - 4 \, a^{4} b - 7 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{5}\right)}}, -\frac{15 \, {\left(a^{4} b \tan\left(f x + e\right)^{7} + a^{5} \tan\left(f x + e\right)^{5}\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) + 2 \, {\left({\left(15 \, a^{4} b - 5 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 56 \, a b^{4} + 48 \, b^{5}\right)} \tan\left(f x + e\right)^{6} + 3 \, a^{5} - 6 \, a^{4} b + 3 \, a^{3} b^{2} + {\left(15 \, a^{5} - 10 \, a^{4} b - a^{3} b^{2} - 28 \, a^{2} b^{3} + 24 \, a b^{4}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{5} - 4 \, a^{4} b - 7 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, {\left({\left(a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} f \tan\left(f x + e\right)^{5}\right)}}\right]"," ",0,"[1/60*(15*(a^4*b*tan(f*x + e)^7 + a^5*tan(f*x + e)^5)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*((15*a^4*b - 5*a^3*b^2 - 2*a^2*b^3 - 56*a*b^4 + 48*b^5)*tan(f*x + e)^6 + 3*a^5 - 6*a^4*b + 3*a^3*b^2 + (15*a^5 - 10*a^4*b - a^3*b^2 - 28*a^2*b^3 + 24*a*b^4)*tan(f*x + e)^4 - (5*a^5 - 4*a^4*b - 7*a^3*b^2 + 6*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5), -1/30*(15*(a^4*b*tan(f*x + e)^7 + a^5*tan(f*x + e)^5)*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*((15*a^4*b - 5*a^3*b^2 - 2*a^2*b^3 - 56*a*b^4 + 48*b^5)*tan(f*x + e)^6 + 3*a^5 - 6*a^4*b + 3*a^3*b^2 + (15*a^5 - 10*a^4*b - a^3*b^2 - 28*a^2*b^3 + 24*a*b^4)*tan(f*x + e)^4 - (5*a^5 - 4*a^4*b - 7*a^3*b^2 + 6*a^2*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5)]","A",0
346,1,608,0,0.551997," ","integrate(tan(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{4} \tan\left(f x + e\right)^{4} + 2 \, a b^{3} \tan\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(2 \, a^{4} - 7 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{3} - 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f\right)}}, \frac{3 \, {\left(b^{4} \tan\left(f x + e\right)^{4} + 2 \, a b^{3} \tan\left(f x + e\right)^{2} + a^{2} b^{2}\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) - 2 \, {\left(2 \, a^{4} - 7 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{3} - 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[1/12*(3*(b^4*tan(f*x + e)^4 + 2*a*b^3*tan(f*x + e)^2 + a^2*b^2)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(2*a^4 - 7*a^3*b + 5*a^2*b^2 + 3*(a^3*b - 3*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*f*tan(f*x + e)^4 + 2*(a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 - a*b^6)*f*tan(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f), 1/6*(3*(b^4*tan(f*x + e)^4 + 2*a*b^3*tan(f*x + e)^2 + a^2*b^2)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) - 2*(2*a^4 - 7*a^3*b + 5*a^2*b^2 + 3*(a^3*b - 3*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*f*tan(f*x + e)^4 + 2*(a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 - a*b^6)*f*tan(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f)]","B",0
347,1,572,0,0.579832," ","integrate(tan(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{3} \tan\left(f x + e\right)^{4} + 2 \, a b^{2} \tan\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left(a^{3} + a^{2} b - 2 \, a b^{2} + 3 \, {\left(a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} f\right)}}, -\frac{3 \, {\left(b^{3} \tan\left(f x + e\right)^{4} + 2 \, a b^{2} \tan\left(f x + e\right)^{2} + a^{2} b\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left(a^{3} + a^{2} b - 2 \, a b^{2} + 3 \, {\left(a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} f\right)}}\right]"," ",0,"[1/12*(3*(b^3*tan(f*x + e)^4 + 2*a*b^2*tan(f*x + e)^2 + a^2*b)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 + 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*(a^3 + a^2*b - 2*a*b^2 + 3*(a*b^2 - b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*f*tan(f*x + e)^4 + 2*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*f*tan(f*x + e)^2 + (a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*f), -1/6*(3*(b^3*tan(f*x + e)^4 + 2*a*b^2*tan(f*x + e)^2 + a^2*b)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(a^3 + a^2*b - 2*a*b^2 + 3*(a*b^2 - b^3)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*f*tan(f*x + e)^4 + 2*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*f*tan(f*x + e)^2 + (a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*f)]","B",0
348,1,544,0,0.564531," ","integrate(tan(f*x+e)/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{a - b} \log\left(-\frac{b^{2} \tan\left(f x + e\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(f x + e\right)^{2} - 4 \, {\left(b \tan\left(f x + e\right)^{2} + 2 \, a - b\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left(3 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, a^{2} - 5 \, a b + b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}, \frac{3 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-a + b} \arctan\left(\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{b \tan\left(f x + e\right)^{2} + 2 \, a - b}\right) + 2 \, {\left(3 \, {\left(a b - b^{2}\right)} \tan\left(f x + e\right)^{2} + 4 \, a^{2} - 5 \, a b + b^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}\right]"," ",0,"[1/12*(3*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*sqrt(a - b)*log(-(b^2*tan(f*x + e)^4 + 2*(4*a*b - 3*b^2)*tan(f*x + e)^2 - 4*(b*tan(f*x + e)^2 + 2*a - b)*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 8*a^2 - 8*a*b + b^2)/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*(3*(a*b - b^2)*tan(f*x + e)^2 + 4*a^2 - 5*a*b + b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f), 1/6*(3*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*sqrt(-a + b)*arctan(2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(b*tan(f*x + e)^2 + 2*a - b)) + 2*(3*(a*b - b^2)*tan(f*x + e)^2 + 4*a^2 - 5*a*b + b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)]","B",0
349,1,1649,0,0.482818," ","integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{4} b \tan\left(f x + e\right)^{2} + a^{5}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + 3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f\right)}}, \frac{6 \, {\left(a^{3} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{4} b \tan\left(f x + e\right)^{2} + a^{5}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + 3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f\right)}}, \frac{6 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 3 \, {\left(a^{3} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{4} b \tan\left(f x + e\right)^{2} + a^{5}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, {\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f\right)}}, \frac{3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 3 \, {\left(a^{3} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{4} b \tan\left(f x + e\right)^{2} + a^{5}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) - {\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{3 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f\right)}}\right]"," ",0,"[1/6*(3*(a^3*b^2*tan(f*x + e)^4 + 2*a^4*b*tan(f*x + e)^2 + a^5)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^2 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f), 1/6*(6*(a^3*b^2*tan(f*x + e)^4 + 2*a^4*b*tan(f*x + e)^2 + a^5)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^2 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f), 1/6*(6*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 3*(a^3*b^2*tan(f*x + e)^4 + 2*a^4*b*tan(f*x + e)^2 + a^5)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - 2*(7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^2 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f), 1/3*(3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 3*(a^3*b^2*tan(f*x + e)^4 + 2*a^4*b*tan(f*x + e)^2 + a^5)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^2 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f)]","B",0
350,1,2083,0,0.630425," ","integrate(cot(f*x+e)^3/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{4} b^{2} \tan\left(f x + e\right)^{6} + 2 \, a^{5} b \tan\left(f x + e\right)^{4} + a^{6} \tan\left(f x + e\right)^{2}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + 3 \, {\left({\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 9 \, a^{2} b^{4} + 13 \, a b^{5} - 5 \, b^{6}\right)} \tan\left(f x + e\right)^{6} + 2 \, {\left(2 \, a^{5} b - a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{6} - a^{5} b - 9 \, a^{4} b^{2} + 13 \, a^{3} b^{3} - 5 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(3 \, a^{6} - 9 \, a^{5} b + 9 \, a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, {\left(a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{5} b - 19 \, a^{4} b^{2} + 26 \, a^{3} b^{3} - 10 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{6} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{2}\right)}}, -\frac{12 \, {\left(a^{4} b^{2} \tan\left(f x + e\right)^{6} + 2 \, a^{5} b \tan\left(f x + e\right)^{4} + a^{6} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) - 3 \, {\left({\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 9 \, a^{2} b^{4} + 13 \, a b^{5} - 5 \, b^{6}\right)} \tan\left(f x + e\right)^{6} + 2 \, {\left(2 \, a^{5} b - a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{6} - a^{5} b - 9 \, a^{4} b^{2} + 13 \, a^{3} b^{3} - 5 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) + 2 \, {\left(3 \, a^{6} - 9 \, a^{5} b + 9 \, a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, {\left(a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{5} b - 19 \, a^{4} b^{2} + 26 \, a^{3} b^{3} - 10 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{6} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{2}\right)}}, -\frac{3 \, {\left({\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 9 \, a^{2} b^{4} + 13 \, a b^{5} - 5 \, b^{6}\right)} \tan\left(f x + e\right)^{6} + 2 \, {\left(2 \, a^{5} b - a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{6} - a^{5} b - 9 \, a^{4} b^{2} + 13 \, a^{3} b^{3} - 5 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) - 3 \, {\left(a^{4} b^{2} \tan\left(f x + e\right)^{6} + 2 \, a^{5} b \tan\left(f x + e\right)^{4} + a^{6} \tan\left(f x + e\right)^{2}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + {\left(3 \, a^{6} - 9 \, a^{5} b + 9 \, a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, {\left(a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{5} b - 19 \, a^{4} b^{2} + 26 \, a^{3} b^{3} - 10 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{6} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{2}\right)}}, -\frac{3 \, {\left({\left(2 \, a^{4} b^{2} - a^{3} b^{3} - 9 \, a^{2} b^{4} + 13 \, a b^{5} - 5 \, b^{6}\right)} \tan\left(f x + e\right)^{6} + 2 \, {\left(2 \, a^{5} b - a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + {\left(2 \, a^{6} - a^{5} b - 9 \, a^{4} b^{2} + 13 \, a^{3} b^{3} - 5 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 6 \, {\left(a^{4} b^{2} \tan\left(f x + e\right)^{6} + 2 \, a^{5} b \tan\left(f x + e\right)^{4} + a^{6} \tan\left(f x + e\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + {\left(3 \, a^{6} - 9 \, a^{5} b + 9 \, a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, {\left(a^{4} b^{2} - 9 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - 5 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{5} b - 19 \, a^{4} b^{2} + 26 \, a^{3} b^{3} - 10 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{6} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{4} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/12*(6*(a^4*b^2*tan(f*x + e)^6 + 2*a^5*b*tan(f*x + e)^4 + a^6*tan(f*x + e)^2)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + 3*((2*a^4*b^2 - a^3*b^3 - 9*a^2*b^4 + 13*a*b^5 - 5*b^6)*tan(f*x + e)^6 + 2*(2*a^5*b - a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + (2*a^6 - a^5*b - 9*a^4*b^2 + 13*a^3*b^3 - 5*a^2*b^4)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(3*a^6 - 9*a^5*b + 9*a^4*b^2 - 3*a^3*b^3 + 3*(a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + 2*(3*a^5*b - 19*a^4*b^2 + 26*a^3*b^3 - 10*a^2*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^6 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^4 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^2), -1/12*(12*(a^4*b^2*tan(f*x + e)^6 + 2*a^5*b*tan(f*x + e)^4 + a^6*tan(f*x + e)^2)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - 3*((2*a^4*b^2 - a^3*b^3 - 9*a^2*b^4 + 13*a*b^5 - 5*b^6)*tan(f*x + e)^6 + 2*(2*a^5*b - a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + (2*a^6 - a^5*b - 9*a^4*b^2 + 13*a^3*b^3 - 5*a^2*b^4)*tan(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(3*a^6 - 9*a^5*b + 9*a^4*b^2 - 3*a^3*b^3 + 3*(a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + 2*(3*a^5*b - 19*a^4*b^2 + 26*a^3*b^3 - 10*a^2*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^6 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^4 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^2), -1/6*(3*((2*a^4*b^2 - a^3*b^3 - 9*a^2*b^4 + 13*a*b^5 - 5*b^6)*tan(f*x + e)^6 + 2*(2*a^5*b - a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + (2*a^6 - a^5*b - 9*a^4*b^2 + 13*a^3*b^3 - 5*a^2*b^4)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - 3*(a^4*b^2*tan(f*x + e)^6 + 2*a^5*b*tan(f*x + e)^4 + a^6*tan(f*x + e)^2)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + (3*a^6 - 9*a^5*b + 9*a^4*b^2 - 3*a^3*b^3 + 3*(a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + 2*(3*a^5*b - 19*a^4*b^2 + 26*a^3*b^3 - 10*a^2*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^6 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^4 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^2), -1/6*(3*((2*a^4*b^2 - a^3*b^3 - 9*a^2*b^4 + 13*a*b^5 - 5*b^6)*tan(f*x + e)^6 + 2*(2*a^5*b - a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + (2*a^6 - a^5*b - 9*a^4*b^2 + 13*a^3*b^3 - 5*a^2*b^4)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 6*(a^4*b^2*tan(f*x + e)^6 + 2*a^5*b*tan(f*x + e)^4 + a^6*tan(f*x + e)^2)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + (3*a^6 - 9*a^5*b + 9*a^4*b^2 - 3*a^3*b^3 + 3*(a^4*b^2 - 9*a^3*b^3 + 13*a^2*b^4 - 5*a*b^5)*tan(f*x + e)^4 + 2*(3*a^5*b - 19*a^4*b^2 + 26*a^3*b^3 - 10*a^2*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^6 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^4 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^2)]","B",0
351,1,2433,0,0.565910," ","integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{24 \, {\left(a^{5} b^{2} \tan\left(f x + e\right)^{8} + 2 \, a^{6} b \tan\left(f x + e\right)^{6} + a^{7} \tan\left(f x + e\right)^{4}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) + 3 \, {\left({\left(8 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4} - 53 \, a^{2} b^{5} + 85 \, a b^{6} - 35 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 2 \, {\left(8 \, a^{6} b - 4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{7} - 4 \, a^{6} b - a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(6 \, a^{7} - 18 \, a^{6} b + 18 \, a^{5} b^{2} - 6 \, a^{4} b^{3} - 3 \, {\left(4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} - 4 \, {\left(6 \, a^{6} b - 3 \, a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{48 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{8} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{4}\right)}}, \frac{48 \, {\left(a^{5} b^{2} \tan\left(f x + e\right)^{8} + 2 \, a^{6} b \tan\left(f x + e\right)^{6} + a^{7} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) + 3 \, {\left({\left(8 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4} - 53 \, a^{2} b^{5} + 85 \, a b^{6} - 35 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 2 \, {\left(8 \, a^{6} b - 4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{7} - 4 \, a^{6} b - a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{a} \log\left(\frac{b \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a} + 2 \, a}{\tan\left(f x + e\right)^{2}}\right) - 2 \, {\left(6 \, a^{7} - 18 \, a^{6} b + 18 \, a^{5} b^{2} - 6 \, a^{4} b^{3} - 3 \, {\left(4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} - 4 \, {\left(6 \, a^{6} b - 3 \, a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{48 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{8} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{4}\right)}}, \frac{3 \, {\left({\left(8 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4} - 53 \, a^{2} b^{5} + 85 \, a b^{6} - 35 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 2 \, {\left(8 \, a^{6} b - 4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{7} - 4 \, a^{6} b - a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 12 \, {\left(a^{5} b^{2} \tan\left(f x + e\right)^{8} + 2 \, a^{6} b \tan\left(f x + e\right)^{6} + a^{7} \tan\left(f x + e\right)^{4}\right)} \sqrt{a - b} \log\left(\frac{b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} + 2 \, a - b}{\tan\left(f x + e\right)^{2} + 1}\right) - {\left(6 \, a^{7} - 18 \, a^{6} b + 18 \, a^{5} b^{2} - 6 \, a^{4} b^{3} - 3 \, {\left(4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} - 4 \, {\left(6 \, a^{6} b - 3 \, a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{24 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{8} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{4}\right)}}, \frac{3 \, {\left({\left(8 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4} - 53 \, a^{2} b^{5} + 85 \, a b^{6} - 35 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 2 \, {\left(8 \, a^{6} b - 4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + {\left(8 \, a^{7} - 4 \, a^{6} b - a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a}}{a}\right) + 24 \, {\left(a^{5} b^{2} \tan\left(f x + e\right)^{8} + 2 \, a^{6} b \tan\left(f x + e\right)^{6} + a^{7} \tan\left(f x + e\right)^{4}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{a - b}\right) - {\left(6 \, a^{7} - 18 \, a^{6} b + 18 \, a^{5} b^{2} - 6 \, a^{4} b^{3} - 3 \, {\left(4 \, a^{5} b^{2} - a^{4} b^{3} - 53 \, a^{3} b^{4} + 85 \, a^{2} b^{5} - 35 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} - 4 \, {\left(6 \, a^{6} b - 3 \, a^{5} b^{2} - 53 \, a^{4} b^{3} + 85 \, a^{3} b^{4} - 35 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - 3 \, {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{24 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{8} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{6} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{4}\right)}}\right]"," ",0,"[1/48*(24*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + e)^4)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) + 3*((8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 1/48*(48*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) + 3*((8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 1/24*(3*((8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 12*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + e)^4)*sqrt(a - b)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - (6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4), 1/24*(3*((8*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 - 53*a^2*b^5 + 85*a*b^6 - 35*b^7)*tan(f*x + e)^8 + 2*(8*a^6*b - 4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 + (8*a^7 - 4*a^6*b - a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) + 24*(a^5*b^2*tan(f*x + e)^8 + 2*a^6*b*tan(f*x + e)^6 + a^7*tan(f*x + e)^4)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (6*a^7 - 18*a^6*b + 18*a^5*b^2 - 6*a^4*b^3 - 3*(4*a^5*b^2 - a^4*b^3 - 53*a^3*b^4 + 85*a^2*b^5 - 35*a*b^6)*tan(f*x + e)^6 - 4*(6*a^6*b - 3*a^5*b^2 - 53*a^4*b^3 + 85*a^3*b^4 - 35*a^2*b^5)*tan(f*x + e)^4 - 3*(4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^8 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^6 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^4)]","B",0
352,1,1714,0,2.581684," ","integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) - 3 \, {\left(b^{5} \tan\left(f x + e\right)^{4} + 2 \, a b^{4} \tan\left(f x + e\right)^{2} + a^{2} b^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, {\left({\left(4 \, a^{3} b^{2} - 11 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{5} - 3 \, a^{2} b^{6} + 3 \, a b^{7} - b^{8}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f\right)}}, -\frac{6 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + 3 \, {\left(b^{5} \tan\left(f x + e\right)^{4} + 2 \, a b^{4} \tan\left(f x + e\right)^{2} + a^{2} b^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(4 \, a^{3} b^{2} - 11 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{5} - 3 \, a^{2} b^{6} + 3 \, a b^{7} - b^{8}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f\right)}}, -\frac{6 \, {\left(b^{5} \tan\left(f x + e\right)^{4} + 2 \, a b^{4} \tan\left(f x + e\right)^{2} + a^{2} b^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - 3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(2 \, b \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{b} \tan\left(f x + e\right) + a\right) + 2 \, {\left({\left(4 \, a^{3} b^{2} - 11 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{5} - 3 \, a^{2} b^{6} + 3 \, a b^{7} - b^{8}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f\right)}}, -\frac{3 \, {\left(b^{5} \tan\left(f x + e\right)^{4} + 2 \, a b^{4} \tan\left(f x + e\right)^{2} + a^{2} b^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + 3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} + {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-b}}{b \tan\left(f x + e\right)}\right) + {\left({\left(4 \, a^{3} b^{2} - 11 \, a^{2} b^{3} + 7 \, a b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{3 \, {\left({\left(a^{3} b^{5} - 3 \, a^{2} b^{6} + 3 \, a b^{7} - b^{8}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} f\right)}}\right]"," ",0,"[1/6*(3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) - 3*(b^5*tan(f*x + e)^4 + 2*a*b^4*tan(f*x + e)^2 + a^2*b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*((4*a^3*b^2 - 11*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^3 + 3*(a^4*b - 3*a^3*b^2 + 2*a^2*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^5 - 3*a^2*b^6 + 3*a*b^7 - b^8)*f*tan(f*x + e)^4 + 2*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^2 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f), -1/6*(6*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + 3*(b^5*tan(f*x + e)^4 + 2*a*b^4*tan(f*x + e)^2 + a^2*b^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*((4*a^3*b^2 - 11*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^3 + 3*(a^4*b - 3*a^3*b^2 + 2*a^2*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^5 - 3*a^2*b^6 + 3*a*b^7 - b^8)*f*tan(f*x + e)^4 + 2*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^2 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f), -1/6*(6*(b^5*tan(f*x + e)^4 + 2*a*b^4*tan(f*x + e)^2 + a^2*b^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(b)*log(2*b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(b)*tan(f*x + e) + a) + 2*((4*a^3*b^2 - 11*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^3 + 3*(a^4*b - 3*a^3*b^2 + 2*a^2*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^5 - 3*a^2*b^6 + 3*a*b^7 - b^8)*f*tan(f*x + e)^4 + 2*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^2 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f), -1/3*(3*(b^5*tan(f*x + e)^4 + 2*a*b^4*tan(f*x + e)^2 + a^2*b^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + 3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(f*x + e)^2)*sqrt(-b)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-b)/(b*tan(f*x + e))) + ((4*a^3*b^2 - 11*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^3 + 3*(a^4*b - 3*a^3*b^2 + 2*a^2*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^5 - 3*a^2*b^6 + 3*a*b^7 - b^8)*f*tan(f*x + e)^4 + 2*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^2 + (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f)]","B",0
353,1,498,0,0.498840," ","integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, {\left({\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(a^{2} - a b\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}, \frac{3 \, {\left(b^{2} \tan\left(f x + e\right)^{4} + 2 \, a b \tan\left(f x + e\right)^{2} + a^{2}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) + {\left({\left(a^{2} - 5 \, a b + 4 \, b^{2}\right)} \tan\left(f x + e\right)^{3} - 3 \, {\left(a^{2} - a b\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{3 \, {\left({\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/6*(3*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*((a^2 - 5*a*b + 4*b^2)*tan(f*x + e)^3 - 3*(a^2 - a*b)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f), 1/3*(3*(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) + ((a^2 - 5*a*b + 4*b^2)*tan(f*x + e)^3 - 3*(a^2 - a*b)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*f*tan(f*x + e)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*f*tan(f*x + e)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*f)]","A",0
354,1,529,0,0.550412," ","integrate(tan(f*x+e)^2/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{2} b \tan\left(f x + e\right)^{2} + a^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} + 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) - 2 \, {\left({\left(2 \, a^{2} b - a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{3} - a^{2} b\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} f\right)}}, -\frac{3 \, {\left(a b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{2} b \tan\left(f x + e\right)^{2} + a^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left({\left(2 \, a^{2} b - a b^{2} - b^{3}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(a^{3} - a^{2} b\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{3 \, {\left({\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/6*(3*(a*b^2*tan(f*x + e)^4 + 2*a^2*b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) - 2*((2*a^2*b - a*b^2 - b^3)*tan(f*x + e)^3 + 3*(a^3 - a^2*b)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*f*tan(f*x + e)^4 + 2*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*f*tan(f*x + e)^2 + (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*f), -1/3*(3*(a*b^2*tan(f*x + e)^4 + 2*a^2*b*tan(f*x + e)^2 + a^3)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - ((2*a^2*b - a*b^2 - b^3)*tan(f*x + e)^3 + 3*(a^3 - a^2*b)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*f*tan(f*x + e)^4 + 2*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*f*tan(f*x + e)^2 + (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*f)]","B",0
355,1,561,0,0.636291," ","integrate(1/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{3} b \tan\left(f x + e\right)^{2} + a^{4}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - 2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b} \tan\left(f x + e\right) - a}{\tan\left(f x + e\right)^{2} + 1}\right) + 2 \, {\left({\left(5 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(2 \, a^{3} b - 3 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}, \frac{3 \, {\left(a^{2} b^{2} \tan\left(f x + e\right)^{4} + 2 \, a^{3} b \tan\left(f x + e\right)^{2} + a^{4}\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{b \tan\left(f x + e\right)^{2} + a}}{\sqrt{a - b} \tan\left(f x + e\right)}\right) - {\left({\left(5 \, a^{2} b^{2} - 7 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(f x + e\right)^{3} + 3 \, {\left(2 \, a^{3} b - 3 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{3 \, {\left({\left(a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} f \tan\left(f x + e\right)^{4} + 2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} f \tan\left(f x + e\right)^{2} + {\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} f\right)}}\right]"," ",0,"[-1/6*(3*(a^2*b^2*tan(f*x + e)^4 + 2*a^3*b*tan(f*x + e)^2 + a^4)*sqrt(-a + b)*log(-((a - 2*b)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b)*tan(f*x + e) - a)/(tan(f*x + e)^2 + 1)) + 2*((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*tan(f*x + e)^3 + 3*(2*a^3*b - 3*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f), 1/3*(3*(a^2*b^2*tan(f*x + e)^4 + 2*a^3*b*tan(f*x + e)^2 + a^4)*sqrt(a - b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)/(sqrt(a - b)*tan(f*x + e))) - ((5*a^2*b^2 - 7*a*b^3 + 2*b^4)*tan(f*x + e)^3 + 3*(2*a^3*b - 3*a^2*b^2 + a*b^3)*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a))/((a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f*tan(f*x + e)^4 + 2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*f*tan(f*x + e)^2 + (a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*f)]","B",0
356,1,753,0,0.683505," ","integrate(cot(f*x+e)^2/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} b^{2} \tan\left(f x + e\right)^{5} + 2 \, a^{4} b \tan\left(f x + e\right)^{3} + a^{5} \tan\left(f x + e\right)\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left(3 \, a^{5} - 9 \, a^{4} b + 9 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + {\left(3 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 22 \, a b^{4} - 8 \, b^{5}\right)} \tan\left(f x + e\right)^{4} + 3 \, {\left(2 \, a^{4} b - 9 \, a^{3} b^{2} + 11 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{5} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f \tan\left(f x + e\right)\right)}}, -\frac{3 \, {\left(a^{3} b^{2} \tan\left(f x + e\right)^{5} + 2 \, a^{4} b \tan\left(f x + e\right)^{3} + a^{5} \tan\left(f x + e\right)\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) + 2 \, {\left(3 \, a^{5} - 9 \, a^{4} b + 9 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + {\left(3 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 22 \, a b^{4} - 8 \, b^{5}\right)} \tan\left(f x + e\right)^{4} + 3 \, {\left(2 \, a^{4} b - 9 \, a^{3} b^{2} + 11 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right)} f \tan\left(f x + e\right)^{5} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} f \tan\left(f x + e\right)^{3} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} f \tan\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(3*(a^3*b^2*tan(f*x + e)^5 + 2*a^4*b*tan(f*x + e)^3 + a^5*tan(f*x + e))*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*(3*a^5 - 9*a^4*b + 9*a^3*b^2 - 3*a^2*b^3 + (3*a^3*b^2 - 17*a^2*b^3 + 22*a*b^4 - 8*b^5)*tan(f*x + e)^4 + 3*(2*a^4*b - 9*a^3*b^2 + 11*a^2*b^3 - 4*a*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^5 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^3 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f*tan(f*x + e)), -1/6*(3*(a^3*b^2*tan(f*x + e)^5 + 2*a^4*b*tan(f*x + e)^3 + a^5*tan(f*x + e))*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*(3*a^5 - 9*a^4*b + 9*a^3*b^2 - 3*a^2*b^3 + (3*a^3*b^2 - 17*a^2*b^3 + 22*a*b^4 - 8*b^5)*tan(f*x + e)^4 + 3*(2*a^4*b - 9*a^3*b^2 + 11*a^2*b^3 - 4*a*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f*tan(f*x + e)^5 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*f*tan(f*x + e)^3 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*f*tan(f*x + e))]","B",0
357,1,879,0,0.729568," ","integrate(cot(f*x+e)^4/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{4} b^{2} \tan\left(f x + e\right)^{7} + 2 \, a^{5} b \tan\left(f x + e\right)^{5} + a^{6} \tan\left(f x + e\right)^{3}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} - 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(3 \, a^{4} b^{2} - a^{3} b^{3} - 26 \, a^{2} b^{4} + 40 \, a b^{5} - 16 \, b^{6}\right)} \tan\left(f x + e\right)^{6} - a^{6} + 3 \, a^{5} b - 3 \, a^{4} b^{2} + a^{3} b^{3} + 3 \, {\left(2 \, a^{5} b - a^{4} b^{2} - 13 \, a^{3} b^{3} + 20 \, a^{2} b^{4} - 8 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 3 \, {\left(a^{6} - a^{5} b - 3 \, a^{4} b^{2} + 5 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{12 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{7} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{3}\right)}}, \frac{3 \, {\left(a^{4} b^{2} \tan\left(f x + e\right)^{7} + 2 \, a^{5} b \tan\left(f x + e\right)^{5} + a^{6} \tan\left(f x + e\right)^{3}\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) + 2 \, {\left({\left(3 \, a^{4} b^{2} - a^{3} b^{3} - 26 \, a^{2} b^{4} + 40 \, a b^{5} - 16 \, b^{6}\right)} \tan\left(f x + e\right)^{6} - a^{6} + 3 \, a^{5} b - 3 \, a^{4} b^{2} + a^{3} b^{3} + 3 \, {\left(2 \, a^{5} b - a^{4} b^{2} - 13 \, a^{3} b^{3} + 20 \, a^{2} b^{4} - 8 \, a b^{5}\right)} \tan\left(f x + e\right)^{4} + 3 \, {\left(a^{6} - a^{5} b - 3 \, a^{4} b^{2} + 5 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{6 \, {\left({\left(a^{7} b^{2} - 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} - a^{4} b^{5}\right)} f \tan\left(f x + e\right)^{7} + 2 \, {\left(a^{8} b - 3 \, a^{7} b^{2} + 3 \, a^{6} b^{3} - a^{5} b^{4}\right)} f \tan\left(f x + e\right)^{5} + {\left(a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right)} f \tan\left(f x + e\right)^{3}\right)}}\right]"," ",0,"[-1/12*(3*(a^4*b^2*tan(f*x + e)^7 + 2*a^5*b*tan(f*x + e)^5 + a^6*tan(f*x + e)^3)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 - 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) - 4*((3*a^4*b^2 - a^3*b^3 - 26*a^2*b^4 + 40*a*b^5 - 16*b^6)*tan(f*x + e)^6 - a^6 + 3*a^5*b - 3*a^4*b^2 + a^3*b^3 + 3*(2*a^5*b - a^4*b^2 - 13*a^3*b^3 + 20*a^2*b^4 - 8*a*b^5)*tan(f*x + e)^4 + 3*(a^6 - a^5*b - 3*a^4*b^2 + 5*a^3*b^3 - 2*a^2*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^7 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^3), 1/6*(3*(a^4*b^2*tan(f*x + e)^7 + 2*a^5*b*tan(f*x + e)^5 + a^6*tan(f*x + e)^3)*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*((3*a^4*b^2 - a^3*b^3 - 26*a^2*b^4 + 40*a*b^5 - 16*b^6)*tan(f*x + e)^6 - a^6 + 3*a^5*b - 3*a^4*b^2 + a^3*b^3 + 3*(2*a^5*b - a^4*b^2 - 13*a^3*b^3 + 20*a^2*b^4 - 8*a*b^5)*tan(f*x + e)^4 + 3*(a^6 - a^5*b - 3*a^4*b^2 + 5*a^3*b^3 - 2*a^2*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^7*b^2 - 3*a^6*b^3 + 3*a^5*b^4 - a^4*b^5)*f*tan(f*x + e)^7 + 2*(a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4)*f*tan(f*x + e)^5 + (a^9 - 3*a^8*b + 3*a^7*b^2 - a^6*b^3)*f*tan(f*x + e)^3)]","A",0
358,1,1023,0,0.983537," ","integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{5} b^{2} \tan\left(f x + e\right)^{9} + 2 \, a^{6} b \tan\left(f x + e\right)^{7} + a^{7} \tan\left(f x + e\right)^{5}\right)} \sqrt{-a + b} \log\left(-\frac{{\left(a^{2} - 8 \, a b + 8 \, b^{2}\right)} \tan\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(f x + e\right)^{2} + a^{2} + 4 \, {\left({\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{3} - a \tan\left(f x + e\right)\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{-a + b}}{\tan\left(f x + e\right)^{4} + 2 \, \tan\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(15 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - 184 \, a^{2} b^{5} + 304 \, a b^{6} - 128 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 3 \, a^{7} - 9 \, a^{6} b + 9 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, {\left(10 \, a^{6} b - 5 \, a^{5} b^{2} - a^{4} b^{3} - 92 \, a^{3} b^{4} + 152 \, a^{2} b^{5} - 64 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + 3 \, {\left(5 \, a^{7} - 5 \, a^{6} b + a^{5} b^{2} - 23 \, a^{4} b^{3} + 38 \, a^{3} b^{4} - 16 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{7} - 7 \, a^{6} b - 9 \, a^{5} b^{2} + 19 \, a^{4} b^{3} - 8 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{60 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{9} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{5}\right)}}, -\frac{15 \, {\left(a^{5} b^{2} \tan\left(f x + e\right)^{9} + 2 \, a^{6} b \tan\left(f x + e\right)^{7} + a^{7} \tan\left(f x + e\right)^{5}\right)} \sqrt{a - b} \arctan\left(-\frac{2 \, \sqrt{b \tan\left(f x + e\right)^{2} + a} \sqrt{a - b} \tan\left(f x + e\right)}{{\left(a - 2 \, b\right)} \tan\left(f x + e\right)^{2} - a}\right) + 2 \, {\left({\left(15 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4} - 184 \, a^{2} b^{5} + 304 \, a b^{6} - 128 \, b^{7}\right)} \tan\left(f x + e\right)^{8} + 3 \, a^{7} - 9 \, a^{6} b + 9 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, {\left(10 \, a^{6} b - 5 \, a^{5} b^{2} - a^{4} b^{3} - 92 \, a^{3} b^{4} + 152 \, a^{2} b^{5} - 64 \, a b^{6}\right)} \tan\left(f x + e\right)^{6} + 3 \, {\left(5 \, a^{7} - 5 \, a^{6} b + a^{5} b^{2} - 23 \, a^{4} b^{3} + 38 \, a^{3} b^{4} - 16 \, a^{2} b^{5}\right)} \tan\left(f x + e\right)^{4} - {\left(5 \, a^{7} - 7 \, a^{6} b - 9 \, a^{5} b^{2} + 19 \, a^{4} b^{3} - 8 \, a^{3} b^{4}\right)} \tan\left(f x + e\right)^{2}\right)} \sqrt{b \tan\left(f x + e\right)^{2} + a}}{30 \, {\left({\left(a^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right)} f \tan\left(f x + e\right)^{9} + 2 \, {\left(a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right)} f \tan\left(f x + e\right)^{7} + {\left(a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right)} f \tan\left(f x + e\right)^{5}\right)}}\right]"," ",0,"[-1/60*(15*(a^5*b^2*tan(f*x + e)^9 + 2*a^6*b*tan(f*x + e)^7 + a^7*tan(f*x + e)^5)*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3*a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f*x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)) + 4*((15*a^5*b^2 - 5*a^4*b^3 - 2*a^3*b^4 - 184*a^2*b^5 + 304*a*b^6 - 128*b^7)*tan(f*x + e)^8 + 3*a^7 - 9*a^6*b + 9*a^5*b^2 - 3*a^4*b^3 + 3*(10*a^6*b - 5*a^5*b^2 - a^4*b^3 - 92*a^3*b^4 + 152*a^2*b^5 - 64*a*b^6)*tan(f*x + e)^6 + 3*(5*a^7 - 5*a^6*b + a^5*b^2 - 23*a^4*b^3 + 38*a^3*b^4 - 16*a^2*b^5)*tan(f*x + e)^4 - (5*a^7 - 7*a^6*b - 9*a^5*b^2 + 19*a^4*b^3 - 8*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^9 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^7 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^5), -1/30*(15*(a^5*b^2*tan(f*x + e)^9 + 2*a^6*b*tan(f*x + e)^7 + a^7*tan(f*x + e)^5)*sqrt(a - b)*arctan(-2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e)^2 - a)) + 2*((15*a^5*b^2 - 5*a^4*b^3 - 2*a^3*b^4 - 184*a^2*b^5 + 304*a*b^6 - 128*b^7)*tan(f*x + e)^8 + 3*a^7 - 9*a^6*b + 9*a^5*b^2 - 3*a^4*b^3 + 3*(10*a^6*b - 5*a^5*b^2 - a^4*b^3 - 92*a^3*b^4 + 152*a^2*b^5 - 64*a*b^6)*tan(f*x + e)^6 + 3*(5*a^7 - 5*a^6*b + a^5*b^2 - 23*a^4*b^3 + 38*a^3*b^4 - 16*a^2*b^5)*tan(f*x + e)^4 - (5*a^7 - 7*a^6*b - 9*a^5*b^2 + 19*a^4*b^3 - 8*a^3*b^4)*tan(f*x + e)^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^8*b^2 - 3*a^7*b^3 + 3*a^6*b^4 - a^5*b^5)*f*tan(f*x + e)^9 + 2*(a^9*b - 3*a^8*b^2 + 3*a^7*b^3 - a^6*b^4)*f*tan(f*x + e)^7 + (a^10 - 3*a^9*b + 3*a^8*b^2 - a^7*b^3)*f*tan(f*x + e)^5)]","A",0
359,0,0,0,0.544995," ","integrate((d*tan(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p*(d*tan(f*x + e))^m, x)","F",0
360,0,0,0,0.608988," ","integrate((d*tan(f*x+e))^m*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*(d*tan(f*x + e))^m, x)","F",0
361,0,0,0,0.603237," ","integrate(tan(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{5}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^5, x)","F",0
362,0,0,0,0.699078," ","integrate(tan(f*x+e)^3*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^3, x)","F",0
363,0,0,0,0.786710," ","integrate(tan(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right), x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e), x)","F",0
364,0,0,0,0.536015," ","integrate(cot(f*x+e)*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right), x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e), x)","F",0
365,0,0,0,0.680479," ","integrate(cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^3, x)","F",0
366,0,0,0,0.756166," ","integrate(cot(f*x+e)^5*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{5}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^5, x)","F",0
367,0,0,0,0.625661," ","integrate(tan(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{6}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^6, x)","F",0
368,0,0,0,0.576889," ","integrate(tan(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^4, x)","F",0
369,0,0,0,0.698214," ","integrate(tan(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*tan(f*x + e)^2, x)","F",0
370,0,0,0,0.561800," ","integrate((a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p, x)","F",0
371,0,0,0,0.579720," ","integrate(cot(f*x+e)^2*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^2, x)","F",0
372,0,0,0,0.670108," ","integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^4, x)","F",0
373,0,0,0,0.762393," ","integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \cot\left(f x + e\right)^{6}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^6, x)","F",0
374,1,225,0,0.600230," ","integrate((a+b*tan(d*x+c)^3)^4,x, algorithm=""fricas"")","\frac{630 \, b^{4} \tan\left(d x + c\right)^{11} - 770 \, b^{4} \tan\left(d x + c\right)^{9} + 3465 \, a b^{3} \tan\left(d x + c\right)^{8} + 990 \, b^{4} \tan\left(d x + c\right)^{7} - 4620 \, a b^{3} \tan\left(d x + c\right)^{6} + 6930 \, a b^{3} \tan\left(d x + c\right)^{4} + 1386 \, {\left(6 \, a^{2} b^{2} - b^{4}\right)} \tan\left(d x + c\right)^{5} - 2310 \, {\left(6 \, a^{2} b^{2} - b^{4}\right)} \tan\left(d x + c\right)^{3} + 6930 \, {\left(a^{4} - 6 \, a^{2} b^{2} + b^{4}\right)} d x + 13860 \, {\left(a^{3} b - a b^{3}\right)} \tan\left(d x + c\right)^{2} + 13860 \, {\left(a^{3} b - a b^{3}\right)} \log\left(\frac{1}{\tan\left(d x + c\right)^{2} + 1}\right) + 6930 \, {\left(6 \, a^{2} b^{2} - b^{4}\right)} \tan\left(d x + c\right)}{6930 \, d}"," ",0,"1/6930*(630*b^4*tan(d*x + c)^11 - 770*b^4*tan(d*x + c)^9 + 3465*a*b^3*tan(d*x + c)^8 + 990*b^4*tan(d*x + c)^7 - 4620*a*b^3*tan(d*x + c)^6 + 6930*a*b^3*tan(d*x + c)^4 + 1386*(6*a^2*b^2 - b^4)*tan(d*x + c)^5 - 2310*(6*a^2*b^2 - b^4)*tan(d*x + c)^3 + 6930*(a^4 - 6*a^2*b^2 + b^4)*d*x + 13860*(a^3*b - a*b^3)*tan(d*x + c)^2 + 13860*(a^3*b - a*b^3)*log(1/(tan(d*x + c)^2 + 1)) + 6930*(6*a^2*b^2 - b^4)*tan(d*x + c))/d","A",0
375,1,148,0,0.475176," ","integrate((a+b*tan(d*x+c)^3)^3,x, algorithm=""fricas"")","\frac{15 \, b^{3} \tan\left(d x + c\right)^{8} - 20 \, b^{3} \tan\left(d x + c\right)^{6} + 72 \, a b^{2} \tan\left(d x + c\right)^{5} + 30 \, b^{3} \tan\left(d x + c\right)^{4} - 120 \, a b^{2} \tan\left(d x + c\right)^{3} + 360 \, a b^{2} \tan\left(d x + c\right) + 120 \, {\left(a^{3} - 3 \, a b^{2}\right)} d x + 60 \, {\left(3 \, a^{2} b - b^{3}\right)} \tan\left(d x + c\right)^{2} + 60 \, {\left(3 \, a^{2} b - b^{3}\right)} \log\left(\frac{1}{\tan\left(d x + c\right)^{2} + 1}\right)}{120 \, d}"," ",0,"1/120*(15*b^3*tan(d*x + c)^8 - 20*b^3*tan(d*x + c)^6 + 72*a*b^2*tan(d*x + c)^5 + 30*b^3*tan(d*x + c)^4 - 120*a*b^2*tan(d*x + c)^3 + 360*a*b^2*tan(d*x + c) + 120*(a^3 - 3*a*b^2)*d*x + 60*(3*a^2*b - b^3)*tan(d*x + c)^2 + 60*(3*a^2*b - b^3)*log(1/(tan(d*x + c)^2 + 1)))/d","A",0
376,1,85,0,0.465466," ","integrate((a+b*tan(d*x+c)^3)^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} \tan\left(d x + c\right)^{5} - 5 \, b^{2} \tan\left(d x + c\right)^{3} + 15 \, a b \tan\left(d x + c\right)^{2} + 15 \, {\left(a^{2} - b^{2}\right)} d x + 15 \, a b \log\left(\frac{1}{\tan\left(d x + c\right)^{2} + 1}\right) + 15 \, b^{2} \tan\left(d x + c\right)}{15 \, d}"," ",0,"1/15*(3*b^2*tan(d*x + c)^5 - 5*b^2*tan(d*x + c)^3 + 15*a*b*tan(d*x + c)^2 + 15*(a^2 - b^2)*d*x + 15*a*b*log(1/(tan(d*x + c)^2 + 1)) + 15*b^2*tan(d*x + c))/d","A",0
377,1,36,0,0.531864," ","integrate(a+b*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{2 \, a d x + b \tan\left(d x + c\right)^{2} + b \log\left(\frac{1}{\tan\left(d x + c\right)^{2} + 1}\right)}{2 \, d}"," ",0,"1/2*(2*a*d*x + b*tan(d*x + c)^2 + b*log(1/(tan(d*x + c)^2 + 1)))/d","A",0
378,1,4817,0,2.095251," ","integrate(1/(a+b*tan(d*x+c)^3),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} + b^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d \log\left(-\frac{4 \, b^{2} \tan\left(d x + c\right)^{2} - {\left({\left(a^{4} + a^{2} b^{2}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{4} + a^{2} b^{2}\right)} d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} + 2 \, {\left(a^{2} b d \tan\left(d x + c\right)^{2} - a^{2} b d + 2 \, {\left(a^{3} - a b^{2}\right)} d \tan\left(d x + c\right)\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} - 4 \, a^{2}}{4 \, {\left(\tan\left(d x + c\right)^{2} + 1\right)}}\right) - 24 \, a d x - {\left({\left(a^{2} + b^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d - 3 \, \sqrt{\frac{1}{3}} {\left(a^{2} + b^{2}\right)} d \sqrt{-\frac{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} d^{2} - 4 \, {\left(a^{2} b + b^{3}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d - 12 \, b^{2}}{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 6 \, b\right)} \log\left(\frac{8 \, a^{4} - 16 \, a^{2} b^{2} - {\left({\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} + 8 \, {\left(2 \, a^{2} b^{2} - b^{4}\right)} \tan\left(d x + c\right)^{2} + 2 \, {\left({\left(a^{4} b + a^{2} b^{3}\right)} d \tan\left(d x + c\right)^{2} + 2 \, {\left(a^{5} - a b^{4}\right)} d \tan\left(d x + c\right) - {\left(a^{4} b + a^{2} b^{3}\right)} d\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} + 3 \, \sqrt{\frac{1}{3}} {\left(4 \, {\left(a^{4} b + a^{2} b^{3}\right)} d \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{5} - a b^{4}\right)} d \tan\left(d x + c\right) - {\left({\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} - 4 \, {\left(a^{4} b + a^{2} b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} d^{2} - 4 \, {\left(a^{2} b + b^{3}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d - 12 \, b^{2}}{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 24 \, {\left(a^{3} b - a b^{3}\right)} \tan\left(d x + c\right)}{4 \, {\left(\tan\left(d x + c\right)^{2} + 1\right)}}\right) - {\left({\left(a^{2} + b^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d + 3 \, \sqrt{\frac{1}{3}} {\left(a^{2} + b^{2}\right)} d \sqrt{-\frac{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} d^{2} - 4 \, {\left(a^{2} b + b^{3}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d - 12 \, b^{2}}{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 6 \, b\right)} \log\left(-\frac{8 \, a^{4} - 16 \, a^{2} b^{2} - {\left({\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} + 8 \, {\left(2 \, a^{2} b^{2} - b^{4}\right)} \tan\left(d x + c\right)^{2} + 2 \, {\left({\left(a^{4} b + a^{2} b^{3}\right)} d \tan\left(d x + c\right)^{2} + 2 \, {\left(a^{5} - a b^{4}\right)} d \tan\left(d x + c\right) - {\left(a^{4} b + a^{2} b^{3}\right)} d\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} - 3 \, \sqrt{\frac{1}{3}} {\left(4 \, {\left(a^{4} b + a^{2} b^{3}\right)} d \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{5} - a b^{4}\right)} d \tan\left(d x + c\right) - {\left({\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} - 4 \, {\left(a^{4} b + a^{2} b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)}^{2} d^{2} - 4 \, {\left(a^{2} b + b^{3}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d + b^{2} d\right)}^{2} {\left(\frac{b}{a^{4} d^{3} + a^{2} b^{2} d^{3}} - \frac{2 \, b^{3}}{{\left(a^{2} d + b^{2} d\right)}^{3}} - \frac{{\left(a^{2} - b^{2}\right)} b}{{\left(a^{2} + b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{2 \, b}{a^{2} d + b^{2} d}\right)} d - 12 \, b^{2}}{{\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 24 \, {\left(a^{3} b - a b^{3}\right)} \tan\left(d x + c\right)}{4 \, {\left(\tan\left(d x + c\right)^{2} + 1\right)}}\right)}{24 \, {\left(a^{2} + b^{2}\right)} d}"," ",0,"-1/24*(2*(a^2 + b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d*log(-1/4*(4*b^2*tan(d*x + c)^2 - ((a^4 + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + a^2*b^2)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2 + 2*(a^2*b*d*tan(d*x + c)^2 - a^2*b*d + 2*(a^3 - a*b^2)*d*tan(d*x + c))*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d)) - 4*a^2)/(tan(d*x + c)^2 + 1)) - 24*a*d*x - ((a^2 + b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 3*sqrt(1/3)*(a^2 + b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2*d^2 - 4*(a^2*b + b^3)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 12*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - 6*b)*log(1/4*(8*a^4 - 16*a^2*b^2 - ((a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*tan(d*x + c)^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2 + 8*(2*a^2*b^2 - b^4)*tan(d*x + c)^2 + 2*((a^4*b + a^2*b^3)*d*tan(d*x + c)^2 + 2*(a^5 - a*b^4)*d*tan(d*x + c) - (a^4*b + a^2*b^3)*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d)) + 3*sqrt(1/3)*(4*(a^4*b + a^2*b^3)*d*tan(d*x + c)^2 - 4*(a^5 - a*b^4)*d*tan(d*x + c) - ((a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*tan(d*x + c)^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d)) - 4*(a^4*b + a^2*b^3)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2*d^2 - 4*(a^2*b + b^3)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 12*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - 24*(a^3*b - a*b^3)*tan(d*x + c))/(tan(d*x + c)^2 + 1)) - ((a^2 + b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d + 3*sqrt(1/3)*(a^2 + b^2)*d*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2*d^2 - 4*(a^2*b + b^3)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 12*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - 6*b)*log(-1/4*(8*a^4 - 16*a^2*b^2 - ((a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*tan(d*x + c)^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2 + 8*(2*a^2*b^2 - b^4)*tan(d*x + c)^2 + 2*((a^4*b + a^2*b^3)*d*tan(d*x + c)^2 + 2*(a^5 - a*b^4)*d*tan(d*x + c) - (a^4*b + a^2*b^3)*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d)) - 3*sqrt(1/3)*(4*(a^4*b + a^2*b^3)*d*tan(d*x + c)^2 - 4*(a^5 - a*b^4)*d*tan(d*x + c) - ((a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*tan(d*x + c)^2 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d)) - 4*(a^4*b + a^2*b^3)*d)*sqrt(-((a^4 + 2*a^2*b^2 + b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))^2*d^2 - 4*(a^2*b + b^3)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3) + 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/((a^2*d + b^2*d)^2*(b/(a^4*d^3 + a^2*b^2*d^3) - 2*b^3/(a^2*d + b^2*d)^3 - (a^2 - b^2)*b/((a^2 + b^2)^2*a^2*d^3))^(1/3)) + 2*b/(a^2*d + b^2*d))*d - 12*b^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - 24*(a^3*b - a*b^3)*tan(d*x + c))/(tan(d*x + c)^2 + 1)))/((a^2 + b^2)*d)","C",0
379,1,11554,0,3.401545," ","integrate(1/(a+b*tan(d*x+c)^3)^2,x, algorithm=""fricas"")","-\frac{216 \, a^{3} b - 432 \, a b^{3} + 216 \, {\left(2 \, a^{2} b^{2} - b^{4} - 3 \, {\left(a^{3} b - a b^{3}\right)} d x\right)} \tan\left(d x + c\right)^{3} - 648 \, {\left(a^{4} - a^{2} b^{2}\right)} d x + 2 \, {\left({\left(a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} d \tan\left(d x + c\right)^{3} + {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} \log\left(\frac{10368 \, a^{6} - 12960 \, a^{4} b^{2} - 3888 \, a^{2} b^{4} + {\left({\left(2 \, a^{10} + 5 \, a^{8} b^{2} + 4 \, a^{6} b^{4} + a^{4} b^{6}\right)} d^{2} \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right)} d^{2} \tan\left(d x + c\right) - {\left(2 \, a^{10} + 5 \, a^{8} b^{2} + 4 \, a^{6} b^{4} + a^{4} b^{6}\right)} d^{2}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} - 1296 \, {\left(18 \, a^{4} b^{2} + 7 \, a^{2} b^{4} + b^{6}\right)} \tan\left(d x + c\right)^{2} - 36 \, {\left({\left(8 \, a^{7} b - 2 \, a^{5} b^{3} - a^{3} b^{5}\right)} d \tan\left(d x + c\right)^{2} + 2 \, {\left(4 \, a^{8} - 20 \, a^{6} b^{2} - 7 \, a^{4} b^{4} - a^{2} b^{6}\right)} d \tan\left(d x + c\right) - {\left(8 \, a^{7} b - 2 \, a^{5} b^{3} - a^{3} b^{5}\right)} d\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} + 2592 \, {\left(4 \, a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} \tan\left(d x + c\right)}{324 \, {\left(\tan\left(d x + c\right)^{2} + 1\right)}}\right) + 216 \, {\left(a^{3} b + a b^{3}\right)} \tan\left(d x + c\right)^{2} + {\left(324 \, a^{2} b^{2} \tan\left(d x + c\right)^{3} + 324 \, a^{3} b - {\left({\left(a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} d \tan\left(d x + c\right)^{3} + {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} + 3 \, \sqrt{\frac{1}{3}} {\left({\left(a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} d \tan\left(d x + c\right)^{3} + {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d\right)} \sqrt{\frac{29808 \, a^{4} b^{2} - 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}}\right)} \log\left(\frac{20736 \, a^{8} - 106272 \, a^{6} b^{2} - 22032 \, a^{4} b^{4} - {\left({\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2} \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right)} d^{2} \tan\left(d x + c\right) - {\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} + 1296 \, {\left(42 \, a^{6} b^{2} - 59 \, a^{4} b^{4} - 22 \, a^{2} b^{6} - 2 \, b^{8}\right)} \tan\left(d x + c\right)^{2} + 36 \, {\left({\left(8 \, a^{9} b + 6 \, a^{7} b^{3} - 3 \, a^{5} b^{5} - a^{3} b^{7}\right)} d \tan\left(d x + c\right)^{2} + 2 \, {\left(4 \, a^{10} - 16 \, a^{8} b^{2} - 27 \, a^{6} b^{4} - 8 \, a^{4} b^{6} - a^{2} b^{8}\right)} d \tan\left(d x + c\right) - {\left(8 \, a^{9} b + 6 \, a^{7} b^{3} - 3 \, a^{5} b^{5} - a^{3} b^{7}\right)} d\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} + 3 \, \sqrt{\frac{1}{3}} {\left(36 \, {\left(10 \, a^{9} b + 21 \, a^{7} b^{3} + 12 \, a^{5} b^{5} + a^{3} b^{7}\right)} d \tan\left(d x + c\right)^{2} - 72 \, {\left(4 \, a^{10} + 2 \, a^{8} b^{2} - 9 \, a^{6} b^{4} - 8 \, a^{4} b^{6} - a^{2} b^{8}\right)} d \tan\left(d x + c\right) - {\left({\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2} \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right)} d^{2} \tan\left(d x + c\right) - {\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} - 36 \, {\left(10 \, a^{9} b + 21 \, a^{7} b^{3} + 12 \, a^{5} b^{5} + a^{3} b^{7}\right)} d\right)} \sqrt{\frac{29808 \, a^{4} b^{2} - 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}} - 2592 \, {\left(28 \, a^{7} b - 78 \, a^{5} b^{3} - 27 \, a^{3} b^{5} - 2 \, a b^{7}\right)} \tan\left(d x + c\right)}{324 \, {\left(\tan\left(d x + c\right)^{2} + 1\right)}}\right) + {\left(324 \, a^{2} b^{2} \tan\left(d x + c\right)^{3} + 324 \, a^{3} b - {\left({\left(a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} d \tan\left(d x + c\right)^{3} + {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} - 3 \, \sqrt{\frac{1}{3}} {\left({\left(a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} d \tan\left(d x + c\right)^{3} + {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d\right)} \sqrt{\frac{29808 \, a^{4} b^{2} - 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}}\right)} \log\left(-\frac{20736 \, a^{8} - 106272 \, a^{6} b^{2} - 22032 \, a^{4} b^{4} - {\left({\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2} \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right)} d^{2} \tan\left(d x + c\right) - {\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} + 1296 \, {\left(42 \, a^{6} b^{2} - 59 \, a^{4} b^{4} - 22 \, a^{2} b^{6} - 2 \, b^{8}\right)} \tan\left(d x + c\right)^{2} + 36 \, {\left({\left(8 \, a^{9} b + 6 \, a^{7} b^{3} - 3 \, a^{5} b^{5} - a^{3} b^{7}\right)} d \tan\left(d x + c\right)^{2} + 2 \, {\left(4 \, a^{10} - 16 \, a^{8} b^{2} - 27 \, a^{6} b^{4} - 8 \, a^{4} b^{6} - a^{2} b^{8}\right)} d \tan\left(d x + c\right) - {\left(8 \, a^{9} b + 6 \, a^{7} b^{3} - 3 \, a^{5} b^{5} - a^{3} b^{7}\right)} d\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} - 3 \, \sqrt{\frac{1}{3}} {\left(36 \, {\left(10 \, a^{9} b + 21 \, a^{7} b^{3} + 12 \, a^{5} b^{5} + a^{3} b^{7}\right)} d \tan\left(d x + c\right)^{2} - 72 \, {\left(4 \, a^{10} + 2 \, a^{8} b^{2} - 9 \, a^{6} b^{4} - 8 \, a^{4} b^{6} - a^{2} b^{8}\right)} d \tan\left(d x + c\right) - {\left({\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2} \tan\left(d x + c\right)^{2} - 4 \, {\left(a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right)} d^{2} \tan\left(d x + c\right) - {\left(2 \, a^{12} + 7 \, a^{10} b^{2} + 9 \, a^{8} b^{4} + 5 \, a^{6} b^{6} + a^{4} b^{8}\right)} d^{2}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} - 36 \, {\left(10 \, a^{9} b + 21 \, a^{7} b^{3} + 12 \, a^{5} b^{5} + a^{3} b^{7}\right)} d\right)} \sqrt{\frac{29808 \, a^{4} b^{2} - 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{4 \, {\left(8 \, a^{2} b + b^{3}\right)}}{729 \, {\left(a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} - \frac{4 \, {\left(8 \, a^{6} - 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} - b^{6}\right)} b}{729 \, {\left(a^{2} + b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}} - 2592 \, {\left(28 \, a^{7} b - 78 \, a^{5} b^{3} - 27 \, a^{3} b^{5} - 2 \, a b^{7}\right)} \tan\left(d x + c\right)}{324 \, {\left(\tan\left(d x + c\right)^{2} + 1\right)}}\right) - 216 \, {\left(a^{2} b^{2} + b^{4}\right)} \tan\left(d x + c\right)}{648 \, {\left({\left(a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right)} d \tan\left(d x + c\right)^{3} + {\left(a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d\right)}}"," ",0,"-1/648*(216*a^3*b - 432*a*b^3 + 216*(2*a^2*b^2 - b^4 - 3*(a^3*b - a*b^3)*d*x)*tan(d*x + c)^3 - 648*(a^4 - a^2*b^2)*d*x + 2*((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*log(1/324*(10368*a^6 - 12960*a^4*b^2 - 3888*a^2*b^4 + ((2*a^10 + 5*a^8*b^2 + 4*a^6*b^4 + a^4*b^6)*d^2*tan(d*x + c)^2 - 4*(a^9*b + 2*a^7*b^3 + a^5*b^5)*d^2*tan(d*x + c) - (2*a^10 + 5*a^8*b^2 + 4*a^6*b^4 + a^4*b^6)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2 - 1296*(18*a^4*b^2 + 7*a^2*b^4 + b^6)*tan(d*x + c)^2 - 36*((8*a^7*b - 2*a^5*b^3 - a^3*b^5)*d*tan(d*x + c)^2 + 2*(4*a^8 - 20*a^6*b^2 - 7*a^4*b^4 - a^2*b^6)*d*tan(d*x + c) - (8*a^7*b - 2*a^5*b^3 - a^3*b^5)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) + 2592*(4*a^5*b + 2*a^3*b^3 + a*b^5)*tan(d*x + c))/(tan(d*x + c)^2 + 1)) + 216*(a^3*b + a*b^3)*tan(d*x + c)^2 + (324*a^2*b^2*tan(d*x + c)^3 + 324*a^3*b - ((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) + 3*sqrt(1/3)*((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*d^2)))*log(1/324*(20736*a^8 - 106272*a^6*b^2 - 22032*a^4*b^4 - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2 + 1296*(42*a^6*b^2 - 59*a^4*b^4 - 22*a^2*b^6 - 2*b^8)*tan(d*x + c)^2 + 36*((8*a^9*b + 6*a^7*b^3 - 3*a^5*b^5 - a^3*b^7)*d*tan(d*x + c)^2 + 2*(4*a^10 - 16*a^8*b^2 - 27*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - (8*a^9*b + 6*a^7*b^3 - 3*a^5*b^5 - a^3*b^7)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) + 3*sqrt(1/3)*(36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d*tan(d*x + c)^2 - 72*(4*a^10 + 2*a^8*b^2 - 9*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) - 36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*d^2)) - 2592*(28*a^7*b - 78*a^5*b^3 - 27*a^3*b^5 - 2*a*b^7)*tan(d*x + c))/(tan(d*x + c)^2 + 1)) + (324*a^2*b^2*tan(d*x + c)^3 + 324*a^3*b - ((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) - 3*sqrt(1/3)*((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*d^2)))*log(-1/324*(20736*a^8 - 106272*a^6*b^2 - 22032*a^4*b^4 - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2 + 1296*(42*a^6*b^2 - 59*a^4*b^4 - 22*a^2*b^6 - 2*b^8)*tan(d*x + c)^2 + 36*((8*a^9*b + 6*a^7*b^3 - 3*a^5*b^5 - a^3*b^7)*d*tan(d*x + c)^2 + 2*(4*a^10 - 16*a^8*b^2 - 27*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - (8*a^9*b + 6*a^7*b^3 - 3*a^5*b^5 - a^3*b^7)*d)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) - 3*sqrt(1/3)*(36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d*tan(d*x + c)^2 - 72*(4*a^10 + 2*a^8*b^2 - 9*a^6*b^4 - 8*a^4*b^6 - a^2*b^8)*d*tan(d*x + c) - ((2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2*tan(d*x + c)^2 - 4*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d^2*tan(d*x + c) - (2*a^12 + 7*a^10*b^2 + 9*a^8*b^4 + 5*a^6*b^6 + a^4*b^8)*d^2)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d)) - 36*(10*a^9*b + 21*a^7*b^3 + 12*a^5*b^5 + a^3*b^7)*d)*sqrt((29808*a^4*b^2 - 10368*a^2*b^4 - 5184*b^6 - (a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b + 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d + 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d + 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 + 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d + 2*a^2*b^2*d + b^4*d)) + 4/729*(8*a^2*b + b^3)/(a^9*d^3 + 2*a^7*b^2*d^3 + a^5*b^4*d^3) - 4/729*(8*a^6 - 28*a^4*b^2 - 10*a^2*b^4 - b^6)*b/((a^2 + b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d))*d)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8)*d^2)) - 2592*(28*a^7*b - 78*a^5*b^3 - 27*a^3*b^5 - 2*a*b^7)*tan(d*x + c))/(tan(d*x + c)^2 + 1)) - 216*(a^2*b^2 + b^4)*tan(d*x + c))/((a^5*b + 2*a^3*b^3 + a*b^5)*d*tan(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d)","C",0
380,1,48,0,0.536160," ","integrate(1/(1+tan(x)^3),x, algorithm=""fricas"")","\frac{1}{2} \, x + \frac{1}{12} \, \log\left(\frac{\tan\left(x\right)^{2} + 2 \, \tan\left(x\right) + 1}{\tan\left(x\right)^{2} + 1}\right) - \frac{1}{3} \, \log\left(\frac{\tan\left(x\right)^{2} - \tan\left(x\right) + 1}{\tan\left(x\right)^{2} + 1}\right)"," ",0,"1/2*x + 1/12*log((tan(x)^2 + 2*tan(x) + 1)/(tan(x)^2 + 1)) - 1/3*log((tan(x)^2 - tan(x) + 1)/(tan(x)^2 + 1))","A",0
381,1,225,0,0.677522," ","integrate((a+tan(d*x+c)^4*b)^4,x, algorithm=""fricas"")","\frac{3003 \, b^{4} \tan\left(d x + c\right)^{15} - 3465 \, b^{4} \tan\left(d x + c\right)^{13} + 4095 \, {\left(4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)^{11} - 5005 \, {\left(4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)^{9} + 6435 \, {\left(6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)^{7} - 9009 \, {\left(6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)^{5} + 15015 \, {\left(4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)^{3} + 45045 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d x - 45045 \, {\left(4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \tan\left(d x + c\right)}{45045 \, d}"," ",0,"1/45045*(3003*b^4*tan(d*x + c)^15 - 3465*b^4*tan(d*x + c)^13 + 4095*(4*a*b^3 + b^4)*tan(d*x + c)^11 - 5005*(4*a*b^3 + b^4)*tan(d*x + c)^9 + 6435*(6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c)^7 - 9009*(6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c)^5 + 15015*(4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c)^3 + 45045*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x - 45045*(4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*tan(d*x + c))/d","A",0
382,1,145,0,0.868454," ","integrate((a+tan(d*x+c)^4*b)^3,x, algorithm=""fricas"")","\frac{315 \, b^{3} \tan\left(d x + c\right)^{11} - 385 \, b^{3} \tan\left(d x + c\right)^{9} + 495 \, {\left(3 \, a b^{2} + b^{3}\right)} \tan\left(d x + c\right)^{7} - 693 \, {\left(3 \, a b^{2} + b^{3}\right)} \tan\left(d x + c\right)^{5} + 1155 \, {\left(3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \tan\left(d x + c\right)^{3} + 3465 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x - 3465 \, {\left(3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \tan\left(d x + c\right)}{3465 \, d}"," ",0,"1/3465*(315*b^3*tan(d*x + c)^11 - 385*b^3*tan(d*x + c)^9 + 495*(3*a*b^2 + b^3)*tan(d*x + c)^7 - 693*(3*a*b^2 + b^3)*tan(d*x + c)^5 + 1155*(3*a^2*b + 3*a*b^2 + b^3)*tan(d*x + c)^3 + 3465*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x - 3465*(3*a^2*b + 3*a*b^2 + b^3)*tan(d*x + c))/d","A",0
383,1,81,0,0.618372," ","integrate((a+tan(d*x+c)^4*b)^2,x, algorithm=""fricas"")","\frac{15 \, b^{2} \tan\left(d x + c\right)^{7} - 21 \, b^{2} \tan\left(d x + c\right)^{5} + 35 \, {\left(2 \, a b + b^{2}\right)} \tan\left(d x + c\right)^{3} + 105 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d x - 105 \, {\left(2 \, a b + b^{2}\right)} \tan\left(d x + c\right)}{105 \, d}"," ",0,"1/105*(15*b^2*tan(d*x + c)^7 - 21*b^2*tan(d*x + c)^5 + 35*(2*a*b + b^2)*tan(d*x + c)^3 + 105*(a^2 + 2*a*b + b^2)*d*x - 105*(2*a*b + b^2)*tan(d*x + c))/d","A",0
384,1,32,0,0.640295," ","integrate(a+tan(d*x+c)^4*b,x, algorithm=""fricas"")","\frac{b \tan\left(d x + c\right)^{3} + 3 \, {\left(a + b\right)} d x - 3 \, b \tan\left(d x + c\right)}{3 \, d}"," ",0,"1/3*(b*tan(d*x + c)^3 + 3*(a + b)*d*x - 3*b*tan(d*x + c))/d","A",0
385,1,1541,0,0.736307," ","integrate(1/(a+tan(d*x+c)^4*b),x, algorithm=""fricas"")","\frac{{\left(a + b\right)} \sqrt{\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(\frac{2 \, {\left(a^{3} - a b^{2}\right)} d \sqrt{\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \tan\left(d x + c\right) + {\left(a b - b^{2}\right)} \tan\left(d x + c\right)^{2} + a^{2} - a b + {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2}\right)} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) - {\left(a + b\right)} \sqrt{\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(-\frac{2 \, {\left(a^{3} - a b^{2}\right)} d \sqrt{\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \tan\left(d x + c\right) - {\left(a b - b^{2}\right)} \tan\left(d x + c\right)^{2} - a^{2} + a b - {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2}\right)} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) + {\left(a + b\right)} \sqrt{-\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(-\frac{2 \, {\left(a^{3} - a b^{2}\right)} d \sqrt{-\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \tan\left(d x + c\right) + {\left(a b - b^{2}\right)} \tan\left(d x + c\right)^{2} + a^{2} - a b - {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2}\right)} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) - {\left(a + b\right)} \sqrt{-\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(\frac{2 \, {\left(a^{3} - a b^{2}\right)} d \sqrt{-\frac{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \tan\left(d x + c\right) - {\left(a b - b^{2}\right)} \tan\left(d x + c\right)^{2} - a^{2} + a b + {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d^{2}\right)} \sqrt{-\frac{a^{2} b - 2 \, a b^{2} + b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) + 8 \, x}{8 \, {\left(a + b\right)}}"," ",0,"1/8*((a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) + (a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log(-(2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) - (a*b - b^2)*tan(d*x + c)^2 - a^2 + a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + (a + b)*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log(-(2*(a^3 - a*b^2)*d*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) + (a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a*b^2)*d*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) - (a*b - b^2)*tan(d*x + c)^2 - a^2 + a*b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + 8*x)/(a + b)","B",0
386,1,4291,0,1.154183," ","integrate(1/(a+tan(d*x+c)^4*b)^2,x, algorithm=""fricas"")","\frac{32 \, a b d x \tan\left(d x + c\right)^{4} + 32 \, a^{2} d x - 8 \, {\left(a b + b^{2}\right)} \tan\left(d x + c\right)^{3} + {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{4} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} + 70 \, a^{2} b + 44 \, a b^{2} + 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} \log\left(\frac{625 \, a^{5} - 750 \, a^{4} b - 1376 \, a^{3} b^{2} - 594 \, a^{2} b^{3} - 81 \, a b^{4} + {\left(625 \, a^{4} b - 750 \, a^{3} b^{2} - 1376 \, a^{2} b^{3} - 594 \, a b^{4} - 81 \, b^{5}\right)} \tan\left(d x + c\right)^{2} + 2 \, {\left(2 \, {\left(a^{11} + 5 \, a^{10} b + 10 \, a^{9} b^{2} + 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} + a^{6} b^{5}\right)} d^{3} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} \tan\left(d x + c\right) + {\left(125 \, a^{7} + 5 \, a^{6} b - 442 \, a^{5} b^{2} - 490 \, a^{4} b^{3} - 195 \, a^{3} b^{4} - 27 \, a^{2} b^{5}\right)} d \tan\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} + 70 \, a^{2} b + 44 \, a b^{2} + 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} + {\left({\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2}\right)} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) - {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{4} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} + 70 \, a^{2} b + 44 \, a b^{2} + 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} \log\left(\frac{625 \, a^{5} - 750 \, a^{4} b - 1376 \, a^{3} b^{2} - 594 \, a^{2} b^{3} - 81 \, a b^{4} + {\left(625 \, a^{4} b - 750 \, a^{3} b^{2} - 1376 \, a^{2} b^{3} - 594 \, a b^{4} - 81 \, b^{5}\right)} \tan\left(d x + c\right)^{2} - 2 \, {\left(2 \, {\left(a^{11} + 5 \, a^{10} b + 10 \, a^{9} b^{2} + 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} + a^{6} b^{5}\right)} d^{3} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} \tan\left(d x + c\right) + {\left(125 \, a^{7} + 5 \, a^{6} b - 442 \, a^{5} b^{2} - 490 \, a^{4} b^{3} - 195 \, a^{3} b^{4} - 27 \, a^{2} b^{5}\right)} d \tan\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} + 70 \, a^{2} b + 44 \, a b^{2} + 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} + {\left({\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2}\right)} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) - {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{4} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} - 70 \, a^{2} b - 44 \, a b^{2} - 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} \log\left(-\frac{625 \, a^{5} - 750 \, a^{4} b - 1376 \, a^{3} b^{2} - 594 \, a^{2} b^{3} - 81 \, a b^{4} + {\left(625 \, a^{4} b - 750 \, a^{3} b^{2} - 1376 \, a^{2} b^{3} - 594 \, a b^{4} - 81 \, b^{5}\right)} \tan\left(d x + c\right)^{2} + 2 \, {\left(2 \, {\left(a^{11} + 5 \, a^{10} b + 10 \, a^{9} b^{2} + 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} + a^{6} b^{5}\right)} d^{3} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} \tan\left(d x + c\right) - {\left(125 \, a^{7} + 5 \, a^{6} b - 442 \, a^{5} b^{2} - 490 \, a^{4} b^{3} - 195 \, a^{3} b^{4} - 27 \, a^{2} b^{5}\right)} d \tan\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} - 70 \, a^{2} b - 44 \, a b^{2} - 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} - {\left({\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2}\right)} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) + {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{4} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} - 70 \, a^{2} b - 44 \, a b^{2} - 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} \log\left(-\frac{625 \, a^{5} - 750 \, a^{4} b - 1376 \, a^{3} b^{2} - 594 \, a^{2} b^{3} - 81 \, a b^{4} + {\left(625 \, a^{4} b - 750 \, a^{3} b^{2} - 1376 \, a^{2} b^{3} - 594 \, a b^{4} - 81 \, b^{5}\right)} \tan\left(d x + c\right)^{2} - 2 \, {\left(2 \, {\left(a^{11} + 5 \, a^{10} b + 10 \, a^{9} b^{2} + 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} + a^{6} b^{5}\right)} d^{3} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} \tan\left(d x + c\right) - {\left(125 \, a^{7} + 5 \, a^{6} b - 442 \, a^{5} b^{2} - 490 \, a^{4} b^{3} - 195 \, a^{3} b^{4} - 27 \, a^{2} b^{5}\right)} d \tan\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}} - 70 \, a^{2} b - 44 \, a b^{2} - 6 \, b^{3}}{{\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}}} - {\left({\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2} \tan\left(d x + c\right)^{2} - {\left(25 \, a^{9} + 109 \, a^{8} b + 186 \, a^{7} b^{2} + 154 \, a^{6} b^{3} + 61 \, a^{5} b^{4} + 9 \, a^{4} b^{5}\right)} d^{2}\right)} \sqrt{-\frac{625 \, a^{6} b - 1950 \, a^{5} b^{2} - 529 \, a^{4} b^{3} + 2748 \, a^{3} b^{4} + 2383 \, a^{2} b^{5} + 738 \, a b^{6} + 81 \, b^{7}}{{\left(a^{15} + 8 \, a^{14} b + 28 \, a^{13} b^{2} + 56 \, a^{12} b^{3} + 70 \, a^{11} b^{4} + 56 \, a^{10} b^{5} + 28 \, a^{9} b^{6} + 8 \, a^{8} b^{7} + a^{7} b^{8}\right)} d^{4}}}}{\tan\left(d x + c\right)^{2} + 1}\right) + 8 \, {\left(a b + b^{2}\right)} \tan\left(d x + c\right)}{32 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \tan\left(d x + c\right)^{4} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}"," ",0,"1/32*(32*a*b*d*x*tan(d*x + c)^4 + 32*a^2*d*x - 8*(a*b + b^2)*tan(d*x + c)^3 + ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) + 70*a^2*b + 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log((625*a^5 - 750*a^4*b - 1376*a^3*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2 + 2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) + (125*a^7 + 5*a^6*b - 442*a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) + 70*a^2*b + 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) + ((25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) - ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) + 70*a^2*b + 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log((625*a^5 - 750*a^4*b - 1376*a^3*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2 - 2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) + (125*a^7 + 5*a^6*b - 442*a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) + 70*a^2*b + 44*a*b^2 + 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) + ((25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) - ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) - 70*a^2*b - 44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log(-(625*a^5 - 750*a^4*b - 1376*a^3*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2 + 2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) - (125*a^7 + 5*a^6*b - 442*a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) - 70*a^2*b - 44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) - ((25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) + ((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) - 70*a^2*b - 44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2))*log(-(625*a^5 - 750*a^4*b - 1376*a^3*b^2 - 594*a^2*b^3 - 81*a*b^4 + (625*a^4*b - 750*a^3*b^2 - 1376*a^2*b^3 - 594*a*b^4 - 81*b^5)*tan(d*x + c)^2 - 2*(2*(a^11 + 5*a^10*b + 10*a^9*b^2 + 10*a^8*b^3 + 5*a^7*b^4 + a^6*b^5)*d^3*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4))*tan(d*x + c) - (125*a^7 + 5*a^6*b - 442*a^5*b^2 - 490*a^4*b^3 - 195*a^3*b^4 - 27*a^2*b^5)*d*tan(d*x + c))*sqrt(-((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)) - 70*a^2*b - 44*a*b^2 - 6*b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^2)) - ((25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2*tan(d*x + c)^2 - (25*a^9 + 109*a^8*b + 186*a^7*b^2 + 154*a^6*b^3 + 61*a^5*b^4 + 9*a^4*b^5)*d^2)*sqrt(-(625*a^6*b - 1950*a^5*b^2 - 529*a^4*b^3 + 2748*a^3*b^4 + 2383*a^2*b^5 + 738*a*b^6 + 81*b^7)/((a^15 + 8*a^14*b + 28*a^13*b^2 + 56*a^12*b^3 + 70*a^11*b^4 + 56*a^10*b^5 + 28*a^9*b^6 + 8*a^8*b^7 + a^7*b^8)*d^4)))/(tan(d*x + c)^2 + 1)) + 8*(a*b + b^2)*tan(d*x + c))/((a^3*b + 2*a^2*b^2 + a*b^3)*d*tan(d*x + c)^4 + (a^4 + 2*a^3*b + a^2*b^2)*d)","B",0
387,0,0,0,1.148515," ","integrate((a+tan(d*x+c)^4*b)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \tan\left(d x + c\right)^{4} + a}, x\right)"," ",0,"integral(sqrt(b*tan(d*x + c)^4 + a), x)","F",0
388,-1,0,0,0.000000," ","integrate(1/(a+tan(d*x+c)^4*b)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
389,1,555,0,0.698617," ","integrate((a+b*tan(x)^4)^(1/2)*tan(x)^3,x, algorithm=""fricas"")","\left[\frac{{\left(a + 2 \, b\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + 2 \, \sqrt{a + b} b \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}}{8 \, b}, -\frac{{\left(a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) - \sqrt{a + b} b \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}}{4 \, b}, \frac{4 \, \sqrt{-a - b} b \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + {\left(a + 2 \, b\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}}{8 \, b}, \frac{2 \, \sqrt{-a - b} b \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) - {\left(a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}}{4 \, b}\right]"," ",0,"[1/8*((a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 2*sqrt(a + b)*b*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b))/b, -1/4*((a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) - sqrt(a + b)*b*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) - sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b))/b, 1/8*(4*sqrt(-a - b)*b*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + (a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b))/b, 1/4*(2*sqrt(-a - b)*b*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) - (a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b))/b]","A",0
390,1,475,0,0.659576," ","integrate((a+b*tan(x)^4)^(1/2)*tan(x),x, algorithm=""fricas"")","\left[\frac{1}{4} \, \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{2} \, \sqrt{b \tan\left(x\right)^{4} + a}, \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{2} \, \sqrt{b \tan\left(x\right)^{4} + a}, -\frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + \frac{1}{4} \, \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + \frac{1}{2} \, \sqrt{b \tan\left(x\right)^{4} + a}, -\frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{2} \, \sqrt{b \tan\left(x\right)^{4} + a}\right]"," ",0,"[1/4*sqrt(b)*log(-2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/2*sqrt(b*tan(x)^4 + a), 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/2*sqrt(b*tan(x)^4 + a), -1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/4*sqrt(b)*log(-2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 1/2*sqrt(b*tan(x)^4 + a), -1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/2*sqrt(b*tan(x)^4 + a)]","A",0
391,1,1021,0,0.987526," ","integrate(cot(x)*(a+b*tan(x)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{1}{4} \, \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{4} \, \sqrt{a} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right), -\frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{4} \, \sqrt{a} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right), \frac{1}{2} \, \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \frac{1}{4} \, \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right), \frac{1}{2} \, \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) - \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right), \frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) + \frac{1}{4} \, \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, \sqrt{a} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right), \frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) - \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, \sqrt{a} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right), \frac{1}{2} \, \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) + \frac{1}{4} \, \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right), \frac{1}{2} \, \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) - \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right)\right]"," ",0,"[1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4), -1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) - 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)), 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4), 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2))]","A",0
392,0,0,0,38.639519," ","integrate((a+b*tan(x)^4)^(1/2)*tan(x)^2,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \tan\left(x\right)^{4} + a} \tan\left(x\right)^{2}, x\right)"," ",0,"integral(sqrt(b*tan(x)^4 + a)*tan(x)^2, x)","F",0
393,1,758,0,0.866290," ","integrate(tan(x)^3*(a+b*tan(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + 24 \, {\left(a b + b^{2}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, {\left(6 \, b^{2} \tan\left(x\right)^{6} - 8 \, b^{2} \tan\left(x\right)^{4} + 3 \, {\left(5 \, a b + 4 \, b^{2}\right)} \tan\left(x\right)^{2} - 32 \, a b - 24 \, b^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{96 \, b}, -\frac{3 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) - 12 \, {\left(a b + b^{2}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - {\left(6 \, b^{2} \tan\left(x\right)^{6} - 8 \, b^{2} \tan\left(x\right)^{4} + 3 \, {\left(5 \, a b + 4 \, b^{2}\right)} \tan\left(x\right)^{2} - 32 \, a b - 24 \, b^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{48 \, b}, \frac{48 \, {\left(a b + b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + 3 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + 2 \, {\left(6 \, b^{2} \tan\left(x\right)^{6} - 8 \, b^{2} \tan\left(x\right)^{4} + 3 \, {\left(5 \, a b + 4 \, b^{2}\right)} \tan\left(x\right)^{2} - 32 \, a b - 24 \, b^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{96 \, b}, \frac{24 \, {\left(a b + b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) - 3 \, {\left(3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + {\left(6 \, b^{2} \tan\left(x\right)^{6} - 8 \, b^{2} \tan\left(x\right)^{4} + 3 \, {\left(5 \, a b + 4 \, b^{2}\right)} \tan\left(x\right)^{2} - 32 \, a b - 24 \, b^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{48 \, b}\right]"," ",0,"[1/96*(3*(3*a^2 + 12*a*b + 8*b^2)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 24*(a*b + b^2)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*(6*b^2*tan(x)^6 - 8*b^2*tan(x)^4 + 3*(5*a*b + 4*b^2)*tan(x)^2 - 32*a*b - 24*b^2)*sqrt(b*tan(x)^4 + a))/b, -1/48*(3*(3*a^2 + 12*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) - 12*(a*b + b^2)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) - (6*b^2*tan(x)^6 - 8*b^2*tan(x)^4 + 3*(5*a*b + 4*b^2)*tan(x)^2 - 32*a*b - 24*b^2)*sqrt(b*tan(x)^4 + a))/b, 1/96*(48*(a*b + b^2)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 3*(3*a^2 + 12*a*b + 8*b^2)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 2*(6*b^2*tan(x)^6 - 8*b^2*tan(x)^4 + 3*(5*a*b + 4*b^2)*tan(x)^2 - 32*a*b - 24*b^2)*sqrt(b*tan(x)^4 + a))/b, 1/48*(24*(a*b + b^2)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) - 3*(3*a^2 + 12*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + (6*b^2*tan(x)^6 - 8*b^2*tan(x)^4 + 3*(5*a*b + 4*b^2)*tan(x)^2 - 32*a*b - 24*b^2)*sqrt(b*tan(x)^4 + a))/b]","A",0
394,1,593,0,0.850881," ","integrate(tan(x)*(a+b*tan(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{1}{8} \, {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + \frac{1}{4} \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{12} \, {\left(2 \, b \tan\left(x\right)^{4} - 3 \, b \tan\left(x\right)^{2} + 8 \, a + 6 \, b\right)} \sqrt{b \tan\left(x\right)^{4} + a}, \frac{1}{4} \, {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{12} \, {\left(2 \, b \tan\left(x\right)^{4} - 3 \, b \tan\left(x\right)^{2} + 8 \, a + 6 \, b\right)} \sqrt{b \tan\left(x\right)^{4} + a}, -\frac{1}{2} \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + \frac{1}{8} \, {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + \frac{1}{12} \, {\left(2 \, b \tan\left(x\right)^{4} - 3 \, b \tan\left(x\right)^{2} + 8 \, a + 6 \, b\right)} \sqrt{b \tan\left(x\right)^{4} + a}, -\frac{1}{2} \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + \frac{1}{4} \, {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{12} \, {\left(2 \, b \tan\left(x\right)^{4} - 3 \, b \tan\left(x\right)^{2} + 8 \, a + 6 \, b\right)} \sqrt{b \tan\left(x\right)^{4} + a}\right]"," ",0,"[1/8*(3*a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a), 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a), -1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/8*(3*a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a), -1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/12*(2*b*tan(x)^4 - 3*b*tan(x)^2 + 8*a + 6*b)*sqrt(b*tan(x)^4 + a)]","A",0
395,1,1269,0,37.866246," ","integrate(cot(x)*(a+b*tan(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{1}{8} \, {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{4} \, a^{\frac{3}{2}} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, -\frac{1}{4} \, {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{4} \, a^{\frac{3}{2}} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, \frac{1}{2} \, \sqrt{-a} a \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \frac{1}{8} \, {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, \frac{1}{2} \, \sqrt{-a} a \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) - \frac{1}{4} \, {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, \frac{1}{2} \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) + \frac{1}{8} \, {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, a^{\frac{3}{2}} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, \frac{1}{2} \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) - \frac{1}{4} \, {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, a^{\frac{3}{2}} \log\left(\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, \frac{1}{2} \, \sqrt{-a} a \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \frac{1}{2} \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) + \frac{1}{8} \, {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(2 \, b \tan\left(x\right)^{4} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} + a\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}, \frac{1}{2} \, \sqrt{-a} a \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \frac{1}{2} \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a - b}}{b \tan\left(x\right)^{2} - a}\right) - \frac{1}{4} \, {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) + \frac{1}{4} \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - 2 \, b\right)}\right]"," ",0,"[1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), -1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) - 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) + 1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/2*(a + b)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/(b*tan(x)^2 - a)) - 1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b)]","A",0
396,1,483,0,0.711828," ","integrate(tan(x)^3/(a+b*tan(x)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a + b\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right) + \sqrt{a + b} b \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right)}{4 \, {\left(a b + b^{2}\right)}}, -\frac{2 \, {\left(a + b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right) - \sqrt{a + b} b \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right)}{4 \, {\left(a b + b^{2}\right)}}, \frac{2 \, \sqrt{-a - b} b \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + {\left(a + b\right)} \sqrt{b} \log\left(-2 \, b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{b} \tan\left(x\right)^{2} - a\right)}{4 \, {\left(a b + b^{2}\right)}}, \frac{\sqrt{-a - b} b \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) - {\left(a + b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-b}}{b \tan\left(x\right)^{2}}\right)}{2 \, {\left(a b + b^{2}\right)}}\right]"," ",0,"[1/4*((a + b)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + sqrt(a + b)*b*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)))/(a*b + b^2), -1/4*(2*(a + b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) - sqrt(a + b)*b*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)))/(a*b + b^2), 1/4*(2*sqrt(-a - b)*b*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + (a + b)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a))/(a*b + b^2), 1/2*(sqrt(-a - b)*b*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) - (a + b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)))/(a*b + b^2)]","A",0
397,1,150,0,0.637372," ","integrate(tan(x)/(a+b*tan(x)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right)}{4 \, \sqrt{a + b}}, -\frac{\sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right)}{2 \, {\left(a + b\right)}}\right]"," ",0,"[1/4*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1))/sqrt(a + b), -1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b))/(a + b)]","A",0
398,1,475,0,0.751853," ","integrate(cot(x)/(a+b*tan(x)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a + b} a \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + {\left(a + b\right)} \sqrt{a} \log\left(-\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right)}{4 \, {\left(a^{2} + a b\right)}}, \frac{2 \, \sqrt{-a} {\left(a + b\right)} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \sqrt{a + b} a \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right)}{4 \, {\left(a^{2} + a b\right)}}, \frac{2 \, a \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + {\left(a + b\right)} \sqrt{a} \log\left(-\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right)}{4 \, {\left(a^{2} + a b\right)}}, \frac{a \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + \sqrt{-a} {\left(a + b\right)} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right)}{2 \, {\left(a^{2} + a b\right)}}\right]"," ",0,"[1/4*(sqrt(a + b)*a*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + (a + b)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4))/(a^2 + a*b), 1/4*(2*sqrt(-a)*(a + b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + sqrt(a + b)*a*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)))/(a^2 + a*b), 1/4*(2*a*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + (a + b)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4))/(a^2 + a*b), 1/2*(a*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + sqrt(-a)*(a + b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a))/(a^2 + a*b)]","A",0
399,0,0,0,0.854881," ","integrate(tan(x)^2/(a+b*tan(x)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(x\right)^{2}}{\sqrt{b \tan\left(x\right)^{4} + a}}, x\right)"," ",0,"integral(tan(x)^2/sqrt(b*tan(x)^4 + a), x)","F",0
400,1,292,0,0.777013," ","integrate(tan(x)^3/(a+b*tan(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(b \tan\left(x\right)^{4} + a\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left({\left(a + b\right)} \tan\left(x\right)^{2} - a - b\right)}}{4 \, {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(x\right)^{4} + a^{3} + 2 \, a^{2} b + a b^{2}\right)}}, \frac{{\left(b \tan\left(x\right)^{4} + a\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + \sqrt{b \tan\left(x\right)^{4} + a} {\left({\left(a + b\right)} \tan\left(x\right)^{2} - a - b\right)}}{2 \, {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(x\right)^{4} + a^{3} + 2 \, a^{2} b + a b^{2}\right)}}\right]"," ",0,"[1/4*((b*tan(x)^4 + a)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*sqrt(b*tan(x)^4 + a)*((a + b)*tan(x)^2 - a - b))/((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2), 1/2*((b*tan(x)^4 + a)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + sqrt(b*tan(x)^4 + a)*((a + b)*tan(x)^2 - a - b))/((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)]","B",0
401,1,319,0,0.745702," ","integrate(tan(x)/(a+b*tan(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b \tan\left(x\right)^{4} + a^{2}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left({\left(a b + b^{2}\right)} \tan\left(x\right)^{2} + a^{2} + a b\right)}}{4 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(x\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)}}, -\frac{{\left(a b \tan\left(x\right)^{4} + a^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) - \sqrt{b \tan\left(x\right)^{4} + a} {\left({\left(a b + b^{2}\right)} \tan\left(x\right)^{2} + a^{2} + a b\right)}}{2 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(x\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)}}\right]"," ",0,"[1/4*((a*b*tan(x)^4 + a^2)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*sqrt(b*tan(x)^4 + a)*((a*b + b^2)*tan(x)^2 + a^2 + a*b))/((a^3*b + 2*a^2*b^2 + a*b^3)*tan(x)^4 + a^4 + 2*a^3*b + a^2*b^2), -1/2*((a*b*tan(x)^4 + a^2)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) - sqrt(b*tan(x)^4 + a)*((a*b + b^2)*tan(x)^2 + a^2 + a*b))/((a^3*b + 2*a^2*b^2 + a*b^3)*tan(x)^4 + a^4 + 2*a^3*b + a^2*b^2)]","B",0
402,1,954,0,0.991690," ","integrate(cot(x)/(a+b*tan(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} b \tan\left(x\right)^{4} + a^{3}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(x\right)^{4} + a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{a} \log\left(-\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)}}{4 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{4}\right)}}, \frac{2 \, {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(x\right)^{4} + a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + {\left(a^{2} b \tan\left(x\right)^{4} + a^{3}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)}}{4 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{4}\right)}}, \frac{2 \, {\left(a^{2} b \tan\left(x\right)^{4} + a^{3}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(x\right)^{4} + a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{a} \log\left(-\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)}}{4 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{4}\right)}}, \frac{{\left(a^{2} b \tan\left(x\right)^{4} + a^{3}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(x\right)^{4} + a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + \sqrt{b \tan\left(x\right)^{4} + a} {\left(a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)}}{2 \, {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{4}\right)}}\right]"," ",0,"[1/4*((a^2*b*tan(x)^4 + a^3)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + ((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 2*sqrt(b*tan(x)^4 + a)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*tan(x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*tan(x)^4), 1/4*(2*((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + (a^2*b*tan(x)^4 + a^3)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*sqrt(b*tan(x)^4 + a)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*tan(x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*tan(x)^4), 1/4*(2*(a^2*b*tan(x)^4 + a^3)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + ((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 2*sqrt(b*tan(x)^4 + a)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*tan(x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*tan(x)^4), 1/2*((a^2*b*tan(x)^4 + a^3)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + ((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + sqrt(b*tan(x)^4 + a)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*tan(x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*tan(x)^4)]","B",0
403,1,556,0,0.796406," ","integrate(tan(x)^3/(a+b*tan(x)^4)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a b^{2} \tan\left(x\right)^{8} + 2 \, a^{2} b \tan\left(x\right)^{4} + a^{3}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, {\left({\left(2 \, a^{2} b + a b^{2} - b^{3}\right)} \tan\left(x\right)^{6} - 3 \, {\left(a^{2} b + a b^{2}\right)} \tan\left(x\right)^{4} - 4 \, a^{3} - 5 \, a^{2} b - a b^{2} + 3 \, {\left(a^{3} + a^{2} b\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{12 \, {\left({\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} \tan\left(x\right)^{8} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + 2 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \tan\left(x\right)^{4}\right)}}, \frac{3 \, {\left(a b^{2} \tan\left(x\right)^{8} + 2 \, a^{2} b \tan\left(x\right)^{4} + a^{3}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + {\left({\left(2 \, a^{2} b + a b^{2} - b^{3}\right)} \tan\left(x\right)^{6} - 3 \, {\left(a^{2} b + a b^{2}\right)} \tan\left(x\right)^{4} - 4 \, a^{3} - 5 \, a^{2} b - a b^{2} + 3 \, {\left(a^{3} + a^{2} b\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{6 \, {\left({\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} \tan\left(x\right)^{8} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + 2 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \tan\left(x\right)^{4}\right)}}\right]"," ",0,"[1/12*(3*(a*b^2*tan(x)^8 + 2*a^2*b*tan(x)^4 + a^3)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*((2*a^2*b + a*b^2 - b^3)*tan(x)^6 - 3*(a^2*b + a*b^2)*tan(x)^4 - 4*a^3 - 5*a^2*b - a*b^2 + 3*(a^3 + a^2*b)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*tan(x)^8 + a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 + 2*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*tan(x)^4), 1/6*(3*(a*b^2*tan(x)^8 + 2*a^2*b*tan(x)^4 + a^3)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + ((2*a^2*b + a*b^2 - b^3)*tan(x)^6 - 3*(a^2*b + a*b^2)*tan(x)^4 - 4*a^3 - 5*a^2*b - a*b^2 + 3*(a^3 + a^2*b)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*tan(x)^8 + a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 + 2*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*tan(x)^4)]","B",0
404,1,599,0,0.960159," ","integrate(tan(x)/(a+b*tan(x)^4)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b^{2} \tan\left(x\right)^{8} + 2 \, a^{3} b \tan\left(x\right)^{4} + a^{4}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} + 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 2 \, {\left({\left(5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(x\right)^{6} + 3 \, {\left(a^{3} b + a^{2} b^{2}\right)} \tan\left(x\right)^{4} + 4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2} + 3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{12 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} \tan\left(x\right)^{8} + a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} \tan\left(x\right)^{4}\right)}}, -\frac{3 \, {\left(a^{2} b^{2} \tan\left(x\right)^{8} + 2 \, a^{3} b \tan\left(x\right)^{4} + a^{4}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) - {\left({\left(5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right)} \tan\left(x\right)^{6} + 3 \, {\left(a^{3} b + a^{2} b^{2}\right)} \tan\left(x\right)^{4} + 4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2} + 3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{6 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} \tan\left(x\right)^{8} + a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} \tan\left(x\right)^{4}\right)}}\right]"," ",0,"[1/12*(3*(a^2*b^2*tan(x)^8 + 2*a^3*b*tan(x)^4 + a^4)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*((5*a^2*b^2 + 7*a*b^3 + 2*b^4)*tan(x)^6 + 3*(a^3*b + a^2*b^2)*tan(x)^4 + 4*a^4 + 5*a^3*b + a^2*b^2 + 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*tan(x)^8 + a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*tan(x)^4), -1/6*(3*(a^2*b^2*tan(x)^8 + 2*a^3*b*tan(x)^4 + a^4)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) - ((5*a^2*b^2 + 7*a*b^3 + 2*b^4)*tan(x)^6 + 3*(a^3*b + a^2*b^2)*tan(x)^4 + 4*a^4 + 5*a^3*b + a^2*b^2 + 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*tan(x)^8 + a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*tan(x)^4)]","B",0
405,1,1749,0,1.192288," ","integrate(cot(x)/(a+b*tan(x)^4)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} b^{2} \tan\left(x\right)^{8} + 2 \, a^{4} b \tan\left(x\right)^{4} + a^{5}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 3 \, {\left({\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} \tan\left(x\right)^{8} + a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4}\right)} \sqrt{a} \log\left(-\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) - 2 \, {\left({\left(5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \tan\left(x\right)^{6} - 7 \, a^{4} b - 11 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{12 \, {\left({\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} \tan\left(x\right)^{8} + a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} \tan\left(x\right)^{4}\right)}}, \frac{6 \, {\left({\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} \tan\left(x\right)^{8} + a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) + 3 \, {\left(a^{3} b^{2} \tan\left(x\right)^{8} + 2 \, a^{4} b \tan\left(x\right)^{4} + a^{5}\right)} \sqrt{a + b} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \tan\left(x\right)^{4} - 2 \, a b \tan\left(x\right)^{2} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{a + b} + 2 \, a^{2} + a b}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - 2 \, {\left({\left(5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \tan\left(x\right)^{6} - 7 \, a^{4} b - 11 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{12 \, {\left({\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} \tan\left(x\right)^{8} + a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} \tan\left(x\right)^{4}\right)}}, \frac{6 \, {\left(a^{3} b^{2} \tan\left(x\right)^{8} + 2 \, a^{4} b \tan\left(x\right)^{4} + a^{5}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + 3 \, {\left({\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} \tan\left(x\right)^{8} + a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4}\right)} \sqrt{a} \log\left(-\frac{b \tan\left(x\right)^{4} - 2 \, \sqrt{b \tan\left(x\right)^{4} + a} \sqrt{a} + 2 \, a}{\tan\left(x\right)^{4}}\right) - 2 \, {\left({\left(5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \tan\left(x\right)^{6} - 7 \, a^{4} b - 11 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{12 \, {\left({\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} \tan\left(x\right)^{8} + a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} \tan\left(x\right)^{4}\right)}}, \frac{3 \, {\left(a^{3} b^{2} \tan\left(x\right)^{8} + 2 \, a^{4} b \tan\left(x\right)^{4} + a^{5}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} {\left(b \tan\left(x\right)^{2} - a\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \tan\left(x\right)^{4} + a^{2} + a b}\right) + 3 \, {\left({\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} \tan\left(x\right)^{8} + a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \tan\left(x\right)^{4} + a} \sqrt{-a}}{a}\right) - {\left({\left(5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + 2 \, a b^{4}\right)} \tan\left(x\right)^{6} - 7 \, a^{4} b - 11 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \tan\left(x\right)^{2}\right)} \sqrt{b \tan\left(x\right)^{4} + a}}{6 \, {\left({\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} \tan\left(x\right)^{8} + a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3} + 2 \, {\left(a^{7} b + 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} + a^{4} b^{4}\right)} \tan\left(x\right)^{4}\right)}}\right]"," ",0,"[1/12*(3*(a^3*b^2*tan(x)^8 + 2*a^4*b*tan(x)^4 + a^5)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 3*((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*tan(x)^8 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) - 2*((5*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*tan(x)^6 - 7*a^4*b - 11*a^3*b^2 - 4*a^2*b^3 - 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4 + 3*(2*a^4*b + 3*a^3*b^2 + a^2*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*tan(x)^8 + a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*tan(x)^4), 1/12*(6*((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*tan(x)^8 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4)*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 3*(a^3*b^2*tan(x)^8 + 2*a^4*b*tan(x)^4 + a^5)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) - 2*((5*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*tan(x)^6 - 7*a^4*b - 11*a^3*b^2 - 4*a^2*b^3 - 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4 + 3*(2*a^4*b + 3*a^3*b^2 + a^2*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*tan(x)^8 + a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*tan(x)^4), 1/12*(6*(a^3*b^2*tan(x)^8 + 2*a^4*b*tan(x)^4 + a^5)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 3*((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*tan(x)^8 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) - 2*((5*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*tan(x)^6 - 7*a^4*b - 11*a^3*b^2 - 4*a^2*b^3 - 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4 + 3*(2*a^4*b + 3*a^3*b^2 + a^2*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*tan(x)^8 + a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*tan(x)^4), 1/6*(3*(a^3*b^2*tan(x)^8 + 2*a^4*b*tan(x)^4 + a^5)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 3*((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*tan(x)^8 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4)*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) - ((5*a^3*b^2 + 7*a^2*b^3 + 2*a*b^4)*tan(x)^6 - 7*a^4*b - 11*a^3*b^2 - 4*a^2*b^3 - 3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4 + 3*(2*a^4*b + 3*a^3*b^2 + a^2*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*tan(x)^8 + a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^7*b + 3*a^6*b^2 + 3*a^5*b^3 + a^4*b^4)*tan(x)^4)]","B",0
406,0,0,0,0.433301," ","integrate((a+b*(c*tan(f*x+e))^(1/2))^2*(d*tan(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(2 \, \sqrt{c \tan\left(f x + e\right)} \left(d \tan\left(f x + e\right)\right)^{m} a b + {\left(b^{2} c \tan\left(f x + e\right) + a^{2}\right)} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(2*sqrt(c*tan(f*x + e))*(d*tan(f*x + e))^m*a*b + (b^2*c*tan(f*x + e) + a^2)*(d*tan(f*x + e))^m, x)","F",0
407,0,0,0,0.520853," ","integrate((a+b*(c*tan(f*x+e))^(1/2))*(d*tan(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{c \tan\left(f x + e\right)} \left(d \tan\left(f x + e\right)\right)^{m} b + \left(d \tan\left(f x + e\right)\right)^{m} a, x\right)"," ",0,"integral(sqrt(c*tan(f*x + e))*(d*tan(f*x + e))^m*b + (d*tan(f*x + e))^m*a, x)","F",0
408,0,0,0,0.437978," ","integrate((d*tan(f*x+e))^m/(a+b*(c*tan(f*x+e))^(1/2)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{c \tan\left(f x + e\right)} \left(d \tan\left(f x + e\right)\right)^{m} b - \left(d \tan\left(f x + e\right)\right)^{m} a}{b^{2} c \tan\left(f x + e\right) - a^{2}}, x\right)"," ",0,"integral((sqrt(c*tan(f*x + e))*(d*tan(f*x + e))^m*b - (d*tan(f*x + e))^m*a)/(b^2*c*tan(f*x + e) - a^2), x)","F",0
409,0,0,0,0.716926," ","integrate((d*tan(f*x+e))^m/(a+b*(c*tan(f*x+e))^(1/2))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{2 \, \sqrt{c \tan\left(f x + e\right)} \left(d \tan\left(f x + e\right)\right)^{m} a b - {\left(b^{2} c \tan\left(f x + e\right) + a^{2}\right)} \left(d \tan\left(f x + e\right)\right)^{m}}{b^{4} c^{2} \tan\left(f x + e\right)^{2} - 2 \, a^{2} b^{2} c \tan\left(f x + e\right) + a^{4}}, x\right)"," ",0,"integral(-(2*sqrt(c*tan(f*x + e))*(d*tan(f*x + e))^m*a*b - (b^2*c*tan(f*x + e) + a^2)*(d*tan(f*x + e))^m)/(b^4*c^2*tan(f*x + e)^2 - 2*a^2*b^2*c*tan(f*x + e) + a^4), x)","F",0
410,0,0,0,0.525329," ","integrate((d*tan(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*(d*tan(f*x + e))^m, x)","F",0
411,0,0,0,0.571229," ","integrate(tan(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*tan(f*x + e)^2, x)","F",0
412,0,0,0,0.414909," ","integrate((b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p, x)","F",0
413,0,0,0,0.409602," ","integrate(cot(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cot(f*x + e)^2, x)","F",0
414,0,0,0,0.416053," ","integrate(cot(f*x+e)^4*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cot\left(f x + e\right)^{4}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cot(f*x + e)^4, x)","F",0
415,0,0,0,0.449863," ","integrate(cot(f*x+e)^6*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cot\left(f x + e\right)^{6}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cot(f*x + e)^6, x)","F",0
416,0,0,0,0.449513," ","integrate(tan(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \tan\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*tan(f*x + e)^3, x)","F",0
417,0,0,0,0.421550," ","integrate(tan(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \tan\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*tan(f*x + e), x)","F",0
418,0,0,0,0.508777," ","integrate(cot(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cot\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cot(f*x + e), x)","F",0
419,0,0,0,0.528789," ","integrate(cot(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cot\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cot(f*x + e)^3, x)","F",0
420,0,0,0,0.545890," ","integrate((d*tan(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*(d*tan(f*x + e))^m, x)","F",0
421,0,0,0,0.530192," ","integrate((d*cot(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p} \left(d \cot\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p*(d*cot(f*x + e))^m, x)","F",0
422,0,0,0,0.645518," ","integrate((d*cot(f*x+e))^m*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \left(d \cot\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*(d*cot(f*x + e))^m, x)","F",0
423,0,0,0,0.451846," ","integrate((d*cot(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \left(d \cot\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*(d*cot(f*x + e))^m, x)","F",0
424,0,0,0,0.500467," ","integrate((d*cot(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \left(d \cot\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*(d*cot(f*x + e))^m, x)","F",0
425,1,95,0,0.514217," ","integrate(sec(d*x+c)^3*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(4 \, a - b\right)} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(4 \, a - b\right)} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left({\left(4 \, a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, b\right)} \sin\left(d x + c\right)}{16 \, d \cos\left(d x + c\right)^{4}}"," ",0,"1/16*((4*a - b)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - (4*a - b)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*((4*a - b)*cos(d*x + c)^2 + 2*b)*sin(d*x + c))/(d*cos(d*x + c)^4)","A",0
426,1,76,0,0.518284," ","integrate(sec(d*x+c)*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, b \sin\left(d x + c\right)}{4 \, d \cos\left(d x + c\right)^{2}}"," ",0,"1/4*((2*a - b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a - b)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*b*sin(d*x + c))/(d*cos(d*x + c)^2)","A",0
427,1,44,0,0.557197," ","integrate(cos(d*x+c)*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{b \log\left(\sin\left(d x + c\right) + 1\right) - b \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(a - b\right)} \sin\left(d x + c\right)}{2 \, d}"," ",0,"1/2*(b*log(sin(d*x + c) + 1) - b*log(-sin(d*x + c) + 1) + 2*(a - b)*sin(d*x + c))/d","A",0
428,1,30,0,0.455598," ","integrate(cos(d*x+c)^3*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a + b\right)} \sin\left(d x + c\right)}{3 \, d}"," ",0,"1/3*((a - b)*cos(d*x + c)^2 + 2*a + b)*sin(d*x + c)/d","A",0
429,1,47,0,0.549482," ","integrate(cos(d*x+c)^5*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(3 \, {\left(a - b\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a + b\right)} \cos\left(d x + c\right)^{2} + 8 \, a + 2 \, b\right)} \sin\left(d x + c\right)}{15 \, d}"," ",0,"1/15*(3*(a - b)*cos(d*x + c)^4 + (4*a + b)*cos(d*x + c)^2 + 8*a + 2*b)*sin(d*x + c)/d","A",0
430,1,63,0,0.543287," ","integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(15 \, {\left(a - b\right)} \cos\left(d x + c\right)^{6} + 3 \, {\left(6 \, a + b\right)} \cos\left(d x + c\right)^{4} + 4 \, {\left(6 \, a + b\right)} \cos\left(d x + c\right)^{2} + 48 \, a + 8 \, b\right)} \sin\left(d x + c\right)}{105 \, d}"," ",0,"1/105*(15*(a - b)*cos(d*x + c)^6 + 3*(6*a + b)*cos(d*x + c)^4 + 4*(6*a + b)*cos(d*x + c)^2 + 48*a + 8*b)*sin(d*x + c)/d","A",0
431,1,74,0,0.465633," ","integrate(sec(d*x+c)^6*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(8 \, {\left(7 \, a - b\right)} \cos\left(d x + c\right)^{6} + 4 \, {\left(7 \, a - b\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(7 \, a - b\right)} \cos\left(d x + c\right)^{2} + 15 \, b\right)} \sin\left(d x + c\right)}{105 \, d \cos\left(d x + c\right)^{7}}"," ",0,"1/105*(8*(7*a - b)*cos(d*x + c)^6 + 4*(7*a - b)*cos(d*x + c)^4 + 3*(7*a - b)*cos(d*x + c)^2 + 15*b)*sin(d*x + c)/(d*cos(d*x + c)^7)","A",0
432,1,56,0,0.487045," ","integrate(sec(d*x+c)^4*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(5 \, a - b\right)} \cos\left(d x + c\right)^{4} + {\left(5 \, a - b\right)} \cos\left(d x + c\right)^{2} + 3 \, b\right)} \sin\left(d x + c\right)}{15 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/15*(2*(5*a - b)*cos(d*x + c)^4 + (5*a - b)*cos(d*x + c)^2 + 3*b)*sin(d*x + c)/(d*cos(d*x + c)^5)","A",0
433,1,37,0,0.502631," ","integrate(sec(d*x+c)^2*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left({\left(3 \, a - b\right)} \cos\left(d x + c\right)^{2} + b\right)} \sin\left(d x + c\right)}{3 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/3*((3*a - b)*cos(d*x + c)^2 + b)*sin(d*x + c)/(d*cos(d*x + c)^3)","A",0
434,1,30,0,0.511557," ","integrate(cos(d*x+c)^2*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(a + b\right)} d x + {\left(a - b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, d}"," ",0,"1/2*((a + b)*d*x + (a - b)*cos(d*x + c)*sin(d*x + c))/d","A",0
435,1,49,0,0.444876," ","integrate(cos(d*x+c)^4*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(3 \, a + b\right)} d x + {\left(2 \, {\left(a - b\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a + b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, d}"," ",0,"1/8*((3*a + b)*d*x + (2*(a - b)*cos(d*x + c)^3 + (3*a + b)*cos(d*x + c))*sin(d*x + c))/d","A",0
436,1,66,0,0.529762," ","integrate(cos(d*x+c)^6*(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(5 \, a + b\right)} d x + {\left(8 \, {\left(a - b\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(5 \, a + b\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(5 \, a + b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{48 \, d}"," ",0,"1/48*(3*(5*a + b)*d*x + (8*(a - b)*cos(d*x + c)^5 + 2*(5*a + b)*cos(d*x + c)^3 + 3*(5*a + b)*cos(d*x + c))*sin(d*x + c))/d","A",0
437,1,137,0,0.533208," ","integrate(sec(d*x+c)^3*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(8 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{6} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left(8 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{6} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(3 \, {\left(8 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(12 \, a b - 7 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 8 \, b^{2}\right)} \sin\left(d x + c\right)}{96 \, d \cos\left(d x + c\right)^{6}}"," ",0,"1/96*(3*(8*a^2 - 4*a*b + b^2)*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 3*(8*a^2 - 4*a*b + b^2)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(3*(8*a^2 - 4*a*b + b^2)*cos(d*x + c)^4 + 2*(12*a*b - 7*b^2)*cos(d*x + c)^2 + 8*b^2)*sin(d*x + c))/(d*cos(d*x + c)^6)","A",0
438,1,116,0,0.458602," ","integrate(sec(d*x+c)*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(8 \, a^{2} - 8 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left({\left(8 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, b^{2}\right)} \sin\left(d x + c\right)}{16 \, d \cos\left(d x + c\right)^{4}}"," ",0,"1/16*((8*a^2 - 8*a*b + 3*b^2)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - (8*a^2 - 8*a*b + 3*b^2)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*((8*a*b - 5*b^2)*cos(d*x + c)^2 + 2*b^2)*sin(d*x + c))/(d*cos(d*x + c)^4)","A",0
439,1,106,0,0.467137," ","integrate(cos(d*x+c)*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}\right)} \sin\left(d x + c\right)}{4 \, d \cos\left(d x + c\right)^{2}}"," ",0,"1/4*((4*a*b - 3*b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (4*a*b - 3*b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(2*(a^2 - 2*a*b + b^2)*cos(d*x + c)^2 + b^2)*sin(d*x + c))/(d*cos(d*x + c)^2)","A",0
440,1,79,0,0.511789," ","integrate(cos(d*x+c)^3*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, b^{2} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a^{2} + 2 \, a b - 4 \, b^{2}\right)} \sin\left(d x + c\right)}{6 \, d}"," ",0,"1/6*(3*b^2*log(sin(d*x + c) + 1) - 3*b^2*log(-sin(d*x + c) + 1) + 2*((a^2 - 2*a*b + b^2)*cos(d*x + c)^2 + 2*a^2 + 2*a*b - 4*b^2)*sin(d*x + c))/d","A",0
441,1,71,0,0.490477," ","integrate(cos(d*x+c)^5*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(2 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 8 \, a^{2} + 4 \, a b + 3 \, b^{2}\right)} \sin\left(d x + c\right)}{15 \, d}"," ",0,"1/15*(3*(a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(2*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 + 8*a^2 + 4*a*b + 3*b^2)*sin(d*x + c)/d","A",0
442,1,95,0,0.509152," ","integrate(cos(d*x+c)^7*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(15 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(3 \, a^{2} + a b - 4 \, b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(24 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 48 \, a^{2} + 16 \, a b + 6 \, b^{2}\right)} \sin\left(d x + c\right)}{105 \, d}"," ",0,"1/105*(15*(a^2 - 2*a*b + b^2)*cos(d*x + c)^6 + 6*(3*a^2 + a*b - 4*b^2)*cos(d*x + c)^4 + (24*a^2 + 8*a*b + 3*b^2)*cos(d*x + c)^2 + 48*a^2 + 16*a*b + 6*b^2)*sin(d*x + c)/d","A",0
443,1,117,0,0.514219," ","integrate(cos(d*x+c)^9*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(35 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{8} + 10 \, {\left(4 \, a^{2} + a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{6} + 3 \, {\left(16 \, a^{2} + 4 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 4 \, {\left(16 \, a^{2} + 4 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 128 \, a^{2} + 32 \, a b + 8 \, b^{2}\right)} \sin\left(d x + c\right)}{315 \, d}"," ",0,"1/315*(35*(a^2 - 2*a*b + b^2)*cos(d*x + c)^8 + 10*(4*a^2 + a*b - 5*b^2)*cos(d*x + c)^6 + 3*(16*a^2 + 4*a*b + b^2)*cos(d*x + c)^4 + 4*(16*a^2 + 4*a*b + b^2)*cos(d*x + c)^2 + 128*a^2 + 32*a*b + 8*b^2)*sin(d*x + c)/d","A",0
444,1,114,0,0.506526," ","integrate(sec(d*x+c)^6*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, {\left(21 \, a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{8} + 4 \, {\left(21 \, a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{6} + 3 \, {\left(21 \, a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 10 \, {\left(9 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 35 \, b^{2}\right)} \sin\left(d x + c\right)}{315 \, d \cos\left(d x + c\right)^{9}}"," ",0,"1/315*(8*(21*a^2 - 6*a*b + b^2)*cos(d*x + c)^8 + 4*(21*a^2 - 6*a*b + b^2)*cos(d*x + c)^6 + 3*(21*a^2 - 6*a*b + b^2)*cos(d*x + c)^4 + 10*(9*a*b - 5*b^2)*cos(d*x + c)^2 + 35*b^2)*sin(d*x + c)/(d*cos(d*x + c)^9)","A",0
445,1,94,0,0.555309," ","integrate(sec(d*x+c)^4*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(35 \, a^{2} - 14 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{6} + {\left(35 \, a^{2} - 14 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} + 6 \, {\left(7 \, a b - 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 15 \, b^{2}\right)} \sin\left(d x + c\right)}{105 \, d \cos\left(d x + c\right)^{7}}"," ",0,"1/105*(2*(35*a^2 - 14*a*b + 3*b^2)*cos(d*x + c)^6 + (35*a^2 - 14*a*b + 3*b^2)*cos(d*x + c)^4 + 6*(7*a*b - 4*b^2)*cos(d*x + c)^2 + 15*b^2)*sin(d*x + c)/(d*cos(d*x + c)^7)","A",0
446,1,69,0,0.509828," ","integrate(sec(d*x+c)^2*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left({\left(15 \, a^{2} - 10 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(5 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, b^{2}\right)} \sin\left(d x + c\right)}{15 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/15*((15*a^2 - 10*a*b + 3*b^2)*cos(d*x + c)^4 + 2*(5*a*b - 3*b^2)*cos(d*x + c)^2 + 3*b^2)*sin(d*x + c)/(d*cos(d*x + c)^5)","A",0
447,1,69,0,0.624856," ","integrate(cos(d*x+c)^2*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(a^{2} + 2 \, a b - 3 \, b^{2}\right)} d x \cos\left(d x + c\right) + {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, b^{2}\right)} \sin\left(d x + c\right)}{2 \, d \cos\left(d x + c\right)}"," ",0,"1/2*((a^2 + 2*a*b - 3*b^2)*d*x*cos(d*x + c) + ((a^2 - 2*a*b + b^2)*cos(d*x + c)^2 + 2*b^2)*sin(d*x + c))/(d*cos(d*x + c))","A",0
448,1,75,0,0.528855," ","integrate(cos(d*x+c)^4*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(3 \, a^{2} + 2 \, a b + 3 \, b^{2}\right)} d x + {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{2} + 2 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, d}"," ",0,"1/8*((3*a^2 + 2*a*b + 3*b^2)*d*x + (2*(a^2 - 2*a*b + b^2)*cos(d*x + c)^3 + (3*a^2 + 2*a*b - 5*b^2)*cos(d*x + c))*sin(d*x + c))/d","A",0
449,1,98,0,0.486029," ","integrate(cos(d*x+c)^6*(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(5 \, a^{2} + 2 \, a b + b^{2}\right)} d x + {\left(8 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(5 \, a^{2} + 2 \, a b - 7 \, b^{2}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(5 \, a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{48 \, d}"," ",0,"1/48*(3*(5*a^2 + 2*a*b + b^2)*d*x + (8*(a^2 - 2*a*b + b^2)*cos(d*x + c)^5 + 2*(5*a^2 + 2*a*b - 7*b^2)*cos(d*x + c)^3 + 3*(5*a^2 + 2*a*b + b^2)*cos(d*x + c))*sin(d*x + c))/d","A",0
450,1,292,0,0.496426," ","integrate(sec(d*x+c)^5/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a - b\right)} \sqrt{\frac{a - b}{a}} \cos\left(d x + c\right)^{2} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + {\left(2 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(2 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, b \sin\left(d x + c\right)}{4 \, b^{2} d \cos\left(d x + c\right)^{2}}, -\frac{4 \, {\left(a - b\right)} \sqrt{-\frac{a - b}{a}} \arctan\left(\sqrt{-\frac{a - b}{a}} \sin\left(d x + c\right)\right) \cos\left(d x + c\right)^{2} + {\left(2 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(2 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, b \sin\left(d x + c\right)}{4 \, b^{2} d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[-1/4*(2*(a - b)*sqrt((a - b)/a)*cos(d*x + c)^2*log(-((a - b)*cos(d*x + c)^2 + 2*a*sqrt((a - b)/a)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + (2*a - 3*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a - 3*b)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*b*sin(d*x + c))/(b^2*d*cos(d*x + c)^2), -1/4*(4*(a - b)*sqrt(-(a - b)/a)*arctan(sqrt(-(a - b)/a)*sin(d*x + c))*cos(d*x + c)^2 + (2*a - 3*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a - 3*b)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*b*sin(d*x + c))/(b^2*d*cos(d*x + c)^2)]","A",0
451,1,169,0,0.488848," ","integrate(sec(d*x+c)^3/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{\frac{a - b}{a}} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + \log\left(\sin\left(d x + c\right) + 1\right) - \log\left(-\sin\left(d x + c\right) + 1\right)}{2 \, b d}, \frac{2 \, \sqrt{-\frac{a - b}{a}} \arctan\left(\sqrt{-\frac{a - b}{a}} \sin\left(d x + c\right)\right) + \log\left(\sin\left(d x + c\right) + 1\right) - \log\left(-\sin\left(d x + c\right) + 1\right)}{2 \, b d}\right]"," ",0,"[1/2*(sqrt((a - b)/a)*log(-((a - b)*cos(d*x + c)^2 + 2*a*sqrt((a - b)/a)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + log(sin(d*x + c) + 1) - log(-sin(d*x + c) + 1))/(b*d), 1/2*(2*sqrt(-(a - b)/a)*arctan(sqrt(-(a - b)/a)*sin(d*x + c)) + log(sin(d*x + c) + 1) - log(-sin(d*x + c) + 1))/(b*d)]","A",0
452,1,122,0,0.436707," ","integrate(sec(d*x+c)/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right)}{2 \, \sqrt{a^{2} - a b} d}, -\frac{\sqrt{-a^{2} + a b} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right)}{{\left(a^{2} - a b\right)} d}\right]"," ",0,"[1/2*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b))/(sqrt(a^2 - a*b)*d), -sqrt(-a^2 + a*b)*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a)/((a^2 - a*b)*d)]","A",0
453,1,182,0,0.459621," ","integrate(cos(d*x+c)/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{a^{2} - a b} b \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) - 2 \, {\left(a^{2} - a b\right)} \sin\left(d x + c\right)}{2 \, {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d}, \frac{\sqrt{-a^{2} + a b} b \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) + {\left(a^{2} - a b\right)} \sin\left(d x + c\right)}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d}\right]"," ",0,"[-1/2*(sqrt(a^2 - a*b)*b*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) - 2*(a^2 - a*b)*sin(d*x + c))/((a^3 - 2*a^2*b + a*b^2)*d), (sqrt(-a^2 + a*b)*b*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) + (a^2 - a*b)*sin(d*x + c))/((a^3 - 2*a^2*b + a*b^2)*d)]","A",0
454,1,276,0,0.497865," ","integrate(cos(d*x+c)^3/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a^{2} - a b} b^{2} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + 2 \, {\left(2 \, a^{3} - 7 \, a^{2} b + 5 \, a b^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d}, -\frac{3 \, \sqrt{-a^{2} + a b} b^{2} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) - {\left(2 \, a^{3} - 7 \, a^{2} b + 5 \, a b^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{3 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d}\right]"," ",0,"[1/6*(3*sqrt(a^2 - a*b)*b^2*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + 2*(2*a^3 - 7*a^2*b + 5*a*b^2 + (a^3 - 2*a^2*b + a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d), -1/3*(3*sqrt(-a^2 + a*b)*b^2*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) - (2*a^3 - 7*a^2*b + 5*a*b^2 + (a^3 - 2*a^2*b + a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d)]","A",0
455,1,395,0,0.549609," ","integrate(cos(d*x+c)^5/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{15 \, \sqrt{a^{2} - a b} b^{3} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) - 2 \, {\left(3 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{4} + 8 \, a^{4} - 34 \, a^{3} b + 59 \, a^{2} b^{2} - 33 \, a b^{3} + {\left(4 \, a^{4} - 17 \, a^{3} b + 22 \, a^{2} b^{2} - 9 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{30 \, {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d}, \frac{15 \, \sqrt{-a^{2} + a b} b^{3} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) + {\left(3 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{4} + 8 \, a^{4} - 34 \, a^{3} b + 59 \, a^{2} b^{2} - 33 \, a b^{3} + {\left(4 \, a^{4} - 17 \, a^{3} b + 22 \, a^{2} b^{2} - 9 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{15 \, {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d}\right]"," ",0,"[-1/30*(15*sqrt(a^2 - a*b)*b^3*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) - 2*(3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(d*x + c)^4 + 8*a^4 - 34*a^3*b + 59*a^2*b^2 - 33*a*b^3 + (4*a^4 - 17*a^3*b + 22*a^2*b^2 - 9*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d), 1/15*(15*sqrt(-a^2 + a*b)*b^3*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) + (3*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(d*x + c)^4 + 8*a^4 - 34*a^3*b + 59*a^2*b^2 - 33*a*b^3 + (4*a^4 - 17*a^3*b + 22*a^2*b^2 - 9*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)]","A",0
456,1,425,0,0.562216," ","integrate(sec(d*x+c)^8/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \sqrt{-a b} \cos\left(d x + c\right)^{5} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) + 4 \, {\left({\left(15 \, a^{3} b - 40 \, a^{2} b^{2} + 33 \, a b^{3}\right)} \cos\left(d x + c\right)^{4} + 3 \, a b^{3} - {\left(5 \, a^{2} b^{2} - 9 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{60 \, a b^{4} d \cos\left(d x + c\right)^{5}}, \frac{15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{5} + 2 \, {\left({\left(15 \, a^{3} b - 40 \, a^{2} b^{2} + 33 \, a b^{3}\right)} \cos\left(d x + c\right)^{4} + 3 \, a b^{3} - {\left(5 \, a^{2} b^{2} - 9 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{30 \, a b^{4} d \cos\left(d x + c\right)^{5}}\right]"," ",0,"[1/60*(15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sqrt(-a*b)*cos(d*x + c)^5*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) + 4*((15*a^3*b - 40*a^2*b^2 + 33*a*b^3)*cos(d*x + c)^4 + 3*a*b^3 - (5*a^2*b^2 - 9*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^4*d*cos(d*x + c)^5), 1/30*(15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c)^5 + 2*((15*a^3*b - 40*a^2*b^2 + 33*a*b^3)*cos(d*x + c)^4 + 3*a*b^3 - (5*a^2*b^2 - 9*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^4*d*cos(d*x + c)^5)]","A",0
457,1,339,0,0.491950," ","integrate(sec(d*x+c)^6/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{-a b} \cos\left(d x + c\right)^{3} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) - 4 \, {\left(a b^{2} - {\left(3 \, a^{2} b - 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, a b^{3} d \cos\left(d x + c\right)^{3}}, -\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} - 2 \, {\left(a b^{2} - {\left(3 \, a^{2} b - 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, a b^{3} d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[-1/12*(3*(a^2 - 2*a*b + b^2)*sqrt(-a*b)*cos(d*x + c)^3*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) - 4*(a*b^2 - (3*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^3*d*cos(d*x + c)^3), -1/6*(3*(a^2 - 2*a*b + b^2)*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c)^3 - 2*(a*b^2 - (3*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^3*d*cos(d*x + c)^3)]","A",0
458,1,267,0,0.555121," ","integrate(sec(d*x+c)^4/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a b} {\left(a - b\right)} \cos\left(d x + c\right) \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) + 4 \, a b \sin\left(d x + c\right)}{4 \, a b^{2} d \cos\left(d x + c\right)}, \frac{\sqrt{a b} {\left(a - b\right)} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right) + 2 \, a b \sin\left(d x + c\right)}{2 \, a b^{2} d \cos\left(d x + c\right)}\right]"," ",0,"[1/4*(sqrt(-a*b)*(a - b)*cos(d*x + c)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) + 4*a*b*sin(d*x + c))/(a*b^2*d*cos(d*x + c)), 1/2*(sqrt(a*b)*(a - b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c) + 2*a*b*sin(d*x + c))/(a*b^2*d*cos(d*x + c))]","B",0
459,1,205,0,0.537404," ","integrate(sec(d*x+c)^2/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right)}{4 \, a b d}, -\frac{\sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{2 \, a b d}\right]"," ",0,"[-1/4*sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2))/(a*b*d), -1/2*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c)))/(a*b*d)]","B",0
460,1,290,0,0.560458," ","integrate(cos(d*x+c)^2/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a - 3 \, b\right)} d x + 2 \, {\left(a - b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + b \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - a b \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right)}{4 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d}, \frac{{\left(a - 3 \, b\right)} d x + {\left(a - b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - b \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d}\right]"," ",0,"[1/4*(2*(a - 3*b)*d*x + 2*(a - b)*cos(d*x + c)*sin(d*x + c) + b*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 - 4*((a^2 + a*b)*cos(d*x + c)^3 - a*b*cos(d*x + c))*sqrt(-b/a)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)))/((a^2 - 2*a*b + b^2)*d), 1/2*((a - 3*b)*d*x + (a - b)*cos(d*x + c)*sin(d*x + c) - b*sqrt(b/a)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(b/a)/(b*cos(d*x + c)*sin(d*x + c))))/((a^2 - 2*a*b + b^2)*d)]","A",0
461,1,401,0,0.565008," ","integrate(cos(d*x+c)^4/(a+b*tan(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{2 \, b^{2} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - a b \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) - {\left(3 \, a^{2} - 10 \, a b + 15 \, b^{2}\right)} d x - {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{2} - 10 \, a b + 7 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d}, \frac{4 \, b^{2} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + {\left(3 \, a^{2} - 10 \, a b + 15 \, b^{2}\right)} d x + {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{2} - 10 \, a b + 7 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d}\right]"," ",0,"[-1/8*(2*b^2*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 - 4*((a^2 + a*b)*cos(d*x + c)^3 - a*b*cos(d*x + c))*sqrt(-b/a)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) - (3*a^2 - 10*a*b + 15*b^2)*d*x - (2*(a^2 - 2*a*b + b^2)*cos(d*x + c)^3 + (3*a^2 - 10*a*b + 7*b^2)*cos(d*x + c))*sin(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d), 1/8*(4*b^2*sqrt(b/a)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(b/a)/(b*cos(d*x + c)*sin(d*x + c))) + (3*a^2 - 10*a*b + 15*b^2)*d*x + (2*(a^2 - 2*a*b + b^2)*cos(d*x + c)^3 + (3*a^2 - 10*a*b + 7*b^2)*cos(d*x + c))*sin(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)]","A",0
462,1,635,0,0.610874," ","integrate(sec(d*x+c)^7/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(4 \, a^{3} - 7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{2} b - 3 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a - b}{a}} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + {\left({\left(4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{2} b - 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left({\left(4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{2} b - 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a b^{2} + {\left(2 \, a^{2} b - 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(a b^{4} d \cos\left(d x + c\right)^{2} + {\left(a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{4}\right)}}, -\frac{2 \, {\left({\left(4 \, a^{3} - 7 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{2} b - 3 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{a - b}{a}} \arctan\left(\sqrt{-\frac{a - b}{a}} \sin\left(d x + c\right)\right) + {\left({\left(4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{2} b - 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left({\left(4 \, a^{3} - 9 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{2} b - 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a b^{2} + {\left(2 \, a^{2} b - 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(a b^{4} d \cos\left(d x + c\right)^{2} + {\left(a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{4}\right)}}\right]"," ",0,"[-1/4*(((4*a^3 - 7*a^2*b + 2*a*b^2 + b^3)*cos(d*x + c)^4 + (4*a^2*b - 3*a*b^2 - b^3)*cos(d*x + c)^2)*sqrt((a - b)/a)*log(-((a - b)*cos(d*x + c)^2 + 2*a*sqrt((a - b)/a)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(a*b^2 + (2*a^2*b - 3*a*b^2 + b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^4*d*cos(d*x + c)^2 + (a^2*b^3 - a*b^4)*d*cos(d*x + c)^4), -1/4*(2*((4*a^3 - 7*a^2*b + 2*a*b^2 + b^3)*cos(d*x + c)^4 + (4*a^2*b - 3*a*b^2 - b^3)*cos(d*x + c)^2)*sqrt(-(a - b)/a)*arctan(sqrt(-(a - b)/a)*sin(d*x + c)) + ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - ((4*a^3 - 9*a^2*b + 5*a*b^2)*cos(d*x + c)^4 + (4*a^2*b - 5*a*b^2)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(a*b^2 + (2*a^2*b - 3*a*b^2 + b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^4*d*cos(d*x + c)^2 + (a^2*b^3 - a*b^4)*d*cos(d*x + c)^4)]","A",0
463,1,407,0,0.659148," ","integrate(sec(d*x+c)^5/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(2 \, a^{2} - a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b + b^{2}\right)} \sqrt{\frac{a - b}{a}} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a \sqrt{\frac{a - b}{a}} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + 2 \, {\left({\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{2} + a b\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 2 \, {\left({\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{2} + a b\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a b - b^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(a b^{3} d + {\left(a^{2} b^{2} - a b^{3}\right)} d \cos\left(d x + c\right)^{2}\right)}}, \frac{{\left({\left(2 \, a^{2} - a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b + b^{2}\right)} \sqrt{-\frac{a - b}{a}} \arctan\left(\sqrt{-\frac{a - b}{a}} \sin\left(d x + c\right)\right) + {\left({\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{2} + a b\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left({\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{2} + a b\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - {\left(a b - b^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left(a b^{3} d + {\left(a^{2} b^{2} - a b^{3}\right)} d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[1/4*(((2*a^2 - a*b - b^2)*cos(d*x + c)^2 + 2*a*b + b^2)*sqrt((a - b)/a)*log(-((a - b)*cos(d*x + c)^2 + 2*a*sqrt((a - b)/a)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + 2*((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(sin(d*x + c) + 1) - 2*((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(-sin(d*x + c) + 1) - 2*(a*b - b^2)*sin(d*x + c))/(a*b^3*d + (a^2*b^2 - a*b^3)*d*cos(d*x + c)^2), 1/2*(((2*a^2 - a*b - b^2)*cos(d*x + c)^2 + 2*a*b + b^2)*sqrt(-(a - b)/a)*arctan(sqrt(-(a - b)/a)*sin(d*x + c)) + ((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(sin(d*x + c) + 1) - ((a^2 - a*b)*cos(d*x + c)^2 + a*b)*log(-sin(d*x + c) + 1) - (a*b - b^2)*sin(d*x + c))/(a*b^3*d + (a^2*b^2 - a*b^3)*d*cos(d*x + c)^2)]","A",0
464,1,266,0,0.661843," ","integrate(sec(d*x+c)^3/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} + b\right)} \sqrt{a^{2} - a b} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + 2 \, {\left(a^{2} - a b\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{3} b - a^{2} b^{2}\right)} d\right)}}, -\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} + b\right)} \sqrt{-a^{2} + a b} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) - {\left(a^{2} - a b\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{3} b - a^{2} b^{2}\right)} d\right)}}\right]"," ",0,"[1/4*(((a - b)*cos(d*x + c)^2 + b)*sqrt(a^2 - a*b)*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + 2*(a^2 - a*b)*sin(d*x + c))/((a^4 - 2*a^3*b + a^2*b^2)*d*cos(d*x + c)^2 + (a^3*b - a^2*b^2)*d), -1/2*(((a - b)*cos(d*x + c)^2 + b)*sqrt(-a^2 + a*b)*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) - (a^2 - a*b)*sin(d*x + c))/((a^4 - 2*a^3*b + a^2*b^2)*d*cos(d*x + c)^2 + (a^3*b - a^2*b^2)*d)]","A",0
465,1,337,0,0.593691," ","integrate(sec(d*x+c)/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b - b^{2}\right)} \sqrt{a^{2} - a b} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) - 2 \, {\left(a^{2} b - a b^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} d\right)}}, -\frac{{\left({\left(2 \, a^{2} - 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b - b^{2}\right)} \sqrt{-a^{2} + a b} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) + {\left(a^{2} b - a b^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} d\right)}}\right]"," ",0,"[1/4*(((2*a^2 - 3*a*b + b^2)*cos(d*x + c)^2 + 2*a*b - b^2)*sqrt(a^2 - a*b)*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) - 2*(a^2*b - a*b^2)*sin(d*x + c))/((a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cos(d*x + c)^2 + (a^4*b - 2*a^3*b^2 + a^2*b^3)*d), -1/2*(((2*a^2 - 3*a*b + b^2)*cos(d*x + c)^2 + 2*a*b - b^2)*sqrt(-a^2 + a*b)*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) + (a^2*b - a*b^2)*sin(d*x + c))/((a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cos(d*x + c)^2 + (a^4*b - 2*a^3*b^2 + a^2*b^3)*d)]","A",0
466,1,451,0,0.569453," ","integrate(cos(d*x+c)/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{{\left(4 \, a b^{2} - b^{3} + {\left(4 \, a^{2} b - 5 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} - a b} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) - 2 \, {\left(2 \, a^{3} b - a^{2} b^{2} - a b^{3} + 2 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d\right)}}, \frac{{\left(4 \, a b^{2} - b^{3} + {\left(4 \, a^{2} b - 5 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} + a b} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) + {\left(2 \, a^{3} b - a^{2} b^{2} - a b^{3} + 2 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d\right)}}\right]"," ",0,"[-1/4*((4*a*b^2 - b^3 + (4*a^2*b - 5*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(a^2 - a*b)*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) - 2*(2*a^3*b - a^2*b^2 - a*b^3 + 2*(a^4 - 2*a^3*b + a^2*b^2)*cos(d*x + c)^2)*sin(d*x + c))/((a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^2 + (a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d), 1/2*((4*a*b^2 - b^3 + (4*a^2*b - 5*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(-a^2 + a*b)*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) + (2*a^3*b - a^2*b^2 - a*b^3 + 2*(a^4 - 2*a^3*b + a^2*b^2)*cos(d*x + c)^2)*sin(d*x + c))/((a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^2 + (a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d)]","B",0
467,1,600,0,0.700236," ","integrate(cos(d*x+c)^3/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(6 \, a b^{3} - b^{4} + {\left(6 \, a^{2} b^{2} - 7 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} - a b} \log\left(-\frac{{\left(a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - a b} \sin\left(d x + c\right) - 2 \, a + b}{{\left(a - b\right)} \cos\left(d x + c\right)^{2} + b}\right) + 2 \, {\left(4 \, a^{4} b - 20 \, a^{3} b^{2} + 13 \, a^{2} b^{3} + 3 \, a b^{4} + 2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(2 \, a^{5} - 11 \, a^{4} b + 16 \, a^{3} b^{2} - 7 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left({\left(a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} b - 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} - 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} d\right)}}, -\frac{3 \, {\left(6 \, a b^{3} - b^{4} + {\left(6 \, a^{2} b^{2} - 7 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} + a b} \arctan\left(\frac{\sqrt{-a^{2} + a b} \sin\left(d x + c\right)}{a}\right) - {\left(4 \, a^{4} b - 20 \, a^{3} b^{2} + 13 \, a^{2} b^{3} + 3 \, a b^{4} + 2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(2 \, a^{5} - 11 \, a^{4} b + 16 \, a^{3} b^{2} - 7 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left({\left(a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} b - 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} - 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} d\right)}}\right]"," ",0,"[1/12*(3*(6*a*b^3 - b^4 + (6*a^2*b^2 - 7*a*b^3 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - a*b)*log(-((a - b)*cos(d*x + c)^2 - 2*sqrt(a^2 - a*b)*sin(d*x + c) - 2*a + b)/((a - b)*cos(d*x + c)^2 + b)) + 2*(4*a^4*b - 20*a^3*b^2 + 13*a^2*b^3 + 3*a*b^4 + 2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*cos(d*x + c)^4 + 2*(2*a^5 - 11*a^4*b + 16*a^3*b^2 - 7*a^2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^7 - 5*a^6*b + 10*a^5*b^2 - 10*a^4*b^3 + 5*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)^2 + (a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d), -1/6*(3*(6*a*b^3 - b^4 + (6*a^2*b^2 - 7*a*b^3 + b^4)*cos(d*x + c)^2)*sqrt(-a^2 + a*b)*arctan(sqrt(-a^2 + a*b)*sin(d*x + c)/a) - (4*a^4*b - 20*a^3*b^2 + 13*a^2*b^3 + 3*a*b^4 + 2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*cos(d*x + c)^4 + 2*(2*a^5 - 11*a^4*b + 16*a^3*b^2 - 7*a^2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^7 - 5*a^6*b + 10*a^5*b^2 - 10*a^4*b^3 + 5*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)^2 + (a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*d)]","B",0
468,1,597,0,0.624071," ","integrate(sec(d*x+c)^8/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(5 \, a^{4} - 14 \, a^{3} b + 12 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{5} + {\left(5 \, a^{3} b - 9 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{3}\right)} \sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) - 4 \, {\left(2 \, a^{2} b^{3} - {\left(15 \, a^{4} b - 37 \, a^{3} b^{2} + 25 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(5 \, a^{3} b^{2} - 7 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{24 \, {\left(a^{2} b^{5} d \cos\left(d x + c\right)^{3} + {\left(a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{5}\right)}}, -\frac{3 \, {\left({\left(5 \, a^{4} - 14 \, a^{3} b + 12 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{5} + {\left(5 \, a^{3} b - 9 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{3}\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left(2 \, a^{2} b^{3} - {\left(15 \, a^{4} b - 37 \, a^{3} b^{2} + 25 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(5 \, a^{3} b^{2} - 7 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left(a^{2} b^{5} d \cos\left(d x + c\right)^{3} + {\left(a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{5}\right)}}\right]"," ",0,"[-1/24*(3*((5*a^4 - 14*a^3*b + 12*a^2*b^2 - 2*a*b^3 - b^4)*cos(d*x + c)^5 + (5*a^3*b - 9*a^2*b^2 + 3*a*b^3 + b^4)*cos(d*x + c)^3)*sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) - 4*(2*a^2*b^3 - (15*a^4*b - 37*a^3*b^2 + 25*a^2*b^3 - 3*a*b^4)*cos(d*x + c)^4 - 2*(5*a^3*b^2 - 7*a^2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^2*b^5*d*cos(d*x + c)^3 + (a^3*b^4 - a^2*b^5)*d*cos(d*x + c)^5), -1/12*(3*((5*a^4 - 14*a^3*b + 12*a^2*b^2 - 2*a*b^3 - b^4)*cos(d*x + c)^5 + (5*a^3*b - 9*a^2*b^2 + 3*a*b^3 + b^4)*cos(d*x + c)^3)*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c))) - 2*(2*a^2*b^3 - (15*a^4*b - 37*a^3*b^2 + 25*a^2*b^3 - 3*a*b^4)*cos(d*x + c)^4 - 2*(5*a^3*b^2 - 7*a^2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^2*b^5*d*cos(d*x + c)^3 + (a^3*b^4 - a^2*b^5)*d*cos(d*x + c)^5)]","B",0
469,1,479,0,0.536583," ","integrate(sec(d*x+c)^6/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) + 4 \, {\left(2 \, a^{2} b^{2} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{8 \, {\left(a^{2} b^{4} d \cos\left(d x + c\right) + {\left(a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{3}\right)}}, \frac{{\left({\left(3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 2 \, {\left(2 \, a^{2} b^{2} + {\left(3 \, a^{3} b - 4 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(a^{2} b^{4} d \cos\left(d x + c\right) + {\left(a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[1/8*(((3*a^3 - 5*a^2*b + a*b^2 + b^3)*cos(d*x + c)^3 + (3*a^2*b - 2*a*b^2 - b^3)*cos(d*x + c))*sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) + 4*(2*a^2*b^2 + (3*a^3*b - 4*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^2*b^4*d*cos(d*x + c) + (a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^3), 1/4*(((3*a^3 - 5*a^2*b + a*b^2 + b^3)*cos(d*x + c)^3 + (3*a^2*b - 2*a*b^2 - b^3)*cos(d*x + c))*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c))) + 2*(2*a^2*b^2 + (3*a^3*b - 4*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^2*b^4*d*cos(d*x + c) + (a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^3)]","B",0
470,1,367,0,0.545697," ","integrate(sec(d*x+c)^4/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a^{2} b - a b^{2}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} + a b + b^{2}\right)} \sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right)}{8 \, {\left(a^{2} b^{3} d + {\left(a^{3} b^{2} - a^{2} b^{3}\right)} d \cos\left(d x + c\right)^{2}\right)}}, -\frac{2 \, {\left(a^{2} b - a b^{2}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} + a b + b^{2}\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{4 \, {\left(a^{2} b^{3} d + {\left(a^{3} b^{2} - a^{2} b^{3}\right)} d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[-1/8*(4*(a^2*b - a*b^2)*cos(d*x + c)*sin(d*x + c) + ((a^2 - b^2)*cos(d*x + c)^2 + a*b + b^2)*sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)))/(a^2*b^3*d + (a^3*b^2 - a^2*b^3)*d*cos(d*x + c)^2), -1/4*(2*(a^2*b - a*b^2)*cos(d*x + c)*sin(d*x + c) + ((a^2 - b^2)*cos(d*x + c)^2 + a*b + b^2)*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c))))/(a^2*b^3*d + (a^3*b^2 - a^2*b^3)*d*cos(d*x + c)^2)]","B",0
471,1,327,0,0.527415," ","integrate(sec(d*x+c)^2/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} + b\right)} \sqrt{-a b} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{3} - b \cos\left(d x + c\right)\right)} \sqrt{-a b} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right)}{8 \, {\left(a^{2} b^{2} d + {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2}\right)}}, \frac{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} + b\right)} \sqrt{a b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{a b}}{2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{4 \, {\left(a^{2} b^{2} d + {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[1/8*(4*a*b*cos(d*x + c)*sin(d*x + c) - ((a - b)*cos(d*x + c)^2 + b)*sqrt(-a*b)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 + 4*((a + b)*cos(d*x + c)^3 - b*cos(d*x + c))*sqrt(-a*b)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)))/(a^2*b^2*d + (a^3*b - a^2*b^2)*d*cos(d*x + c)^2), 1/4*(2*a*b*cos(d*x + c)*sin(d*x + c) - ((a - b)*cos(d*x + c)^2 + b)*sqrt(a*b)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(a*b)/(a*b*cos(d*x + c)*sin(d*x + c))))/(a^2*b^2*d + (a^3*b - a^2*b^2)*d*cos(d*x + c)^2)]","B",0
472,1,614,0,0.573232," ","integrate(cos(d*x+c)^2/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} - 6 \, a^{2} b + 5 \, a b^{2}\right)} d x \cos\left(d x + c\right)^{2} + 4 \, {\left(a^{2} b - 5 \, a b^{2}\right)} d x + {\left(5 \, a b^{2} - b^{3} + {\left(5 \, a^{2} b - 6 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - a b \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) + 4 \, {\left({\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(a^{2} b - b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, {\left({\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d\right)}}, \frac{2 \, {\left(a^{3} - 6 \, a^{2} b + 5 \, a b^{2}\right)} d x \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{2} b - 5 \, a b^{2}\right)} d x - {\left(5 \, a b^{2} - b^{3} + {\left(5 \, a^{2} b - 6 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 2 \, {\left({\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(a^{2} b - b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d\right)}}\right]"," ",0,"[1/8*(4*(a^3 - 6*a^2*b + 5*a*b^2)*d*x*cos(d*x + c)^2 + 4*(a^2*b - 5*a*b^2)*d*x + (5*a*b^2 - b^3 + (5*a^2*b - 6*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 - 4*((a^2 + a*b)*cos(d*x + c)^3 - a*b*cos(d*x + c))*sqrt(-b/a)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) + 4*((a^3 - 2*a^2*b + a*b^2)*cos(d*x + c)^3 + (a^2*b - b^3)*cos(d*x + c))*sin(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cos(d*x + c)^2 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d), 1/4*(2*(a^3 - 6*a^2*b + 5*a*b^2)*d*x*cos(d*x + c)^2 + 2*(a^2*b - 5*a*b^2)*d*x - (5*a*b^2 - b^3 + (5*a^2*b - 6*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(b/a)/(b*cos(d*x + c)*sin(d*x + c))) + 2*((a^3 - 2*a^2*b + a*b^2)*cos(d*x + c)^3 + (a^2*b - b^3)*cos(d*x + c))*sin(d*x + c))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d*cos(d*x + c)^2 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d)]","A",0
473,1,801,0,0.595126," ","integrate(cos(d*x+c)^4/(a+b*tan(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{4} - 17 \, a^{3} b + 49 \, a^{2} b^{2} - 35 \, a b^{3}\right)} d x \cos\left(d x + c\right)^{2} + {\left(3 \, a^{3} b - 14 \, a^{2} b^{2} + 35 \, a b^{3}\right)} d x - {\left(7 \, a b^{3} - b^{4} + {\left(7 \, a^{2} b^{2} - 8 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - a b \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(d x + c\right) + b^{2}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}}\right) + {\left(2 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{5} + 3 \, {\left(a^{4} - 5 \, a^{3} b + 7 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{3} b - 14 \, a^{2} b^{2} + 7 \, a b^{3} + 4 \, b^{4}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, {\left({\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)}}, \frac{{\left(3 \, a^{4} - 17 \, a^{3} b + 49 \, a^{2} b^{2} - 35 \, a b^{3}\right)} d x \cos\left(d x + c\right)^{2} + {\left(3 \, a^{3} b - 14 \, a^{2} b^{2} + 35 \, a b^{3}\right)} d x + 2 \, {\left(7 \, a b^{3} - b^{4} + {\left(7 \, a^{2} b^{2} - 8 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - b\right)} \sqrt{\frac{b}{a}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + {\left(2 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{5} + 3 \, {\left(a^{4} - 5 \, a^{3} b + 7 \, a^{2} b^{2} - 3 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{3} b - 14 \, a^{2} b^{2} + 7 \, a b^{3} + 4 \, b^{4}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, {\left({\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)}}\right]"," ",0,"[1/8*((3*a^4 - 17*a^3*b + 49*a^2*b^2 - 35*a*b^3)*d*x*cos(d*x + c)^2 + (3*a^3*b - 14*a^2*b^2 + 35*a*b^3)*d*x - (7*a*b^3 - b^4 + (7*a^2*b^2 - 8*a*b^3 + b^4)*cos(d*x + c)^2)*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(d*x + c)^4 - 2*(3*a*b + b^2)*cos(d*x + c)^2 - 4*((a^2 + a*b)*cos(d*x + c)^3 - a*b*cos(d*x + c))*sqrt(-b/a)*sin(d*x + c) + b^2)/((a^2 - 2*a*b + b^2)*cos(d*x + c)^4 + 2*(a*b - b^2)*cos(d*x + c)^2 + b^2)) + (2*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(d*x + c)^5 + 3*(a^4 - 5*a^3*b + 7*a^2*b^2 - 3*a*b^3)*cos(d*x + c)^3 + (3*a^3*b - 14*a^2*b^2 + 7*a*b^3 + 4*b^4)*cos(d*x + c))*sin(d*x + c))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d), 1/8*((3*a^4 - 17*a^3*b + 49*a^2*b^2 - 35*a*b^3)*d*x*cos(d*x + c)^2 + (3*a^3*b - 14*a^2*b^2 + 35*a*b^3)*d*x + 2*(7*a*b^3 - b^4 + (7*a^2*b^2 - 8*a*b^3 + b^4)*cos(d*x + c)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(d*x + c)^2 - b)*sqrt(b/a)/(b*cos(d*x + c)*sin(d*x + c))) + (2*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(d*x + c)^5 + 3*(a^4 - 5*a^3*b + 7*a^2*b^2 - 3*a*b^3)*cos(d*x + c)^3 + (3*a^3*b - 14*a^2*b^2 + 7*a*b^3 + 4*b^4)*cos(d*x + c))*sin(d*x + c))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)]","A",0
474,0,0,0,0.422313," ","integrate((d*sec(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p} \left(d \sec\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p*(d*sec(f*x + e))^m, x)","F",0
475,0,0,0,0.532639," ","integrate((d*sec(f*x+e))^m*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \left(d \sec\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*(d*sec(f*x + e))^m, x)","F",0
476,0,0,0,0.473789," ","integrate((d*sec(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \left(d \sec\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*(d*sec(f*x + e))^m, x)","F",0
477,1,107,0,0.446878," ","integrate(sec(f*x+e)^6*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","\frac{{\left(n^{2} p^{2} + 8 \, \cos\left(f x + e\right)^{4} + 4 \, {\left(n p + 1\right)} \cos\left(f x + e\right)^{2} + 4 \, n p + 3\right)} e^{\left(n p \log\left(\frac{c \sin\left(f x + e\right)}{\cos\left(f x + e\right)}\right) + p \log\left(b\right)\right)} \sin\left(f x + e\right)}{{\left(f n^{3} p^{3} + 9 \, f n^{2} p^{2} + 23 \, f n p + 15 \, f\right)} \cos\left(f x + e\right)^{5}}"," ",0,"(n^2*p^2 + 8*cos(f*x + e)^4 + 4*(n*p + 1)*cos(f*x + e)^2 + 4*n*p + 3)*e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))*sin(f*x + e)/((f*n^3*p^3 + 9*f*n^2*p^2 + 23*f*n*p + 15*f)*cos(f*x + e)^5)","A",0
478,1,75,0,0.510082," ","integrate(sec(f*x+e)^4*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","\frac{{\left(n p + 2 \, \cos\left(f x + e\right)^{2} + 1\right)} e^{\left(n p \log\left(\frac{c \sin\left(f x + e\right)}{\cos\left(f x + e\right)}\right) + p \log\left(b\right)\right)} \sin\left(f x + e\right)}{{\left(f n^{2} p^{2} + 4 \, f n p + 3 \, f\right)} \cos\left(f x + e\right)^{3}}"," ",0,"(n*p + 2*cos(f*x + e)^2 + 1)*e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))*sin(f*x + e)/((f*n^2*p^2 + 4*f*n*p + 3*f)*cos(f*x + e)^3)","A",0
479,1,49,0,0.546088," ","integrate(sec(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","\frac{e^{\left(n p \log\left(\frac{c \sin\left(f x + e\right)}{\cos\left(f x + e\right)}\right) + p \log\left(b\right)\right)} \sin\left(f x + e\right)}{{\left(f n p + f\right)} \cos\left(f x + e\right)}"," ",0,"e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))*sin(f*x + e)/((f*n*p + f)*cos(f*x + e))","A",0
480,0,0,0,0.566847," ","integrate((b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p, x)","F",0
481,0,0,0,0.557908," ","integrate(cos(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cos(f*x + e)^2, x)","F",0
482,0,0,0,0.651217," ","integrate(sec(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*sec(f*x + e)^3, x)","F",0
483,0,0,0,0.496454," ","integrate(sec(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \sec\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*sec(f*x + e), x)","F",0
484,0,0,0,0.470997," ","integrate(cos(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cos\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cos(f*x + e), x)","F",0
485,0,0,0,0.459108," ","integrate(cos(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*cos(f*x + e)^3, x)","F",0
486,0,0,0,0.502259," ","integrate((d*sec(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \left(d \sec\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*(d*sec(f*x + e))^m, x)","F",0
487,0,0,0,0.585355," ","integrate(sec(f*x+e)^3*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^3, x)","F",0
488,0,0,0,0.525919," ","integrate(sec(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \sec\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e), x)","F",0
489,0,0,0,0.544723," ","integrate(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \cos\left(f x + e\right), x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*cos(f*x + e), x)","F",0
490,0,0,0,0.522606," ","integrate(cos(f*x+e)^3*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*cos(f*x + e)^3, x)","F",0
491,0,0,0,0.501547," ","integrate(sec(f*x+e)^6*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \sec\left(f x + e\right)^{6}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^6, x)","F",0
492,0,0,0,0.488198," ","integrate(sec(f*x+e)^4*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^4, x)","F",0
493,0,0,0,0.484629," ","integrate(sec(f*x+e)^2*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^2, x)","F",0
494,0,0,0,0.441655," ","integrate((a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p, x)","F",0
495,0,0,0,0.523172," ","integrate(cos(f*x+e)^2*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*cos(f*x + e)^2, x)","F",0
496,0,0,0,0.499803," ","integrate((d*csc(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \tan\left(f x + e\right)^{2}\right)^{p} \left(d \csc\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2)^p*(d*csc(f*x + e))^m, x)","F",0
497,0,0,0,0.701287," ","integrate((d*csc(f*x+e))^m*(a+b*tan(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \tan\left(f x + e\right)^{2} + a\right)}^{p} \left(d \csc\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*tan(f*x + e)^2 + a)^p*(d*csc(f*x + e))^m, x)","F",0
498,0,0,0,0.505058," ","integrate((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(c \tan\left(f x + e\right)\right)^{n} b\right)^{p} \left(d \csc\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b)^p*(d*csc(f*x + e))^m, x)","F",0
499,0,0,0,0.629849," ","integrate((d*csc(f*x+e))^m*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\left(c \tan\left(f x + e\right)\right)^{n} b + a\right)}^{p} \left(d \csc\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(((c*tan(f*x + e))^n*b + a)^p*(d*csc(f*x + e))^m, x)","F",0
