1,1,45,0,3.740669," ","integrate((a*sin(x)^2)^(5/2),x, algorithm=""giac"")","\frac{1}{15} \, {\left(8 \, a^{2} \mathrm{sgn}\left(\sin\left(x\right)\right) - {\left(3 \, a^{2} \cos\left(x\right)^{5} - 10 \, a^{2} \cos\left(x\right)^{3} + 15 \, a^{2} \cos\left(x\right)\right)} \mathrm{sgn}\left(\sin\left(x\right)\right)\right)} \sqrt{a}"," ",0,"1/15*(8*a^2*sgn(sin(x)) - (3*a^2*cos(x)^5 - 10*a^2*cos(x)^3 + 15*a^2*cos(x))*sgn(sin(x)))*sqrt(a)","A",0
2,1,24,0,0.129540," ","integrate((a*sin(x)^2)^(3/2),x, algorithm=""giac"")","\frac{1}{3} \, {\left({\left(\cos\left(x\right)^{3} - 3 \, \cos\left(x\right)\right)} \mathrm{sgn}\left(\sin\left(x\right)\right) + 2 \, \mathrm{sgn}\left(\sin\left(x\right)\right)\right)} a^{\frac{3}{2}}"," ",0,"1/3*((cos(x)^3 - 3*cos(x))*sgn(sin(x)) + 2*sgn(sin(x)))*a^(3/2)","A",0
3,1,17,0,0.133748," ","integrate((a*sin(x)^2)^(1/2),x, algorithm=""giac"")","-{\left(\cos\left(x\right) \mathrm{sgn}\left(\sin\left(x\right)\right) - \mathrm{sgn}\left(\sin\left(x\right)\right)\right)} \sqrt{a}"," ",0,"-(cos(x)*sgn(sin(x)) - sgn(sin(x)))*sqrt(a)","A",0
4,1,15,0,0.159478," ","integrate(1/(a*sin(x)^2)^(1/2),x, algorithm=""giac"")","\frac{\log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{\sqrt{a} \mathrm{sgn}\left(\sin\left(x\right)\right)}"," ",0,"log(abs(tan(1/2*x)))/(sqrt(a)*sgn(sin(x)))","A",0
5,1,61,0,0.227417," ","integrate(1/(a*sin(x)^2)^(3/2),x, algorithm=""giac"")","\frac{\frac{\tan\left(\frac{1}{2} \, x\right)^{2}}{\mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right)} + \frac{2 \, \log\left(\tan\left(\frac{1}{2} \, x\right)^{2}\right)}{\mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right)} - \frac{2 \, \tan\left(\frac{1}{2} \, x\right)^{2} + 1}{\mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right) \tan\left(\frac{1}{2} \, x\right)^{2}}}{8 \, a^{\frac{3}{2}}}"," ",0,"1/8*(tan(1/2*x)^2/sgn(tan(1/2*x)) + 2*log(tan(1/2*x)^2)/sgn(tan(1/2*x)) - (2*tan(1/2*x)^2 + 1)/(sgn(tan(1/2*x))*tan(1/2*x)^2))/a^(3/2)","A",0
6,1,80,0,0.246892," ","integrate(1/(a*sin(x)^2)^(5/2),x, algorithm=""giac"")","\frac{\mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right) \tan\left(\frac{1}{2} \, x\right)^{4} + 8 \, \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right) \tan\left(\frac{1}{2} \, x\right)^{2} + \frac{12 \, \log\left(\tan\left(\frac{1}{2} \, x\right)^{2}\right)}{\mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right)} - \frac{18 \, \tan\left(\frac{1}{2} \, x\right)^{4} + 8 \, \tan\left(\frac{1}{2} \, x\right)^{2} + 1}{\mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right) \tan\left(\frac{1}{2} \, x\right)^{4}}}{64 \, a^{\frac{5}{2}}}"," ",0,"1/64*(sgn(tan(1/2*x))*tan(1/2*x)^4 + 8*sgn(tan(1/2*x))*tan(1/2*x)^2 + 12*log(tan(1/2*x)^2)/sgn(tan(1/2*x)) - (18*tan(1/2*x)^4 + 8*tan(1/2*x)^2 + 1)/(sgn(tan(1/2*x))*tan(1/2*x)^4))/a^(5/2)","A",0
7,0,0,0,0.000000," ","integrate((a*sin(x)^3)^(5/2),x, algorithm=""giac"")","\int \left(a \sin\left(x\right)^{3}\right)^{\frac{5}{2}}\,{d x}"," ",0,"integrate((a*sin(x)^3)^(5/2), x)","F",0
8,0,0,0,0.000000," ","integrate((a*sin(x)^3)^(3/2),x, algorithm=""giac"")","\int \left(a \sin\left(x\right)^{3}\right)^{\frac{3}{2}}\,{d x}"," ",0,"integrate((a*sin(x)^3)^(3/2), x)","F",0
9,0,0,0,0.000000," ","integrate((a*sin(x)^3)^(1/2),x, algorithm=""giac"")","\int \sqrt{a \sin\left(x\right)^{3}}\,{d x}"," ",0,"integrate(sqrt(a*sin(x)^3), x)","F",0
10,0,0,0,0.000000," ","integrate(1/(a*sin(x)^3)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{a \sin\left(x\right)^{3}}}\,{d x}"," ",0,"integrate(1/sqrt(a*sin(x)^3), x)","F",0
11,0,0,0,0.000000," ","integrate(1/(a*sin(x)^3)^(3/2),x, algorithm=""giac"")","\int \frac{1}{\left(a \sin\left(x\right)^{3}\right)^{\frac{3}{2}}}\,{d x}"," ",0,"integrate((a*sin(x)^3)^(-3/2), x)","F",0
12,0,0,0,0.000000," ","integrate(1/(a*sin(x)^3)^(5/2),x, algorithm=""giac"")","\int \frac{1}{\left(a \sin\left(x\right)^{3}\right)^{\frac{5}{2}}}\,{d x}"," ",0,"integrate((a*sin(x)^3)^(-5/2), x)","F",0
13,1,57,0,0.145742," ","integrate((a*sin(x)^4)^(5/2),x, algorithm=""giac"")","\frac{1}{10240} \, {\left(2520 \, a^{2} x - 2 \, a^{2} \sin\left(10 \, x\right) + 25 \, a^{2} \sin\left(8 \, x\right) - 150 \, a^{2} \sin\left(6 \, x\right) + 600 \, a^{2} \sin\left(4 \, x\right) - 2100 \, a^{2} \sin\left(2 \, x\right)\right)} \sqrt{a}"," ",0,"1/10240*(2520*a^2*x - 2*a^2*sin(10*x) + 25*a^2*sin(8*x) - 150*a^2*sin(6*x) + 600*a^2*sin(4*x) - 2100*a^2*sin(2*x))*sqrt(a)","A",0
14,1,27,0,0.127755," ","integrate((a*sin(x)^4)^(3/2),x, algorithm=""giac"")","\frac{1}{192} \, a^{\frac{3}{2}} {\left(60 \, x - \sin\left(6 \, x\right) + 9 \, \sin\left(4 \, x\right) - 45 \, \sin\left(2 \, x\right)\right)}"," ",0,"1/192*a^(3/2)*(60*x - sin(6*x) + 9*sin(4*x) - 45*sin(2*x))","A",0
15,1,15,0,0.143247," ","integrate((a*sin(x)^4)^(1/2),x, algorithm=""giac"")","\frac{1}{4} \, \sqrt{a} {\left(2 \, x - \sin\left(2 \, x\right)\right)}"," ",0,"1/4*sqrt(a)*(2*x - sin(2*x))","A",0
16,1,9,0,0.129810," ","integrate(1/(a*sin(x)^4)^(1/2),x, algorithm=""giac"")","-\frac{1}{\sqrt{a} \tan\left(x\right)}"," ",0,"-1/(sqrt(a)*tan(x))","A",0
17,-2,0,0,0.000000," ","integrate(1/(a*sin(x)^4)^(3/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value","F(-2)",0
18,-2,0,0,0.000000," ","integrate(1/(a*sin(x)^4)^(5/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value","F(-2)",0
19,0,0,0,0.000000," ","integrate((c*sin(b*x+a)^m)^(5/2),x, algorithm=""giac"")","\int \left(c \sin\left(b x + a\right)^{m}\right)^{\frac{5}{2}}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^m)^(5/2), x)","F",0
20,0,0,0,0.000000," ","integrate((c*sin(b*x+a)^m)^(3/2),x, algorithm=""giac"")","\int \left(c \sin\left(b x + a\right)^{m}\right)^{\frac{3}{2}}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^m)^(3/2), x)","F",0
21,0,0,0,0.000000," ","integrate((c*sin(b*x+a)^m)^(1/2),x, algorithm=""giac"")","\int \sqrt{c \sin\left(b x + a\right)^{m}}\,{d x}"," ",0,"integrate(sqrt(c*sin(b*x + a)^m), x)","F",0
22,0,0,0,0.000000," ","integrate(1/(c*sin(b*x+a)^m)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{c \sin\left(b x + a\right)^{m}}}\,{d x}"," ",0,"integrate(1/sqrt(c*sin(b*x + a)^m), x)","F",0
23,0,0,0,0.000000," ","integrate(1/(c*sin(b*x+a)^m)^(3/2),x, algorithm=""giac"")","\int \frac{1}{\left(c \sin\left(b x + a\right)^{m}\right)^{\frac{3}{2}}}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^m)^(-3/2), x)","F",0
24,0,0,0,0.000000," ","integrate(1/(c*sin(b*x+a)^m)^(5/2),x, algorithm=""giac"")","\int \frac{1}{\left(c \sin\left(b x + a\right)^{m}\right)^{\frac{5}{2}}}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^m)^(-5/2), x)","F",0
25,0,0,0,0.000000," ","integrate((b*sin(d*x+c)^n)^p,x, algorithm=""giac"")","\int \left(b \sin\left(d x + c\right)^{n}\right)^{p}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n)^p, x)","F",0
26,0,0,0,0.000000," ","integrate((c*sin(b*x+a)^2)^p,x, algorithm=""giac"")","\int \left(c \sin\left(b x + a\right)^{2}\right)^{p}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^2)^p, x)","F",0
27,0,0,0,0.000000," ","integrate((c*sin(b*x+a)^3)^p,x, algorithm=""giac"")","\int \left(c \sin\left(b x + a\right)^{3}\right)^{p}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^3)^p, x)","F",0
28,0,0,0,0.000000," ","integrate((c*sin(b*x+a)^4)^p,x, algorithm=""giac"")","\int \left(c \sin\left(b x + a\right)^{4}\right)^{p}\,{d x}"," ",0,"integrate((c*sin(b*x + a)^4)^p, x)","F",0
29,1,384,0,3.425696," ","integrate((c*sin(b*x+a)^n)^(1/n),x, algorithm=""giac"")","\frac{{\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right)^{2} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{4} - 2 \, {\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right)^{2} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{2} + 4 \, {\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right) \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{3} - {\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{4} + {\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right)^{2} - 4 \, {\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right) \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right) + 2 \, {\left| c \right|}^{\left(\frac{1}{n}\right)} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{2} - {\left| c \right|}^{\left(\frac{1}{n}\right)}}{b \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right)^{2} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{4} + 2 \, b \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right)^{2} \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{2} + b \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{4} + b \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left(c\right)}{4 \, n} - \frac{\pi}{4 \, n}\right)^{2} + 2 \, b \tan\left(\frac{1}{2} \, b x + \frac{1}{2} \, a\right)^{2} + b}"," ",0,"(abs(c)^(1/n)*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)^2*tan(1/2*b*x + 1/2*a)^4 - 2*abs(c)^(1/n)*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)^2*tan(1/2*b*x + 1/2*a)^2 + 4*abs(c)^(1/n)*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)*tan(1/2*b*x + 1/2*a)^3 - abs(c)^(1/n)*tan(1/2*b*x + 1/2*a)^4 + abs(c)^(1/n)*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)^2 - 4*abs(c)^(1/n)*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)*tan(1/2*b*x + 1/2*a) + 2*abs(c)^(1/n)*tan(1/2*b*x + 1/2*a)^2 - abs(c)^(1/n))/(b*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)^2*tan(1/2*b*x + 1/2*a)^4 + 2*b*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)^2*tan(1/2*b*x + 1/2*a)^2 + b*tan(1/2*b*x + 1/2*a)^4 + b*tan(1/2*b*x + 1/2*a + 1/4*pi*sgn(c)/n - 1/4*pi/n)^2 + 2*b*tan(1/2*b*x + 1/2*a)^2 + b)","B",0
30,0,0,0,0.000000," ","integrate((a*(b*sin(d*x+c))^p)^n,x, algorithm=""giac"")","\int \left(\left(b \sin\left(d x + c\right)\right)^{p} a\right)^{n}\,{d x}"," ",0,"integrate(((b*sin(d*x + c))^p*a)^n, x)","F",0
31,1,17,0,0.120843," ","integrate(a-a*sin(x)^2,x, algorithm=""giac"")","-\frac{1}{4} \, a {\left(2 \, x - \sin\left(2 \, x\right)\right)} + a x"," ",0,"-1/4*a*(2*x - sin(2*x)) + a*x","A",0
32,1,25,0,0.139051," ","integrate((a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{3}{8} \, a^{2} x + \frac{1}{32} \, a^{2} \sin\left(4 \, x\right) + \frac{1}{4} \, a^{2} \sin\left(2 \, x\right)"," ",0,"3/8*a^2*x + 1/32*a^2*sin(4*x) + 1/4*a^2*sin(2*x)","A",0
33,1,34,0,0.127984," ","integrate((a-a*sin(x)^2)^3,x, algorithm=""giac"")","\frac{5}{16} \, a^{3} x + \frac{1}{192} \, a^{3} \sin\left(6 \, x\right) + \frac{3}{64} \, a^{3} \sin\left(4 \, x\right) + \frac{15}{64} \, a^{3} \sin\left(2 \, x\right)"," ",0,"5/16*a^3*x + 1/192*a^3*sin(6*x) + 3/64*a^3*sin(4*x) + 15/64*a^3*sin(2*x)","A",0
34,1,43,0,0.122312," ","integrate((a-a*sin(x)^2)^4,x, algorithm=""giac"")","\frac{35}{128} \, a^{4} x + \frac{1}{1024} \, a^{4} \sin\left(8 \, x\right) + \frac{1}{96} \, a^{4} \sin\left(6 \, x\right) + \frac{7}{128} \, a^{4} \sin\left(4 \, x\right) + \frac{7}{32} \, a^{4} \sin\left(2 \, x\right)"," ",0,"35/128*a^4*x + 1/1024*a^4*sin(8*x) + 1/96*a^4*sin(6*x) + 7/128*a^4*sin(4*x) + 7/32*a^4*sin(2*x)","A",0
35,1,149,0,0.148602," ","integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{2 \, {\left(\frac{5}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}} + \frac{\frac{50 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{80 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{30 \, {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} - \frac{5 \, {\left(\cos\left(d x + c\right) - 1\right)}^{4}}{{\left(\cos\left(d x + c\right) + 1\right)}^{4}} - 11}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1\right)}^{5}}\right)}}{5 \, d}"," ",0,"2/5*(5/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) + (50*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 80*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 30*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 5*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 11)/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5))/d","B",0
36,1,105,0,0.142159," ","integrate(sin(d*x+c)^5/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{2 \, {\left(\frac{3}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}} + \frac{\frac{12 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - 5}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1\right)}^{3}}\right)}}{3 \, d}"," ",0,"2/3*(3/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) + (12*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 5)/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^3))/d","B",0
37,1,29,0,0.134474," ","integrate(sin(d*x+c)^3/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\cos\left(d x + c\right)}{a d} + \frac{1}{a d \cos\left(d x + c\right)}"," ",0,"cos(d*x + c)/(a*d) + 1/(a*d*cos(d*x + c))","A",0
38,1,15,0,0.126253," ","integrate(sin(d*x+c)/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{1}{a d \cos\left(d x + c\right)}"," ",0,"1/(a*d*cos(d*x + c))","A",0
39,1,62,0,0.145998," ","integrate(csc(d*x+c)/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{\log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a} + \frac{4}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}}}{2 \, d}"," ",0,"1/2*(log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a + 4/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)))/d","B",0
40,1,149,0,0.212717," ","integrate(csc(d*x+c)^3/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{6 \, \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a} + \frac{\frac{14 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + 1}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + \frac{{\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}} - \frac{\cos\left(d x + c\right) - 1}{a {\left(\cos\left(d x + c\right) + 1\right)}}}{8 \, d}"," ",0,"1/8*(6*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a + (14*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 1)/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + (cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)) - (cos(d*x + c) - 1)/(a*(cos(d*x + c) + 1)))/d","B",0
41,1,181,0,0.174459," ","integrate(csc(d*x+c)^5/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{{\left(\frac{16 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{90 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - 1\right)} {\left(\cos\left(d x + c\right) + 1\right)}^{2}}{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}} + \frac{60 \, \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a} - \frac{\frac{16 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}}{a^{2}} + \frac{128}{a {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}}}{64 \, d}"," ",0,"1/64*((16*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 90*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1)*(cos(d*x + c) + 1)^2/(a*(cos(d*x + c) - 1)^2) + 60*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a - (16*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/a^2 + 128/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)))/d","B",0
42,1,63,0,0.148097," ","integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{15 \, {\left(d x + c\right)}}{a} - \frac{8 \, \tan\left(d x + c\right)}{a} - \frac{9 \, \tan\left(d x + c\right)^{3} + 7 \, \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)}^{2} a}}{8 \, d}"," ",0,"-1/8*(15*(d*x + c)/a - 8*tan(d*x + c)/a - (9*tan(d*x + c)^3 + 7*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*a))/d","A",0
43,1,50,0,0.137100," ","integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{3 \, {\left(d x + c\right)}}{a} - \frac{2 \, \tan\left(d x + c\right)}{a} - \frac{\tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)} a}}{2 \, d}"," ",0,"-1/2*(3*(d*x + c)/a - 2*tan(d*x + c)/a - tan(d*x + c)/((tan(d*x + c)^2 + 1)*a))/d","A",0
44,1,26,0,0.154452," ","integrate(sin(d*x+c)^2/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{d x + c}{a} - \frac{\tan\left(d x + c\right)}{a}}{d}"," ",0,"-((d*x + c)/a - tan(d*x + c)/a)/d","A",0
45,1,13,0,0.142403," ","integrate(1/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\tan\left(d x + c\right)}{a d}"," ",0,"tan(d*x + c)/(a*d)","A",0
46,1,19,0,0.159355," ","integrate(csc(d*x+c)^2/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{2}{a d \tan\left(2 \, d x + 2 \, c\right)}"," ",0,"-2/(a*d*tan(2*d*x + 2*c))","A",0
47,1,42,0,0.152971," ","integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{3 \, \tan\left(d x + c\right)}{a} - \frac{6 \, \tan\left(d x + c\right)^{2} + 1}{a \tan\left(d x + c\right)^{3}}}{3 \, d}"," ",0,"1/3*(3*tan(d*x + c)/a - (6*tan(d*x + c)^2 + 1)/(a*tan(d*x + c)^3))/d","A",0
48,1,52,0,0.167859," ","integrate(csc(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{5 \, \tan\left(d x + c\right)}{a} - \frac{15 \, \tan\left(d x + c\right)^{4} + 5 \, \tan\left(d x + c\right)^{2} + 1}{a \tan\left(d x + c\right)^{5}}}{5 \, d}"," ",0,"1/5*(5*tan(d*x + c)/a - (15*tan(d*x + c)^4 + 5*tan(d*x + c)^2 + 1)/(a*tan(d*x + c)^5))/d","A",0
49,1,57,0,0.175390," ","integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{32 \, {\left(\frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - 1\right)}}{3 \, a^{2} d {\left(\frac{{\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - 1\right)}^{3}}"," ",0,"-32/3*(3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1)/(a^2*d*((cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 1)^3)","A",0
50,1,106,0,0.162313," ","integrate(sin(d*x+c)^5/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{2 \, {\left(\frac{3}{a^{2} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1\right)}} - \frac{\frac{12 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + 5}{a^{2} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}^{3}}\right)}}{3 \, d}"," ",0,"2/3*(3/(a^2*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) - (12*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 5)/(a^2*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^3))/d","B",0
51,1,28,0,0.144305," ","integrate(sin(d*x+c)^3/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{3 \, \cos\left(d x + c\right)^{2} - 1}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/3*(3*cos(d*x + c)^2 - 1)/(a^2*d*cos(d*x + c)^3)","A",0
52,1,16,0,0.139335," ","integrate(sin(d*x+c)/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{1}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"1/3/(a^2*d*cos(d*x + c)^3)","A",0
53,1,107,0,0.176578," ","integrate(csc(d*x+c)/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{3 \, \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a^{2}} + \frac{8 \, {\left(\frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + 2\right)}}{a^{2} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}^{3}}}{6 \, d}"," ",0,"1/6*(3*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2 + 8*(3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 2)/(a^2*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^3))/d","B",0
54,1,175,0,0.166728," ","integrate(csc(d*x+c)^3/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{\frac{3 \, {\left(\frac{10 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - 1\right)} {\left(\cos\left(d x + c\right) + 1\right)}}{a^{2} {\left(\cos\left(d x + c\right) - 1\right)}} - \frac{30 \, \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a^{2}} + \frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}}{a^{2} {\left(\cos\left(d x + c\right) + 1\right)}} - \frac{16 \, {\left(\frac{12 \, {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{9 \, {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + 7\right)}}{a^{2} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}^{3}}}{24 \, d}"," ",0,"-1/24*(3*(10*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)*(cos(d*x + c) + 1)/(a^2*(cos(d*x + c) - 1)) - 30*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2 + 3*(cos(d*x + c) - 1)/(a^2*(cos(d*x + c) + 1)) - 16*(12*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 7)/(a^2*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^3))/d","B",0
55,1,68,0,0.168746," ","integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{15 \, {\left(d x + c\right)}}{a^{2}} - \frac{3 \, \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)} a^{2}} + \frac{2 \, {\left(a^{4} \tan\left(d x + c\right)^{3} - 6 \, a^{4} \tan\left(d x + c\right)\right)}}{a^{6}}}{6 \, d}"," ",0,"1/6*(15*(d*x + c)/a^2 - 3*tan(d*x + c)/((tan(d*x + c)^2 + 1)*a^2) + 2*(a^4*tan(d*x + c)^3 - 6*a^4*tan(d*x + c))/a^6)/d","A",0
56,1,44,0,0.149624," ","integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(d x + c\right)}}{a^{2}} + \frac{a^{4} \tan\left(d x + c\right)^{3} - 3 \, a^{4} \tan\left(d x + c\right)}{a^{6}}}{3 \, d}"," ",0,"1/3*(3*(d*x + c)/a^2 + (a^4*tan(d*x + c)^3 - 3*a^4*tan(d*x + c))/a^6)/d","A",0
57,1,16,0,0.145923," ","integrate(sin(d*x+c)^2/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\tan\left(d x + c\right)^{3}}{3 \, a^{2} d}"," ",0,"1/3*tan(d*x + c)^3/(a^2*d)","A",0
58,1,25,0,0.126578," ","integrate(1/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\tan\left(d x + c\right)^{3} + 3 \, \tan\left(d x + c\right)}{3 \, a^{2} d}"," ",0,"1/3*(tan(d*x + c)^3 + 3*tan(d*x + c))/(a^2*d)","A",0
59,1,48,0,0.146519," ","integrate(csc(d*x+c)^2/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{\frac{3}{a^{2} \tan\left(d x + c\right)} - \frac{a^{4} \tan\left(d x + c\right)^{3} + 6 \, a^{4} \tan\left(d x + c\right)}{a^{6}}}{3 \, d}"," ",0,"-1/3*(3/(a^2*tan(d*x + c)) - (a^4*tan(d*x + c)^3 + 6*a^4*tan(d*x + c))/a^6)/d","A",0
60,1,34,0,0.152358," ","integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{8 \, {\left(3 \, \tan\left(2 \, d x + 2 \, c\right)^{2} + 1\right)}}{3 \, a^{2} d \tan\left(2 \, d x + 2 \, c\right)^{3}}"," ",0,"-8/3*(3*tan(2*d*x + 2*c)^2 + 1)/(a^2*d*tan(2*d*x + 2*c)^3)","A",0
61,1,22,0,0.131423," ","integrate(1/(a-a*sin(x)^2)^3,x, algorithm=""giac"")","\frac{3 \, \tan\left(x\right)^{5} + 10 \, \tan\left(x\right)^{3} + 15 \, \tan\left(x\right)}{15 \, a^{3}}"," ",0,"1/15*(3*tan(x)^5 + 10*tan(x)^3 + 15*tan(x))/a^3","A",0
62,1,28,0,0.121976," ","integrate(1/(a-a*sin(x)^2)^4,x, algorithm=""giac"")","\frac{5 \, \tan\left(x\right)^{7} + 21 \, \tan\left(x\right)^{5} + 35 \, \tan\left(x\right)^{3} + 35 \, \tan\left(x\right)}{35 \, a^{4}}"," ",0,"1/35*(5*tan(x)^7 + 21*tan(x)^5 + 35*tan(x)^3 + 35*tan(x))/a^4","A",0
63,1,34,0,0.124667," ","integrate(1/(a-a*sin(x)^2)^5,x, algorithm=""giac"")","\frac{35 \, \tan\left(x\right)^{9} + 180 \, \tan\left(x\right)^{7} + 378 \, \tan\left(x\right)^{5} + 420 \, \tan\left(x\right)^{3} + 315 \, \tan\left(x\right)}{315 \, a^{5}}"," ",0,"1/315*(35*tan(x)^9 + 180*tan(x)^7 + 378*tan(x)^5 + 420*tan(x)^3 + 315*tan(x))/a^5","A",0
64,1,67,0,0.130840," ","integrate(sin(d*x+c)^3*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{b \cos\left(d x + c\right)^{5}}{5 \, d} + \frac{a \cos\left(d x + c\right)^{3}}{3 \, d} + \frac{2 \, b \cos\left(d x + c\right)^{3}}{3 \, d} - \frac{a \cos\left(d x + c\right)}{d} - \frac{b \cos\left(d x + c\right)}{d}"," ",0,"-1/5*b*cos(d*x + c)^5/d + 1/3*a*cos(d*x + c)^3/d + 2/3*b*cos(d*x + c)^3/d - a*cos(d*x + c)/d - b*cos(d*x + c)/d","A",0
65,1,40,0,0.127507," ","integrate(sin(d*x+c)*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{1}{3} \, {\left(\frac{\cos\left(d x + c\right)^{3}}{d} - \frac{3 \, \cos\left(d x + c\right)}{d}\right)} b - \frac{a \cos\left(d x + c\right)}{d}"," ",0,"1/3*(cos(d*x + c)^3/d - 3*cos(d*x + c)/d)*b - a*cos(d*x + c)/d","A",0
66,1,58,0,0.163838," ","integrate(csc(d*x+c)*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{a \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right) + \frac{4 \, b}{\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1}}{2 \, d}"," ",0,"1/2*(a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) + 4*b/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/d","B",0
67,1,121,0,0.163693," ","integrate(csc(d*x+c)^3*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{2 \, {\left(a + 2 \, b\right)} \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right) + \frac{{\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1}\right)} {\left(\cos\left(d x + c\right) + 1\right)}}{\cos\left(d x + c\right) - 1} - \frac{a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1}}{8 \, d}"," ",0,"1/8*(2*(a + 2*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) + (a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/(cos(d*x + c) - 1) - a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/d","B",0
68,1,68,0,0.133968," ","integrate(sin(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{1}{16} \, {\left(6 \, a + 5 \, b\right)} x - \frac{b \sin\left(6 \, d x + 6 \, c\right)}{192 \, d} + \frac{{\left(2 \, a + 3 \, b\right)} \sin\left(4 \, d x + 4 \, c\right)}{64 \, d} - \frac{{\left(16 \, a + 15 \, b\right)} \sin\left(2 \, d x + 2 \, c\right)}{64 \, d}"," ",0,"1/16*(6*a + 5*b)*x - 1/192*b*sin(6*d*x + 6*c)/d + 1/64*(2*a + 3*b)*sin(4*d*x + 4*c)/d - 1/64*(16*a + 15*b)*sin(2*d*x + 2*c)/d","A",0
69,1,43,0,0.132392," ","integrate(sin(d*x+c)^2*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{1}{8} \, {\left(4 \, a + 3 \, b\right)} x + \frac{b \sin\left(4 \, d x + 4 \, c\right)}{32 \, d} - \frac{{\left(a + b\right)} \sin\left(2 \, d x + 2 \, c\right)}{4 \, d}"," ",0,"1/8*(4*a + 3*b)*x + 1/32*b*sin(4*d*x + 4*c)/d - 1/4*(a + b)*sin(2*d*x + 2*c)/d","A",0
70,1,25,0,0.121570," ","integrate(a+b*sin(d*x+c)^2,x, algorithm=""giac"")","\frac{1}{4} \, b {\left(2 \, x - \frac{\sin\left(2 \, d x + 2 \, c\right)}{d}\right)} + a x"," ",0,"1/4*b*(2*x - sin(2*d*x + 2*c)/d) + a*x","A",0
71,1,39,0,0.132868," ","integrate(csc(d*x+c)^2*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{2 \, {\left(d x + c\right)} b + a \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) - \frac{a}{\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)}}{2 \, d}"," ",0,"1/2*(2*(d*x + c)*b + a*tan(1/2*d*x + 1/2*c) - a/tan(1/2*d*x + 1/2*c))/d","B",0
72,1,37,0,0.146017," ","integrate(csc(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{3 \, a \tan\left(d x + c\right)^{2} + 3 \, b \tan\left(d x + c\right)^{2} + a}{3 \, d \tan\left(d x + c\right)^{3}}"," ",0,"-1/3*(3*a*tan(d*x + c)^2 + 3*b*tan(d*x + c)^2 + a)/(d*tan(d*x + c)^3)","A",0
73,1,61,0,0.154458," ","integrate(csc(d*x+c)^6*(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{15 \, a \tan\left(d x + c\right)^{4} + 15 \, b \tan\left(d x + c\right)^{4} + 10 \, a \tan\left(d x + c\right)^{2} + 5 \, b \tan\left(d x + c\right)^{2} + 3 \, a}{15 \, d \tan\left(d x + c\right)^{5}}"," ",0,"-1/15*(15*a*tan(d*x + c)^4 + 15*b*tan(d*x + c)^4 + 10*a*tan(d*x + c)^2 + 5*b*tan(d*x + c)^2 + 3*a)/(d*tan(d*x + c)^5)","A",0
74,1,17,0,0.133098," ","integrate(a+b*sin(x)^2,x, algorithm=""giac"")","\frac{1}{4} \, b {\left(2 \, x - \sin\left(2 \, x\right)\right)} + a x"," ",0,"1/4*b*(2*x - sin(2*x)) + a*x","A",0
75,1,42,0,0.130646," ","integrate((a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{1}{32} \, b^{2} \sin\left(4 \, x\right) + \frac{1}{8} \, {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} x - \frac{1}{4} \, {\left(2 \, a b + b^{2}\right)} \sin\left(2 \, x\right)"," ",0,"1/32*b^2*sin(4*x) + 1/8*(8*a^2 + 8*a*b + 3*b^2)*x - 1/4*(2*a*b + b^2)*sin(2*x)","A",0
76,1,76,0,0.146569," ","integrate((a+b*sin(x)^2)^3,x, algorithm=""giac"")","-\frac{1}{192} \, b^{3} \sin\left(6 \, x\right) + \frac{1}{16} \, {\left(16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right)} x + \frac{3}{64} \, {\left(2 \, a b^{2} + b^{3}\right)} \sin\left(4 \, x\right) - \frac{3}{64} \, {\left(16 \, a^{2} b + 16 \, a b^{2} + 5 \, b^{3}\right)} \sin\left(2 \, x\right)"," ",0,"-1/192*b^3*sin(6*x) + 1/16*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*x + 3/64*(2*a*b^2 + b^3)*sin(4*x) - 3/64*(16*a^2*b + 16*a*b^2 + 5*b^3)*sin(2*x)","A",0
77,1,118,0,0.136425," ","integrate((a+b*sin(x)^2)^4,x, algorithm=""giac"")","\frac{1}{1024} \, b^{4} \sin\left(8 \, x\right) + \frac{1}{128} \, {\left(128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right)} x - \frac{1}{96} \, {\left(2 \, a b^{3} + b^{4}\right)} \sin\left(6 \, x\right) + \frac{1}{128} \, {\left(24 \, a^{2} b^{2} + 24 \, a b^{3} + 7 \, b^{4}\right)} \sin\left(4 \, x\right) - \frac{1}{32} \, {\left(32 \, a^{3} b + 48 \, a^{2} b^{2} + 30 \, a b^{3} + 7 \, b^{4}\right)} \sin\left(2 \, x\right)"," ",0,"1/1024*b^4*sin(8*x) + 1/128*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*x - 1/96*(2*a*b^3 + b^4)*sin(6*x) + 1/128*(24*a^2*b^2 + 24*a*b^3 + 7*b^4)*sin(4*x) - 1/32*(32*a^3*b + 48*a^2*b^2 + 30*a*b^3 + 7*b^4)*sin(2*x)","A",0
78,1,332,0,0.177169," ","integrate(sin(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{15 \, a^{3} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} b^{3}} - \frac{2 \, {\left(15 \, a^{2} - 10 \, a b + 8 \, b^{2} - \frac{60 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{50 \, a b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{40 \, b^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{90 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{70 \, a b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{80 \, b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{60 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{30 \, a b {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{15 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{4}}{{\left(\cos\left(d x + c\right) + 1\right)}^{4}}\right)}}{b^{3} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1\right)}^{5}}}{15 \, d}"," ",0,"-1/15*(15*a^3*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/(sqrt(-a*b - b^2)*b^3) - 2*(15*a^2 - 10*a*b + 8*b^2 - 60*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 50*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 40*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 90*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 70*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 80*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 60*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 30*a*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 15*a^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)/(b^3*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^5))/d","B",0
79,1,173,0,0.154244," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{3 \, a^{2} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} b^{2}} - \frac{2 \, {\left(3 \, a - 2 \, b - \frac{6 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{6 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}}{b^{2} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1\right)}^{3}}}{3 \, d}"," ",0,"1/3*(3*a^2*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/(sqrt(-a*b - b^2)*b^2) - 2*(3*a - 2*b - 6*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(b^2*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^3))/d","B",0
80,1,57,0,0.169198," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{a \arctan\left(\frac{b \cos\left(d x + c\right)}{\sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} b d} - \frac{\cos\left(d x + c\right)}{b d}"," ",0,"-a*arctan(b*cos(d*x + c)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*b*d) - cos(d*x + c)/(b*d)","A",0
81,1,37,0,0.171217," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{b \cos\left(d x + c\right)}{\sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} d}"," ",0,"arctan(b*cos(d*x + c)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*d)","A",0
82,1,100,0,0.163400," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{2 \, b \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} a} - \frac{\log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a}}{2 \, d}"," ",0,"-1/2*(2*b*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/(sqrt(-a*b - b^2)*a) - log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a)/d","B",0
83,1,196,0,0.207078," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{8 \, b^{2} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} a^{2}} + \frac{2 \, {\left(a - 2 \, b\right)} \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a^{2}} + \frac{{\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1}\right)} {\left(\cos\left(d x + c\right) + 1\right)}}{a^{2} {\left(\cos\left(d x + c\right) - 1\right)}} - \frac{\cos\left(d x + c\right) - 1}{a {\left(\cos\left(d x + c\right) + 1\right)}}}{8 \, d}"," ",0,"1/8*(8*b^2*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/(sqrt(-a*b - b^2)*a^2) + 2*(a - 2*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2 + (a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/(a^2*(cos(d*x + c) - 1)) - (cos(d*x + c) - 1)/(a*(cos(d*x + c) + 1)))/d","B",0
84,1,334,0,0.245855," ","integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{64 \, b^{3} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{\sqrt{-a b - b^{2}} a^{3}} + \frac{\frac{8 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{8 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}}{a^{2}} - \frac{4 \, {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a^{3}} + \frac{{\left(a^{2} - \frac{8 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{8 \, a b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{18 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{24 \, a b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{48 \, b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)} {\left(\cos\left(d x + c\right) + 1\right)}^{2}}{a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}}{64 \, d}"," ",0,"-1/64*(64*b^3*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/(sqrt(-a*b - b^2)*a^3) + (8*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 8*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/a^2 - 4*(3*a^2 - 4*a*b + 8*b^2)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 + (a^2 - 8*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 8*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 18*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 24*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 48*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)^2/(a^3*(cos(d*x + c) - 1)^2))/d","B",0
85,1,233,0,0.179285," ","integrate(sin(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{48 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a^{4}}{\sqrt{a^{2} + a b} b^{4}} - \frac{3 \, {\left(16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right)} {\left(d x + c\right)}}{b^{4}} - \frac{24 \, a^{2} \tan\left(d x + c\right)^{5} - 30 \, a b \tan\left(d x + c\right)^{5} + 33 \, b^{2} \tan\left(d x + c\right)^{5} + 48 \, a^{2} \tan\left(d x + c\right)^{3} - 48 \, a b \tan\left(d x + c\right)^{3} + 40 \, b^{2} \tan\left(d x + c\right)^{3} + 24 \, a^{2} \tan\left(d x + c\right) - 18 \, a b \tan\left(d x + c\right) + 15 \, b^{2} \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)}^{3} b^{3}}}{48 \, d}"," ",0,"1/48*(48*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a^4/(sqrt(a^2 + a*b)*b^4) - 3*(16*a^3 - 8*a^2*b + 6*a*b^2 - 5*b^3)*(d*x + c)/b^4 - (24*a^2*tan(d*x + c)^5 - 30*a*b*tan(d*x + c)^5 + 33*b^2*tan(d*x + c)^5 + 48*a^2*tan(d*x + c)^3 - 48*a*b*tan(d*x + c)^3 + 40*b^2*tan(d*x + c)^3 + 24*a^2*tan(d*x + c) - 18*a*b*tan(d*x + c) + 15*b^2*tan(d*x + c))/((tan(d*x + c)^2 + 1)^3*b^3))/d","A",0
86,1,157,0,0.156945," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{8 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a^{3}}{\sqrt{a^{2} + a b} b^{3}} - \frac{{\left(8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right)} {\left(d x + c\right)}}{b^{3}} - \frac{4 \, a \tan\left(d x + c\right)^{3} - 5 \, b \tan\left(d x + c\right)^{3} + 4 \, a \tan\left(d x + c\right) - 3 \, b \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)}^{2} b^{2}}}{8 \, d}"," ",0,"-1/8*(8*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a^3/(sqrt(a^2 + a*b)*b^3) - (8*a^2 - 4*a*b + 3*b^2)*(d*x + c)/b^3 - (4*a*tan(d*x + c)^3 - 5*b*tan(d*x + c)^3 + 4*a*tan(d*x + c) - 3*b*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*b^2))/d","A",0
87,1,114,0,0.151563," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{2 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a^{2}}{\sqrt{a^{2} + a b} b^{2}} - \frac{{\left(d x + c\right)} {\left(2 \, a - b\right)}}{b^{2}} - \frac{\tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)} b}}{2 \, d}"," ",0,"1/2*(2*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a^2/(sqrt(a^2 + a*b)*b^2) - (d*x + c)*(2*a - b)/b^2 - tan(d*x + c)/((tan(d*x + c)^2 + 1)*b))/d","A",0
88,1,81,0,0.148784," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a}{\sqrt{a^{2} + a b} b} - \frac{d x + c}{b}}{d}"," ",0,"-((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a/(sqrt(a^2 + a*b)*b) - (d*x + c)/b)/d","B",0
89,1,64,0,0.133615," ","integrate(1/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)}{\sqrt{a^{2} + a b} d}"," ",0,"(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*d)","B",0
90,1,83,0,0.189583," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} b}{\sqrt{a^{2} + a b} a} + \frac{1}{a \tan\left(d x + c\right)}}{d}"," ",0,"-((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*b/(sqrt(a^2 + a*b)*a) + 1/(a*tan(d*x + c)))/d","A",0
91,1,111,0,0.189037," ","integrate(csc(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{3 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} b^{2}}{\sqrt{a^{2} + a b} a^{2}} - \frac{3 \, a \tan\left(d x + c\right)^{2} - 3 \, b \tan\left(d x + c\right)^{2} + a}{a^{2} \tan\left(d x + c\right)^{3}}}{3 \, d}"," ",0,"1/3*(3*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*b^2/(sqrt(a^2 + a*b)*a^2) - (3*a*tan(d*x + c)^2 - 3*b*tan(d*x + c)^2 + a)/(a^2*tan(d*x + c)^3))/d","A",0
92,1,155,0,0.168426," ","integrate(csc(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{15 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} b^{3}}{\sqrt{a^{2} + a b} a^{3}} + \frac{15 \, a^{2} \tan\left(d x + c\right)^{4} - 15 \, a b \tan\left(d x + c\right)^{4} + 15 \, b^{2} \tan\left(d x + c\right)^{4} + 10 \, a^{2} \tan\left(d x + c\right)^{2} - 5 \, a b \tan\left(d x + c\right)^{2} + 3 \, a^{2}}{a^{3} \tan\left(d x + c\right)^{5}}}{15 \, d}"," ",0,"-1/15*(15*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*b^3/(sqrt(a^2 + a*b)*a^3) + (15*a^2*tan(d*x + c)^4 - 15*a*b*tan(d*x + c)^4 + 15*b^2*tan(d*x + c)^4 + 10*a^2*tan(d*x + c)^2 - 5*a*b*tan(d*x + c)^2 + 3*a^2)/(a^3*tan(d*x + c)^5))/d","A",0
93,1,215,0,0.184889," ","integrate(csc(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{105 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} b^{4}}{\sqrt{a^{2} + a b} a^{4}} - \frac{105 \, a^{3} \tan\left(d x + c\right)^{6} - 105 \, a^{2} b \tan\left(d x + c\right)^{6} + 105 \, a b^{2} \tan\left(d x + c\right)^{6} - 105 \, b^{3} \tan\left(d x + c\right)^{6} + 105 \, a^{3} \tan\left(d x + c\right)^{4} - 70 \, a^{2} b \tan\left(d x + c\right)^{4} + 35 \, a b^{2} \tan\left(d x + c\right)^{4} + 63 \, a^{3} \tan\left(d x + c\right)^{2} - 21 \, a^{2} b \tan\left(d x + c\right)^{2} + 15 \, a^{3}}{a^{4} \tan\left(d x + c\right)^{7}}}{105 \, d}"," ",0,"1/105*(105*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*b^4/(sqrt(a^2 + a*b)*a^4) - (105*a^3*tan(d*x + c)^6 - 105*a^2*b*tan(d*x + c)^6 + 105*a*b^2*tan(d*x + c)^6 - 105*b^3*tan(d*x + c)^6 + 105*a^3*tan(d*x + c)^4 - 70*a^2*b*tan(d*x + c)^4 + 35*a*b^2*tan(d*x + c)^4 + 63*a^3*tan(d*x + c)^2 - 21*a^2*b*tan(d*x + c)^2 + 15*a^3)/(a^4*tan(d*x + c)^7))/d","A",0
94,1,322,0,0.205577," ","integrate(sin(d*x+c)^7/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(5 \, a^{3} + 6 \, a^{2} b\right)} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{{\left(a b^{3} + b^{4}\right)} \sqrt{-a b - b^{2}}} + \frac{6 \, {\left(a^{3} - \frac{a^{3} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{2 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1}\right)}}{{\left(a b^{3} + b^{4}\right)} {\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}} - \frac{8 \, {\left(3 \, a - b - \frac{6 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}}{b^{3} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1\right)}^{3}}}{6 \, d}"," ",0,"1/6*(3*(5*a^3 + 6*a^2*b)*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/((a*b^3 + b^4)*sqrt(-a*b - b^2)) + 6*(a^3 - a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*a^2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a*b^3 + b^4)*(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)) - 8*(3*a - b - 6*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(b^3*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^3))/d","B",0
95,1,342,0,0.210226," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{\frac{{\left(3 \, a^{2} + 4 \, a b\right)} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{{\left(a b^{2} + b^{3}\right)} \sqrt{-a b - b^{2}}} + \frac{2 \, {\left(3 \, a^{2} + 2 \, a b - \frac{6 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{14 \, a b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{8 \, b^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{4 \, a b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}}{{\left(a b^{2} + b^{3}\right)} {\left(a - \frac{3 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}}\right)}}}{2 \, d}"," ",0,"-1/2*((3*a^2 + 4*a*b)*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/((a*b^2 + b^3)*sqrt(-a*b - b^2)) + 2*(3*a^2 + 2*a*b - 6*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 14*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 8*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 4*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((a*b^2 + b^3)*(a - 3*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 4*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)))/d","B",0
96,1,93,0,0.175630," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{{\left(a + 2 \, b\right)} \arctan\left(\frac{b \cos\left(d x + c\right)}{\sqrt{-a b - b^{2}}}\right)}{2 \, {\left(a b + b^{2}\right)} \sqrt{-a b - b^{2}} d} - \frac{a \cos\left(d x + c\right)}{2 \, {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} {\left(a b + b^{2}\right)} d}"," ",0,"1/2*(a + 2*b)*arctan(b*cos(d*x + c)/sqrt(-a*b - b^2))/((a*b + b^2)*sqrt(-a*b - b^2)*d) - 1/2*a*cos(d*x + c)/((b*cos(d*x + c)^2 - a - b)*(a*b + b^2)*d)","A",0
97,1,79,0,0.156694," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\arctan\left(\frac{b \cos\left(d x + c\right)}{\sqrt{-a b - b^{2}}}\right)}{2 \, \sqrt{-a b - b^{2}} {\left(a + b\right)} d} + \frac{\cos\left(d x + c\right)}{2 \, {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} {\left(a + b\right)} d}"," ",0,"1/2*arctan(b*cos(d*x + c)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*(a + b)*d) + 1/2*cos(d*x + c)/((b*cos(d*x + c)^2 - a - b)*(a + b)*d)","A",0
98,1,246,0,0.193686," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{\frac{{\left(3 \, a b + 2 \, b^{2}\right)} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{{\left(a^{3} + a^{2} b\right)} \sqrt{-a b - b^{2}}} - \frac{2 \, {\left(a b - \frac{a b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{2 \, b^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1}\right)}}{{\left(a^{3} + a^{2} b\right)} {\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}} - \frac{\log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a^{2}}}{2 \, d}"," ",0,"-1/2*((3*a*b + 2*b^2)*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/((a^3 + a^2*b)*sqrt(-a*b - b^2)) - 2*(a*b - a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a^3 + a^2*b)*(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)) - log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^2)/d","B",0
99,1,512,0,0.201181," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{12 \, {\left(5 \, a b^{2} + 4 \, b^{3}\right)} \arctan\left(\frac{b \cos\left(d x + c\right) + a + b}{\sqrt{-a b - b^{2}} \cos\left(d x + c\right) + \sqrt{-a b - b^{2}}}\right)}{{\left(a^{4} + a^{3} b\right)} \sqrt{-a b - b^{2}}} + \frac{3 \, a^{3} + 3 \, a^{2} b - \frac{8 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{12 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{28 \, a b^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{7 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{16 \, a b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{16 \, b^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{2 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{6 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{8 \, a b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}}}{{\left(a^{4} + a^{3} b\right)} {\left(\frac{a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}}\right)}} + \frac{6 \, {\left(a - 4 \, b\right)} \log\left(\frac{{\left| -\cos\left(d x + c\right) + 1 \right|}}{{\left| \cos\left(d x + c\right) + 1 \right|}}\right)}{a^{3}} - \frac{3 \, {\left(\cos\left(d x + c\right) - 1\right)}}{a^{2} {\left(\cos\left(d x + c\right) + 1\right)}}}{24 \, d}"," ",0,"1/24*(12*(5*a*b^2 + 4*b^3)*arctan((b*cos(d*x + c) + a + b)/(sqrt(-a*b - b^2)*cos(d*x + c) + sqrt(-a*b - b^2)))/((a^4 + a^3*b)*sqrt(-a*b - b^2)) + (3*a^3 + 3*a^2*b - 8*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*a^2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 28*a*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 7*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - a^2*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 16*a*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 16*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 2*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 6*a^2*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 8*a*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)/((a^4 + a^3*b)*(a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 4*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3)) + 6*(a - 4*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/a^3 - 3*(cos(d*x + c) - 1)/(a^2*(cos(d*x + c) + 1)))/d","B",0
100,1,223,0,0.183172," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{{\left(4 \, a^{3} + 5 \, a^{2} b\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a b^{3} + b^{4}\right)} \sqrt{a^{2} + a b}} - \frac{2 \, a^{2} \tan\left(d x + c\right)^{3} + 2 \, a b \tan\left(d x + c\right)^{3} + b^{2} \tan\left(d x + c\right)^{3} + 2 \, a^{2} \tan\left(d x + c\right) + a b \tan\left(d x + c\right)}{{\left(a \tan\left(d x + c\right)^{4} + b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)} {\left(a b^{2} + b^{3}\right)}} - \frac{{\left(d x + c\right)} {\left(4 \, a - b\right)}}{b^{3}}}{2 \, d}"," ",0,"1/2*((4*a^3 + 5*a^2*b)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a*b^3 + b^4)*sqrt(a^2 + a*b)) - (2*a^2*tan(d*x + c)^3 + 2*a*b*tan(d*x + c)^3 + b^2*tan(d*x + c)^3 + 2*a^2*tan(d*x + c) + a*b*tan(d*x + c))/((a*tan(d*x + c)^4 + b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)*(a*b^2 + b^3)) - (d*x + c)*(4*a - b)/b^3)/d","A",0
101,1,140,0,0.192490," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(2 \, a^{2} + 3 \, a b\right)}}{{\left(a b^{2} + b^{3}\right)} \sqrt{a^{2} + a b}} - \frac{a \tan\left(d x + c\right)}{{\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)} {\left(a b + b^{2}\right)}} - \frac{2 \, {\left(d x + c\right)}}{b^{2}}}{2 \, d}"," ",0,"-1/2*((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(2*a^2 + 3*a*b)/((a*b^2 + b^3)*sqrt(a^2 + a*b)) - a*tan(d*x + c)/((a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)*(a*b + b^2)) - 2*(d*x + c)/b^2)/d","A",0
102,1,109,0,0.181847," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)}{\sqrt{a^{2} + a b} {\left(a + b\right)}} - \frac{\tan\left(d x + c\right)}{{\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)} {\left(a + b\right)}}}{2 \, d}"," ",0,"1/2*((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*(a + b)) - tan(d*x + c)/((a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)*(a + b)))/d","A",0
103,1,113,0,0.154963," ","integrate(1/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(2 \, a + b\right)}}{{\left(a^{2} + a b\right)}^{\frac{3}{2}}} + \frac{b \tan\left(d x + c\right)}{{\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)} {\left(a^{2} + a b\right)}}}{2 \, d}"," ",0,"1/2*((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(2*a + b)/(a^2 + a*b)^(3/2) + b*tan(d*x + c)/((a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)*(a^2 + a*b)))/d","A",0
104,1,179,0,0.178383," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","-\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(4 \, a b + 3 \, b^{2}\right)}}{{\left(a^{3} + a^{2} b\right)} \sqrt{a^{2} + a b}} + \frac{2 \, a^{2} \tan\left(d x + c\right)^{2} + 4 \, a b \tan\left(d x + c\right)^{2} + 3 \, b^{2} \tan\left(d x + c\right)^{2} + 2 \, a^{2} + 2 \, a b}{{\left(a \tan\left(d x + c\right)^{3} + b \tan\left(d x + c\right)^{3} + a \tan\left(d x + c\right)\right)} {\left(a^{3} + a^{2} b\right)}}}{2 \, d}"," ",0,"-1/2*((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(4*a*b + 3*b^2)/((a^3 + a^2*b)*sqrt(a^2 + a*b)) + (2*a^2*tan(d*x + c)^2 + 4*a*b*tan(d*x + c)^2 + 3*b^2*tan(d*x + c)^2 + 2*a^2 + 2*a*b)/((a*tan(d*x + c)^3 + b*tan(d*x + c)^3 + a*tan(d*x + c))*(a^3 + a^2*b)))/d","A",0
105,1,174,0,0.193640," ","integrate(csc(d*x+c)^4/(a+b*sin(d*x+c)^2)^2,x, algorithm=""giac"")","\frac{\frac{3 \, b^{3} \tan\left(d x + c\right)}{{\left(a^{4} + a^{3} b\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}} + \frac{3 \, {\left(6 \, a b^{2} + 5 \, b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a^{4} + a^{3} b\right)} \sqrt{a^{2} + a b}} - \frac{2 \, {\left(3 \, a \tan\left(d x + c\right)^{2} - 6 \, b \tan\left(d x + c\right)^{2} + a\right)}}{a^{3} \tan\left(d x + c\right)^{3}}}{6 \, d}"," ",0,"1/6*(3*b^3*tan(d*x + c)/((a^4 + a^3*b)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)) + 3*(6*a*b^2 + 5*b^3)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^4 + a^3*b)*sqrt(a^2 + a*b)) - 2*(3*a*tan(d*x + c)^2 - 6*b*tan(d*x + c)^2 + a)/(a^3*tan(d*x + c)^3))/d","A",0
106,1,224,0,0.242817," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^3,x, algorithm=""giac"")","-\frac{\frac{{\left(8 \, a^{3} + 20 \, a^{2} b + 15 \, a b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} \sqrt{a^{2} + a b}} - \frac{4 \, a^{3} \tan\left(d x + c\right)^{3} + 13 \, a^{2} b \tan\left(d x + c\right)^{3} + 9 \, a b^{2} \tan\left(d x + c\right)^{3} + 4 \, a^{3} \tan\left(d x + c\right) + 7 \, a^{2} b \tan\left(d x + c\right)}{{\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{2}} - \frac{8 \, {\left(d x + c\right)}}{b^{3}}}{8 \, d}"," ",0,"-1/8*((8*a^3 + 20*a^2*b + 15*a*b^2)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^2*b^3 + 2*a*b^4 + b^5)*sqrt(a^2 + a*b)) - (4*a^3*tan(d*x + c)^3 + 13*a^2*b*tan(d*x + c)^3 + 9*a*b^2*tan(d*x + c)^3 + 4*a^3*tan(d*x + c) + 7*a^2*b*tan(d*x + c))/((a^2*b^2 + 2*a*b^3 + b^4)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2) - 8*(d*x + c)/b^3)/d","A",0
107,1,152,0,0.204200," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^2)^3,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a^{2} + a b}} - \frac{5 \, a \tan\left(d x + c\right)^{3} + 5 \, b \tan\left(d x + c\right)^{3} + 3 \, a \tan\left(d x + c\right)}{{\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{2} {\left(a^{2} + 2 \, a b + b^{2}\right)}}}{8 \, d}"," ",0,"1/8*(3*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^2 + 2*a*b + b^2)*sqrt(a^2 + a*b)) - (5*a*tan(d*x + c)^3 + 5*b*tan(d*x + c)^3 + 3*a*tan(d*x + c))/((a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2*(a^2 + 2*a*b + b^2)))/d","A",0
108,1,191,0,0.197536," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^2)^3,x, algorithm=""giac"")","\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(4 \, a + b\right)}}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{a^{2} + a b}} - \frac{4 \, a^{2} \tan\left(d x + c\right)^{3} + 3 \, a b \tan\left(d x + c\right)^{3} - b^{2} \tan\left(d x + c\right)^{3} + 4 \, a^{2} \tan\left(d x + c\right) + a b \tan\left(d x + c\right)}{{\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{2}}}{8 \, d}"," ",0,"1/8*((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(4*a + b)/((a^3 + 2*a^2*b + a*b^2)*sqrt(a^2 + a*b)) - (4*a^2*tan(d*x + c)^3 + 3*a*b*tan(d*x + c)^3 - b^2*tan(d*x + c)^3 + 4*a^2*tan(d*x + c) + a*b*tan(d*x + c))/((a^3 + 2*a^2*b + a*b^2)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2))/d","A",0
109,1,211,0,0.137542," ","integrate(1/(a+b*sin(d*x+c)^2)^3,x, algorithm=""giac"")","\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)}}{{\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \sqrt{a^{2} + a b}} + \frac{8 \, a^{2} b \tan\left(d x + c\right)^{3} + 11 \, a b^{2} \tan\left(d x + c\right)^{3} + 3 \, b^{3} \tan\left(d x + c\right)^{3} + 8 \, a^{2} b \tan\left(d x + c\right) + 5 \, a b^{2} \tan\left(d x + c\right)}{{\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{2}}}{8 \, d}"," ",0,"1/8*((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(8*a^2 + 8*a*b + 3*b^2)/((a^4 + 2*a^3*b + a^2*b^2)*sqrt(a^2 + a*b)) + (8*a^2*b*tan(d*x + c)^3 + 11*a*b^2*tan(d*x + c)^3 + 3*b^3*tan(d*x + c)^3 + 8*a^2*b*tan(d*x + c) + 5*a*b^2*tan(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2))/d","A",0
110,1,232,0,0.209219," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^2)^3,x, algorithm=""giac"")","-\frac{\frac{3 \, {\left(8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \sqrt{a^{2} + a b}} + \frac{12 \, a^{2} b^{2} \tan\left(d x + c\right)^{3} + 19 \, a b^{3} \tan\left(d x + c\right)^{3} + 7 \, b^{4} \tan\left(d x + c\right)^{3} + 12 \, a^{2} b^{2} \tan\left(d x + c\right) + 9 \, a b^{3} \tan\left(d x + c\right)}{{\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{2}} + \frac{8}{a^{3} \tan\left(d x + c\right)}}{8 \, d}"," ",0,"-1/8*(3*(8*a^2*b + 12*a*b^2 + 5*b^3)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^5 + 2*a^4*b + a^3*b^2)*sqrt(a^2 + a*b)) + (12*a^2*b^2*tan(d*x + c)^3 + 19*a*b^3*tan(d*x + c)^3 + 7*b^4*tan(d*x + c)^3 + 12*a^2*b^2*tan(d*x + c) + 9*a*b^3*tan(d*x + c))/((a^5 + 2*a^4*b + a^3*b^2)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^2) + 8/(a^3*tan(d*x + c)))/d","A",0
111,1,344,0,0.171200," ","integrate(1/(a+b*sin(d*x+c)^2)^4,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \sqrt{a^{2} + a b}} + \frac{72 \, a^{4} b \tan\left(d x + c\right)^{5} + 198 \, a^{3} b^{2} \tan\left(d x + c\right)^{5} + 195 \, a^{2} b^{3} \tan\left(d x + c\right)^{5} + 84 \, a b^{4} \tan\left(d x + c\right)^{5} + 15 \, b^{5} \tan\left(d x + c\right)^{5} + 144 \, a^{4} b \tan\left(d x + c\right)^{3} + 288 \, a^{3} b^{2} \tan\left(d x + c\right)^{3} + 184 \, a^{2} b^{3} \tan\left(d x + c\right)^{3} + 40 \, a b^{4} \tan\left(d x + c\right)^{3} + 72 \, a^{4} b \tan\left(d x + c\right) + 90 \, a^{3} b^{2} \tan\left(d x + c\right) + 33 \, a^{2} b^{3} \tan\left(d x + c\right)}{{\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{3}}}{48 \, d}"," ",0,"1/48*(3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sqrt(a^2 + a*b)) + (72*a^4*b*tan(d*x + c)^5 + 198*a^3*b^2*tan(d*x + c)^5 + 195*a^2*b^3*tan(d*x + c)^5 + 84*a*b^4*tan(d*x + c)^5 + 15*b^5*tan(d*x + c)^5 + 144*a^4*b*tan(d*x + c)^3 + 288*a^3*b^2*tan(d*x + c)^3 + 184*a^2*b^3*tan(d*x + c)^3 + 40*a*b^4*tan(d*x + c)^3 + 72*a^4*b*tan(d*x + c) + 90*a^3*b^2*tan(d*x + c) + 33*a^2*b^3*tan(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^3))/d","A",0
112,1,524,0,0.168200," ","integrate(1/(a+b*sin(d*x+c)^2)^5,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{{\left(a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right)} \sqrt{a^{2} + a b}} + \frac{768 \, a^{6} b \tan\left(d x + c\right)^{7} + 3168 \, a^{5} b^{2} \tan\left(d x + c\right)^{7} + 5376 \, a^{4} b^{3} \tan\left(d x + c\right)^{7} + 4905 \, a^{3} b^{4} \tan\left(d x + c\right)^{7} + 2619 \, a^{2} b^{5} \tan\left(d x + c\right)^{7} + 795 \, a b^{6} \tan\left(d x + c\right)^{7} + 105 \, b^{7} \tan\left(d x + c\right)^{7} + 2304 \, a^{6} b \tan\left(d x + c\right)^{5} + 7776 \, a^{5} b^{2} \tan\left(d x + c\right)^{5} + 10400 \, a^{4} b^{3} \tan\left(d x + c\right)^{5} + 7073 \, a^{3} b^{4} \tan\left(d x + c\right)^{5} + 2530 \, a^{2} b^{5} \tan\left(d x + c\right)^{5} + 385 \, a b^{6} \tan\left(d x + c\right)^{5} + 2304 \, a^{6} b \tan\left(d x + c\right)^{3} + 6048 \, a^{5} b^{2} \tan\left(d x + c\right)^{3} + 6080 \, a^{4} b^{3} \tan\left(d x + c\right)^{3} + 2847 \, a^{3} b^{4} \tan\left(d x + c\right)^{3} + 511 \, a^{2} b^{5} \tan\left(d x + c\right)^{3} + 768 \, a^{6} b \tan\left(d x + c\right) + 1440 \, a^{5} b^{2} \tan\left(d x + c\right) + 1056 \, a^{4} b^{3} \tan\left(d x + c\right) + 279 \, a^{3} b^{4} \tan\left(d x + c\right)}{{\left(a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right)} {\left(a \tan\left(d x + c\right)^{2} + b \tan\left(d x + c\right)^{2} + a\right)}^{4}}}{384 \, d}"," ",0,"1/384*(3*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4)*sqrt(a^2 + a*b)) + (768*a^6*b*tan(d*x + c)^7 + 3168*a^5*b^2*tan(d*x + c)^7 + 5376*a^4*b^3*tan(d*x + c)^7 + 4905*a^3*b^4*tan(d*x + c)^7 + 2619*a^2*b^5*tan(d*x + c)^7 + 795*a*b^6*tan(d*x + c)^7 + 105*b^7*tan(d*x + c)^7 + 2304*a^6*b*tan(d*x + c)^5 + 7776*a^5*b^2*tan(d*x + c)^5 + 10400*a^4*b^3*tan(d*x + c)^5 + 7073*a^3*b^4*tan(d*x + c)^5 + 2530*a^2*b^5*tan(d*x + c)^5 + 385*a*b^6*tan(d*x + c)^5 + 2304*a^6*b*tan(d*x + c)^3 + 6048*a^5*b^2*tan(d*x + c)^3 + 6080*a^4*b^3*tan(d*x + c)^3 + 2847*a^3*b^4*tan(d*x + c)^3 + 511*a^2*b^5*tan(d*x + c)^3 + 768*a^6*b*tan(d*x + c) + 1440*a^5*b^2*tan(d*x + c) + 1056*a^4*b^3*tan(d*x + c) + 279*a^3*b^4*tan(d*x + c))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^4))/d","B",0
113,1,10,0,0.197047," ","integrate(sin(x)/(1+sin(x)^2)^(1/2),x, algorithm=""giac"")","-\arcsin\left(\frac{1}{2} \, \sqrt{2} \cos\left(x\right)\right)"," ",0,"-arcsin(1/2*sqrt(2)*cos(x))","A",0
114,1,25,0,0.149288," ","integrate(sin(x)*(1+sin(x)^2)^(1/2),x, algorithm=""giac"")","-\frac{1}{2} \, \sqrt{-\cos\left(x\right)^{2} + 2} \cos\left(x\right) - \arcsin\left(\frac{1}{2} \, \sqrt{2} \cos\left(x\right)\right)"," ",0,"-1/2*sqrt(-cos(x)^2 + 2)*cos(x) - arcsin(1/2*sqrt(2)*cos(x))","A",0
115,1,11,0,0.378601," ","integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm=""giac"")","-\frac{1}{3} \, \arcsin\left(\frac{1}{2} \, \cos\left(3 \, x + 7\right)\right)"," ",0,"-1/3*arcsin(1/2*cos(3*x + 7))","A",0
116,1,84,0,0.206228," ","integrate((a-a*sin(x)^2)^(5/2),x, algorithm=""giac"")","-\frac{2 \, {\left(15 \, a^{\frac{5}{2}} {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{4} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right) - 40 \, a^{\frac{5}{2}} {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{2} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right) + 48 \, a^{\frac{5}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)\right)}}{15 \, {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{5}}"," ",0,"-2/15*(15*a^(5/2)*(1/tan(1/2*x) + tan(1/2*x))^4*sgn(tan(1/2*x)^4 - 1) - 40*a^(5/2)*(1/tan(1/2*x) + tan(1/2*x))^2*sgn(tan(1/2*x)^4 - 1) + 48*a^(5/2)*sgn(tan(1/2*x)^4 - 1))/(1/tan(1/2*x) + tan(1/2*x))^5","B",0
117,1,57,0,0.181579," ","integrate((a-a*sin(x)^2)^(3/2),x, algorithm=""giac"")","-\frac{2 \, {\left(3 \, a^{\frac{3}{2}} {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{2} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right) - 4 \, a^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)\right)}}{3 \, {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{3}}"," ",0,"-2/3*(3*a^(3/2)*(1/tan(1/2*x) + tan(1/2*x))^2*sgn(tan(1/2*x)^4 - 1) - 4*a^(3/2)*sgn(tan(1/2*x)^4 - 1))/(1/tan(1/2*x) + tan(1/2*x))^3","B",0
118,1,27,0,0.145941," ","integrate((a-a*sin(x)^2)^(1/2),x, algorithm=""giac"")","-\frac{2 \, \sqrt{a} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)}{\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)}"," ",0,"-2*sqrt(a)*sgn(tan(1/2*x)^4 - 1)/(1/tan(1/2*x) + tan(1/2*x))","B",0
119,0,0,0,0.000000," ","integrate(1/(a-a*sin(x)^2)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{-a \sin\left(x\right)^{2} + a}}\,{d x}"," ",0,"integrate(1/sqrt(-a*sin(x)^2 + a), x)","F",0
120,1,44,0,0.233439," ","integrate(1/(a-a*sin(x)^2)^(3/2),x, algorithm=""giac"")","-\frac{\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)}{{\left({\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{2} - 4\right)} a^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)}"," ",0,"-(1/tan(1/2*x) + tan(1/2*x))/(((1/tan(1/2*x) + tan(1/2*x))^2 - 4)*a^(3/2)*sgn(tan(1/2*x)^4 - 1))","A",0
121,1,129,0,0.271958," ","integrate(1/(a-a*sin(x)^2)^(5/2),x, algorithm=""giac"")","-\frac{3 \, \log\left({\left| \frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right) + 2 \right|}\right)}{16 \, a^{\frac{5}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)} + \frac{3 \, \log\left({\left| \frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right) - 2 \right|}\right)}{16 \, a^{\frac{5}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)} - \frac{5 \, \sqrt{a} {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{3} - 12 \, \sqrt{a} {\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}}{4 \, {\left({\left(\frac{1}{\tan\left(\frac{1}{2} \, x\right)} + \tan\left(\frac{1}{2} \, x\right)\right)}^{2} - 4\right)}^{2} a^{3} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{4} - 1\right)}"," ",0,"-3/16*log(abs(1/tan(1/2*x) + tan(1/2*x) + 2))/(a^(5/2)*sgn(tan(1/2*x)^4 - 1)) + 3/16*log(abs(1/tan(1/2*x) + tan(1/2*x) - 2))/(a^(5/2)*sgn(tan(1/2*x)^4 - 1)) - 1/4*(5*sqrt(a)*(1/tan(1/2*x) + tan(1/2*x))^3 - 12*sqrt(a)*(1/tan(1/2*x) + tan(1/2*x)))/(((1/tan(1/2*x) + tan(1/2*x))^2 - 4)^2*a^3*sgn(tan(1/2*x)^4 - 1))","B",0
122,-2,0,0,0.000000," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-2*(-8*b^2*f^6/64/b^2/f^4*cos(f*x+exp(1))/f*cos(f*x+exp(1))/f+(20*b^2*f^4+4*b*f^4*a)/64/b^2/f^4)*cos(f*x+exp(1))/f*sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+2*(a^2-2*a*b-3*b^2)/16/b/sqrt(-b)/abs(f)*ln(abs(sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
123,-2,0,0,0.000000," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-1/2*cos(f*x+exp(1))/f*sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+2*(-a-b)/4/sqrt(-b)/abs(f)*ln(abs(sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
124,-2,0,0,0.000000," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[63]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[-14]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[-42]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[9]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[65]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regular value [0] was discarded and replaced randomly by 0=[-56]Evaluation time: 0.7index.cc index_m operator + Error: Bad Argument Value","F(-2)",0
125,-2,0,0,0.000000," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to divide, perhaps due to rounding error%%%{16,[4,4]%%%}+%%%{%%%{32,[1]%%%},[4,3]%%%}+%%%{%%%{16,[2]%%%},[4,2]%%%}+%%%{%%%{-32,[1]%%%},[2,4]%%%}+%%%{%%%{-64,[2]%%%},[2,3]%%%}+%%%{%%%{-32,[3]%%%},[2,2]%%%}+%%%{%%%{16,[2]%%%},[0,4]%%%}+%%%{%%%{32,[3]%%%},[0,3]%%%}+%%%{%%%{16,[4]%%%},[0,2]%%%} / %%%{%%%{1,[1]%%%},[4,0]%%%}+%%%{%%%{-2,[2]%%%},[2,0]%%%}+%%%{%%%{1,[3]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
126,1,962,0,0.547704," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\frac{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} {\left(\tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{7 \, a + 2 \, b}{a}\right)} + \frac{8 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a} - \frac{4 \, {\left(3 \, a^{\frac{5}{2}} + 2 \, a^{\frac{3}{2}} b - \sqrt{a} b^{2}\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{2}} + \frac{4 \, {\left(4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} + 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{2} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} - 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{2} - 3 \, a^{\frac{7}{2}}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - a\right)}^{2} a}}{64 \, f}"," ",0,"1/64*(sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)*(tan(1/2*f*x + 1/2*e)^2 + (7*a + 2*b)/a) + 8*(3*a^2 + 2*a*b - b^2)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a) - 4*(3*a^(5/2) + 2*a^(3/2)*b - sqrt(a)*b^2)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^2 + 4*(4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2 + 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2 - 3*a^(7/2))/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^2*a))/f","B",0
127,0,0,0,0.000000," ","integrate(sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sin\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sin(f*x + e)^4, x)","F",0
128,0,0,0,0.000000," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sin\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sin(f*x + e)^2, x)","F",0
129,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a), x)","F",0
130,0,0,0,0.000000," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \csc\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*csc(f*x + e)^2, x)","F",0
131,0,0,0,0.000000," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \csc\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*csc(f*x + e)^4, x)","F",0
132,-2,0,0,0.000000," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-2*((192*b^5*f^12/2304/b^4/f^8*cos(f*x+exp(1))/f*cos(f*x+exp(1))/f+(-624*b^5*f^10-336*b^4*f^10*a)/2304/b^4/f^8)*cos(f*x+exp(1))/f*cos(f*x+exp(1))/f+(792*b^5*f^8+864*b^4*f^8*a+72*b^3*f^8*a^2)/2304/b^4/f^8)*cos(f*x+exp(1))/f*sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+2*(a^3-3*a^2*b-9*a*b^2-5*b^3)/32/b/sqrt(-b)/abs(f)*ln(abs(sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
133,-2,0,0,0.000000," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-2*(-4*b^3*f^6/32/b^2/f^4*cos(f*x+exp(1))/f*cos(f*x+exp(1))/f+(10*b^3*f^4+10*b^2*f^4*a)/32/b^2/f^4)*cos(f*x+exp(1))/f*sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+2*(-3*a^2-6*a*b-3*b^2)/16/sqrt(-b)/abs(f)*ln(abs(sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
134,-2,0,0,0.000000," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Error: Bad Argument Type","F(-2)",0
135,-2,0,0,0.000000," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Error: Bad Argument Type","F(-2)",0
136,1,960,0,0.695208," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} {\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{7 \, a^{2} + 10 \, a b}{a}\right)} + \frac{24 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a}} - \frac{12 \, {\left(a^{\frac{5}{2}} + 2 \, a^{\frac{3}{2}} b + \sqrt{a} b^{2}\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a} + \frac{4 \, {\left(4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} + 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b + 10 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{2} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} + 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} - 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b - 6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{2} - 3 \, a^{\frac{7}{2}} - 4 \, a^{\frac{5}{2}} b\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - a\right)}^{2}}}{64 \, f}"," ",0,"1/64*(sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)*(a*tan(1/2*f*x + 1/2*e)^2 + (7*a^2 + 10*a*b)/a) + 24*(a^2 + 2*a*b + b^2)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/sqrt(-a) - 12*(a^(5/2) + 2*a^(3/2)*b + sqrt(a)*b^2)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a + 4*(4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2 + 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + 10*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) + 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b - 6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2 - 3*a^(7/2) - 4*a^(5/2)*b)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^2)/f","B",0
137,1,1708,0,1.139165," ","integrate(csc(f*x+e)^7*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} {\left({\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{8 \, a^{3} + 7 \, a^{2} b}{a^{2}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{37 \, a^{3} + 51 \, a^{2} b + 6 \, a b^{2}}{a^{2}}\right)} + \frac{24 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a} - \frac{12 \, {\left(5 \, a^{\frac{7}{2}} + 9 \, a^{\frac{5}{2}} b + 3 \, a^{\frac{3}{2}} b^{2} - \sqrt{a} b^{3}\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{2}} + \frac{2 \, {\left(45 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{3} + 132 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{2} b + 108 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b^{2} + 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{3} + 63 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{7}{2}} + 120 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{5}{2}} b + 48 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b^{2} - 50 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{4} - 156 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{3} b - 96 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b^{2} + 32 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{3} - 78 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{9}{2}} - 108 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{7}{2}} b + 21 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{5} + 72 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{4} b + 36 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b^{2} - 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{3} + 31 \, a^{\frac{11}{2}} + 36 \, a^{\frac{9}{2}} b\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - a\right)}^{3} a}}{384 \, f}"," ",0,"1/384*(sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)*((a*tan(1/2*f*x + 1/2*e)^2 + (8*a^3 + 7*a^2*b)/a^2)*tan(1/2*f*x + 1/2*e)^2 + (37*a^3 + 51*a^2*b + 6*a*b^2)/a^2) + 24*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a) - 12*(5*a^(7/2) + 9*a^(5/2)*b + 3*a^(3/2)*b^2 - sqrt(a)*b^3)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^2 + 2*(45*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 + 132*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b + 108*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b^2 + 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 63*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) + 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*b + 48*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b^2 - 50*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^4 - 156*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^3*b - 96*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b^2 + 32*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^3 - 78*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2) - 108*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*b + 21*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^5 + 72*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*b + 36*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b^2 - 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^3 + 31*a^(11/2) + 36*a^(9/2)*b)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^3*a))/f","B",0
138,0,0,0,0.000000," ","integrate(sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sin\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sin(f*x + e)^4, x)","F",0
139,0,0,0,0.000000," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sin\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sin(f*x + e)^2, x)","F",0
140,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
141,0,0,0,0.000000," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \csc\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*csc(f*x + e)^2, x)","F",0
142,0,0,0,0.000000," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \csc\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*csc(f*x + e)^4, x)","F",0
143,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^2)^(5/2),x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{\frac{5}{2}}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^(5/2), x)","F",0
144,-2,0,0,0.000000," ","integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-1/2/b*cos(f*x+exp(1))/f*sqrt(b*(1-(cos(f*x+exp(1))/f*f)^2)+a)+2*(a-b)/4/b/sqrt(-b)/abs(f)*ln(abs(sqrt(b*(1-(cos(f*x+exp(1))/f*f)^2)+a)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
145,-2,0,0,0.000000," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-1/sqrt(-b)/abs(f)*ln(abs(sqrt(a-b*f^2*(-cos(f*x+exp(1))/f)^2+b)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
146,-2,0,0,0.000000," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-81,22]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[53,42]Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Evaluation time: 0.41Error: Bad Argument Type","F(-2)",0
147,-2,0,0,0.000000," ","integrate(csc(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-81,22]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[53,42]Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Evaluation time: 0.5Unable to divide, perhaps due to rounding error%%%{1,[4,0]%%%}+%%%{%%%{-2,[1]%%%},[2,0]%%%}+%%%{%%%{1,[2]%%%},[0,0]%%%} / %%%{%%%{1,[1]%%%},[4,0]%%%}+%%%{%%%{-2,[2]%%%},[2,0]%%%}+%%%{%%%{1,[3]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
148,0,0,0,0.000000," ","integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{4}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(sin(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
149,0,0,0,0.000000," ","integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{2}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(sin(f*x + e)^2/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
150,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(1/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
151,0,0,0,0.000000," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\csc\left(f x + e\right)^{2}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(csc(f*x + e)^2/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
152,0,0,0,0.000000," ","integrate(csc(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\csc\left(f x + e\right)^{4}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(csc(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
153,-2,0,0,0.000000," ","integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-4*a/(-4*b^2-4*b*a)*cos(f*x+exp(1))/f*sqrt(b*(1-(cos(f*x+exp(1))/f*f)^2)+a)/(b*(1-(cos(f*x+exp(1))/f*f)^2)+a)-1/b/sqrt(-b)/abs(f)*ln(abs(sqrt(b*(1-(cos(f*x+exp(1))/f*f)^2)+a)+sqrt(-b*f^2)*cos(f*x+exp(1))/f))","F(-2)",0
154,1,53,0,0.580316," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\sqrt{-{\left(\cos\left(f x + e\right)^{2} - 1\right)} b + a} \cos\left(f x + e\right)}{{\left({\left(\cos\left(f x + e\right)^{2} - 1\right)} b - a\right)} {\left(a + b\right)} f}"," ",0,"sqrt(-(cos(f*x + e)^2 - 1)*b + a)*cos(f*x + e)/(((cos(f*x + e)^2 - 1)*b - a)*(a + b)*f)","A",0
155,-2,0,0,0.000000," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-81,22]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[53,42]Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Evaluation time: 0.62Error: Bad Argument Type","F(-2)",0
156,1,520,0,0.864647," ","integrate(csc(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\frac{{\left(\frac{{\left(a^{5} b + a^{4} b^{2}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{6} b + a^{5} b^{2}} + \frac{2 \, {\left(a^{5} b + 3 \, a^{4} b^{2} + 6 \, a^{3} b^{3}\right)}}{a^{6} b + a^{5} b^{2}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{a^{5} b + a^{4} b^{2} - 8 \, a^{3} b^{3}}{a^{6} b + a^{5} b^{2}}}{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}} - \frac{2 \, {\left(a^{\frac{3}{2}} - 3 \, \sqrt{a} b\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{3}} + \frac{2 \, {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{\frac{3}{2}} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} b + a^{2}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} - a^{\frac{3}{2}}\right)} a^{2}}}{8 \, f}"," ",0,"1/8*((((a^5*b + a^4*b^2)*tan(1/2*f*x + 1/2*e)^2/(a^6*b + a^5*b^2) + 2*(a^5*b + 3*a^4*b^2 + 6*a^3*b^3)/(a^6*b + a^5*b^2))*tan(1/2*f*x + 1/2*e)^2 + (a^5*b + a^4*b^2 - 8*a^3*b^3)/(a^6*b + a^5*b^2))/sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - 2*(a^(3/2) - 3*sqrt(a)*b)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^3 + 2*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^(3/2) + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a)*b + a^2)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a) - a^(3/2))*a^2))/f","B",0
157,0,0,0,0.000000," ","integrate(sin(f*x+e)^6/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{6}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(sin(f*x + e)^6/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
158,0,0,0,0.000000," ","integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(sin(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
159,0,0,0,0.000000," ","integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(sin(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
160,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(-3/2), x)","F",0
161,0,0,0,0.000000," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\csc\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(csc(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
162,1,338,0,0.951190," ","integrate(sin(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{\frac{{\left({\left(\frac{{\left(3 \, a^{3} b^{8} + 5 \, a^{2} b^{9}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{2} b^{10} + 2 \, a b^{11} + b^{12}} + \frac{3 \, {\left(a^{3} b^{8} + 7 \, a^{2} b^{9} + 8 \, a b^{10}\right)}}{a^{2} b^{10} + 2 \, a b^{11} + b^{12}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{3 \, {\left(a^{3} b^{8} + 7 \, a^{2} b^{9} + 8 \, a b^{10}\right)}}{a^{2} b^{10} + 2 \, a b^{11} + b^{12}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{3 \, a^{3} b^{8} + 5 \, a^{2} b^{9}}{a^{2} b^{10} + 2 \, a b^{11} + b^{12}}}{{\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a\right)}^{\frac{3}{2}}} - \frac{6 \, \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} + \sqrt{a}}{2 \, \sqrt{b}}\right)}{b^{\frac{5}{2}}}}{3 \, f}"," ",0,"-1/3*(((((3*a^3*b^8 + 5*a^2*b^9)*tan(1/2*f*x + 1/2*e)^2/(a^2*b^10 + 2*a*b^11 + b^12) + 3*(a^3*b^8 + 7*a^2*b^9 + 8*a*b^10)/(a^2*b^10 + 2*a*b^11 + b^12))*tan(1/2*f*x + 1/2*e)^2 - 3*(a^3*b^8 + 7*a^2*b^9 + 8*a*b^10)/(a^2*b^10 + 2*a*b^11 + b^12))*tan(1/2*f*x + 1/2*e)^2 - (3*a^3*b^8 + 5*a^2*b^9)/(a^2*b^10 + 2*a*b^11 + b^12))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) - 6*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) + sqrt(a))/sqrt(b))/b^(5/2))/f","B",0
163,1,149,0,0.790011," ","integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{\sqrt{-{\left(\cos\left(f x + e\right)^{2} - 1\right)} b + a} {\left(\frac{3 \, {\left(a b f^{2} + b^{2} f^{2}\right)}}{a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}} - \frac{{\left(a b f^{4} + 3 \, b^{2} f^{4}\right)} \cos\left(f x + e\right)^{2}}{{\left(a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}\right)} f^{2}}\right)} \cos\left(f x + e\right)}{3 \, {\left({\left(\cos\left(f x + e\right)^{2} - 1\right)} b - a\right)}^{2} f}"," ",0,"-1/3*sqrt(-(cos(f*x + e)^2 - 1)*b + a)*(3*(a*b*f^2 + b^2*f^2)/(a^2*b*f^2 + 2*a*b^2*f^2 + b^3*f^2) - (a*b*f^4 + 3*b^2*f^4)*cos(f*x + e)^2/((a^2*b*f^2 + 2*a*b^2*f^2 + b^3*f^2)*f^2))*cos(f*x + e)/(((cos(f*x + e)^2 - 1)*b - a)^2*f)","B",0
164,1,137,0,0.704945," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\frac{{\left(\frac{2 \, b^{2} f^{2} \cos\left(f x + e\right)^{2}}{a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}} - \frac{3 \, {\left(a b f^{2} + b^{2} f^{2}\right)}}{a^{2} b f^{2} + 2 \, a b^{2} f^{2} + b^{3} f^{2}}\right)} \sqrt{-{\left(\cos\left(f x + e\right)^{2} - 1\right)} b + a} \cos\left(f x + e\right)}{3 \, {\left({\left(\cos\left(f x + e\right)^{2} - 1\right)} b - a\right)}^{2} f}"," ",0,"1/3*(2*b^2*f^2*cos(f*x + e)^2/(a^2*b*f^2 + 2*a*b^2*f^2 + b^3*f^2) - 3*(a*b*f^2 + b^2*f^2)/(a^2*b*f^2 + 2*a*b^2*f^2 + b^3*f^2))*sqrt(-(cos(f*x + e)^2 - 1)*b + a)*cos(f*x + e)/(((cos(f*x + e)^2 - 1)*b - a)^2*f)","B",0
165,-2,0,0,0.000000," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-81,22]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[53,42]Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Evaluation time: 0.66Error: Bad Argument Type","F(-2)",0
166,0,0,0,0.000000," ","integrate(sin(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{6}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(sin(f*x + e)^6/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
167,0,0,0,0.000000," ","integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(sin(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
168,0,0,0,0.000000," ","integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\sin\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(sin(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
169,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(-5/2), x)","F",0
170,0,0,0,0.000000," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\csc\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(csc(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
171,0,0,0,0.000000," ","integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \left(d \sin\left(f x + e\right)\right)^{m}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*(d*sin(f*x + e))^m, x)","F",0
172,0,0,0,0.000000," ","integrate(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^5, x)","F",0
173,0,0,0,0.000000," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^3, x)","F",0
174,0,0,0,0.000000," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e), x)","F",0
175,0,0,0,0.000000," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*csc(f*x + e), x)","F",0
176,0,0,0,0.000000," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*csc(f*x + e)^3, x)","F",0
177,0,0,0,0.000000," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*csc(f*x + e)^5, x)","F",0
178,0,0,0,0.000000," ","integrate(sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^4, x)","F",0
179,0,0,0,0.000000," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sin\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^2, x)","F",0
180,0,0,0,0.000000," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*csc(f*x + e)^2, x)","F",0
181,0,0,0,0.000000," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \csc\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*csc(f*x + e)^4, x)","F",0
182,0,0,0,0.000000," ","integrate(sin(d*x+c)^7/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{7}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)^7/(b*sin(d*x + c)^3 + a), x)","F",0
183,0,0,0,0.000000," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{5}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)^5/(b*sin(d*x + c)^3 + a), x)","F",0
184,0,0,0,0.000000," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{3}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)^3/(b*sin(d*x + c)^3 + a), x)","F",0
185,0,0,0,0.000000," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)/(b*sin(d*x + c)^3 + a), x)","F",0
186,0,0,0,0.000000," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(csc(d*x + c)/(b*sin(d*x + c)^3 + a), x)","F",0
187,0,0,0,0.000000," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)^{3}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(csc(d*x + c)^3/(b*sin(d*x + c)^3 + a), x)","F",0
188,0,0,0,0.000000," ","integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)^{5}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(csc(d*x + c)^5/(b*sin(d*x + c)^3 + a), x)","F",0
189,0,0,0,0.000000," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{6}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)^6/(b*sin(d*x + c)^3 + a), x)","F",0
190,0,0,0,0.000000," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{4}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)^4/(b*sin(d*x + c)^3 + a), x)","F",0
191,0,0,0,0.000000," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{2}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sin(d*x + c)^2/(b*sin(d*x + c)^3 + a), x)","F",0
192,0,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{1}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(1/(b*sin(d*x + c)^3 + a), x)","F",0
193,0,0,0,0.000000," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)^{2}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(csc(d*x + c)^2/(b*sin(d*x + c)^3 + a), x)","F",0
194,0,0,0,0.000000," ","integrate(csc(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)^{4}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(csc(d*x + c)^4/(b*sin(d*x + c)^3 + a), x)","F",0
195,-2,0,0,0.000000," ","integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-42,-63]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-2,-75]-2/d*(-15*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a-60*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a-90*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-80*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b-60*(1-cos(c+d*x))/(1+cos(c+d*x))*a-40*(1-cos(c+d*x))/(1+cos(c+d*x))*b-15*a-8*b)*1/15/b^2/((1-cos(c+d*x))/(1+cos(c+d*x))+1)^5-2/d/b^2*2/d*((-2*a^4*b+12*a^3*b^2-6*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^3*a*b-3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-10*a^2*b^3+12*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-12*a^2*b*a*b+6*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a*b^2*a*b+a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(-8*a+sqrt(8*a*8*a-4*(4*a-4*b)*4*a))/2/(4*a-4*b)))+pi*floor((c+d*x)/pi+1/2))-(-2*a^4*b+12*a^3*b^2+6*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^3*a*b-3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-10*a^2*b^3-12*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-12*a^2*b*a*b+6*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a*b^2*a*b+a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(-8*a-sqrt(8*a*8*a-4*(4*a-4*b)*4*a))/2/(4*a-4*b)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
196,-2,0,0,0.000000," ","integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[76,51]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[11,92]-2/d*(-6*(1-cos(c+d*x))/(1+cos(c+d*x))-2)*1/3/b/((1-cos(c+d*x))/(1+cos(c+d*x))+1)^3+2/d*4*a/b*2/d*(1/(8*b*2)*(c+d*x)+((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)*b^2+(-4*a^4*b^2-3*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+8*a^3*b^3+9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a^3*b*a*b-4*a^2*b^4-5*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*a^2*b^2*a*b-a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b+sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2))-((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(-4*a^4*b^2+3*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*a^3*b^3-9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a^3*b*a*b-4*a^2*b^4+5*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-8*a^2*b^2*a*b+a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b-sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
197,-2,0,0,0.000000," ","integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-42,-63]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-2,-75]2/d/b/((1-cos(c+d*x))/(1+cos(c+d*x))+1)-2/d/b*2/d*((-2*a^3*b+12*a^2*b^2-6*a^2*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^2*a*b-3*a^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-10*a*b^3+12*a*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-12*a*b*a*b+6*a*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*b^2*a*b+b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(-8*a+sqrt(8*a*8*a-4*(4*a-4*b)*4*a))/2/(4*a-4*b)))+pi*floor((c+d*x)/pi+1/2))-(-2*a^3*b+12*a^2*b^2+6*a^2*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^2*a*b-3*a^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-10*a*b^3-12*a*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-12*a*b*a*b+6*a*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*b^2*a*b+b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(-8*a-sqrt(8*a*8*a-4*(4*a-4*b)*4*a))/2/(4*a-4*b)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
198,1,166,0,0.735175," ","integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\sqrt{-b^{2} - \sqrt{a b} b} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{b d^{2} + \sqrt{{\left(a - b\right)} b d^{4} + b^{2} d^{4}}}{b d^{4}}}}\right)}{2 \, {\left(b + \sqrt{a b}\right)} d {\left| b \right|}} + \frac{\sqrt{-b^{2} + \sqrt{a b} b} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{b d^{2} - \sqrt{{\left(a - b\right)} b d^{4} + b^{2} d^{4}}}{b d^{4}}}}\right)}{2 \, {\left(b - \sqrt{a b}\right)} d {\left| b \right|}}"," ",0,"1/2*sqrt(-b^2 - sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(b*d^2 + sqrt((a - b)*b*d^4 + b^2*d^4))/(b*d^4))))/((b + sqrt(a*b))*d*abs(b)) + 1/2*sqrt(-b^2 + sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(b*d^2 - sqrt((a - b)*b*d^4 + b^2*d^4))/(b*d^4))))/((b - sqrt(a*b))*d*abs(b))","B",0
199,1,183,0,0.755750," ","integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\sqrt{a b} \sqrt{-b^{2} - \sqrt{a b} b} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{b d^{2} + \sqrt{{\left(a - b\right)} b d^{4} + b^{2} d^{4}}}{b d^{4}}}}\right)}{2 \, {\left(a b + \sqrt{a b} a\right)} d {\left| b \right|}} + \frac{\sqrt{a b} \sqrt{-b^{2} + \sqrt{a b} b} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{b d^{2} - \sqrt{{\left(a - b\right)} b d^{4} + b^{2} d^{4}}}{b d^{4}}}}\right)}{2 \, {\left(a b - \sqrt{a b} a\right)} d {\left| b \right|}}"," ",0,"-1/2*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(b*d^2 + sqrt((a - b)*b*d^4 + b^2*d^4))/(b*d^4))))/((a*b + sqrt(a*b)*a)*d*abs(b)) + 1/2*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(b*d^2 - sqrt((a - b)*b*d^4 + b^2*d^4))/(b*d^4))))/((a*b - sqrt(a*b)*a)*d*abs(b))","B",0
200,-2,0,0,0.000000," ","integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[76,51]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[11,92]2/d*1/4/a*ln(abs(1-cos(c+d*x))/abs(1+cos(c+d*x)))+2/d*4*b/a*2/d*(1/(8*b*2)*(c+d*x)+((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)*b^2+(-4*a^4*b^2-3*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+8*a^3*b^3+9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a^3*b*a*b-4*a^2*b^4-5*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*a^2*b^2*a*b-a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b+sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2))-((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(-4*a^4*b^2+3*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*a^3*b^3-9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a^3*b*a*b-4*a^2*b^4+5*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-8*a^2*b^2*a*b+a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b-sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
201,-2,0,0,0.000000," ","integrate(csc(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-35,-31]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[87,-13]-2/d*(-(1-cos(c+d*x))/(1+cos(c+d*x))*1/16/a+(2*(1-cos(c+d*x))/(1+cos(c+d*x))+1)*1/16/a/(1-cos(c+d*x))*(1+cos(c+d*x))-1/8/a*ln(abs(1-cos(c+d*x))/abs(1+cos(c+d*x))))-2/d/a*2/d*((2*a^3*b-12*a^2*b^2-6*a^2*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*a^2*a*b+3*a^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a*b^3+12*a*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+12*a*b*a*b-6*a*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-10*b^2*a*b-b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^5-96*a^4*b+112*a^3*b^2-32*a^2*b^3-8*a*b^4)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(4*b-4*a)*4*a))/2/(4*b-4*a)))+pi*floor((c+d*x)/pi+1/2))-(2*a^3*b-12*a^2*b^2+6*a^2*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-2*a^2*a*b+3*a^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a*b^3-12*a*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+12*a*b*a*b-6*a*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-10*b^2*a*b-b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^5-96*a^4*b+112*a^3*b^2-32*a^2*b^3-8*a*b^4)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(4*b-4*a)*4*a))/2/(4*b-4*a)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
202,-2,0,0,0.000000," ","integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[76,51]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[11,92]-2/d*((-32*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-256*(1-cos(c+d*x))/(1+cos(c+d*x))*a)*1/4096/a^2+(18*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a+48*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+8*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)*1/128/a^2/((1-cos(c+d*x))/(1+cos(c+d*x)))^2+(-3*a-8*b)*1/32/a^2*ln(abs(1-cos(c+d*x))/abs(1+cos(c+d*x))))+2/d*4*b^2/a^2*2/d*(1/(8*b*2)*(c+d*x)+((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)*b^2+(-4*a^4*b^2-3*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+8*a^3*b^3+9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a^3*b*a*b-4*a^2*b^4-5*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*a^2*b^2*a*b-a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b+sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2))-((2*a^4*b-8*a^3*b^2-2*a^3*a*b+3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a^2*b^3+8*a^2*b*a*b-12*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-4*a*b^4-10*a*b^2*a*b+11*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*b^3*a*b+2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(-4*a^4*b^2+3*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*a^3*b^3-9*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a^3*b*a*b-4*a^2*b^4+5*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-8*a^2*b^2*a*b+a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+4*a*b^3*a*b)*abs(a-b)*abs(b)+(2*a^4*b^3-4*a^3*b^4-2*a^3*b^2*a*b+3*a^3*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^2*b^5+4*a^2*b^3*a*b-6*a^2*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-2*a*b^4*a*b-a*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((96*a^6*b^2-480*a^5*b^3+832*a^4*b^4-576*a^3*b^5+96*a^2*b^6+32*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(32*a*b-sqrt(32*a*b*32*a*b+4*(-16*a*b+16*b^2)*16*a*b))/2/(-16*a*b+16*b^2)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
203,1,461,0,1.112757," ","integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{4 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b^{2} + \sqrt{a^{2} b^{4} - {\left(a b^{2} - b^{3}\right)} a b^{2}}}{a b^{2} - b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{4} b^{2} - 12 \, a^{3} b^{3} + 14 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6}} + \frac{4 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b^{2} - \sqrt{a^{2} b^{4} - {\left(a b^{2} - b^{3}\right)} a b^{2}}}{a b^{2} - b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{4} b^{2} - 12 \, a^{3} b^{3} + 14 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6}} - \frac{{\left(d x + c\right)} {\left(8 \, a + 3 \, b\right)}}{b^{2}} + \frac{5 \, \tan\left(d x + c\right)^{3} + 3 \, \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)}^{2} b}}{8 \, d}"," ",0,"1/8*(4*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 + sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^4*b^2 - 12*a^3*b^3 + 14*a^2*b^4 - 4*a*b^5 - b^6) + 4*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 - sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^4*b^2 - 12*a^3*b^3 + 14*a^2*b^4 - 4*a*b^5 - b^6) - (d*x + c)*(8*a + 3*b)/b^2 + (5*tan(d*x + c)^3 + 3*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*b))/d","B",0
204,1,695,0,1.022256," ","integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{d x + c}{b} + \frac{{\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b + \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} {\left| b \right|}} - \frac{{\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b - \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} {\left| b \right|}} - \frac{\tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)} b}}{2 \, d}"," ",0,"-1/2*((d*x + c)/b + ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^2*b^5 + 3*a*b^6 + b^7)*abs(b)) - ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b - sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^2*b^5 + 3*a*b^6 + b^7)*abs(b)) - tan(d*x + c)/((tan(d*x + c)^2 + 1)*b))/d","B",0
205,1,912,0,0.982995," ","integrate(sin(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{2 \, {\left(d x + c\right)}}{b} + \frac{{\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b - 9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{3} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left| -a + b \right|} {\left| b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b + \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} {\left| b \right|}} - \frac{{\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b - 9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{3} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left| -a + b \right|} {\left| b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b - \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} {\left| b \right|}}}{2 \, d}"," ",0,"-1/2*(2*(d*x + c)/b + ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^2*b^5 + 3*a*b^6 + b^7)*abs(b)) - ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b - sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^2*b^5 + 3*a*b^6 + b^7)*abs(b)))/d","B",0
206,1,398,0,0.990401," ","integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{{\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(d x + c\right)}{\sqrt{\frac{4 \, a + \sqrt{-16 \, {\left(a - b\right)} a + 16 \, a^{2}}}{a - b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}} - \frac{{\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(d x + c\right)}{\sqrt{\frac{4 \, a - \sqrt{-16 \, {\left(a - b\right)} a + 16 \, a^{2}}}{a - b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}}}{2 \, d}"," ",0,"1/2*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*tan(d*x + c)/sqrt((4*a + sqrt(-16*(a - b)*a + 16*a^2))/(a - b))))*abs(a - b)/(3*a^5*b - 12*a^4*b^2 + 14*a^3*b^3 - 4*a^2*b^4 - a*b^5) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*tan(d*x + c)/sqrt((4*a - sqrt(-16*(a - b)*a + 16*a^2))/(a - b))))*abs(a - b)/(3*a^5*b - 12*a^4*b^2 + 14*a^3*b^3 - 4*a^2*b^4 - a*b^5))/d","B",0
207,1,361,0,0.420412," ","integrate(1/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{{\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(d x + c\right)}{\sqrt{\frac{4 \, a + \sqrt{-16 \, {\left(a - b\right)} a + 16 \, a^{2}}}{a - b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{5} - 12 \, a^{4} b + 14 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - a b^{4}} + \frac{{\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(d x + c\right)}{\sqrt{\frac{4 \, a - \sqrt{-16 \, {\left(a - b\right)} a + 16 \, a^{2}}}{a - b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{5} - 12 \, a^{4} b + 14 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - a b^{4}}}{2 \, d}"," ",0,"1/2*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*tan(d*x + c)/sqrt((4*a + sqrt(-16*(a - b)*a + 16*a^2))/(a - b))))*abs(a - b)/(3*a^5 - 12*a^4*b + 14*a^3*b^2 - 4*a^2*b^3 - a*b^4) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*tan(d*x + c)/sqrt((4*a - sqrt(-16*(a - b)*a + 16*a^2))/(a - b))))*abs(a - b)/(3*a^5 - 12*a^4*b + 14*a^3*b^2 - 4*a^2*b^3 - a*b^4))/d","B",0
208,1,672,0,1.158847," ","integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{{\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} a^{2} {\left| a - b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2}\right)} {\left| a - b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} + \sqrt{a^{4} - {\left(a^{2} - a b\right)} a^{2}}}{a^{2} - a b}}}\right)\right)}}{{\left(3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} {\left| a \right|}} - \frac{{\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} a^{2} {\left| a - b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2}\right)} {\left| a - b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} - \sqrt{a^{4} - {\left(a^{2} - a b\right)} a^{2}}}{a^{2} - a b}}}\right)\right)}}{{\left(3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} {\left| a \right|}} + \frac{2}{a \tan\left(d x + c\right)}}{2 \, d}"," ",0,"-1/2*(((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 + sqrt(a^4 - (a^2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) - ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - sqrt(a^4 - (a^2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) + 2/(a*tan(d*x + c)))/d","B",0
209,1,938,0,1.039981," ","integrate(csc(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{3 \, {\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} a^{2} {\left| a - b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b - 9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{4}\right)} {\left| a - b \right|} {\left| a \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3}\right)} {\left| a - b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} + \sqrt{a^{4} - {\left(a^{2} - a b\right)} a^{2}}}{a^{2} - a b}}}\right)\right)}}{{\left(3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} {\left| a \right|}} - \frac{3 \, {\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} a^{2} {\left| a - b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b - 9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{4}\right)} {\left| a - b \right|} {\left| a \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3}\right)} {\left| a - b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} - \sqrt{a^{4} - {\left(a^{2} - a b\right)} a^{2}}}{a^{2} - a b}}}\right)\right)}}{{\left(3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} {\left| a \right|}} + \frac{2 \, {\left(3 \, \tan\left(d x + c\right)^{2} + 1\right)}}{a \tan\left(d x + c\right)^{3}}}{6 \, d}"," ",0,"-1/6*(3*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4)*abs(a - b)*abs(a) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 + sqrt(a^4 - (a^2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) - 3*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4)*abs(a - b)*abs(a) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - sqrt(a^4 - (a^2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) + 2*(3*tan(d*x + c)^2 + 1)/(a*tan(d*x + c)^3))/d","B",0
210,1,471,0,0.994174," ","integrate(csc(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{15 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} + \sqrt{a^{6} - {\left(a^{3} - a^{2} b\right)} a^{3}}}{a^{3} - a^{2} b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac{15 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} - \sqrt{a^{6} - {\left(a^{3} - a^{2} b\right)} a^{3}}}{a^{3} - a^{2} b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac{2 \, {\left(15 \, a \tan\left(d x + c\right)^{4} + 15 \, b \tan\left(d x + c\right)^{4} + 10 \, a \tan\left(d x + c\right)^{2} + 3 \, a\right)}}{a^{2} \tan\left(d x + c\right)^{5}}}{30 \, d}"," ",0,"1/30*(15*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 + sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b + 14*a^5*b^2 - 4*a^4*b^3 - a^3*b^4) - 15*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b + 14*a^5*b^2 - 4*a^4*b^3 - a^3*b^4) - 2*(15*a*tan(d*x + c)^4 + 15*b*tan(d*x + c)^4 + 10*a*tan(d*x + c)^2 + 3*a)/(a^2*tan(d*x + c)^5))/d","B",0
211,1,467,0,1.022051," ","integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{105 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} + \sqrt{a^{6} - {\left(a^{3} - a^{2} b\right)} a^{3}}}{a^{3} - a^{2} b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} + \frac{105 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} - \sqrt{a^{6} - {\left(a^{3} - a^{2} b\right)} a^{3}}}{a^{3} - a^{2} b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{7} - 12 \, a^{6} b + 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - a^{3} b^{4}} - \frac{2 \, {\left(105 \, a \tan\left(d x + c\right)^{6} + 105 \, b \tan\left(d x + c\right)^{6} + 105 \, a \tan\left(d x + c\right)^{4} + 35 \, b \tan\left(d x + c\right)^{4} + 63 \, a \tan\left(d x + c\right)^{2} + 15 \, a\right)}}{a^{2} \tan\left(d x + c\right)^{7}}}{210 \, d}"," ",0,"1/210*(105*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 + sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b + 14*a^5*b^2 - 4*a^4*b^3 - a^3*b^4) + 105*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - sqrt(a^6 - (a^3 - a^2*b)*a^3))/(a^3 - a^2*b))))*abs(a - b)/(3*a^7 - 12*a^6*b + 14*a^5*b^2 - 4*a^4*b^3 - a^3*b^4) - 2*(105*a*tan(d*x + c)^6 + 105*b*tan(d*x + c)^6 + 105*a*tan(d*x + c)^4 + 35*b*tan(d*x + c)^4 + 63*a*tan(d*x + c)^2 + 15*a)/(a^2*tan(d*x + c)^7))/d","B",0
212,-2,0,0,0.000000," ","integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-54,3]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[65,-69]2/d*(5*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2-6*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b+20*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2-26*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b+30*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2-94*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b+64*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b^2+20*(1-cos(c+d*x))/(1+cos(c+d*x))*a^2-14*(1-cos(c+d*x))/(1+cos(c+d*x))*a*b+5*a^2-4*a*b)/(-4*a*b^2+4*b^3)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a+5*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+10*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*b+10*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+5*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)+2/d/(-4*a*b^2+4*b^3)*2/d*(-a/4*(c+d*x)+(-10*a^4*b+72*a^3*b^2+33*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+10*a^3*a*b-15*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-118*a^2*b^3-102*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-72*a^2*b*a*b+42*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+56*a*b^4+61*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+118*a*b^2*a*b-19*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+12*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-56*b^3*a*b-4*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2))-(-10*a^4*b+72*a^3*b^2-33*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+10*a^3*a*b-15*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-118*a^2*b^3+102*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-72*a^2*b*a*b+42*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+56*a*b^4-61*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+118*a*b^2*a*b-19*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-12*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-56*b^3*a*b-4*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^4*b-96*a^3*b^2+112*a^2*b^3-32*a*b^4-8*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
213,-2,0,0,0.000000," ","integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[54,-78]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-23,24]2/d*(((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a-3*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a+8*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b-5*(1-cos(c+d*x))/(1+cos(c+d*x))*a-a)/(4*a*b-4*b^2)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)+2/d/(4*a*b-4*b^2)*2/d*(-(3*a-4*b)/(2*b*2)*(c+d*x)+((-6*a^5*b+30*a^4*b^2+6*a^4*a*b-9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-58*a^3*b^3-30*a^3*b*a*b+45*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+50*a^2*b^4+58*a^2*b^2*a*b-75*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-16*a*b^5-50*a*b^3*a*b+39*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+16*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)*b^2+(12*a^5*b^2+9*a^5*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-32*a^4*b^3-33*a^4*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-12*a^4*b*a*b+28*a^3*b^4+33*a^3*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+32*a^3*b^2*a*b-8*a^2*b^5-7*a^2*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-28*a^2*b^3*a*b-2*a*b^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+8*a*b^4*a*b)*abs(a-b)*abs(b)+(-8*a^5*b^3+28*a^4*b^4+8*a^4*b^2*a*b-12*a^4*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-32*a^3*b^5-28*a^3*b^3*a*b+42*a^3*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+12*a^2*b^6+32*a^2*b^4*a*b-32*a^2*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-12*a*b^5*a*b-6*a*b^5*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b))/((24*a^6*b^2-120*a^5*b^3+208*a^4*b^4-144*a^3*b^5+24*a^2*b^6+8*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(8*a*b+sqrt(8*a*b*8*a*b+4*(-4*a*b+4*b^2)*4*a*b))/2/(-4*a*b+4*b^2)))+pi*floor((c+d*x)/pi+1/2))-((-6*a^5*b+30*a^4*b^2+6*a^4*a*b-9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-58*a^3*b^3-30*a^3*b*a*b+45*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+50*a^2*b^4+58*a^2*b^2*a*b-75*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-16*a*b^5-50*a*b^3*a*b+39*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+16*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(12*a^5*b^2-9*a^5*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-32*a^4*b^3+33*a^4*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-12*a^4*b*a*b+28*a^3*b^4-33*a^3*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+32*a^3*b^2*a*b-8*a^2*b^5+7*a^2*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-28*a^2*b^3*a*b+2*a*b^5*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*a*b^4*a*b)*abs(a-b)*abs(b)+(-8*a^5*b^3+28*a^4*b^4+8*a^4*b^2*a*b-12*a^4*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-32*a^3*b^5-28*a^3*b^3*a*b+42*a^3*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+12*a^2*b^6+32*a^2*b^4*a*b-32*a^2*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-12*a*b^5*a*b-6*a*b^5*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((24*a^6*b^2-120*a^5*b^3+208*a^4*b^4-144*a^3*b^5+24*a^2*b^6+8*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(8*a*b-sqrt(8*a*b*8*a*b+4*(-4*a*b+4*b^2)*4*a*b))/2/(-4*a*b+4*b^2)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
214,-2,0,0,0.000000," ","integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-57,-84]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[27,61]2/d*(((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a-2*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*b+3*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-8*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+3*(1-cos(c+d*x))/(1+cos(c+d*x))*a+2*(1-cos(c+d*x))/(1+cos(c+d*x))*b+a)/(-4*a*b+4*b^2)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)+2/d/(-4*a*b+4*b^2)*2/d*(-1/4*(c+d*x)+(-2*a^4*b+16*a^3*b^2+9*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^3*a*b-3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-30*a^2*b^3-30*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-16*a^2*b*a*b+6*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+16*a*b^4+21*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+30*a*b^2*a*b+a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+4*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-16*b^3*a*b)*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2))-(-2*a^4*b+16*a^3*b^2-9*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^3*a*b-3*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-30*a^2*b^3+30*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-16*a^2*b*a*b+6*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+16*a*b^4-21*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+30*a*b^2*a*b+a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-4*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-16*b^3*a*b)*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
215,1,605,0,1.037353," ","integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","-\frac{\frac{\cos\left(d x + c\right)^{3}}{d} - \frac{2 \, \cos\left(d x + c\right)}{d}}{4 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} - a + b\right)} {\left(a - b\right)}} + \frac{{\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \sqrt{a b} \sqrt{-b^{2} + \sqrt{a b} b} d^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{-b^{2} + \sqrt{a b} b} d^{2} {\left| -a d^{2} + b d^{2} \right|} + {\left(a d^{2} - b d^{2}\right)}^{2} \sqrt{a b} \sqrt{-b^{2} + \sqrt{a b} b} a\right)} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a b d^{2} - b^{2} d^{2} + \sqrt{{\left(a b d^{2} - b^{2} d^{2}\right)}^{2} + {\left(a b d^{4} - b^{2} d^{4}\right)} {\left(a^{2} - 2 \, a b + b^{2}\right)}}}{a b d^{4} - b^{2} d^{4}}}}\right)}{8 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{3} {\left| -a d^{2} + b d^{2} \right|} {\left| b \right|}} - \frac{{\left({\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \sqrt{a b} \sqrt{-b^{2} - \sqrt{a b} b} d^{4} + 2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{-b^{2} - \sqrt{a b} b} d^{2} {\left| -a d^{2} + b d^{2} \right|} + {\left(a d^{2} - b d^{2}\right)}^{2} \sqrt{a b} \sqrt{-b^{2} - \sqrt{a b} b} a\right)} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a b d^{2} - b^{2} d^{2} - \sqrt{{\left(a b d^{2} - b^{2} d^{2}\right)}^{2} + {\left(a b d^{4} - b^{2} d^{4}\right)} {\left(a^{2} - 2 \, a b + b^{2}\right)}}}{a b d^{4} - b^{2} d^{4}}}}\right)}{8 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{3} {\left| -a d^{2} + b d^{2} \right|} {\left| b \right|}}"," ",0,"-1/4*(cos(d*x + c)^3/d - 2*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)*(a - b)) + 1/8*((a^2*b - 2*a*b^2 + b^3)*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*d^4 - 2*(a^2*b - a*b^2)*sqrt(-b^2 + sqrt(a*b)*b)*d^2*abs(-a*d^2 + b*d^2) + (a*d^2 - b*d^2)^2*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a)*arctan(cos(d*x + c)/(d*sqrt(-(a*b*d^2 - b^2*d^2 + sqrt((a*b*d^2 - b^2*d^2)^2 + (a*b*d^4 - b^2*d^4)*(a^2 - 2*a*b + b^2)))/(a*b*d^4 - b^2*d^4))))/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^3*abs(-a*d^2 + b*d^2)*abs(b)) - 1/8*((a^2*b - 2*a*b^2 + b^3)*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*d^4 + 2*(a^2*b - a*b^2)*sqrt(-b^2 - sqrt(a*b)*b)*d^2*abs(-a*d^2 + b*d^2) + (a*d^2 - b*d^2)^2*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a)*arctan(cos(d*x + c)/(d*sqrt(-(a*b*d^2 - b^2*d^2 - sqrt((a*b*d^2 - b^2*d^2)^2 + (a*b*d^4 - b^2*d^4)*(a^2 - 2*a*b + b^2)))/(a*b*d^4 - b^2*d^4))))/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^3*abs(-a*d^2 + b*d^2)*abs(b))","B",0
216,1,693,0,0.892996," ","integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","-\frac{\frac{b \cos\left(d x + c\right)^{3}}{d} - \frac{a \cos\left(d x + c\right)}{d} - \frac{b \cos\left(d x + c\right)}{d}}{4 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} - a + b\right)} {\left(a^{2} - a b\right)}} + \frac{{\left({\left(3 \, a^{4} b - 8 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 2 \, a b^{4}\right)} \sqrt{-b^{2} + \sqrt{a b} b} d^{4} - {\left(3 \, a^{2} - 4 \, a b + b^{2}\right)} \sqrt{a b} \sqrt{-b^{2} + \sqrt{a b} b} d^{2} {\left| -a^{2} d^{2} + a b d^{2} \right|} + {\left(a^{2} d^{2} - a b d^{2}\right)}^{2} \sqrt{-b^{2} + \sqrt{a b} b} b\right)} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a^{2} b d^{2} - a b^{2} d^{2} + \sqrt{{\left(a^{2} b d^{2} - a b^{2} d^{2}\right)}^{2} + {\left(a^{2} b d^{4} - a b^{2} d^{4}\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{a^{2} b d^{4} - a b^{2} d^{4}}}}\right)}{8 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \sqrt{a b} d^{3} {\left| -a^{2} d^{2} + a b d^{2} \right|} {\left| b \right|}} - \frac{{\left({\left(3 \, a^{4} b - 8 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 2 \, a b^{4}\right)} \sqrt{-b^{2} - \sqrt{a b} b} d^{4} + {\left(3 \, a^{2} - 4 \, a b + b^{2}\right)} \sqrt{a b} \sqrt{-b^{2} - \sqrt{a b} b} d^{2} {\left| -a^{2} d^{2} + a b d^{2} \right|} + {\left(a^{2} d^{2} - a b d^{2}\right)}^{2} \sqrt{-b^{2} - \sqrt{a b} b} b\right)} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a^{2} b d^{2} - a b^{2} d^{2} - \sqrt{{\left(a^{2} b d^{2} - a b^{2} d^{2}\right)}^{2} + {\left(a^{2} b d^{4} - a b^{2} d^{4}\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{a^{2} b d^{4} - a b^{2} d^{4}}}}\right)}{8 \, {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} \sqrt{a b} d^{3} {\left| -a^{2} d^{2} + a b d^{2} \right|} {\left| b \right|}}"," ",0,"-1/4*(b*cos(d*x + c)^3/d - a*cos(d*x + c)/d - b*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)*(a^2 - a*b)) + 1/8*((3*a^4*b - 8*a^3*b^2 + 7*a^2*b^3 - 2*a*b^4)*sqrt(-b^2 + sqrt(a*b)*b)*d^4 - (3*a^2 - 4*a*b + b^2)*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*d^2*abs(-a^2*d^2 + a*b*d^2) + (a^2*d^2 - a*b*d^2)^2*sqrt(-b^2 + sqrt(a*b)*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^2*b*d^2 - a*b^2*d^2 + sqrt((a^2*b*d^2 - a*b^2*d^2)^2 + (a^2*b*d^4 - a*b^2*d^4)*(a^3 - 2*a^2*b + a*b^2)))/(a^2*b*d^4 - a*b^2*d^4))))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*sqrt(a*b)*d^3*abs(-a^2*d^2 + a*b*d^2)*abs(b)) - 1/8*((3*a^4*b - 8*a^3*b^2 + 7*a^2*b^3 - 2*a*b^4)*sqrt(-b^2 - sqrt(a*b)*b)*d^4 + (3*a^2 - 4*a*b + b^2)*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*d^2*abs(-a^2*d^2 + a*b*d^2) + (a^2*d^2 - a*b*d^2)^2*sqrt(-b^2 - sqrt(a*b)*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^2*b*d^2 - a*b^2*d^2 - sqrt((a^2*b*d^2 - a*b^2*d^2)^2 + (a^2*b*d^4 - a*b^2*d^4)*(a^3 - 2*a^2*b + a*b^2)))/(a^2*b*d^4 - a*b^2*d^4))))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*sqrt(a*b)*d^3*abs(-a^2*d^2 + a*b*d^2)*abs(b))","B",0
217,-2,0,0,0.000000," ","integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-89,-82]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[91,55]2/d*((((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b-3*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b+8*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b^2-5*(1-cos(c+d*x))/(1+cos(c+d*x))*a*b-a*b)/(4*a^3-4*a^2*b)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)+1/4/a^2*ln(abs(1-cos(c+d*x))/abs(1+cos(c+d*x))))+2/d/(4*a^3-4*a^2*b)*2/d*((-4*b+5*a)/4*(c+d*x)+(-2*a^4*b-15*a^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*a^3*b^2+48*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+2*a^3*a*b+27*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+26*a^2*b^3-31*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+8*a^2*b*a*b-78*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-16*a*b^4-6*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-26*a*b^2*a*b+39*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+16*b^3*a*b+8*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^5-96*a^4*b+112*a^3*b^2-32*a^2*b^3-8*a*b^4)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(4*b-4*a)*4*a))/2/(4*b-4*a)))+pi*floor((c+d*x)/pi+1/2))-(-2*a^4*b+15*a^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-8*a^3*b^2-48*a^3*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+2*a^3*a*b+27*a^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+26*a^2*b^3+31*a^2*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*a^2*b*a*b-78*a^2*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-16*a*b^4+6*a*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-26*a*b^2*a*b+39*a*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+16*b^3*a*b+8*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^5-96*a^4*b+112*a^3*b^2-32*a^2*b^3-8*a*b^4)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(4*b-4*a)*4*a))/2/(4*b-4*a)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
218,1,1563,0,1.173204," ","integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\frac{\frac{{\left(2 \, {\left(6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 21 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 16 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a b^{2} - b^{3}\right)}^{2} {\left| -a + b \right|} - {\left(12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{2} - 63 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{3} + 116 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{4} - 86 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{5} + 16 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{6} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{7}\right)} {\left| -a b^{2} + b^{3} \right|} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{4} - 51 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{5} + 102 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{6} - 82 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{7} + 17 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{8} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{9}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} b^{2} - a b^{3} + \sqrt{{\left(a^{2} b^{2} - a b^{3}\right)}^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)}}}{a^{2} b^{2} - 2 \, a b^{3} + b^{4}}}}\right)\right)}}{{\left(3 \, a^{7} b^{4} - 21 \, a^{6} b^{5} + 59 \, a^{5} b^{6} - 85 \, a^{4} b^{7} + 65 \, a^{3} b^{8} - 23 \, a^{2} b^{9} + a b^{10} + b^{11}\right)} {\left| -a b^{2} + b^{3} \right|}} - \frac{{\left(2 \, {\left(6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 21 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 16 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a b^{2} - b^{3}\right)}^{2} {\left| -a + b \right|} + {\left(12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{2} - 63 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{3} + 116 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{4} - 86 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{5} + 16 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{6} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{7}\right)} {\left| -a b^{2} + b^{3} \right|} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{4} - 51 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{5} + 102 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{6} - 82 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{7} + 17 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{8} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{9}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} b^{2} - a b^{3} - \sqrt{{\left(a^{2} b^{2} - a b^{3}\right)}^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)}}}{a^{2} b^{2} - 2 \, a b^{3} + b^{4}}}}\right)\right)}}{{\left(3 \, a^{7} b^{4} - 21 \, a^{6} b^{5} + 59 \, a^{5} b^{6} - 85 \, a^{4} b^{7} + 65 \, a^{3} b^{8} - 23 \, a^{2} b^{9} + a b^{10} + b^{11}\right)} {\left| -a b^{2} + b^{3} \right|}} - \frac{2 \, {\left(2 \, a \tan\left(d x + c\right)^{3} + a \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)} {\left(a b - b^{2}\right)}} + \frac{8 \, {\left(d x + c\right)}}{b^{2}}}{8 \, d}"," ",0,"1/8*((2*(6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 21*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a*b^2 - b^3)^2*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^2 - 63*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^3 + 116*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^4 - 86*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^7)*abs(-a*b^2 + b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 - 51*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 102*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 82*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2*b^2 - a*b^3 + sqrt((a^2*b^2 - a*b^3)^2 - (a^2*b^2 - a*b^3)*(a^2*b^2 - 2*a*b^3 + b^4)))/(a^2*b^2 - 2*a*b^3 + b^4))))/((3*a^7*b^4 - 21*a^6*b^5 + 59*a^5*b^6 - 85*a^4*b^7 + 65*a^3*b^8 - 23*a^2*b^9 + a*b^10 + b^11)*abs(-a*b^2 + b^3)) - (2*(6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 21*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a*b^2 - b^3)^2*abs(-a + b) + (12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^2 - 63*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^3 + 116*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^4 - 86*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^7)*abs(-a*b^2 + b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 - 51*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 102*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 82*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 17*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2*b^2 - a*b^3 - sqrt((a^2*b^2 - a*b^3)^2 - (a^2*b^2 - a*b^3)*(a^2*b^2 - 2*a*b^3 + b^4)))/(a^2*b^2 - 2*a*b^3 + b^4))))/((3*a^7*b^4 - 21*a^6*b^5 + 59*a^5*b^6 - 85*a^4*b^7 + 65*a^3*b^8 - 23*a^2*b^9 + a*b^10 + b^11)*abs(-a*b^2 + b^3)) - 2*(2*a*tan(d*x + c)^3 + a*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)*(a*b - b^2)) + 8*(d*x + c)/b^2)/d","B",0
219,1,1481,0,1.201556," ","integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\frac{\frac{{\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 17 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a b - b^{2}\right)}^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{2} + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{3} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{5}\right)} {\left| -a b + b^{2} \right|} {\left| -a + b \right|} - 2 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b - 18 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{2} + 38 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{3} - 32 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{4} + 7 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{5} + 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{6}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} b - a b^{2} + \sqrt{{\left(a^{2} b - a b^{2}\right)}^{2} - {\left(a^{2} b - a b^{2}\right)} {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)}}}{a^{2} b - 2 \, a b^{2} + b^{3}}}}\right)\right)}}{{\left(3 \, a^{8} b^{2} - 21 \, a^{7} b^{3} + 59 \, a^{6} b^{4} - 85 \, a^{5} b^{5} + 65 \, a^{4} b^{6} - 23 \, a^{3} b^{7} + a^{2} b^{8} + a b^{9}\right)} {\left| -a b + b^{2} \right|}} - \frac{{\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 17 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a b - b^{2}\right)}^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{2} + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{3} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{5}\right)} {\left| -a b + b^{2} \right|} {\left| -a + b \right|} - 2 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b - 18 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{2} + 38 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{3} - 32 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{4} + 7 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{5} + 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{6}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} b - a b^{2} - \sqrt{{\left(a^{2} b - a b^{2}\right)}^{2} - {\left(a^{2} b - a b^{2}\right)} {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)}}}{a^{2} b - 2 \, a b^{2} + b^{3}}}}\right)\right)}}{{\left(3 \, a^{8} b^{2} - 21 \, a^{7} b^{3} + 59 \, a^{6} b^{4} - 85 \, a^{5} b^{5} + 65 \, a^{4} b^{6} - 23 \, a^{3} b^{7} + a^{2} b^{8} + a b^{9}\right)} {\left| -a b + b^{2} \right|}} - \frac{2 \, {\left(a \tan\left(d x + c\right)^{3} + b \tan\left(d x + c\right)^{3} + a \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)} {\left(a b - b^{2}\right)}}}{8 \, d}"," ",0,"1/8*(((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a*b - b^2)^2*abs(-a + b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^5)*abs(-a*b + b^2)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b - 18*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 + 38*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 - 32*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5 + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^6)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2*b - a*b^2 + sqrt((a^2*b - a*b^2)^2 - (a^2*b - a*b^2)*(a^2*b - 2*a*b^2 + b^3)))/(a^2*b - 2*a*b^2 + b^3))))/((3*a^8*b^2 - 21*a^7*b^3 + 59*a^6*b^4 - 85*a^5*b^5 + 65*a^4*b^6 - 23*a^3*b^7 + a^2*b^8 + a*b^9)*abs(-a*b + b^2)) - ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 17*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a*b - b^2)^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^5)*abs(-a*b + b^2)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b - 18*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 + 38*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 - 32*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5 + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^6)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2*b - a*b^2 - sqrt((a^2*b - a*b^2)^2 - (a^2*b - a*b^2)*(a^2*b - 2*a*b^2 + b^3)))/(a^2*b - 2*a*b^2 + b^3))))/((3*a^8*b^2 - 21*a^7*b^3 + 59*a^6*b^4 - 85*a^5*b^5 + 65*a^4*b^6 - 23*a^3*b^7 + a^2*b^8 + a*b^9)*abs(-a*b + b^2)) - 2*(a*tan(d*x + c)^3 + b*tan(d*x + c)^3 + a*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)*(a*b - b^2)))/d","B",0
220,1,1264,0,1.062183," ","integrate(sin(d*x+c)^4/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","\frac{\frac{{\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} - 9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b + 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} + 10 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5} - 2 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a - b\right)}^{2} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} - a b + \sqrt{{\left(a^{2} - a b\right)}^{2} - {\left(a^{2} - a b\right)} {\left(a^{2} - 2 \, a b + b^{2}\right)}}}{a^{2} - 2 \, a b + b^{2}}}}\right)\right)}}{3 \, a^{8} b - 21 \, a^{7} b^{2} + 59 \, a^{6} b^{3} - 85 \, a^{5} b^{4} + 65 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a^{2} b^{7} + a b^{8}} - \frac{{\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} - 9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b + 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} + 10 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5} - 2 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a - b\right)}^{2} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} - a b - \sqrt{{\left(a^{2} - a b\right)}^{2} - {\left(a^{2} - a b\right)} {\left(a^{2} - 2 \, a b + b^{2}\right)}}}{a^{2} - 2 \, a b + b^{2}}}}\right)\right)}}{3 \, a^{8} b - 21 \, a^{7} b^{2} + 59 \, a^{6} b^{3} - 85 \, a^{5} b^{4} + 65 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a^{2} b^{7} + a b^{8}} - \frac{2 \, {\left(2 \, \tan\left(d x + c\right)^{3} + \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)} {\left(a - b\right)}}}{8 \, d}"," ",0,"1/8*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b + sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b + b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8*b - 21*a^7*b^2 + 59*a^6*b^3 - 85*a^5*b^4 + 65*a^4*b^5 - 23*a^3*b^6 + a^2*b^7 + a*b^8) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b - sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b + b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8*b - 21*a^7*b^2 + 59*a^6*b^3 - 85*a^5*b^4 + 65*a^4*b^5 - 23*a^3*b^6 + a^2*b^7 + a*b^8) - 2*(2*tan(d*x + c)^3 + tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)*(a - b)))/d","B",0
221,1,1407,0,1.120025," ","integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","-\frac{\frac{{\left({\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 21 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{2} - a b\right)}^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{2} + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{3} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{5}\right)} {\left| -a^{2} + a b \right|} {\left| -a + b \right|} - 2 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{2} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} - a^{2} b + \sqrt{{\left(a^{3} - a^{2} b\right)}^{2} - {\left(a^{3} - a^{2} b\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{a^{3} - 2 \, a^{2} b + a b^{2}}}}\right)\right)}}{{\left(3 \, a^{10} b - 21 \, a^{9} b^{2} + 59 \, a^{8} b^{3} - 85 \, a^{7} b^{4} + 65 \, a^{6} b^{5} - 23 \, a^{5} b^{6} + a^{4} b^{7} + a^{3} b^{8}\right)} {\left| -a^{2} + a b \right|}} - \frac{{\left({\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 21 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{2} - a b\right)}^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{2} + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{3} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{5}\right)} {\left| -a^{2} + a b \right|} {\left| -a + b \right|} - 2 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{2} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} - a^{2} b - \sqrt{{\left(a^{3} - a^{2} b\right)}^{2} - {\left(a^{3} - a^{2} b\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{a^{3} - 2 \, a^{2} b + a b^{2}}}}\right)\right)}}{{\left(3 \, a^{10} b - 21 \, a^{9} b^{2} + 59 \, a^{8} b^{3} - 85 \, a^{7} b^{4} + 65 \, a^{6} b^{5} - 23 \, a^{5} b^{6} + a^{4} b^{7} + a^{3} b^{8}\right)} {\left| -a^{2} + a b \right|}} + \frac{2 \, {\left(a \tan\left(d x + c\right)^{3} + b \tan\left(d x + c\right)^{3} + a \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)} {\left(a^{2} - a b\right)}}}{8 \, d}"," ",0,"-1/8*(((9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 21*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^2 - a*b)^2*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^5)*abs(-a^2 + a*b)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8 - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^2 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - a^2*b + sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a^3 - 2*a^2*b + a*b^2))))/((3*a^10*b - 21*a^9*b^2 + 59*a^8*b^3 - 85*a^7*b^4 + 65*a^6*b^5 - 23*a^5*b^6 + a^4*b^7 + a^3*b^8)*abs(-a^2 + a*b)) - ((9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 21*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^2 - a*b)^2*abs(-a + b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^5)*abs(-a^2 + a*b)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8 - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^2 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - a^2*b - sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a^3 - 2*a^2*b + a*b^2))))/((3*a^10*b - 21*a^9*b^2 + 59*a^8*b^3 - 85*a^7*b^4 + 65*a^6*b^5 - 23*a^5*b^6 + a^4*b^7 + a^3*b^8)*abs(-a^2 + a*b)) + 2*(a*tan(d*x + c)^3 + b*tan(d*x + c)^3 + a*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)*(a^2 - a*b)))/d","B",0
222,1,1506,0,0.438486," ","integrate(1/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","-\frac{\frac{{\left(2 \, {\left(6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a^{2} - a b\right)}^{2} {\left| -a + b \right|} - {\left(12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} - 57 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b + 92 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{2} - 58 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{3} + 8 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{4} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{5}\right)} {\left| -a^{2} + a b \right|} {\left| -a + b \right|} - {\left(15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} - 69 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b + 106 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{2} - 62 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{3} + 7 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{4} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} - a^{2} b + \sqrt{{\left(a^{3} - a^{2} b\right)}^{2} - {\left(a^{3} - a^{2} b\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{a^{3} - 2 \, a^{2} b + a b^{2}}}}\right)\right)}}{{\left(3 \, a^{10} - 21 \, a^{9} b + 59 \, a^{8} b^{2} - 85 \, a^{7} b^{3} + 65 \, a^{6} b^{4} - 23 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} {\left| -a^{2} + a b \right|}} - \frac{{\left(2 \, {\left(6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a^{2} - a b\right)}^{2} {\left| -a + b \right|} + {\left(12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} - 57 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b + 92 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{2} - 58 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{3} + 8 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{4} + 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{5}\right)} {\left| -a^{2} + a b \right|} {\left| -a + b \right|} - {\left(15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} - 69 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b + 106 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{2} - 62 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{3} + 7 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{4} + 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} - a^{2} b - \sqrt{{\left(a^{3} - a^{2} b\right)}^{2} - {\left(a^{3} - a^{2} b\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{a^{3} - 2 \, a^{2} b + a b^{2}}}}\right)\right)}}{{\left(3 \, a^{10} - 21 \, a^{9} b + 59 \, a^{8} b^{2} - 85 \, a^{7} b^{3} + 65 \, a^{6} b^{4} - 23 \, a^{5} b^{5} + a^{4} b^{6} + a^{3} b^{7}\right)} {\left| -a^{2} + a b \right|}} + \frac{2 \, {\left(2 \, b \tan\left(d x + c\right)^{3} + b \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)} {\left(a^{2} - a b\right)}}}{8 \, d}"," ",0,"-1/8*((2*(6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a^2 - a*b)^2*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6 - 57*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b + 92*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^2 - 58*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^3 + 8*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^4 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^5)*abs(-a^2 + a*b)*abs(-a + b) - (15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7 - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b + 106*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 - 62*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 + 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - a^2*b + sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a^3 - 2*a^2*b + a*b^2))))/((3*a^10 - 21*a^9*b + 59*a^8*b^2 - 85*a^7*b^3 + 65*a^6*b^4 - 23*a^5*b^5 + a^4*b^6 + a^3*b^7)*abs(-a^2 + a*b)) - (2*(6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a^2 - a*b)^2*abs(-a + b) + (12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6 - 57*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b + 92*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^2 - 58*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^3 + 8*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^4 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^5)*abs(-a^2 + a*b)*abs(-a + b) - (15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7 - 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b + 106*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 - 62*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 + 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - a^2*b - sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a^3 - 2*a^2*b + a*b^2))))/((3*a^10 - 21*a^9*b + 59*a^8*b^2 - 85*a^7*b^3 + 65*a^6*b^4 - 23*a^5*b^5 + a^4*b^6 + a^3*b^7)*abs(-a^2 + a*b)) + 2*(2*b*tan(d*x + c)^3 + b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)*(a^2 - a*b)))/d","B",0
223,1,1545,0,1.127968," ","integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^2,x, algorithm=""giac"")","-\frac{\frac{{\left({\left(21 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 57 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + 23 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} - a^{2} b\right)}^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{7} b - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b^{2} + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{3} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{5}\right)} {\left| -a^{3} + a^{2} b \right|} {\left| -a + b \right|} - 2 \, {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} - 42 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b + 66 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{2} - 40 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{3} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{4} + 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} - a^{3} b + \sqrt{{\left(a^{4} - a^{3} b\right)}^{2} - {\left(a^{4} - a^{3} b\right)} {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}}}{a^{4} - 2 \, a^{3} b + a^{2} b^{2}}}}\right)\right)}}{{\left(3 \, a^{12} - 21 \, a^{11} b + 59 \, a^{10} b^{2} - 85 \, a^{9} b^{3} + 65 \, a^{8} b^{4} - 23 \, a^{7} b^{5} + a^{6} b^{6} + a^{5} b^{7}\right)} {\left| -a^{3} + a^{2} b \right|}} - \frac{{\left({\left(21 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 57 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + 23 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} - a^{2} b\right)}^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{7} b - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b^{2} + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{3} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{5}\right)} {\left| -a^{3} + a^{2} b \right|} {\left| -a + b \right|} - 2 \, {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} - 42 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b + 66 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{2} - 40 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{3} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{4} + 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} - a^{3} b - \sqrt{{\left(a^{4} - a^{3} b\right)}^{2} - {\left(a^{4} - a^{3} b\right)} {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}}}{a^{4} - 2 \, a^{3} b + a^{2} b^{2}}}}\right)\right)}}{{\left(3 \, a^{12} - 21 \, a^{11} b + 59 \, a^{10} b^{2} - 85 \, a^{9} b^{3} + 65 \, a^{8} b^{4} - 23 \, a^{7} b^{5} + a^{6} b^{6} + a^{5} b^{7}\right)} {\left| -a^{3} + a^{2} b \right|}} + \frac{2 \, {\left(4 \, a^{2} \tan\left(d x + c\right)^{4} - 7 \, a b \tan\left(d x + c\right)^{4} + 5 \, b^{2} \tan\left(d x + c\right)^{4} + 8 \, a^{2} \tan\left(d x + c\right)^{2} - 7 \, a b \tan\left(d x + c\right)^{2} + 4 \, a^{2} - 4 \, a b\right)}}{{\left(a \tan\left(d x + c\right)^{5} - b \tan\left(d x + c\right)^{5} + 2 \, a \tan\left(d x + c\right)^{3} + a \tan\left(d x + c\right)\right)} {\left(a^{3} - a^{2} b\right)}}}{8 \, d}"," ",0,"-1/8*(((21*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 57*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 23*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - a^2*b)^2*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^5)*abs(-a^3 + a^2*b)*abs(-a + b) - 2*(9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10 - 42*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b + 66*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 - 40*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - a^3*b + sqrt((a^4 - a^3*b)^2 - (a^4 - a^3*b)*(a^4 - 2*a^3*b + a^2*b^2)))/(a^4 - 2*a^3*b + a^2*b^2))))/((3*a^12 - 21*a^11*b + 59*a^10*b^2 - 85*a^9*b^3 + 65*a^8*b^4 - 23*a^7*b^5 + a^6*b^6 + a^5*b^7)*abs(-a^3 + a^2*b)) - ((21*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 57*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 23*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - a^2*b)^2*abs(-a + b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^5)*abs(-a^3 + a^2*b)*abs(-a + b) - 2*(9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10 - 42*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b + 66*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 - 40*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - a^3*b - sqrt((a^4 - a^3*b)^2 - (a^4 - a^3*b)*(a^4 - 2*a^3*b + a^2*b^2)))/(a^4 - 2*a^3*b + a^2*b^2))))/((3*a^12 - 21*a^11*b + 59*a^10*b^2 - 85*a^9*b^3 + 65*a^8*b^4 - 23*a^7*b^5 + a^6*b^6 + a^5*b^7)*abs(-a^3 + a^2*b)) + 2*(4*a^2*tan(d*x + c)^4 - 7*a*b*tan(d*x + c)^4 + 5*b^2*tan(d*x + c)^4 + 8*a^2*tan(d*x + c)^2 - 7*a*b*tan(d*x + c)^2 + 4*a^2 - 4*a*b)/((a*tan(d*x + c)^5 - b*tan(d*x + c)^5 + 2*a*tan(d*x + c)^3 + a*tan(d*x + c))*(a^3 - a^2*b)))/d","B",0
224,-2,0,0,0.000000," ","integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-82,8]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-37,16]-2/d*(5*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^3-11*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*b+12*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a*b^2+35*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^3-85*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^2*b+104*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a*b^2+105*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^3-407*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^2*b+652*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a*b^2-320*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*b^3+175*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3-865*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b+1696*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b^2-1408*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*b^3+175*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^3-849*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b+756*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b^2+320*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*b^3+105*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^3-383*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2*b+248*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b^2+35*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3-77*(1-cos(c+d*x))/(1+cos(c+d*x))*a^2*b-12*(1-cos(c+d*x))/(1+cos(c+d*x))*a*b^2+5*a^3-11*a^2*b)/(-32*a^2*b^2+64*a*b^3-32*b^4)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)^2-2/d/(32*a^2*b^2-64*a*b^3+32*b^4)*2/d*(-(-2*a+5*b)/2*(c+d*x)+(10*a^5*b-82*a^4*b^2-42*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-10*a^4*a*b+15*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+190*a^3*b^3+180*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+82*a^3*b*a*b-39*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-198*a^2*b^4-250*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-190*a^2*b^2*a*b-11*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+80*a*b^5+112*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+198*a*b^3*a*b+51*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+24*b^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-80*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2))-(10*a^5*b-82*a^4*b^2+42*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-10*a^4*a*b+15*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+190*a^3*b^3-180*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+82*a^3*b*a*b-39*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-198*a^2*b^4+250*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-190*a^2*b^2*a*b-11*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+80*a*b^5-112*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+198*a*b^3*a*b+51*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-24*b^5*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-80*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
225,-2,0,0,0.000000," ","integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[61,-66]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[30,-29]-2/d*(-3*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*b+3*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^3-30*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^2*b+16*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^3-111*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^2*b+80*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a*b^2+35*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3-26*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b-64*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b^2+256*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*b^3+40*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^3+95*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b-336*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b^2+25*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^3+54*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2*b-64*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b^2+8*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3+19*(1-cos(c+d*x))/(1+cos(c+d*x))*a^2*b+a^3+2*a^2*b)/(16*a^3*b-32*a^2*b^2+16*a*b^3)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)^2-2/d/(16*a^2*b-32*a*b^2+16*b^3)*2/d*(-(-3*a+9*b)/(4*b*2)*(c+d*x)+((6*a^5*b-42*a^4*b^2-6*a^4*a*b+9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+114*a^3*b^3+42*a^3*b*a*b-63*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-126*a^2*b^4-114*a^2*b^2*a*b+159*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+48*a*b^5+126*a*b^3*a*b-129*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-48*b^4*a*b-24*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)*b^2+(-12*a^5*b^2-9*a^5*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+60*a^4*b^3+54*a^4*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+12*a^4*b*a*b-132*a^3*b^4-132*a^3*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-60*a^3*b^2*a*b+132*a^2*b^5+150*a^2*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+132*a^2*b^3*a*b-48*a*b^6-51*a*b^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-132*a*b^4*a*b-12*b^6*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+48*b^5*a*b)*abs(a-b)*abs(b)+(6*a^5*b^3-18*a^4*b^4-6*a^4*b^2*a*b+9*a^4*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-6*a^3*b^5+18*a^3*b^3*a*b-27*a^3*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+42*a^2*b^6+6*a^2*b^4*a*b-21*a^2*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-24*a*b^7-42*a*b^5*a*b+75*a*b^5*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+24*b^6*a*b+12*b^6*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b))/((48*a^6*b^2-240*a^5*b^3+416*a^4*b^4-288*a^3*b^5+48*a^2*b^6+16*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(16*a*b+sqrt(16*a*b*16*a*b+4*(-8*a*b+8*b^2)*8*a*b))/2/(-8*a*b+8*b^2)))+pi*floor((c+d*x)/pi+1/2))-((6*a^5*b-42*a^4*b^2-6*a^4*a*b+9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+114*a^3*b^3+42*a^3*b*a*b-63*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-126*a^2*b^4-114*a^2*b^2*a*b+159*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+48*a*b^5+126*a*b^3*a*b-129*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-48*b^4*a*b-24*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)*b^2+(-12*a^5*b^2+9*a^5*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+60*a^4*b^3-54*a^4*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+12*a^4*b*a*b-132*a^3*b^4+132*a^3*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-60*a^3*b^2*a*b+132*a^2*b^5-150*a^2*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+132*a^2*b^3*a*b-48*a*b^6+51*a*b^5*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-132*a*b^4*a*b+12*b^6*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+48*b^5*a*b)*abs(a-b)*abs(b)+(6*a^5*b^3-18*a^4*b^4-6*a^4*b^2*a*b+9*a^4*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-6*a^3*b^5+18*a^3*b^3*a*b-27*a^3*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+42*a^2*b^6+6*a^2*b^4*a*b-21*a^2*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-24*a*b^7-42*a*b^5*a*b+75*a*b^5*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+24*b^6*a*b+12*b^6*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b))/((48*a^6*b^2-240*a^5*b^3+416*a^4*b^4-288*a^3*b^5+48*a^2*b^6+16*a*b^7)*abs(b))*(atan(tan(c+d*x)/sqrt(-(16*a*b-sqrt(16*a*b*16*a*b+4*(-8*a*b+8*b^2)*8*a*b))/2/(-8*a*b+8*b^2)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
226,-2,0,0,0.000000," ","integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[40,31]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[7,-19]-2/d*(-3*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^3+13*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*b-4*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a*b^2-21*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^3+99*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^2*b-24*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a*b^2-63*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^3+225*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^2*b-68*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a*b^2-64*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*b^3-105*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3+183*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b-96*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b^2-384*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*b^3-105*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^3-9*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b+452*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b^2+64*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*b^3-63*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^3-87*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2*b+120*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b^2-21*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3-37*(1-cos(c+d*x))/(1+cos(c+d*x))*a^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a*b^2-3*a^3-3*a^2*b)/(-32*a^3*b+64*a^2*b^2-32*a*b^3)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)^2-2/d/(32*a^3*b-64*a^2*b^2+32*a*b^3)*2/d*(-(2*a+b)/2*(c+d*x)+(-6*a^5*b+62*a^4*b^2+30*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+6*a^4*a*b-9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-178*a^3*b^3-132*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-62*a^3*b*a*b+33*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+170*a^2*b^4+158*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+178*a^2*b^2*a*b-51*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-48*a*b^5-24*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-170*a*b^3*a*b+43*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-8*b^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+48*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a+sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2))-(-6*a^5*b+62*a^4*b^2-30*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+6*a^4*a*b-9*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-178*a^3*b^3+132*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-62*a^3*b*a*b+33*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+170*a^2*b^4-158*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+178*a^2*b^2*a*b-51*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-48*a*b^5+24*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-170*a*b^3*a*b+43*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+8*b^5*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+48*b^4*a*b+8*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(24*a^5*b-96*a^4*b^2+112*a^3*b^3-32*a^2*b^4-8*a*b^5)*(atan(tan(c+d*x)/sqrt(-(8*a-sqrt(8*a*8*a+4*(-4*a+4*b)*4*a))/2/(-4*a+4*b)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
227,1,1076,0,1.861199," ","integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","-\frac{\frac{5 \, a b \cos\left(d x + c\right)^{7}}{d} + \frac{b^{2} \cos\left(d x + c\right)^{7}}{d} - \frac{21 \, a b \cos\left(d x + c\right)^{5}}{d} - \frac{3 \, b^{2} \cos\left(d x + c\right)^{5}}{d} - \frac{9 \, a^{2} \cos\left(d x + c\right)^{3}}{d} + \frac{30 \, a b \cos\left(d x + c\right)^{3}}{d} + \frac{3 \, b^{2} \cos\left(d x + c\right)^{3}}{d} + \frac{19 \, a^{2} \cos\left(d x + c\right)}{d} - \frac{18 \, a b \cos\left(d x + c\right)}{d} - \frac{b^{2} \cos\left(d x + c\right)}{d}}{32 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} - a + b\right)}^{2} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}} - \frac{{\left(2 \, {\left(4 \, a^{6} b - 17 \, a^{5} b^{2} + 28 \, a^{4} b^{3} - 22 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - a b^{6}\right)} \sqrt{a b} \sqrt{-b^{2} + \sqrt{a b} b} d^{4} + {\left(13 \, a^{4} b - 27 \, a^{3} b^{2} + 15 \, a^{2} b^{3} - a b^{4}\right)} \sqrt{-b^{2} + \sqrt{a b} b} d^{2} {\left| a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right|} + {\left(a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2}\right)}^{2} \sqrt{a b} \sqrt{-b^{2} + \sqrt{a b} b} {\left(5 \, a + b\right)}\right)} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2} - \sqrt{{\left(a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2}\right)}^{2} + {\left(a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}\right)} {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)}}}{a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}}}}\right)}{64 \, {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{3} {\left| a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right|} {\left| b \right|}} + \frac{{\left(2 \, {\left(4 \, a^{6} b - 17 \, a^{5} b^{2} + 28 \, a^{4} b^{3} - 22 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - a b^{6}\right)} \sqrt{-b^{2} - \sqrt{a b} b} d^{4} - {\left(13 \, a^{3} - 27 \, a^{2} b + 15 \, a b^{2} - b^{3}\right)} \sqrt{a b} \sqrt{-b^{2} - \sqrt{a b} b} d^{2} {\left| a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right|} + {\left(a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2}\right)}^{2} \sqrt{-b^{2} - \sqrt{a b} b} {\left(5 \, a + b\right)}\right)} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2} + \sqrt{{\left(a^{3} b d^{2} - 2 \, a^{2} b^{2} d^{2} + a b^{3} d^{2}\right)}^{2} + {\left(a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}\right)} {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)}}}{a^{3} b d^{4} - 2 \, a^{2} b^{2} d^{4} + a b^{3} d^{4}}}}\right)}{64 \, {\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)} \sqrt{a b} d^{3} {\left| a^{3} d^{2} - 2 \, a^{2} b d^{2} + a b^{2} d^{2} \right|} {\left| b \right|}}"," ",0,"-1/32*(5*a*b*cos(d*x + c)^7/d + b^2*cos(d*x + c)^7/d - 21*a*b*cos(d*x + c)^5/d - 3*b^2*cos(d*x + c)^5/d - 9*a^2*cos(d*x + c)^3/d + 30*a*b*cos(d*x + c)^3/d + 3*b^2*cos(d*x + c)^3/d + 19*a^2*cos(d*x + c)/d - 18*a*b*cos(d*x + c)/d - b^2*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)^2*(a^3 - 2*a^2*b + a*b^2)) - 1/64*(2*(4*a^6*b - 17*a^5*b^2 + 28*a^4*b^3 - 22*a^3*b^4 + 8*a^2*b^5 - a*b^6)*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*d^4 + (13*a^4*b - 27*a^3*b^2 + 15*a^2*b^3 - a*b^4)*sqrt(-b^2 + sqrt(a*b)*b)*d^2*abs(a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2) + (a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2)^2*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*(5*a + b))*arctan(cos(d*x + c)/(d*sqrt(-(a^3*b*d^2 - 2*a^2*b^2*d^2 + a*b^3*d^2 - sqrt((a^3*b*d^2 - 2*a^2*b^2*d^2 + a*b^3*d^2)^2 + (a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)))/(a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4))))/((a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^3*abs(a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2)*abs(b)) + 1/64*(2*(4*a^6*b - 17*a^5*b^2 + 28*a^4*b^3 - 22*a^3*b^4 + 8*a^2*b^5 - a*b^6)*sqrt(-b^2 - sqrt(a*b)*b)*d^4 - (13*a^3 - 27*a^2*b + 15*a*b^2 - b^3)*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*d^2*abs(a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2) + (a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2)^2*sqrt(-b^2 - sqrt(a*b)*b)*(5*a + b))*arctan(cos(d*x + c)/(d*sqrt(-(a^3*b*d^2 - 2*a^2*b^2*d^2 + a*b^3*d^2 + sqrt((a^3*b*d^2 - 2*a^2*b^2*d^2 + a*b^3*d^2)^2 + (a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)))/(a^3*b*d^4 - 2*a^2*b^2*d^4 + a*b^3*d^4))))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*sqrt(a*b)*d^3*abs(a^3*d^2 - 2*a^2*b*d^2 + a*b^2*d^2)*abs(b))","B",0
228,1,766,0,1.753893," ","integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","-\frac{3 \, {\left(7 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} - {\left(11 \, a^{2} - 11 \, a b + 4 \, b^{2}\right)} \sqrt{a b}\right)} \sqrt{-b^{2} - \sqrt{a b} b} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2} + \sqrt{{\left(a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2}\right)}^{2} + {\left(a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}\right)} {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)}}}{a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}}}}\right)}{64 \, {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d {\left| b \right|}} - \frac{3 \, {\left(7 \, a^{3} - 5 \, a^{2} b + 2 \, a b^{2} + {\left(11 \, a^{2} - 11 \, a b + 4 \, b^{2}\right)} \sqrt{a b}\right)} \sqrt{-b^{2} + \sqrt{a b} b} \arctan\left(\frac{\cos\left(d x + c\right)}{d \sqrt{-\frac{a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2} - \sqrt{{\left(a^{4} b d^{2} - 2 \, a^{3} b^{2} d^{2} + a^{2} b^{3} d^{2}\right)}^{2} + {\left(a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}\right)} {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)}}}{a^{4} b d^{4} - 2 \, a^{3} b^{2} d^{4} + a^{2} b^{3} d^{4}}}}\right)}{64 \, {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d {\left| b \right|}} - \frac{\frac{12 \, a b^{2} \cos\left(d x + c\right)^{7}}{d} - \frac{6 \, b^{3} \cos\left(d x + c\right)^{7}}{d} - \frac{7 \, a^{2} b \cos\left(d x + c\right)^{5}}{d} - \frac{35 \, a b^{2} \cos\left(d x + c\right)^{5}}{d} + \frac{18 \, b^{3} \cos\left(d x + c\right)^{5}}{d} - \frac{2 \, a^{2} b \cos\left(d x + c\right)^{3}}{d} + \frac{44 \, a b^{2} \cos\left(d x + c\right)^{3}}{d} - \frac{18 \, b^{3} \cos\left(d x + c\right)^{3}}{d} + \frac{11 \, a^{3} \cos\left(d x + c\right)}{d} + \frac{4 \, a^{2} b \cos\left(d x + c\right)}{d} - \frac{21 \, a b^{2} \cos\left(d x + c\right)}{d} + \frac{6 \, b^{3} \cos\left(d x + c\right)}{d}}{32 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} - a + b\right)}^{2} {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}}"," ",0,"-3/64*(7*a^3 - 5*a^2*b + 2*a*b^2 - (11*a^2 - 11*a*b + 4*b^2)*sqrt(a*b))*sqrt(-b^2 - sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2 + sqrt((a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2)^2 + (a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4))))/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d*abs(b)) - 3/64*(7*a^3 - 5*a^2*b + 2*a*b^2 + (11*a^2 - 11*a*b + 4*b^2)*sqrt(a*b))*sqrt(-b^2 + sqrt(a*b)*b)*arctan(cos(d*x + c)/(d*sqrt(-(a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2 - sqrt((a^4*b*d^2 - 2*a^3*b^2*d^2 + a^2*b^3*d^2)^2 + (a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^4*b*d^4 - 2*a^3*b^2*d^4 + a^2*b^3*d^4))))/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d*abs(b)) - 1/32*(12*a*b^2*cos(d*x + c)^7/d - 6*b^3*cos(d*x + c)^7/d - 7*a^2*b*cos(d*x + c)^5/d - 35*a*b^2*cos(d*x + c)^5/d + 18*b^3*cos(d*x + c)^5/d - 2*a^2*b*cos(d*x + c)^3/d + 44*a*b^2*cos(d*x + c)^3/d - 18*b^3*cos(d*x + c)^3/d + 11*a^3*cos(d*x + c)/d + 4*a^2*b*cos(d*x + c)/d - 21*a*b^2*cos(d*x + c)/d + 6*b^3*cos(d*x + c)/d)/((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - a + b)^2*(a^4 - 2*a^3*b + a^2*b^2))","B",0
229,-2,0,0,0.000000," ","integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[53,-89]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-57,87]Precision problem choosing root in common_EXT, current precision 14-2/d*((-8*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^3*b+5*((1-cos(c+d*x))/(1+cos(c+d*x)))^7*a^2*b^2-5*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^3*b-86*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a^2*b^2+64*((1-cos(c+d*x))/(1+cos(c+d*x)))^6*a*b^3+104*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^3*b-327*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a^2*b^2+208*((1-cos(c+d*x))/(1+cos(c+d*x)))^5*a*b^3+315*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^3*b-1074*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a^2*b^2+1728*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a*b^3-768*((1-cos(c+d*x))/(1+cos(c+d*x)))^4*b^4+400*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^3*b-1161*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a^2*b^2+560*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a*b^3+257*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^3*b-370*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a^2*b^2+128*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a*b^3+80*(1-cos(c+d*x))/(1+cos(c+d*x))*a^3*b-53*(1-cos(c+d*x))/(1+cos(c+d*x))*a^2*b^2+9*a^3*b-6*a^2*b^2)/(16*a^5-32*a^4*b+16*a^3*b^2)/(((1-cos(c+d*x))/(1+cos(c+d*x)))^4*a+4*((1-cos(c+d*x))/(1+cos(c+d*x)))^3*a+6*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*a-16*((1-cos(c+d*x))/(1+cos(c+d*x)))^2*b+4*(1-cos(c+d*x))/(1+cos(c+d*x))*a+a)^2-1/4/a^3*ln(abs(1-cos(c+d*x))/abs(1+cos(c+d*x))))-2/d/(16*a^5-32*a^4*b+16*a^3*b^2)*2/d*((-32*b^2+71*b*a-45*a^2)/8*(c+d*x)+(32*a^5*b+135*a^5*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-32*a^4*b^2-579*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-32*a^4*a*b-222*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-184*a^3*b^3+729*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+32*a^3*b*a*b+840*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+312*a^2*b^4-209*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+184*a^2*b^2*a*b-910*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-128*a*b^5-52*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(a-b))-312*a*b^3*a*b+252*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b))+128*b^4*a*b+64*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(a-b)))*abs(a-b)/(48*a^5-192*a^4*b+224*a^3*b^2-64*a^2*b^3-16*a*b^4)*(atan(tan(c+d*x)/sqrt(-(16*a+sqrt(16*a*16*a+4*(8*b-8*a)*8*a))/2/(8*b-8*a)))+pi*floor((c+d*x)/pi+1/2))-(32*a^5*b-135*a^5*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-32*a^4*b^2+579*a^4*b*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-32*a^4*a*b-222*a^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-184*a^3*b^3-729*a^3*b^2*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+32*a^3*b*a*b+840*a^3*b*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+312*a^2*b^4+209*a^2*b^3*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+184*a^2*b^2*a*b-910*a^2*b^2*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-128*a*b^5+52*a*b^4*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))-312*a*b^3*a*b+252*a*b^3*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b))+128*b^4*a*b+64*b^4*sqrt(a*b)*sqrt(a^2-a*b+sqrt(a*b)*(-a+b)))*abs(a-b)/(48*a^5-192*a^4*b+224*a^3*b^2-64*a^2*b^3-16*a*b^4)*(atan(tan(c+d*x)/sqrt(-(16*a-sqrt(16*a*16*a+4*(8*b-8*a)*8*a))/2/(8*b-8*a)))+pi*floor((c+d*x)/pi+1/2)))","F(-2)",0
230,1,1989,0,2.162444," ","integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\frac{\frac{{\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 45 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 77 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 13 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)}^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b - 49 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{3} + 112 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{4} - 87 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{5} + 16 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{6} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{7}\right)} {\left| a^{2} b - 2 \, a b^{2} + b^{3} \right|} {\left| -a + b \right|} - {\left(6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b - 63 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{2} + 229 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{3} - 367 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{4} + 233 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{5} + 27 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{6} - 89 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{7} + 19 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{8} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{9}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} b - 2 \, a^{2} b^{2} + a b^{3} + \sqrt{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)}^{2} - {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)}}}{a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}}}}\right)\right)}}{{\left(3 \, a^{10} b^{2} - 27 \, a^{9} b^{3} + 104 \, a^{8} b^{4} - 224 \, a^{7} b^{5} + 294 \, a^{6} b^{6} - 238 \, a^{5} b^{7} + 112 \, a^{4} b^{8} - 24 \, a^{3} b^{9} - a^{2} b^{10} + a b^{11}\right)} {\left| a^{2} b - 2 \, a b^{2} + b^{3} \right|}} - \frac{{\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 45 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 77 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 13 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)}^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b - 49 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{3} + 112 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{4} - 87 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{5} + 16 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{6} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{7}\right)} {\left| a^{2} b - 2 \, a b^{2} + b^{3} \right|} {\left| -a + b \right|} - {\left(6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b - 63 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{2} + 229 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{3} - 367 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{4} + 233 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{5} + 27 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{6} - 89 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{7} + 19 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{8} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{9}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{3} b - 2 \, a^{2} b^{2} + a b^{3} - \sqrt{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)}^{2} - {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)}}}{a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}}}}\right)\right)}}{{\left(3 \, a^{10} b^{2} - 27 \, a^{9} b^{3} + 104 \, a^{8} b^{4} - 224 \, a^{7} b^{5} + 294 \, a^{6} b^{6} - 238 \, a^{5} b^{7} + 112 \, a^{4} b^{8} - 24 \, a^{3} b^{9} - a^{2} b^{10} + a b^{11}\right)} {\left| a^{2} b - 2 \, a b^{2} + b^{3} \right|}} - \frac{2 \, {\left(a^{2} \tan\left(d x + c\right)^{7} + 18 \, a b \tan\left(d x + c\right)^{7} - 19 \, b^{2} \tan\left(d x + c\right)^{7} + 3 \, a^{2} \tan\left(d x + c\right)^{5} + 30 \, a b \tan\left(d x + c\right)^{5} - 9 \, b^{2} \tan\left(d x + c\right)^{5} + 3 \, a^{2} \tan\left(d x + c\right)^{3} + 21 \, a b \tan\left(d x + c\right)^{3} + a^{2} \tan\left(d x + c\right) + 5 \, a b \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)}^{2} {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)}}}{64 \, d}"," ",0,"1/64*(((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 45*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 77*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 13*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a^2*b - 2*a*b^2 + b^3)^2*abs(-a + b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b - 49*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^3 + 112*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^4 - 87*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^7)*abs(a^2*b - 2*a*b^2 + b^3)*abs(-a + b) - (6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b - 63*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^2 + 229*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^3 - 367*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 + 233*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 27*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 89*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3*b - 2*a^2*b^2 + a*b^3 + sqrt((a^3*b - 2*a^2*b^2 + a*b^3)^2 - (a^3*b - 2*a^2*b^2 + a*b^3)*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)))/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4))))/((3*a^10*b^2 - 27*a^9*b^3 + 104*a^8*b^4 - 224*a^7*b^5 + 294*a^6*b^6 - 238*a^5*b^7 + 112*a^4*b^8 - 24*a^3*b^9 - a^2*b^10 + a*b^11)*abs(a^2*b - 2*a*b^2 + b^3)) - ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 45*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 77*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 13*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a^2*b - 2*a*b^2 + b^3)^2*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b - 49*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^3 + 112*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^4 - 87*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^7)*abs(a^2*b - 2*a*b^2 + b^3)*abs(-a + b) - (6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b - 63*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^2 + 229*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^3 - 367*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 + 233*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 27*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 89*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3*b - 2*a^2*b^2 + a*b^3 - sqrt((a^3*b - 2*a^2*b^2 + a*b^3)^2 - (a^3*b - 2*a^2*b^2 + a*b^3)*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)))/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4))))/((3*a^10*b^2 - 27*a^9*b^3 + 104*a^8*b^4 - 224*a^7*b^5 + 294*a^6*b^6 - 238*a^5*b^7 + 112*a^4*b^8 - 24*a^3*b^9 - a^2*b^10 + a*b^11)*abs(a^2*b - 2*a*b^2 + b^3)) - 2*(a^2*tan(d*x + c)^7 + 18*a*b*tan(d*x + c)^7 - 19*b^2*tan(d*x + c)^7 + 3*a^2*tan(d*x + c)^5 + 30*a*b*tan(d*x + c)^5 - 9*b^2*tan(d*x + c)^5 + 3*a^2*tan(d*x + c)^3 + 21*a*b*tan(d*x + c)^3 + a^2*tan(d*x + c) + 5*a*b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^2*b - 2*a*b^2 + b^3)))/d","B",0
231,1,2231,0,2.236785," ","integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\frac{\frac{{\left({\left(6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} - 63 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b + 109 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)}^{2} {\left| -a + b \right|} + 2 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{8} b - 9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{7} b^{2} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b^{3} + 34 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{4} - 33 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{5} + 7 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{6} + 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{7}\right)} {\left| a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right|} {\left| -a + b \right|} - {\left(12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{11} b - 117 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} b^{2} + 431 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b^{3} - 773 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{4} + 703 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{5} - 279 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{6} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{7} + 17 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{8} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{9}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + \sqrt{{\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)}^{2} - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)}}}{a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}}}}\right)\right)}}{{\left(3 \, a^{12} b^{2} - 27 \, a^{11} b^{3} + 104 \, a^{10} b^{4} - 224 \, a^{9} b^{5} + 294 \, a^{8} b^{6} - 238 \, a^{7} b^{7} + 112 \, a^{6} b^{8} - 24 \, a^{5} b^{9} - a^{4} b^{10} + a^{3} b^{11}\right)} {\left| a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right|}} - \frac{{\left({\left(6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} - 63 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b + 109 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)}^{2} {\left| -a + b \right|} - 2 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{8} b - 9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{7} b^{2} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b^{3} + 34 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{4} - 33 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{5} + 7 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{6} + 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{7}\right)} {\left| a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right|} {\left| -a + b \right|} - {\left(12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{11} b - 117 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} b^{2} + 431 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b^{3} - 773 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{4} + 703 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{5} - 279 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{6} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{7} + 17 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{8} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{9}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} - \sqrt{{\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)}^{2} - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)}}}{a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}}}}\right)\right)}}{{\left(3 \, a^{12} b^{2} - 27 \, a^{11} b^{3} + 104 \, a^{10} b^{4} - 224 \, a^{9} b^{5} + 294 \, a^{8} b^{6} - 238 \, a^{7} b^{7} + 112 \, a^{6} b^{8} - 24 \, a^{5} b^{9} - a^{4} b^{10} + a^{3} b^{11}\right)} {\left| a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right|}} - \frac{2 \, {\left(2 \, a^{3} \tan\left(d x + c\right)^{7} + 13 \, a^{2} b \tan\left(d x + c\right)^{7} - 12 \, a b^{2} \tan\left(d x + c\right)^{7} - 3 \, b^{3} \tan\left(d x + c\right)^{7} + 6 \, a^{3} \tan\left(d x + c\right)^{5} + 28 \, a^{2} b \tan\left(d x + c\right)^{5} - 10 \, a b^{2} \tan\left(d x + c\right)^{5} + 6 \, a^{3} \tan\left(d x + c\right)^{3} + 19 \, a^{2} b \tan\left(d x + c\right)^{3} - a b^{2} \tan\left(d x + c\right)^{3} + 2 \, a^{3} \tan\left(d x + c\right) + 4 \, a^{2} b \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)}^{2} {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)}}}{64 \, d}"," ",0,"1/64*(((6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 - 63*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 109*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3*b - 2*a^2*b^2 + a*b^3)^2*abs(-a + b) + 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b^2 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^3 + 34*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^4 - 33*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^5 + 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^6 + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^7)*abs(a^3*b - 2*a^2*b^2 + a*b^3)*abs(-a + b) - (12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b - 117*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^2 + 431*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^3 - 773*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^4 + 703*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^5 - 279*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^7 + 17*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^8 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3 + sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3)^2 - (a^4*b - 2*a^3*b^2 + a^2*b^3)*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)))/(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4))))/((3*a^12*b^2 - 27*a^11*b^3 + 104*a^10*b^4 - 224*a^9*b^5 + 294*a^8*b^6 - 238*a^7*b^7 + 112*a^6*b^8 - 24*a^5*b^9 - a^4*b^10 + a^3*b^11)*abs(a^3*b - 2*a^2*b^2 + a*b^3)) - ((6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 - 63*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 109*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3*b - 2*a^2*b^2 + a*b^3)^2*abs(-a + b) - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^8*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b^2 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^3 + 34*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^4 - 33*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^5 + 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^6 + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^7)*abs(a^3*b - 2*a^2*b^2 + a*b^3)*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b - 117*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^2 + 431*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^3 - 773*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^4 + 703*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^5 - 279*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^7 + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^8 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3 - sqrt((a^4*b - 2*a^3*b^2 + a^2*b^3)^2 - (a^4*b - 2*a^3*b^2 + a^2*b^3)*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)))/(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4))))/((3*a^12*b^2 - 27*a^11*b^3 + 104*a^10*b^4 - 224*a^9*b^5 + 294*a^8*b^6 - 238*a^7*b^7 + 112*a^6*b^8 - 24*a^5*b^9 - a^4*b^10 + a^3*b^11)*abs(a^3*b - 2*a^2*b^2 + a*b^3)) - 2*(2*a^3*tan(d*x + c)^7 + 13*a^2*b*tan(d*x + c)^7 - 12*a*b^2*tan(d*x + c)^7 - 3*b^3*tan(d*x + c)^7 + 6*a^3*tan(d*x + c)^5 + 28*a^2*b*tan(d*x + c)^5 - 10*a*b^2*tan(d*x + c)^5 + 6*a^3*tan(d*x + c)^3 + 19*a^2*b*tan(d*x + c)^3 - a*b^2*tan(d*x + c)^3 + 2*a^3*tan(d*x + c) + 4*a^2*b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^3*b - 2*a^2*b^2 + a*b^3)))/d","B",0
232,1,1986,0,2.225293," ","integrate(sin(d*x+c)^4/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","-\frac{\frac{3 \, {\left({\left(15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 33 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}^{2} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{7} b - 48 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b^{2} + 93 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{3} - 80 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{4} + 27 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{5} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{7}\right)} {\left| a^{3} - 2 \, a^{2} b + a b^{2} \right|} {\left| -a + b \right|} - {\left(6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} - 27 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b + 25 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{2} + 53 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{3} - 131 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{4} + 103 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{5} - 29 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{6} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{7} + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{8}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} - 2 \, a^{3} b + a^{2} b^{2} + \sqrt{{\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)}}}{a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}}}}\right)\right)}}{{\left(3 \, a^{12} b - 27 \, a^{11} b^{2} + 104 \, a^{10} b^{3} - 224 \, a^{9} b^{4} + 294 \, a^{8} b^{5} - 238 \, a^{7} b^{6} + 112 \, a^{6} b^{7} - 24 \, a^{5} b^{8} - a^{4} b^{9} + a^{3} b^{10}\right)} {\left| a^{3} - 2 \, a^{2} b + a b^{2} \right|}} - \frac{3 \, {\left({\left(15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 33 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}^{2} {\left| -a + b \right|} + {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{7} b - 48 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b^{2} + 93 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{3} - 80 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{4} + 27 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{5} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{7}\right)} {\left| a^{3} - 2 \, a^{2} b + a b^{2} \right|} {\left| -a + b \right|} - {\left(6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} - 27 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b + 25 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{2} + 53 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{3} - 131 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{4} + 103 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{5} - 29 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{6} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{7} + \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{8}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} - 2 \, a^{3} b + a^{2} b^{2} - \sqrt{{\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)}}}{a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}}}}\right)\right)}}{{\left(3 \, a^{12} b - 27 \, a^{11} b^{2} + 104 \, a^{10} b^{3} - 224 \, a^{9} b^{4} + 294 \, a^{8} b^{5} - 238 \, a^{7} b^{6} + 112 \, a^{6} b^{7} - 24 \, a^{5} b^{8} - a^{4} b^{9} + a^{3} b^{10}\right)} {\left| a^{3} - 2 \, a^{2} b + a b^{2} \right|}} + \frac{2 \, {\left(17 \, a^{2} \tan\left(d x + c\right)^{7} - 14 \, a b \tan\left(d x + c\right)^{7} - 3 \, b^{2} \tan\left(d x + c\right)^{7} + 43 \, a^{2} \tan\left(d x + c\right)^{5} - 18 \, a b \tan\left(d x + c\right)^{5} - b^{2} \tan\left(d x + c\right)^{5} + 35 \, a^{2} \tan\left(d x + c\right)^{3} - 11 \, a b \tan\left(d x + c\right)^{3} + 9 \, a^{2} \tan\left(d x + c\right) - 3 \, a b \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)}^{2} {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)}}}{64 \, d}"," ",0,"-1/64*(3*((15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 33*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 2*a^2*b + a*b^2)^2*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b - 48*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^2 + 93*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^3 - 80*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^4 + 27*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^5 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^7)*abs(a^3 - 2*a^2*b + a*b^2)*abs(-a + b) - (6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10 - 27*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b + 25*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 53*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 131*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 103*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 29*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^6 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 2*a^3*b + a^2*b^2 + sqrt((a^4 - 2*a^3*b + a^2*b^2)^2 - (a^4 - 2*a^3*b + a^2*b^2)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)))/(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3))))/((3*a^12*b - 27*a^11*b^2 + 104*a^10*b^3 - 224*a^9*b^4 + 294*a^8*b^5 - 238*a^7*b^6 + 112*a^6*b^7 - 24*a^5*b^8 - a^4*b^9 + a^3*b^10)*abs(a^3 - 2*a^2*b + a*b^2)) - 3*((15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 33*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 2*a^2*b + a*b^2)^2*abs(-a + b) + (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b - 48*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^2 + 93*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^3 - 80*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^4 + 27*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^5 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^7)*abs(a^3 - 2*a^2*b + a*b^2)*abs(-a + b) - (6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10 - 27*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b + 25*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 53*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 131*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 103*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 29*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^6 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 2*a^3*b + a^2*b^2 - sqrt((a^4 - 2*a^3*b + a^2*b^2)^2 - (a^4 - 2*a^3*b + a^2*b^2)*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)))/(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3))))/((3*a^12*b - 27*a^11*b^2 + 104*a^10*b^3 - 224*a^9*b^4 + 294*a^8*b^5 - 238*a^7*b^6 + 112*a^6*b^7 - 24*a^5*b^8 - a^4*b^9 + a^3*b^10)*abs(a^3 - 2*a^2*b + a*b^2)) + 2*(17*a^2*tan(d*x + c)^7 - 14*a*b*tan(d*x + c)^7 - 3*b^2*tan(d*x + c)^7 + 43*a^2*tan(d*x + c)^5 - 18*a*b*tan(d*x + c)^5 - b^2*tan(d*x + c)^5 + 35*a^2*tan(d*x + c)^3 - 11*a*b*tan(d*x + c)^3 + 9*a^2*tan(d*x + c) - 3*a*b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^3 - 2*a^2*b + a*b^2)))/d","B",0
233,1,1184,0,2.291100," ","integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\frac{\frac{{\left(30 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b - 72 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} + 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{4} + 36 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} - 105 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b + 69 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 19 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{5} - 2 \, a^{4} b + a^{3} b^{2} + \sqrt{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)}^{2} - {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)}}}{a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{9} b - 18 \, a^{8} b^{2} + 41 \, a^{7} b^{3} - 44 \, a^{6} b^{4} + 21 \, a^{5} b^{5} - 2 \, a^{4} b^{6} - a^{3} b^{7}} + \frac{{\left(30 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b - 72 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} + 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{4} - 36 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} + 105 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 69 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + 19 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{5} - 2 \, a^{4} b + a^{3} b^{2} - \sqrt{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)}^{2} - {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)}}}{a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{9} b - 18 \, a^{8} b^{2} + 41 \, a^{7} b^{3} - 44 \, a^{6} b^{4} + 21 \, a^{5} b^{5} - 2 \, a^{4} b^{6} - a^{3} b^{7}} - \frac{2 \, {\left(10 \, a^{3} \tan\left(d x + c\right)^{7} + 5 \, a^{2} b \tan\left(d x + c\right)^{7} - 20 \, a b^{2} \tan\left(d x + c\right)^{7} + 5 \, b^{3} \tan\left(d x + c\right)^{7} + 30 \, a^{3} \tan\left(d x + c\right)^{5} + 12 \, a^{2} b \tan\left(d x + c\right)^{5} - 18 \, a b^{2} \tan\left(d x + c\right)^{5} + 30 \, a^{3} \tan\left(d x + c\right)^{3} + 3 \, a^{2} b \tan\left(d x + c\right)^{3} - 9 \, a b^{2} \tan\left(d x + c\right)^{3} + 10 \, a^{3} \tan\left(d x + c\right) - 4 \, a^{2} b \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)}^{2} {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}}}{64 \, d}"," ",0,"1/64*((30*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 72*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 + 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 + 36*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 - 105*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^5 - 2*a^4*b + a^3*b^2 + sqrt((a^5 - 2*a^4*b + a^3*b^2)^2 - (a^5 - 2*a^4*b + a^3*b^2)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3))))*abs(-a + b)/(3*a^9*b - 18*a^8*b^2 + 41*a^7*b^3 - 44*a^6*b^4 + 21*a^5*b^5 - 2*a^4*b^6 - a^3*b^7) + (30*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 72*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 + 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - 36*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 + 105*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^5 - 2*a^4*b + a^3*b^2 - sqrt((a^5 - 2*a^4*b + a^3*b^2)^2 - (a^5 - 2*a^4*b + a^3*b^2)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3))))*abs(-a + b)/(3*a^9*b - 18*a^8*b^2 + 41*a^7*b^3 - 44*a^6*b^4 + 21*a^5*b^5 - 2*a^4*b^6 - a^3*b^7) - 2*(10*a^3*tan(d*x + c)^7 + 5*a^2*b*tan(d*x + c)^7 - 20*a*b^2*tan(d*x + c)^7 + 5*b^3*tan(d*x + c)^7 + 30*a^3*tan(d*x + c)^5 + 12*a^2*b*tan(d*x + c)^5 - 18*a*b^2*tan(d*x + c)^5 + 30*a^3*tan(d*x + c)^3 + 3*a^2*b*tan(d*x + c)^3 - 9*a*b^2*tan(d*x + c)^3 + 10*a^3*tan(d*x + c) - 4*a^2*b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^4 - 2*a^3*b + a^2*b^2)))/d","B",0
234,1,1131,0,0.675798," ","integrate(1/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","\frac{\frac{{\left(96 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} - 333 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b + 313 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 79 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{3} - 21 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{4} + 42 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 108 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 34 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 8 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{5} - 2 \, a^{4} b + a^{3} b^{2} + \sqrt{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)}^{2} - {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)}}}{a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{9} - 18 \, a^{8} b + 41 \, a^{7} b^{2} - 44 \, a^{6} b^{3} + 21 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}} + \frac{{\left(96 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} - 333 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b + 313 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 79 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{3} - 21 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{4} - 42 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} + 108 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 34 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - 8 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{5} - 2 \, a^{4} b + a^{3} b^{2} - \sqrt{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)}^{2} - {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)}}}{a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{9} - 18 \, a^{8} b + 41 \, a^{7} b^{2} - 44 \, a^{6} b^{3} + 21 \, a^{5} b^{4} - 2 \, a^{4} b^{5} - a^{3} b^{6}} - \frac{2 \, {\left(33 \, a^{2} b \tan\left(d x + c\right)^{7} - 46 \, a b^{2} \tan\left(d x + c\right)^{7} + 13 \, b^{3} \tan\left(d x + c\right)^{7} + 83 \, a^{2} b \tan\left(d x + c\right)^{5} - 66 \, a b^{2} \tan\left(d x + c\right)^{5} + 7 \, b^{3} \tan\left(d x + c\right)^{5} + 67 \, a^{2} b \tan\left(d x + c\right)^{3} - 43 \, a b^{2} \tan\left(d x + c\right)^{3} + 17 \, a^{2} b \tan\left(d x + c\right) - 11 \, a b^{2} \tan\left(d x + c\right)\right)}}{{\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)}^{2} {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)}}}{64 \, d}"," ",0,"1/64*((96*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4 - 333*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b + 313*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 79*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - 21*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4 + 42*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 108*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 34*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 8*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^5 - 2*a^4*b + a^3*b^2 + sqrt((a^5 - 2*a^4*b + a^3*b^2)^2 - (a^5 - 2*a^4*b + a^3*b^2)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3))))*abs(-a + b)/(3*a^9 - 18*a^8*b + 41*a^7*b^2 - 44*a^6*b^3 + 21*a^5*b^4 - 2*a^4*b^5 - a^3*b^6) + (96*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4 - 333*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b + 313*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 79*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - 21*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4 - 42*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 + 108*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 34*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - 8*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^5 - 2*a^4*b + a^3*b^2 - sqrt((a^5 - 2*a^4*b + a^3*b^2)^2 - (a^5 - 2*a^4*b + a^3*b^2)*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)))/(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3))))*abs(-a + b)/(3*a^9 - 18*a^8*b + 41*a^7*b^2 - 44*a^6*b^3 + 21*a^5*b^4 - 2*a^4*b^5 - a^3*b^6) - 2*(33*a^2*b*tan(d*x + c)^7 - 46*a*b^2*tan(d*x + c)^7 + 13*b^3*tan(d*x + c)^7 + 83*a^2*b*tan(d*x + c)^5 - 66*a*b^2*tan(d*x + c)^5 + 7*b^3*tan(d*x + c)^5 + 67*a^2*b*tan(d*x + c)^3 - 43*a*b^2*tan(d*x + c)^3 + 17*a^2*b*tan(d*x + c) - 11*a*b^2*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^4 - 2*a^3*b + a^2*b^2)))/d","B",0
235,1,2203,0,2.269577," ","integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm=""giac"")","-\frac{\frac{3 \, {\left({\left(78 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b - 267 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} + 241 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 53 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - 15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5}\right)} {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)}^{2} {\left| -a + b \right|} - 2 \, {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{10} b - 51 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{9} b^{2} + 108 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{8} b^{3} - 106 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{7} b^{4} + 45 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b^{5} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{6} - 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{7}\right)} {\left| a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right|} {\left| -a + b \right|} - {\left(60 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{15} - 441 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{14} b + 1339 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{13} b^{2} - 2185 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{12} b^{3} + 2059 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{11} b^{4} - 1091 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} b^{5} + 265 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b^{6} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{7} - 11 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{8}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{6} - 2 \, a^{5} b + a^{4} b^{2} + \sqrt{{\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)}^{2} - {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)}}}{a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}}}}\right)\right)}}{{\left(3 \, a^{16} - 27 \, a^{15} b + 104 \, a^{14} b^{2} - 224 \, a^{13} b^{3} + 294 \, a^{12} b^{4} - 238 \, a^{11} b^{5} + 112 \, a^{10} b^{6} - 24 \, a^{9} b^{7} - a^{8} b^{8} + a^{7} b^{9}\right)} {\left| a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right|}} - \frac{3 \, {\left({\left(78 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b - 267 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} + 241 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 53 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - 15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5}\right)} {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)}^{2} {\left| -a + b \right|} + 2 \, {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{10} b - 51 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{9} b^{2} + 108 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{8} b^{3} - 106 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{7} b^{4} + 45 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b^{5} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{6} - 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{7}\right)} {\left| a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right|} {\left| -a + b \right|} - {\left(60 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{15} - 441 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{14} b + 1339 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{13} b^{2} - 2185 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{12} b^{3} + 2059 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{11} b^{4} - 1091 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} b^{5} + 265 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b^{6} + 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{7} - 11 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{8}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{6} - 2 \, a^{5} b + a^{4} b^{2} - \sqrt{{\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)}^{2} - {\left(a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right)} {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)}}}{a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}}}}\right)\right)}}{{\left(3 \, a^{16} - 27 \, a^{15} b + 104 \, a^{14} b^{2} - 224 \, a^{13} b^{3} + 294 \, a^{12} b^{4} - 238 \, a^{11} b^{5} + 112 \, a^{10} b^{6} - 24 \, a^{9} b^{7} - a^{8} b^{8} + a^{7} b^{9}\right)} {\left| a^{5} - 2 \, a^{4} b + a^{3} b^{2} \right|}} + \frac{2 \, {\left(18 \, a^{3} b \tan\left(d x + c\right)^{7} - 3 \, a^{2} b^{2} \tan\left(d x + c\right)^{7} - 28 \, a b^{3} \tan\left(d x + c\right)^{7} + 13 \, b^{4} \tan\left(d x + c\right)^{7} + 54 \, a^{3} b \tan\left(d x + c\right)^{5} - 4 \, a^{2} b^{2} \tan\left(d x + c\right)^{5} - 26 \, a b^{3} \tan\left(d x + c\right)^{5} + 54 \, a^{3} b \tan\left(d x + c\right)^{3} - 13 \, a^{2} b^{2} \tan\left(d x + c\right)^{3} - 17 \, a b^{3} \tan\left(d x + c\right)^{3} + 18 \, a^{3} b \tan\left(d x + c\right) - 12 \, a^{2} b^{2} \tan\left(d x + c\right)\right)}}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} {\left(a \tan\left(d x + c\right)^{4} - b \tan\left(d x + c\right)^{4} + 2 \, a \tan\left(d x + c\right)^{2} + a\right)}^{2}} + \frac{64}{a^{3} \tan\left(d x + c\right)}}{64 \, d}"," ",0,"-1/64*(3*((78*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 267*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 241*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 53*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^5)*(a^5 - 2*a^4*b + a^3*b^2)^2*abs(-a + b) - 2*(9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^10*b - 51*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^9*b^2 + 108*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b^3 - 106*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b^4 + 45*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^5 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^6 - 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^7)*abs(a^5 - 2*a^4*b + a^3*b^2)*abs(-a + b) - (60*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^15 - 441*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b + 1339*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^13*b^2 - 2185*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 2059*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 1091*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 265*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 - 11*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^6 - 2*a^5*b + a^4*b^2 + sqrt((a^6 - 2*a^5*b + a^4*b^2)^2 - (a^6 - 2*a^5*b + a^4*b^2)*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)))/(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3))))/((3*a^16 - 27*a^15*b + 104*a^14*b^2 - 224*a^13*b^3 + 294*a^12*b^4 - 238*a^11*b^5 + 112*a^10*b^6 - 24*a^9*b^7 - a^8*b^8 + a^7*b^9)*abs(a^5 - 2*a^4*b + a^3*b^2)) - 3*((78*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 267*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 241*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 53*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^5)*(a^5 - 2*a^4*b + a^3*b^2)^2*abs(-a + b) + 2*(9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^10*b - 51*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^9*b^2 + 108*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^8*b^3 - 106*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b^4 + 45*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^5 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^6 - 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^7)*abs(a^5 - 2*a^4*b + a^3*b^2)*abs(-a + b) - (60*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^15 - 441*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b + 1339*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^13*b^2 - 2185*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 2059*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 1091*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 265*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 - 11*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^6 - 2*a^5*b + a^4*b^2 - sqrt((a^6 - 2*a^5*b + a^4*b^2)^2 - (a^6 - 2*a^5*b + a^4*b^2)*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)))/(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3))))/((3*a^16 - 27*a^15*b + 104*a^14*b^2 - 224*a^13*b^3 + 294*a^12*b^4 - 238*a^11*b^5 + 112*a^10*b^6 - 24*a^9*b^7 - a^8*b^8 + a^7*b^9)*abs(a^5 - 2*a^4*b + a^3*b^2)) + 2*(18*a^3*b*tan(d*x + c)^7 - 3*a^2*b^2*tan(d*x + c)^7 - 28*a*b^3*tan(d*x + c)^7 + 13*b^4*tan(d*x + c)^7 + 54*a^3*b*tan(d*x + c)^5 - 4*a^2*b^2*tan(d*x + c)^5 - 26*a*b^3*tan(d*x + c)^5 + 54*a^3*b*tan(d*x + c)^3 - 13*a^2*b^2*tan(d*x + c)^3 - 17*a*b^3*tan(d*x + c)^3 + 18*a^3*b*tan(d*x + c) - 12*a^2*b^2*tan(d*x + c))/((a^5 - 2*a^4*b + a^3*b^2)*(a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2) + 64/(a^3*tan(d*x + c)))/d","B",0
236,1,51,0,0.118166," ","integrate(1/(1-sin(x)^4),x, algorithm=""giac"")","\frac{1}{4} \, \sqrt{2} {\left(x + \arctan\left(-\frac{\sqrt{2} \sin\left(2 \, x\right) - 2 \, \sin\left(2 \, x\right)}{\sqrt{2} \cos\left(2 \, x\right) + \sqrt{2} - 2 \, \cos\left(2 \, x\right) + 2}\right)\right)} + \frac{1}{2} \, \tan\left(x\right)"," ",0,"1/4*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - 2*cos(2*x) + 2))) + 1/2*tan(x)","B",0
237,1,318,0,0.389016," ","integrate(1/(a+b*sin(x)^4),x, algorithm=""giac"")","\frac{{\left(3 \, \sqrt{a^{2} + a b + \sqrt{-a b} {\left(a + b\right)}} a^{2} + 6 \, \sqrt{a^{2} + a b + \sqrt{-a b} {\left(a + b\right)}} a b - \sqrt{a^{2} + a b + \sqrt{-a b} {\left(a + b\right)}} b^{2}\right)} {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(x\right)}{\sqrt{\frac{4 \, a + \sqrt{-16 \, {\left(a + b\right)} a + 16 \, a^{2}}}{a + b}}}\right)\right)} {\left| a + b \right|}}{2 \, {\left(3 \, a^{5} + 12 \, a^{4} b + 14 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - a b^{4}\right)}} + \frac{{\left(3 \, \sqrt{a^{2} + a b - \sqrt{-a b} {\left(a + b\right)}} a^{2} + 6 \, \sqrt{a^{2} + a b - \sqrt{-a b} {\left(a + b\right)}} a b - \sqrt{a^{2} + a b - \sqrt{-a b} {\left(a + b\right)}} b^{2}\right)} {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(x\right)}{\sqrt{\frac{4 \, a - \sqrt{-16 \, {\left(a + b\right)} a + 16 \, a^{2}}}{a + b}}}\right)\right)} {\left| a + b \right|}}{2 \, {\left(3 \, a^{5} + 12 \, a^{4} b + 14 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - a b^{4}\right)}}"," ",0,"1/2*(3*sqrt(a^2 + a*b + sqrt(-a*b)*(a + b))*a^2 + 6*sqrt(a^2 + a*b + sqrt(-a*b)*(a + b))*a*b - sqrt(a^2 + a*b + sqrt(-a*b)*(a + b))*b^2)*(pi*floor(x/pi + 1/2) + arctan(2*tan(x)/sqrt((4*a + sqrt(-16*(a + b)*a + 16*a^2))/(a + b))))*abs(a + b)/(3*a^5 + 12*a^4*b + 14*a^3*b^2 + 4*a^2*b^3 - a*b^4) + 1/2*(3*sqrt(a^2 + a*b - sqrt(-a*b)*(a + b))*a^2 + 6*sqrt(a^2 + a*b - sqrt(-a*b)*(a + b))*a*b - sqrt(a^2 + a*b - sqrt(-a*b)*(a + b))*b^2)*(pi*floor(x/pi + 1/2) + arctan(2*tan(x)/sqrt((4*a - sqrt(-16*(a + b)*a + 16*a^2))/(a + b))))*abs(a + b)/(3*a^5 + 12*a^4*b + 14*a^3*b^2 + 4*a^2*b^3 - a*b^4)","A",0
238,1,170,0,0.361525," ","integrate(1/(1+sin(x)^4),x, algorithm=""giac"")","\frac{1}{4} \, {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{3}{4}} {\left(\left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} + 2 \, \tan\left(x\right)\right)}}{\sqrt{\sqrt{2} + 2}}\right)\right)} \sqrt{\sqrt{2} + 1} + \frac{1}{4} \, {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(-\frac{2 \, \left(\frac{1}{2}\right)^{\frac{3}{4}} {\left(\left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} - 2 \, \tan\left(x\right)\right)}}{\sqrt{\sqrt{2} + 2}}\right)\right)} \sqrt{\sqrt{2} + 1} + \frac{1}{8} \, \sqrt{\sqrt{2} - 1} \log\left(\tan\left(x\right)^{2} + \left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} \tan\left(x\right) + \sqrt{\frac{1}{2}}\right) - \frac{1}{8} \, \sqrt{\sqrt{2} - 1} \log\left(\tan\left(x\right)^{2} - \left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} \tan\left(x\right) + \sqrt{\frac{1}{2}}\right)"," ",0,"1/4*(pi*floor(x/pi + 1/2) + arctan(2*(1/2)^(3/4)*((1/2)^(1/4)*sqrt(-sqrt(2) + 2) + 2*tan(x))/sqrt(sqrt(2) + 2)))*sqrt(sqrt(2) + 1) + 1/4*(pi*floor(x/pi + 1/2) + arctan(-2*(1/2)^(3/4)*((1/2)^(1/4)*sqrt(-sqrt(2) + 2) - 2*tan(x))/sqrt(sqrt(2) + 2)))*sqrt(sqrt(2) + 1) + 1/8*sqrt(sqrt(2) - 1)*log(tan(x)^2 + (1/2)^(1/4)*sqrt(-sqrt(2) + 2)*tan(x) + sqrt(1/2)) - 1/8*sqrt(sqrt(2) - 1)*log(tan(x)^2 - (1/2)^(1/4)*sqrt(-sqrt(2) + 2)*tan(x) + sqrt(1/2))","A",0
239,0,0,0,0.000000," ","integrate(sin(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(d x + c\right)^{4} + a} \sin\left(d x + c\right)\,{d x}"," ",0,"integrate(sqrt(b*sin(d*x + c)^4 + a)*sin(d*x + c), x)","F",0
240,0,0,0,0.000000," ","integrate(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(d x + c\right)^{4} + a} \csc\left(d x + c\right)\,{d x}"," ",0,"integrate(sqrt(b*sin(d*x + c)^4 + a)*csc(d*x + c), x)","F",0
241,0,0,0,0.000000," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{5}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(sin(d*x + c)^5/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
242,-1,0,0,0.000000," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
243,0,0,0,0.000000," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(sin(d*x + c)/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
244,0,0,0,0.000000," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(csc(d*x + c)/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
245,-2,0,0,0.000000," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Evaluation time: 2.48Not invertible Error: Bad Argument Value","F(-2)",0
246,0,0,0,0.000000," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\sin\left(d x + c\right)^{2}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(sin(d*x + c)^2/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
247,0,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(1/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
248,0,0,0,0.000000," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\csc\left(d x + c\right)^{2}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(csc(d*x + c)^2/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
249,0,0,0,0.000000," ","integrate(1/(a+b*sin(x)^5),x, algorithm=""giac"")","\int \frac{1}{b \sin\left(x\right)^{5} + a}\,{d x}"," ",0,"integrate(1/(b*sin(x)^5 + a), x)","F",0
250,0,0,0,0.000000," ","integrate(1/(a+b*sin(x)^6),x, algorithm=""giac"")","\int \frac{1}{b \sin\left(x\right)^{6} + a}\,{d x}"," ",0,"integrate(1/(b*sin(x)^6 + a), x)","F",0
251,0,0,0,0.000000," ","integrate(1/(a+b*sin(x)^8),x, algorithm=""giac"")","\int \frac{1}{b \sin\left(x\right)^{8} + a}\,{d x}"," ",0,"integrate(1/(b*sin(x)^8 + a), x)","F",0
252,0,0,0,0.000000," ","integrate(1/(a-b*sin(x)^5),x, algorithm=""giac"")","\int -\frac{1}{b \sin\left(x\right)^{5} - a}\,{d x}"," ",0,"integrate(-1/(b*sin(x)^5 - a), x)","F",0
253,0,0,0,0.000000," ","integrate(1/(a-b*sin(x)^6),x, algorithm=""giac"")","\int -\frac{1}{b \sin\left(x\right)^{6} - a}\,{d x}"," ",0,"integrate(-1/(b*sin(x)^6 - a), x)","F",0
254,0,0,0,0.000000," ","integrate(1/(a-b*sin(x)^8),x, algorithm=""giac"")","\int -\frac{1}{b \sin\left(x\right)^{8} - a}\,{d x}"," ",0,"integrate(-1/(b*sin(x)^8 - a), x)","F",0
255,0,0,0,0.000000," ","integrate(1/(1+sin(x)^5),x, algorithm=""giac"")","\mathit{sage}_{0} x"," ",0,"sage0*x","F",0
256,1,185,0,0.158080," ","integrate(1/(1+sin(x)^6),x, algorithm=""giac"")","\frac{1}{6} \, \sqrt{3} {\left(x + \arctan\left(-\frac{\sqrt{3} \sin\left(2 \, x\right) + \cos\left(2 \, x\right) - 2 \, \sin\left(2 \, x\right) + 1}{\sqrt{3} \cos\left(2 \, x\right) + \sqrt{3} - 2 \, \cos\left(2 \, x\right) - \sin\left(2 \, x\right) + 2}\right)\right)} + \frac{1}{6} \, \sqrt{3} {\left(x + \arctan\left(-\frac{\sqrt{3} \sin\left(2 \, x\right) - \cos\left(2 \, x\right) - 2 \, \sin\left(2 \, x\right) - 1}{\sqrt{3} \cos\left(2 \, x\right) + \sqrt{3} - 2 \, \cos\left(2 \, x\right) + \sin\left(2 \, x\right) + 2}\right)\right)} + \frac{1}{6} \, \sqrt{2} {\left(x + \arctan\left(-\frac{\sqrt{2} \sin\left(2 \, x\right) - 2 \, \sin\left(2 \, x\right)}{\sqrt{2} \cos\left(2 \, x\right) + \sqrt{2} - 2 \, \cos\left(2 \, x\right) + 2}\right)\right)} + \frac{1}{12} \, \log\left(\tan\left(x\right)^{2} + \tan\left(x\right) + 1\right) - \frac{1}{12} \, \log\left(\tan\left(x\right)^{2} - \tan\left(x\right) + 1\right)"," ",0,"1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) + cos(2*x) - 2*sin(2*x) + 1)/(sqrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) - sin(2*x) + 2))) + 1/6*sqrt(3)*(x + arctan(-(sqrt(3)*sin(2*x) - cos(2*x) - 2*sin(2*x) - 1)/(sqrt(3)*cos(2*x) + sqrt(3) - 2*cos(2*x) + sin(2*x) + 2))) + 1/6*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - 2*cos(2*x) + 2))) + 1/12*log(tan(x)^2 + tan(x) + 1) - 1/12*log(tan(x)^2 - tan(x) + 1)","B",0
257,0,0,0,0.000000," ","integrate(1/(1+sin(x)^8),x, algorithm=""giac"")","\mathit{sage}_{0} x"," ",0,"sage0*x","F",0
258,0,0,0,0.000000," ","integrate(1/(1-sin(x)^5),x, algorithm=""giac"")","\mathit{sage}_{0} x"," ",0,"sage0*x","F",0
259,1,197,0,0.152659," ","integrate(1/(1-sin(x)^6),x, algorithm=""giac"")","\frac{1}{18} \, {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor - \arctan\left(-\frac{3 \, \left(\frac{1}{3}\right)^{\frac{3}{4}} {\left(\left(\frac{1}{3}\right)^{\frac{1}{4}} {\left(\sqrt{6} - \sqrt{2}\right)} + 4 \, \tan\left(x\right)\right)}}{\sqrt{6} + \sqrt{2}}\right)\right)} \sqrt{6 \, \sqrt{3} + 9} + \frac{1}{18} \, {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(-\frac{3 \, \left(\frac{1}{3}\right)^{\frac{3}{4}} {\left(\left(\frac{1}{3}\right)^{\frac{1}{4}} {\left(\sqrt{6} - \sqrt{2}\right)} - 4 \, \tan\left(x\right)\right)}}{\sqrt{6} + \sqrt{2}}\right)\right)} \sqrt{6 \, \sqrt{3} + 9} + \frac{1}{36} \, \sqrt{6 \, \sqrt{3} - 9} \log\left(\frac{1}{2} \, {\left(\sqrt{6} \left(\frac{1}{3}\right)^{\frac{1}{4}} - \sqrt{2} \left(\frac{1}{3}\right)^{\frac{1}{4}}\right)} \tan\left(x\right) + \tan\left(x\right)^{2} + \sqrt{\frac{1}{3}}\right) - \frac{1}{36} \, \sqrt{6 \, \sqrt{3} - 9} \log\left(-\frac{1}{2} \, {\left(\sqrt{6} \left(\frac{1}{3}\right)^{\frac{1}{4}} - \sqrt{2} \left(\frac{1}{3}\right)^{\frac{1}{4}}\right)} \tan\left(x\right) + \tan\left(x\right)^{2} + \sqrt{\frac{1}{3}}\right) + \frac{1}{3} \, \tan\left(x\right)"," ",0,"1/18*(pi*floor(x/pi + 1/2) - arctan(-3*(1/3)^(3/4)*((1/3)^(1/4)*(sqrt(6) - sqrt(2)) + 4*tan(x))/(sqrt(6) + sqrt(2))))*sqrt(6*sqrt(3) + 9) + 1/18*(pi*floor(x/pi + 1/2) + arctan(-3*(1/3)^(3/4)*((1/3)^(1/4)*(sqrt(6) - sqrt(2)) - 4*tan(x))/(sqrt(6) + sqrt(2))))*sqrt(6*sqrt(3) + 9) + 1/36*sqrt(6*sqrt(3) - 9)*log(1/2*(sqrt(6)*(1/3)^(1/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan(x)^2 + sqrt(1/3)) - 1/36*sqrt(6*sqrt(3) - 9)*log(-1/2*(sqrt(6)*(1/3)^(1/4) - sqrt(2)*(1/3)^(1/4))*tan(x) + tan(x)^2 + sqrt(1/3)) + 1/3*tan(x)","B",0
260,1,220,0,0.350858," ","integrate(1/(1-sin(x)^8),x, algorithm=""giac"")","\frac{1}{8} \, \sqrt{2} {\left(x + \arctan\left(-\frac{\sqrt{2} \sin\left(2 \, x\right) - 2 \, \sin\left(2 \, x\right)}{\sqrt{2} \cos\left(2 \, x\right) + \sqrt{2} - 2 \, \cos\left(2 \, x\right) + 2}\right)\right)} + \frac{1}{8} \, {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{3}{4}} {\left(\left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} + 2 \, \tan\left(x\right)\right)}}{\sqrt{\sqrt{2} + 2}}\right)\right)} \sqrt{\sqrt{2} + 1} + \frac{1}{8} \, {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(-\frac{2 \, \left(\frac{1}{2}\right)^{\frac{3}{4}} {\left(\left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} - 2 \, \tan\left(x\right)\right)}}{\sqrt{\sqrt{2} + 2}}\right)\right)} \sqrt{\sqrt{2} + 1} + \frac{1}{16} \, \sqrt{\sqrt{2} - 1} \log\left(\tan\left(x\right)^{2} + \left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} \tan\left(x\right) + \sqrt{\frac{1}{2}}\right) - \frac{1}{16} \, \sqrt{\sqrt{2} - 1} \log\left(\tan\left(x\right)^{2} - \left(\frac{1}{2}\right)^{\frac{1}{4}} \sqrt{-\sqrt{2} + 2} \tan\left(x\right) + \sqrt{\frac{1}{2}}\right) + \frac{1}{4} \, \tan\left(x\right)"," ",0,"1/8*sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - 2*sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - 2*cos(2*x) + 2))) + 1/8*(pi*floor(x/pi + 1/2) + arctan(2*(1/2)^(3/4)*((1/2)^(1/4)*sqrt(-sqrt(2) + 2) + 2*tan(x))/sqrt(sqrt(2) + 2)))*sqrt(sqrt(2) + 1) + 1/8*(pi*floor(x/pi + 1/2) + arctan(-2*(1/2)^(3/4)*((1/2)^(1/4)*sqrt(-sqrt(2) + 2) - 2*tan(x))/sqrt(sqrt(2) + 2)))*sqrt(sqrt(2) + 1) + 1/16*sqrt(sqrt(2) - 1)*log(tan(x)^2 + (1/2)^(1/4)*sqrt(-sqrt(2) + 2)*tan(x) + sqrt(1/2)) - 1/16*sqrt(sqrt(2) - 1)*log(tan(x)^2 - (1/2)^(1/4)*sqrt(-sqrt(2) + 2)*tan(x) + sqrt(1/2)) + 1/4*tan(x)","B",0
261,1,28,0,0.142008," ","integrate(cos(x)^9/(a-a*sin(x)^2),x, algorithm=""giac"")","-\frac{5 \, \sin\left(x\right)^{7} - 21 \, \sin\left(x\right)^{5} + 35 \, \sin\left(x\right)^{3} - 35 \, \sin\left(x\right)}{35 \, a}"," ",0,"-1/35*(5*sin(x)^7 - 21*sin(x)^5 + 35*sin(x)^3 - 35*sin(x))/a","A",0
262,1,22,0,0.121115," ","integrate(cos(x)^7/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{3 \, \sin\left(x\right)^{5} - 10 \, \sin\left(x\right)^{3} + 15 \, \sin\left(x\right)}{15 \, a}"," ",0,"1/15*(3*sin(x)^5 - 10*sin(x)^3 + 15*sin(x))/a","A",0
263,1,14,0,0.136647," ","integrate(cos(x)^5/(a-a*sin(x)^2),x, algorithm=""giac"")","-\frac{\sin\left(x\right)^{3} - 3 \, \sin\left(x\right)}{3 \, a}"," ",0,"-1/3*(sin(x)^3 - 3*sin(x))/a","A",0
264,1,6,0,0.118752," ","integrate(cos(x)^3/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{\sin\left(x\right)}{a}"," ",0,"sin(x)/a","A",0
265,1,23,0,0.125385," ","integrate(cos(x)/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{2 \, a} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{2 \, a}"," ",0,"1/2*log(sin(x) + 1)/a - 1/2*log(-sin(x) + 1)/a","B",0
266,1,47,0,0.122565," ","integrate(sec(x)^3/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{3 \, \log\left(\sin\left(x\right) + 1\right)}{16 \, a} - \frac{3 \, \log\left(-\sin\left(x\right) + 1\right)}{16 \, a} - \frac{3 \, \sin\left(x\right)^{3} - 5 \, \sin\left(x\right)}{8 \, {\left(\sin\left(x\right)^{2} - 1\right)}^{2} a}"," ",0,"3/16*log(sin(x) + 1)/a - 3/16*log(-sin(x) + 1)/a - 1/8*(3*sin(x)^3 - 5*sin(x))/((sin(x)^2 - 1)^2*a)","A",0
267,1,36,0,0.124380," ","integrate(cos(x)^6/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{3 \, \arctan\left(\tan\left(x\right)\right)}{8 \, a} + \frac{\frac{3 \, \tan\left(x\right)^{3}}{a} + \frac{5 \, \tan\left(x\right)}{a}}{8 \, {\left(\tan\left(x\right)^{2} + 1\right)}^{2}}"," ",0,"3/8*arctan(tan(x))/a + 1/8*(3*tan(x)^3/a + 5*tan(x)/a)/(tan(x)^2 + 1)^2","A",0
268,1,24,0,0.124348," ","integrate(cos(x)^4/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{\arctan\left(\tan\left(x\right)\right)}{2 \, a} + \frac{\tan\left(x\right)}{2 \, {\left(\tan\left(x\right)^{2} + 1\right)} a}"," ",0,"1/2*arctan(tan(x))/a + 1/2*tan(x)/((tan(x)^2 + 1)*a)","A",0
269,1,14,0,0.135374," ","integrate(cos(x)^2/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{{\left| a \right|} \tan\left(x\right)}{a}\right)}{{\left| a \right|}}"," ",0,"arctan(abs(a)*tan(x)/a)/abs(a)","B",0
270,1,38,0,0.131342," ","integrate(sec(x)/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{4 \, a} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{4 \, a} - \frac{\sin\left(x\right)}{2 \, {\left(\sin\left(x\right)^{2} - 1\right)} a}"," ",0,"1/4*log(sin(x) + 1)/a - 1/4*log(-sin(x) + 1)/a - 1/2*sin(x)/((sin(x)^2 - 1)*a)","B",0
271,1,14,0,0.141818," ","integrate(sec(x)^2/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{\tan\left(x\right)^{3} + 3 \, \tan\left(x\right)}{3 \, a}"," ",0,"1/3*(tan(x)^3 + 3*tan(x))/a","A",0
272,1,22,0,0.194302," ","integrate(sec(x)^4/(a-a*sin(x)^2),x, algorithm=""giac"")","\frac{3 \, \tan\left(x\right)^{5} + 10 \, \tan\left(x\right)^{3} + 15 \, \tan\left(x\right)}{15 \, a}"," ",0,"1/15*(3*tan(x)^5 + 10*tan(x)^3 + 15*tan(x))/a","A",0
273,1,22,0,0.149544," ","integrate(cos(x)^9/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{3 \, \sin\left(x\right)^{5} - 10 \, \sin\left(x\right)^{3} + 15 \, \sin\left(x\right)}{15 \, a^{2}}"," ",0,"1/15*(3*sin(x)^5 - 10*sin(x)^3 + 15*sin(x))/a^2","A",0
274,1,14,0,0.142002," ","integrate(cos(x)^7/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","-\frac{\sin\left(x\right)^{3} - 3 \, \sin\left(x\right)}{3 \, a^{2}}"," ",0,"-1/3*(sin(x)^3 - 3*sin(x))/a^2","A",0
275,1,6,0,0.116950," ","integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\sin\left(x\right)}{a^{2}}"," ",0,"sin(x)/a^2","A",0
276,1,23,0,0.132100," ","integrate(cos(x)^3/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{2 \, a^{2}} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{2 \, a^{2}}"," ",0,"1/2*log(sin(x) + 1)/a^2 - 1/2*log(-sin(x) + 1)/a^2","B",0
277,1,38,0,0.150794," ","integrate(cos(x)/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{4 \, a^{2}} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{4 \, a^{2}} - \frac{\sin\left(x\right)}{2 \, {\left(\sin\left(x\right)^{2} - 1\right)} a^{2}}"," ",0,"1/4*log(sin(x) + 1)/a^2 - 1/4*log(-sin(x) + 1)/a^2 - 1/2*sin(x)/((sin(x)^2 - 1)*a^2)","B",0
278,1,47,0,0.142373," ","integrate(sec(x)/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{3 \, \log\left(\sin\left(x\right) + 1\right)}{16 \, a^{2}} - \frac{3 \, \log\left(-\sin\left(x\right) + 1\right)}{16 \, a^{2}} - \frac{3 \, \sin\left(x\right)^{3} - 5 \, \sin\left(x\right)}{8 \, {\left(\sin\left(x\right)^{2} - 1\right)}^{2} a^{2}}"," ",0,"3/16*log(sin(x) + 1)/a^2 - 3/16*log(-sin(x) + 1)/a^2 - 1/8*(3*sin(x)^3 - 5*sin(x))/((sin(x)^2 - 1)^2*a^2)","A",0
279,1,31,0,0.122734," ","integrate(cos(x)^8/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{3 \, x}{8 \, a^{2}} + \frac{3 \, \tan\left(x\right)^{3} + 5 \, \tan\left(x\right)}{8 \, {\left(\tan\left(x\right)^{2} + 1\right)}^{2} a^{2}}"," ",0,"3/8*x/a^2 + 1/8*(3*tan(x)^3 + 5*tan(x))/((tan(x)^2 + 1)^2*a^2)","A",0
280,1,22,0,0.120165," ","integrate(cos(x)^6/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{x}{2 \, a^{2}} + \frac{\tan\left(x\right)}{2 \, {\left(\tan\left(x\right)^{2} + 1\right)} a^{2}}"," ",0,"1/2*x/a^2 + 1/2*tan(x)/((tan(x)^2 + 1)*a^2)","A",0
281,1,5,0,0.126841," ","integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{x}{a^{2}}"," ",0,"x/a^2","A",0
282,1,6,0,0.126769," ","integrate(cos(x)^2/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\tan\left(x\right)}{a^{2}}"," ",0,"tan(x)/a^2","A",0
283,1,22,0,0.148694," ","integrate(sec(x)^2/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{3 \, \tan\left(x\right)^{5} + 10 \, \tan\left(x\right)^{3} + 15 \, \tan\left(x\right)}{15 \, a^{2}}"," ",0,"1/15*(3*tan(x)^5 + 10*tan(x)^3 + 15*tan(x))/a^2","A",0
284,1,28,0,0.127919," ","integrate(sec(x)^4/(a-a*sin(x)^2)^2,x, algorithm=""giac"")","\frac{5 \, \tan\left(x\right)^{7} + 21 \, \tan\left(x\right)^{5} + 35 \, \tan\left(x\right)^{3} + 35 \, \tan\left(x\right)}{35 \, a^{2}}"," ",0,"1/35*(5*tan(x)^7 + 21*tan(x)^5 + 35*tan(x)^3 + 35*tan(x))/a^2","A",0
285,1,87,0,0.233224," ","integrate(cos(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","\frac{5}{128} \, {\left(8 \, a + b\right)} x - \frac{b \sin\left(8 \, f x + 8 \, e\right)}{1024 \, f} + \frac{{\left(a - b\right)} \sin\left(6 \, f x + 6 \, e\right)}{192 \, f} + \frac{{\left(6 \, a - b\right)} \sin\left(4 \, f x + 4 \, e\right)}{128 \, f} + \frac{{\left(15 \, a + b\right)} \sin\left(2 \, f x + 2 \, e\right)}{64 \, f}"," ",0,"5/128*(8*a + b)*x - 1/1024*b*sin(8*f*x + 8*e)/f + 1/192*(a - b)*sin(6*f*x + 6*e)/f + 1/128*(6*a - b)*sin(4*f*x + 4*e)/f + 1/64*(15*a + b)*sin(2*f*x + 2*e)/f","A",0
286,1,67,0,0.181007," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","\frac{1}{16} \, {\left(6 \, a + b\right)} x - \frac{b \sin\left(6 \, f x + 6 \, e\right)}{192 \, f} + \frac{{\left(2 \, a - b\right)} \sin\left(4 \, f x + 4 \, e\right)}{64 \, f} + \frac{{\left(16 \, a + b\right)} \sin\left(2 \, f x + 2 \, e\right)}{64 \, f}"," ",0,"1/16*(6*a + b)*x - 1/192*b*sin(6*f*x + 6*e)/f + 1/64*(2*a - b)*sin(4*f*x + 4*e)/f + 1/64*(16*a + b)*sin(2*f*x + 2*e)/f","A",0
287,1,41,0,0.159245," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","\frac{1}{8} \, {\left(4 \, a + b\right)} x - \frac{b \sin\left(4 \, f x + 4 \, e\right)}{32 \, f} + \frac{a \sin\left(2 \, f x + 2 \, e\right)}{4 \, f}"," ",0,"1/8*(4*a + b)*x - 1/32*b*sin(4*f*x + 4*e)/f + 1/4*a*sin(2*f*x + 2*e)/f","A",0
288,1,26,0,0.131328," ","integrate(a+b*sin(f*x+e)^2,x, algorithm=""giac"")","\frac{1}{4} \, b {\left(2 \, x - \frac{\sin\left(2 \, f x + 2 \, e\right)}{f}\right)} + a x"," ",0,"1/4*b*(2*x - sin(2*f*x + 2*e)/f) + a*x","A",0
289,1,49,0,0.178065," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","-\frac{{\left(f x - \pi \left \lfloor \frac{f x + e}{\pi} + \frac{1}{2} \right \rfloor + e - \tan\left(f x + e\right)\right)} b - a \tan\left(f x + e\right)}{f}"," ",0,"-((f*x - pi*floor((f*x + e)/pi + 1/2) + e - tan(f*x + e))*b - a*tan(f*x + e))/f","B",0
290,1,38,0,0.192297," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","\frac{a \tan\left(f x + e\right)^{3} + b \tan\left(f x + e\right)^{3} + 3 \, a \tan\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(a*tan(f*x + e)^3 + b*tan(f*x + e)^3 + 3*a*tan(f*x + e))/f","A",0
291,1,64,0,0.202359," ","integrate(sec(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","\frac{3 \, a \tan\left(f x + e\right)^{5} + 3 \, b \tan\left(f x + e\right)^{5} + 10 \, a \tan\left(f x + e\right)^{3} + 5 \, b \tan\left(f x + e\right)^{3} + 15 \, a \tan\left(f x + e\right)}{15 \, f}"," ",0,"1/15*(3*a*tan(f*x + e)^5 + 3*b*tan(f*x + e)^5 + 10*a*tan(f*x + e)^3 + 5*b*tan(f*x + e)^3 + 15*a*tan(f*x + e))/f","A",0
292,1,88,0,0.222720," ","integrate(sec(f*x+e)^8*(a+b*sin(f*x+e)^2),x, algorithm=""giac"")","\frac{15 \, a \tan\left(f x + e\right)^{7} + 15 \, b \tan\left(f x + e\right)^{7} + 63 \, a \tan\left(f x + e\right)^{5} + 42 \, b \tan\left(f x + e\right)^{5} + 105 \, a \tan\left(f x + e\right)^{3} + 35 \, b \tan\left(f x + e\right)^{3} + 105 \, a \tan\left(f x + e\right)}{105 \, f}"," ",0,"1/105*(15*a*tan(f*x + e)^7 + 15*b*tan(f*x + e)^7 + 63*a*tan(f*x + e)^5 + 42*b*tan(f*x + e)^5 + 105*a*tan(f*x + e)^3 + 35*b*tan(f*x + e)^3 + 105*a*tan(f*x + e))/f","A",0
293,1,108,0,0.241117," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{1}{128} \, {\left(48 \, a^{2} + 16 \, a b + 3 \, b^{2}\right)} x + \frac{b^{2} \sin\left(8 \, f x + 8 \, e\right)}{1024 \, f} - \frac{a b \sin\left(6 \, f x + 6 \, e\right)}{96 \, f} + \frac{{\left(4 \, a^{2} - 4 \, a b - b^{2}\right)} \sin\left(4 \, f x + 4 \, e\right)}{128 \, f} + \frac{{\left(8 \, a^{2} + a b\right)} \sin\left(2 \, f x + 2 \, e\right)}{32 \, f}"," ",0,"1/128*(48*a^2 + 16*a*b + 3*b^2)*x + 1/1024*b^2*sin(8*f*x + 8*e)/f - 1/96*a*b*sin(6*f*x + 6*e)/f + 1/128*(4*a^2 - 4*a*b - b^2)*sin(4*f*x + 4*e)/f + 1/32*(8*a^2 + a*b)*sin(2*f*x + 2*e)/f","A",0
294,1,84,0,0.183785," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{1}{16} \, {\left(8 \, a^{2} + 4 \, a b + b^{2}\right)} x + \frac{b^{2} \sin\left(6 \, f x + 6 \, e\right)}{192 \, f} - \frac{{\left(4 \, a b + b^{2}\right)} \sin\left(4 \, f x + 4 \, e\right)}{64 \, f} + \frac{{\left(16 \, a^{2} - b^{2}\right)} \sin\left(2 \, f x + 2 \, e\right)}{64 \, f}"," ",0,"1/16*(8*a^2 + 4*a*b + b^2)*x + 1/192*b^2*sin(6*f*x + 6*e)/f - 1/64*(4*a*b + b^2)*sin(4*f*x + 4*e)/f + 1/64*(16*a^2 - b^2)*sin(2*f*x + 2*e)/f","A",0
295,1,60,0,0.132323," ","integrate((a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{1}{8} \, {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} x + \frac{b^{2} \sin\left(4 \, f x + 4 \, e\right)}{32 \, f} - \frac{{\left(2 \, a b + b^{2}\right)} \sin\left(2 \, f x + 2 \, e\right)}{4 \, f}"," ",0,"1/8*(8*a^2 + 8*a*b + 3*b^2)*x + 1/32*b^2*sin(4*f*x + 4*e)/f - 1/4*(2*a*b + b^2)*sin(2*f*x + 2*e)/f","A",0
296,1,99,0,0.185567," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{2 \, a^{2} \tan\left(f x + e\right) + 4 \, a b \tan\left(f x + e\right) + 2 \, b^{2} \tan\left(f x + e\right) - {\left(4 \, a b + 3 \, b^{2}\right)} {\left(f x - \pi \left \lfloor \frac{f x + e}{\pi} + \frac{1}{2} \right \rfloor + e\right)} + \frac{b^{2} \tan\left(f x + e\right)}{\tan\left(f x + e\right)^{2} + 1}}{2 \, f}"," ",0,"1/2*(2*a^2*tan(f*x + e) + 4*a*b*tan(f*x + e) + 2*b^2*tan(f*x + e) - (4*a*b + 3*b^2)*(f*x - pi*floor((f*x + e)/pi + 1/2) + e) + b^2*tan(f*x + e)/(tan(f*x + e)^2 + 1))/f","B",0
297,1,80,0,0.220593," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{a^{2} \tan\left(f x + e\right)^{3} + 2 \, a b \tan\left(f x + e\right)^{3} + b^{2} \tan\left(f x + e\right)^{3} + 3 \, {\left(f x + e\right)} b^{2} + 3 \, a^{2} \tan\left(f x + e\right) - 3 \, b^{2} \tan\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(a^2*tan(f*x + e)^3 + 2*a*b*tan(f*x + e)^3 + b^2*tan(f*x + e)^3 + 3*(f*x + e)*b^2 + 3*a^2*tan(f*x + e) - 3*b^2*tan(f*x + e))/f","A",0
298,1,86,0,0.213075," ","integrate(sec(f*x+e)^6*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{3 \, a^{2} \tan\left(f x + e\right)^{5} + 6 \, a b \tan\left(f x + e\right)^{5} + 3 \, b^{2} \tan\left(f x + e\right)^{5} + 10 \, a^{2} \tan\left(f x + e\right)^{3} + 10 \, a b \tan\left(f x + e\right)^{3} + 15 \, a^{2} \tan\left(f x + e\right)}{15 \, f}"," ",0,"1/15*(3*a^2*tan(f*x + e)^5 + 6*a*b*tan(f*x + e)^5 + 3*b^2*tan(f*x + e)^5 + 10*a^2*tan(f*x + e)^3 + 10*a*b*tan(f*x + e)^3 + 15*a^2*tan(f*x + e))/f","A",0
299,1,127,0,0.247967," ","integrate(sec(f*x+e)^8*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{15 \, a^{2} \tan\left(f x + e\right)^{7} + 30 \, a b \tan\left(f x + e\right)^{7} + 15 \, b^{2} \tan\left(f x + e\right)^{7} + 63 \, a^{2} \tan\left(f x + e\right)^{5} + 84 \, a b \tan\left(f x + e\right)^{5} + 21 \, b^{2} \tan\left(f x + e\right)^{5} + 105 \, a^{2} \tan\left(f x + e\right)^{3} + 70 \, a b \tan\left(f x + e\right)^{3} + 105 \, a^{2} \tan\left(f x + e\right)}{105 \, f}"," ",0,"1/105*(15*a^2*tan(f*x + e)^7 + 30*a*b*tan(f*x + e)^7 + 15*b^2*tan(f*x + e)^7 + 63*a^2*tan(f*x + e)^5 + 84*a*b*tan(f*x + e)^5 + 21*b^2*tan(f*x + e)^5 + 105*a^2*tan(f*x + e)^3 + 70*a*b*tan(f*x + e)^3 + 105*a^2*tan(f*x + e))/f","A",0
300,1,168,0,0.288672," ","integrate(sec(f*x+e)^10*(a+b*sin(f*x+e)^2)^2,x, algorithm=""giac"")","\frac{35 \, a^{2} \tan\left(f x + e\right)^{9} + 70 \, a b \tan\left(f x + e\right)^{9} + 35 \, b^{2} \tan\left(f x + e\right)^{9} + 180 \, a^{2} \tan\left(f x + e\right)^{7} + 270 \, a b \tan\left(f x + e\right)^{7} + 90 \, b^{2} \tan\left(f x + e\right)^{7} + 378 \, a^{2} \tan\left(f x + e\right)^{5} + 378 \, a b \tan\left(f x + e\right)^{5} + 63 \, b^{2} \tan\left(f x + e\right)^{5} + 420 \, a^{2} \tan\left(f x + e\right)^{3} + 210 \, a b \tan\left(f x + e\right)^{3} + 315 \, a^{2} \tan\left(f x + e\right)}{315 \, f}"," ",0,"1/315*(35*a^2*tan(f*x + e)^9 + 70*a*b*tan(f*x + e)^9 + 35*b^2*tan(f*x + e)^9 + 180*a^2*tan(f*x + e)^7 + 270*a*b*tan(f*x + e)^7 + 90*b^2*tan(f*x + e)^7 + 378*a^2*tan(f*x + e)^5 + 378*a*b*tan(f*x + e)^5 + 63*b^2*tan(f*x + e)^5 + 420*a^2*tan(f*x + e)^3 + 210*a*b*tan(f*x + e)^3 + 315*a^2*tan(f*x + e))/f","A",0
301,1,98,0,0.138000," ","integrate(cos(x)^7/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{\sqrt{a b} b^{3}} - \frac{3 \, b^{4} \sin\left(x\right)^{5} - 5 \, a b^{3} \sin\left(x\right)^{3} - 15 \, b^{4} \sin\left(x\right)^{3} + 15 \, a^{2} b^{2} \sin\left(x\right) + 45 \, a b^{3} \sin\left(x\right) + 45 \, b^{4} \sin\left(x\right)}{15 \, b^{5}}"," ",0,"(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/15*(3*b^4*sin(x)^5 - 5*a*b^3*sin(x)^3 - 15*b^4*sin(x)^3 + 15*a^2*b^2*sin(x) + 45*a*b^3*sin(x) + 45*b^4*sin(x))/b^5","A",0
302,1,131,0,0.158943," ","integrate(cos(x)^6/(a+b*sin(x)^2),x, algorithm=""giac"")","-\frac{{\left(8 \, a^{2} + 20 \, a b + 15 \, b^{2}\right)} x}{8 \, b^{3}} + \frac{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{\sqrt{a^{2} + a b} b^{3}} - \frac{4 \, a \tan\left(x\right)^{3} + 7 \, b \tan\left(x\right)^{3} + 4 \, a \tan\left(x\right) + 9 \, b \tan\left(x\right)}{8 \, {\left(\tan\left(x\right)^{2} + 1\right)}^{2} b^{2}}"," ",0,"-1/8*(8*a^2 + 20*a*b + 15*b^2)*x/b^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*b^3) - 1/8*(4*a*tan(x)^3 + 7*b*tan(x)^3 + 4*a*tan(x) + 9*b*tan(x))/((tan(x)^2 + 1)^2*b^2)","A",0
303,1,58,0,0.152302," ","integrate(cos(x)^5/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{{\left(a^{2} + 2 \, a b + b^{2}\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{\sqrt{a b} b^{2}} + \frac{b^{2} \sin\left(x\right)^{3} - 3 \, a b \sin\left(x\right) - 6 \, b^{2} \sin\left(x\right)}{3 \, b^{3}}"," ",0,"(a^2 + 2*a*b + b^2)*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/3*(b^2*sin(x)^3 - 3*a*b*sin(x) - 6*b^2*sin(x))/b^3","A",0
304,1,92,0,0.151493," ","integrate(cos(x)^4/(a+b*sin(x)^2),x, algorithm=""giac"")","-\frac{{\left(2 \, a + 3 \, b\right)} x}{2 \, b^{2}} + \frac{{\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(a^{2} + 2 \, a b + b^{2}\right)}}{\sqrt{a^{2} + a b} b^{2}} - \frac{\tan\left(x\right)}{2 \, {\left(\tan\left(x\right)^{2} + 1\right)} b}"," ",0,"-1/2*(2*a + 3*b)*x/b^2 + (pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))*(a^2 + 2*a*b + b^2)/(sqrt(a^2 + a*b)*b^2) - 1/2*tan(x)/((tan(x)^2 + 1)*b)","A",0
305,1,30,0,0.123221," ","integrate(cos(x)^3/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{{\left(a + b\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{\sqrt{a b} b} - \frac{\sin\left(x\right)}{b}"," ",0,"(a + b)*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*b) - sin(x)/b","A",0
306,1,62,0,0.132004," ","integrate(cos(x)^2/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{{\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(a + b\right)}}{\sqrt{a^{2} + a b} b} - \frac{x}{b}"," ",0,"(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))*(a + b)/(sqrt(a^2 + a*b)*b) - x/b","A",0
307,1,16,0,0.136598," ","integrate(cos(x)/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{\sqrt{a b}}"," ",0,"arctan(b*sin(x)/sqrt(a*b))/sqrt(a*b)","A",0
308,1,49,0,0.142875," ","integrate(sec(x)/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{b \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{\sqrt{a b} {\left(a + b\right)}} + \frac{\log\left(\sin\left(x\right) + 1\right)}{2 \, {\left(a + b\right)}} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a + b\right)}}"," ",0,"b*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*(a + b)) + 1/2*log(sin(x) + 1)/(a + b) - 1/2*log(-sin(x) + 1)/(a + b)","A",0
309,1,45,0,0.132136," ","integrate(sec(x)^2/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{b \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)}{\sqrt{a^{2} + a b} {\left(a + b\right)}} + \frac{\tan\left(x\right)}{a + b}"," ",0,"b*arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b))/(sqrt(a^2 + a*b)*(a + b)) + tan(x)/(a + b)","A",0
310,1,102,0,0.121162," ","integrate(sec(x)^3/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{b^{2} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a b}} + \frac{{\left(a + 3 \, b\right)} \log\left(\sin\left(x\right) + 1\right)}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)}} - \frac{{\left(a + 3 \, b\right)} \log\left(-\sin\left(x\right) + 1\right)}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)}} - \frac{\sin\left(x\right)}{2 \, {\left(\sin\left(x\right)^{2} - 1\right)} {\left(a + b\right)}}"," ",0,"b^2*arctan(b*sin(x)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(a*b)) + 1/4*(a + 3*b)*log(sin(x) + 1)/(a^2 + 2*a*b + b^2) - 1/4*(a + 3*b)*log(-sin(x) + 1)/(a^2 + 2*a*b + b^2) - 1/2*sin(x)/((sin(x)^2 - 1)*(a + b))","B",0
311,1,134,0,0.155627," ","integrate(sec(x)^4/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{{\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)} b^{2}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a^{2} + a b}} + \frac{a^{2} \tan\left(x\right)^{3} + 2 \, a b \tan\left(x\right)^{3} + b^{2} \tan\left(x\right)^{3} + 3 \, a^{2} \tan\left(x\right) + 9 \, a b \tan\left(x\right) + 6 \, b^{2} \tan\left(x\right)}{3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)}}"," ",0,"(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))*b^2/((a^2 + 2*a*b + b^2)*sqrt(a^2 + a*b)) + 1/3*(a^2*tan(x)^3 + 2*a*b*tan(x)^3 + b^2*tan(x)^3 + 3*a^2*tan(x) + 9*a*b*tan(x) + 6*b^2*tan(x))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)","B",0
312,1,177,0,0.141549," ","integrate(sec(x)^5/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{b^{3} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{a b}} + \frac{{\left(3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right)} \log\left(\sin\left(x\right) + 1\right)}{16 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)}} - \frac{{\left(3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right)} \log\left(-\sin\left(x\right) + 1\right)}{16 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)}} - \frac{3 \, a \sin\left(x\right)^{3} + 7 \, b \sin\left(x\right)^{3} - 5 \, a \sin\left(x\right) - 9 \, b \sin\left(x\right)}{8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} {\left(\sin\left(x\right)^{2} - 1\right)}^{2}}"," ",0,"b^3*arctan(b*sin(x)/sqrt(a*b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a*b)) + 1/16*(3*a^2 + 10*a*b + 15*b^2)*log(sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/16*(3*a^2 + 10*a*b + 15*b^2)*log(-sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/8*(3*a*sin(x)^3 + 7*b*sin(x)^3 - 5*a*sin(x) - 9*b*sin(x))/((a^2 + 2*a*b + b^2)*(sin(x)^2 - 1)^2)","B",0
313,1,254,0,0.131208," ","integrate(sec(x)^6/(a+b*sin(x)^2),x, algorithm=""giac"")","\frac{{\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)} b^{3}}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{a^{2} + a b}} + \frac{3 \, a^{4} \tan\left(x\right)^{5} + 12 \, a^{3} b \tan\left(x\right)^{5} + 18 \, a^{2} b^{2} \tan\left(x\right)^{5} + 12 \, a b^{3} \tan\left(x\right)^{5} + 3 \, b^{4} \tan\left(x\right)^{5} + 10 \, a^{4} \tan\left(x\right)^{3} + 45 \, a^{3} b \tan\left(x\right)^{3} + 75 \, a^{2} b^{2} \tan\left(x\right)^{3} + 55 \, a b^{3} \tan\left(x\right)^{3} + 15 \, b^{4} \tan\left(x\right)^{3} + 15 \, a^{4} \tan\left(x\right) + 75 \, a^{3} b \tan\left(x\right) + 150 \, a^{2} b^{2} \tan\left(x\right) + 135 \, a b^{3} \tan\left(x\right) + 45 \, b^{4} \tan\left(x\right)}{15 \, {\left(a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right)}}"," ",0,"(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))*b^3/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a^2 + a*b)) + 1/15*(3*a^4*tan(x)^5 + 12*a^3*b*tan(x)^5 + 18*a^2*b^2*tan(x)^5 + 12*a*b^3*tan(x)^5 + 3*b^4*tan(x)^5 + 10*a^4*tan(x)^3 + 45*a^3*b*tan(x)^3 + 75*a^2*b^2*tan(x)^3 + 55*a*b^3*tan(x)^3 + 15*b^4*tan(x)^3 + 15*a^4*tan(x) + 75*a^3*b*tan(x) + 150*a^2*b^2*tan(x) + 135*a*b^3*tan(x) + 45*b^4*tan(x))/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)","B",0
314,1,175,0,0.162286," ","integrate(cos(x)^6/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{{\left(4 \, a + 5 \, b\right)} x}{2 \, b^{3}} - \frac{{\left(4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{2 \, \sqrt{a^{2} + a b} a b^{3}} + \frac{2 \, a^{2} \tan\left(x\right)^{3} + 3 \, a b \tan\left(x\right)^{3} + b^{2} \tan\left(x\right)^{3} + 2 \, a^{2} \tan\left(x\right) + 2 \, a b \tan\left(x\right) + b^{2} \tan\left(x\right)}{2 \, {\left(a \tan\left(x\right)^{4} + b \tan\left(x\right)^{4} + 2 \, a \tan\left(x\right)^{2} + b \tan\left(x\right)^{2} + a\right)} a b^{2}}"," ",0,"1/2*(4*a + 5*b)*x/b^3 - 1/2*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*a*b^3) + 1/2*(2*a^2*tan(x)^3 + 3*a*b*tan(x)^3 + b^2*tan(x)^3 + 2*a^2*tan(x) + 2*a*b*tan(x) + b^2*tan(x))/((a*tan(x)^4 + b*tan(x)^4 + 2*a*tan(x)^2 + b*tan(x)^2 + a)*a*b^2)","A",0
315,1,82,0,0.122670," ","integrate(cos(x)^5/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\sin\left(x\right)}{b^{2}} - \frac{{\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{2 \, \sqrt{a b} a b^{2}} + \frac{a^{2} \sin\left(x\right) + 2 \, a b \sin\left(x\right) + b^{2} \sin\left(x\right)}{2 \, {\left(b \sin\left(x\right)^{2} + a\right)} a b^{2}}"," ",0,"sin(x)/b^2 - 1/2*(3*a^2 + 2*a*b - b^2)*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*a*b^2) + 1/2*(a^2*sin(x) + 2*a*b*sin(x) + b^2*sin(x))/((b*sin(x)^2 + a)*a*b^2)","A",0
316,1,109,0,0.158662," ","integrate(cos(x)^4/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{x}{b^{2}} - \frac{{\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(2 \, a^{2} + a b - b^{2}\right)}}{2 \, \sqrt{a^{2} + a b} a b^{2}} + \frac{a \tan\left(x\right) + b \tan\left(x\right)}{2 \, {\left(a \tan\left(x\right)^{2} + b \tan\left(x\right)^{2} + a\right)} a b}"," ",0,"x/b^2 - 1/2*(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))*(2*a^2 + a*b - b^2)/(sqrt(a^2 + a*b)*a*b^2) + 1/2*(a*tan(x) + b*tan(x))/((a*tan(x)^2 + b*tan(x)^2 + a)*a*b)","A",0
317,1,56,0,0.126192," ","integrate(cos(x)^3/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","-\frac{{\left(a - b\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{2 \, \sqrt{a b} a b} + \frac{a \sin\left(x\right) + b \sin\left(x\right)}{2 \, {\left(b \sin\left(x\right)^{2} + a\right)} a b}"," ",0,"-1/2*(a - b)*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*a*b) + 1/2*(a*sin(x) + b*sin(x))/((b*sin(x)^2 + a)*a*b)","A",0
318,1,77,0,0.153896," ","integrate(cos(x)^2/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)}{2 \, \sqrt{a^{2} + a b} a} + \frac{\tan\left(x\right)}{2 \, {\left(a \tan\left(x\right)^{2} + b \tan\left(x\right)^{2} + a\right)} a}"," ",0,"1/2*(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*a) + 1/2*tan(x)/((a*tan(x)^2 + b*tan(x)^2 + a)*a)","A",0
319,1,38,0,0.142870," ","integrate(cos(x)/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{\arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{2 \, \sqrt{a b} a} + \frac{\sin\left(x\right)}{2 \, {\left(b \sin\left(x\right)^{2} + a\right)} a}"," ",0,"1/2*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*a) + 1/2*sin(x)/((b*sin(x)^2 + a)*a)","A",0
320,1,109,0,0.129763," ","integrate(sec(x)/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{{\left(3 \, a b + b^{2}\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{a b}} + \frac{\log\left(\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)}} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)}} + \frac{b \sin\left(x\right)}{2 \, {\left(b \sin\left(x\right)^{2} + a\right)} {\left(a^{2} + a b\right)}}"," ",0,"1/2*(3*a*b + b^2)*arctan(b*sin(x)/sqrt(a*b))/((a^3 + 2*a^2*b + a*b^2)*sqrt(a*b)) + 1/2*log(sin(x) + 1)/(a^2 + 2*a*b + b^2) - 1/2*log(-sin(x) + 1)/(a^2 + 2*a*b + b^2) + 1/2*b*sin(x)/((b*sin(x)^2 + a)*(a^2 + a*b))","A",0
321,1,113,0,0.132675," ","integrate(sec(x)^2/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{b^{2} \tan\left(x\right)}{2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} {\left(a \tan\left(x\right)^{2} + b \tan\left(x\right)^{2} + a\right)}} + \frac{{\left(4 \, a b + b^{2}\right)} \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)}{2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \sqrt{a^{2} + a b}} + \frac{\tan\left(x\right)}{a^{2} + 2 \, a b + b^{2}}"," ",0,"1/2*b^2*tan(x)/((a^3 + 2*a^2*b + a*b^2)*(a*tan(x)^2 + b*tan(x)^2 + a)) + 1/2*(4*a*b + b^2)*arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b))/((a^3 + 2*a^2*b + a*b^2)*sqrt(a^2 + a*b)) + tan(x)/(a^2 + 2*a*b + b^2)","A",0
322,1,194,0,0.130425," ","integrate(sec(x)^3/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{{\left(a + 5 \, b\right)} \log\left(\sin\left(x\right) + 1\right)}{4 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)}} - \frac{{\left(a + 5 \, b\right)} \log\left(-\sin\left(x\right) + 1\right)}{4 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)}} + \frac{{\left(5 \, a b^{2} + b^{3}\right)} \arctan\left(\frac{b \sin\left(x\right)}{\sqrt{a b}}\right)}{2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \sqrt{a b}} - \frac{a b \sin\left(x\right)^{3} - b^{2} \sin\left(x\right)^{3} + a^{2} \sin\left(x\right) + b^{2} \sin\left(x\right)}{2 \, {\left(b \sin\left(x\right)^{4} + a \sin\left(x\right)^{2} - b \sin\left(x\right)^{2} - a\right)} {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)}}"," ",0,"1/4*(a + 5*b)*log(sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(a + 5*b)*log(-sin(x) + 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/2*(5*a*b^2 + b^3)*arctan(b*sin(x)/sqrt(a*b))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a*b)) - 1/2*(a*b*sin(x)^3 - b^2*sin(x)^3 + a^2*sin(x) + b^2*sin(x))/((b*sin(x)^4 + a*sin(x)^2 - b*sin(x)^2 - a)*(a^3 + 2*a^2*b + a*b^2))","B",0
323,1,270,0,0.142061," ","integrate(sec(x)^4/(a+b*sin(x)^2)^2,x, algorithm=""giac"")","\frac{b^{3} \tan\left(x\right)}{2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} {\left(a \tan\left(x\right)^{2} + b \tan\left(x\right)^{2} + a\right)}} + \frac{{\left(6 \, a b^{2} + b^{3}\right)} {\left(\pi \left \lfloor \frac{x}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(x\right) + b \tan\left(x\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \sqrt{a^{2} + a b}} + \frac{a^{4} \tan\left(x\right)^{3} + 4 \, a^{3} b \tan\left(x\right)^{3} + 6 \, a^{2} b^{2} \tan\left(x\right)^{3} + 4 \, a b^{3} \tan\left(x\right)^{3} + b^{4} \tan\left(x\right)^{3} + 3 \, a^{4} \tan\left(x\right) + 18 \, a^{3} b \tan\left(x\right) + 36 \, a^{2} b^{2} \tan\left(x\right) + 30 \, a b^{3} \tan\left(x\right) + 9 \, b^{4} \tan\left(x\right)}{3 \, {\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)}}"," ",0,"1/2*b^3*tan(x)/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*(a*tan(x)^2 + b*tan(x)^2 + a)) + 1/2*(6*a*b^2 + b^3)*(pi*floor(x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(a^2 + a*b)) + 1/3*(a^4*tan(x)^3 + 4*a^3*b*tan(x)^3 + 6*a^2*b^2*tan(x)^3 + 4*a*b^3*tan(x)^3 + b^4*tan(x)^3 + 3*a^4*tan(x) + 18*a^3*b*tan(x) + 36*a^2*b^2*tan(x) + 30*a*b^3*tan(x) + 9*b^4*tan(x))/(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)","B",0
324,-2,0,0,0.000000," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)1/f*(2*(-8*b^2/64/b^2*sin(f*x+exp(1))*sin(f*x+exp(1))-(-16*b^2+4*b*a)/64/b^2)*sin(f*x+exp(1))*sqrt(a+b*sin(f*x+exp(1))^2)+2*(-a^2-4*a*b)/16/b/sqrt(b)*ln(abs(sqrt(a+b*sin(f*x+exp(1))^2)-sqrt(b)*sin(f*x+exp(1)))))","F(-2)",0
325,-2,0,0,0.000000," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)1/f*(1/2*sin(f*x+exp(1))*sqrt(a+b*sin(f*x+exp(1))^2)-2*a/4/sqrt(b)*ln(abs(sqrt(a+b*sin(f*x+exp(1))^2)-sqrt(b)*sin(f*x+exp(1)))))","F(-2)",0
326,0,0,0,0.000000," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sec(f*x + e), x)","F",0
327,0,0,0,0.000000," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sec(f*x + e)^3, x)","F",0
328,0,0,0,0.000000," ","integrate(sec(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sec(f*x + e)^5, x)","F",0
329,0,0,0,0.000000," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*cos(f*x + e)^4, x)","F",0
330,0,0,0,0.000000," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \cos\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*cos(f*x + e)^2, x)","F",0
331,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a), x)","F",0
332,0,0,0,0.000000," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sec(f*x + e)^2, x)","F",0
333,0,0,0,0.000000," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \sec\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*sec(f*x + e)^4, x)","F",0
334,-2,0,0,0.000000," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)1/f*(2*((-192*b^5/2304/b^4*sin(f*x+exp(1))*sin(f*x+exp(1))+(288*b^5-336*b^4*a)/2304/b^4)*sin(f*x+exp(1))*sin(f*x+exp(1))+(720*b^4*a-72*b^3*a^2)/2304/b^4)*sin(f*x+exp(1))*sqrt(a+b*sin(f*x+exp(1))^2)+2*(-a^3-6*a^2*b)/32/b/sqrt(b)*ln(abs(sqrt(a+b*sin(f*x+exp(1))^2)-sqrt(b)*sin(f*x+exp(1)))))","F(-2)",0
335,-2,0,0,0.000000," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)1/f*(2*(4*b^3/32/b^2*sin(f*x+exp(1))*sin(f*x+exp(1))+10*b^2*a/32/b^2)*sin(f*x+exp(1))*sqrt(a+b*sin(f*x+exp(1))^2)-6*a^2/16/sqrt(b)*ln(abs(sqrt(a+b*sin(f*x+exp(1))^2)-sqrt(b)*sin(f*x+exp(1)))))","F(-2)",0
336,0,0,0,0.000000," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sec\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sec(f*x + e), x)","F",0
337,0,0,0,0.000000," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sec\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^3, x)","F",0
338,0,0,0,0.000000," ","integrate(sec(f*x+e)^5*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sec\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^5, x)","F",0
339,0,0,0,0.000000," ","integrate(sec(f*x+e)^7*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sec\left(f x + e\right)^{7}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^7, x)","F",0
340,0,0,0,0.000000," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \cos\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*cos(f*x + e)^4, x)","F",0
341,0,0,0,0.000000," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \cos\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*cos(f*x + e)^2, x)","F",0
342,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
343,0,0,0,0.000000," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sec\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^2, x)","F",0
344,0,0,0,0.000000," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \sec\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*sec(f*x + e)^4, x)","F",0
345,-2,0,0,0.000000," ","integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)1/f*(-1/2/b*sin(f*x+exp(1))*sqrt(b*sin(f*x+exp(1))^2+a)+2*(-a-2*b)/4/b/sqrt(b)*ln(abs(sqrt(b*sin(f*x+exp(1))^2+a)-sqrt(b)*sin(f*x+exp(1)))))","F(-2)",0
346,-2,0,0,0.000000," ","integrate(cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-1/f/sqrt(b)*ln(abs(sqrt(b*sin(f*x+exp(1))^2+a)-sqrt(b)*sin(f*x+exp(1))))","F(-2)",0
347,0,0,0,0.000000," ","integrate(sec(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(sec(f*x + e)/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
348,0,0,0,0.000000," ","integrate(sec(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)^{3}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(sec(f*x + e)^3/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
349,0,0,0,0.000000," ","integrate(cos(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{4}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(cos(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
350,0,0,0,0.000000," ","integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{2}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(cos(f*x + e)^2/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
351,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(1/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
352,0,0,0,0.000000," ","integrate(sec(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)^{2}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(sec(f*x + e)^2/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
353,0,0,0,0.000000," ","integrate(sec(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)^{4}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(sec(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
354,-2,0,0,0.000000," ","integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)1/f*(-2*(-2*b-2*a)/4/b/a*sin(f*x+exp(1))*sqrt(b*sin(f*x+exp(1))^2+a)/(b*sin(f*x+exp(1))^2+a)+1/b/sqrt(b)*ln(abs(sqrt(b*sin(f*x+exp(1))^2+a)-sqrt(b)*sin(f*x+exp(1)))))","F(-2)",0
355,1,29,0,0.626937," ","integrate(cos(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\sin\left(f x + e\right)}{\sqrt{b \sin\left(f x + e\right)^{2} + a} a f}"," ",0,"sin(f*x + e)/(sqrt(b*sin(f*x + e)^2 + a)*a*f)","A",0
356,0,0,0,0.000000," ","integrate(sec(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(sec(f*x + e)/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
357,0,0,0,0.000000," ","integrate(sec(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)^{3}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(sec(f*x + e)^3/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
358,0,0,0,0.000000," ","integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{6}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^6/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
359,0,0,0,0.000000," ","integrate(cos(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
360,0,0,0,0.000000," ","integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
361,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(-3/2), x)","F",0
362,0,0,0,0.000000," ","integrate(sec(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(sec(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
363,0,0,0,0.000000," ","integrate(cos(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{5}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^5/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
364,1,58,0,0.778111," ","integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{{\left(\frac{{\left(a b - 2 \, b^{2}\right)} \sin\left(f x + e\right)^{2}}{a^{2} b} - \frac{3}{a}\right)} \sin\left(f x + e\right)}{3 \, {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} f}"," ",0,"-1/3*((a*b - 2*b^2)*sin(f*x + e)^2/(a^2*b) - 3/a)*sin(f*x + e)/((b*sin(f*x + e)^2 + a)^(3/2)*f)","A",0
365,1,48,0,0.731571," ","integrate(cos(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\frac{{\left(\frac{2 \, b \sin\left(f x + e\right)^{2}}{a^{2}} + \frac{3}{a}\right)} \sin\left(f x + e\right)}{3 \, {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} f}"," ",0,"1/3*(2*b*sin(f*x + e)^2/a^2 + 3/a)*sin(f*x + e)/((b*sin(f*x + e)^2 + a)^(3/2)*f)","A",0
366,0,0,0,0.000000," ","integrate(sec(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(sec(f*x + e)/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
367,0,0,0,0.000000," ","integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{6}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^6/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
368,0,0,0,0.000000," ","integrate(cos(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
369,0,0,0,0.000000," ","integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\cos\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(cos(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
370,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(-5/2), x)","F",0
371,0,0,0,0.000000," ","integrate(sec(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\sec\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(sec(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
372,0,0,0,0.000000," ","integrate((d*cos(f*x+e))^m*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \left(d \cos\left(f x + e\right)\right)^{m}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*(d*cos(f*x + e))^m, x)","F",0
373,0,0,0,0.000000," ","integrate(cos(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*cos(f*x + e)^5, x)","F",0
374,0,0,0,0.000000," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*cos(f*x + e)^3, x)","F",0
375,0,0,0,0.000000," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*cos(f*x + e), x)","F",0
376,0,0,0,0.000000," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sec(f*x + e), x)","F",0
377,0,0,0,0.000000," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sec(f*x + e)^3, x)","F",0
378,0,0,0,0.000000," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*cos(f*x + e)^4, x)","F",0
379,0,0,0,0.000000," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \cos\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*cos(f*x + e)^2, x)","F",0
380,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p, x)","F",0
381,0,0,0,0.000000," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sec(f*x + e)^2, x)","F",0
382,0,0,0,0.000000," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \sec\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*sec(f*x + e)^4, x)","F",0
383,1,221,0,0.188879," ","integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\frac{\frac{3 \, \sin\left(d x + c\right)^{2}}{b} - \frac{4 \, \log\left({\left| b \sin\left(d x + c\right)^{3} + a \right|}\right)}{b} + \frac{2 \, \sqrt{3} {\left(\left(-a b^{2}\right)^{\frac{1}{3}} b^{2} + \left(-a b^{2}\right)^{\frac{2}{3}} a\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a b^{3}} + \frac{{\left(\left(-a b^{2}\right)^{\frac{1}{3}} b^{2} - \left(-a b^{2}\right)^{\frac{2}{3}} a\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a b^{3}} + \frac{2 \, {\left(a b^{4} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - b^{5}\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a b^{5}}}{6 \, d}"," ",0,"1/6*(3*sin(d*x + c)^2/b - 4*log(abs(b*sin(d*x + c)^3 + a))/b + 2*sqrt(3)*((-a*b^2)^(1/3)*b^2 + (-a*b^2)^(2/3)*a)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a*b^3) + ((-a*b^2)^(1/3)*b^2 - (-a*b^2)^(2/3)*a)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a*b^3) + 2*(a*b^4*(-a/b)^(1/3) - b^5)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a*b^5))/d","A",0
384,1,156,0,0.184843," ","integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","-\frac{\frac{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a} + \frac{2 \, \log\left({\left| b \sin\left(d x + c\right)^{3} + a \right|}\right)}{b} - \frac{2 \, \sqrt{3} \left(-a b^{2}\right)^{\frac{1}{3}} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a b} - \frac{\left(-a b^{2}\right)^{\frac{1}{3}} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a b}}{6 \, d}"," ",0,"-1/6*(2*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/a + 2*log(abs(b*sin(d*x + c)^3 + a))/b - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a*b) - (-a*b^2)^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a*b))/d","A",0
385,1,137,0,0.148841," ","integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","-\frac{\frac{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a} - \frac{2 \, \sqrt{3} \left(-a b^{2}\right)^{\frac{1}{3}} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a b} - \frac{\left(-a b^{2}\right)^{\frac{1}{3}} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a b}}{6 \, d}"," ",0,"-1/6*(2*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/a - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a*b) - (-a*b^2)^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a*b))/d","A",0
386,1,309,0,0.331615," ","integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","-\frac{\frac{2 \, {\left(a^{3} b^{2} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - a b^{4} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - a^{2} b^{3} + b^{5}\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{5} b - 2 \, a^{3} b^{3} + a b^{5}} + \frac{2 \, {\left(\sqrt{3} \left(-a b^{2}\right)^{\frac{1}{3}} b^{2} + \sqrt{3} \left(-a b^{2}\right)^{\frac{2}{3}} a\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{3} b - a b^{3}} + \frac{{\left(\left(-a b^{2}\right)^{\frac{1}{3}} b^{2} - \left(-a b^{2}\right)^{\frac{2}{3}} a\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{3} b - a b^{3}} + \frac{2 \, b \log\left({\left| b \sin\left(d x + c\right)^{3} + a \right|}\right)}{a^{2} - b^{2}} - \frac{3 \, \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a - b} + \frac{3 \, \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a + b}}{6 \, d}"," ",0,"-1/6*(2*(a^3*b^2*(-a/b)^(1/3) - a*b^4*(-a/b)^(1/3) - a^2*b^3 + b^5)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^5*b - 2*a^3*b^3 + a*b^5) + 2*(sqrt(3)*(-a*b^2)^(1/3)*b^2 + sqrt(3)*(-a*b^2)^(2/3)*a)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^3*b - a*b^3) + ((-a*b^2)^(1/3)*b^2 - (-a*b^2)^(2/3)*a)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^3*b - a*b^3) + 2*b*log(abs(b*sin(d*x + c)^3 + a))/(a^2 - b^2) - 3*log(abs(sin(d*x + c) + 1))/(a - b) + 3*log(abs(sin(d*x + c) - 1))/(a + b))/d","A",0
387,1,510,0,0.233554," ","integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\frac{\frac{4 \, {\left(3 \, a^{5} b^{4} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 6 \, a^{3} b^{6} \left(-\frac{a}{b}\right)^{\frac{1}{3}} + 3 \, a b^{8} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 2 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - b^{9}\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}} + \frac{4 \, {\left(3 \, \sqrt{3} \left(-a b^{2}\right)^{\frac{2}{3}} a b + {\left(2 \, \sqrt{3} a^{2} b + \sqrt{3} b^{3}\right)} \left(-a b^{2}\right)^{\frac{1}{3}}\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} - \frac{2 \, {\left(3 \, \left(-a b^{2}\right)^{\frac{2}{3}} a b - {\left(2 \, a^{2} b + b^{3}\right)} \left(-a b^{2}\right)^{\frac{1}{3}}\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac{4 \, {\left(a^{2} b + 2 \, b^{3}\right)} \log\left({\left| b \sin\left(d x + c\right)^{3} + a \right|}\right)}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{3 \, {\left(a - 4 \, b\right)} \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a^{2} - 2 \, a b + b^{2}} - \frac{3 \, {\left(a + 4 \, b\right)} \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a^{2} + 2 \, a b + b^{2}} + \frac{6 \, {\left(a^{2} b \sin\left(d x + c\right)^{2} + 2 \, b^{3} \sin\left(d x + c\right)^{2} - a^{3} \sin\left(d x + c\right) + a b^{2} \sin\left(d x + c\right) - 3 \, b^{3}\right)}}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\sin\left(d x + c\right)^{2} - 1\right)}}}{12 \, d}"," ",0,"1/12*(4*(3*a^5*b^4*(-a/b)^(1/3) - 6*a^3*b^6*(-a/b)^(1/3) + 3*a*b^8*(-a/b)^(1/3) - 2*a^6*b^3 + 3*a^4*b^5 - b^9)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9) + 4*(3*sqrt(3)*(-a*b^2)^(2/3)*a*b + (2*sqrt(3)*a^2*b + sqrt(3)*b^3)*(-a*b^2)^(1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^5 - 2*a^3*b^2 + a*b^4) - 2*(3*(-a*b^2)^(2/3)*a*b - (2*a^2*b + b^3)*(-a*b^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^5 - 2*a^3*b^2 + a*b^4) + 4*(a^2*b + 2*b^3)*log(abs(b*sin(d*x + c)^3 + a))/(a^4 - 2*a^2*b^2 + b^4) + 3*(a - 4*b)*log(abs(sin(d*x + c) + 1))/(a^2 - 2*a*b + b^2) - 3*(a + 4*b)*log(abs(sin(d*x + c) - 1))/(a^2 + 2*a*b + b^2) + 6*(a^2*b*sin(d*x + c)^2 + 2*b^3*sin(d*x + c)^2 - a^3*sin(d*x + c) + a*b^2*sin(d*x + c) - 3*b^3)/((a^4 - 2*a^2*b^2 + b^4)*(sin(d*x + c)^2 - 1)))/d","A",0
388,0,0,0,0.000000," ","integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\cos\left(d x + c\right)^{4}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(cos(d*x + c)^4/(b*sin(d*x + c)^3 + a), x)","F",0
389,0,0,0,0.000000," ","integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\cos\left(d x + c\right)^{2}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(cos(d*x + c)^2/(b*sin(d*x + c)^3 + a), x)","F",0
390,0,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{1}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(1/(b*sin(d*x + c)^3 + a), x)","F",0
391,0,0,0,0.000000," ","integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sec\left(d x + c\right)^{2}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sec(d*x + c)^2/(b*sin(d*x + c)^3 + a), x)","F",0
392,0,0,0,0.000000," ","integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""giac"")","\int \frac{\sec\left(d x + c\right)^{4}}{b \sin\left(d x + c\right)^{3} + a}\,{d x}"," ",0,"integrate(sec(d*x + c)^4/(b*sin(d*x + c)^3 + a), x)","F",0
393,1,277,0,0.213657," ","integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","-\frac{\frac{9 \, \sin\left(d x + c\right)}{b^{2}} + \frac{2 \, {\left(3 \, a b \left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, a^{2} + b^{2}\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{2} b^{2}} + \frac{2 \, \sqrt{3} {\left(3 \, \left(-a b^{2}\right)^{\frac{2}{3}} a - \left(-a b^{2}\right)^{\frac{1}{3}} {\left(2 \, a^{2} + b^{2}\right)}\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{2} b^{3}} + \frac{3 \, {\left(3 \, a b \sin\left(d x + c\right)^{2} + a^{2} \sin\left(d x + c\right) - b^{2} \sin\left(d x + c\right) - 3 \, a b\right)}}{{\left(b \sin\left(d x + c\right)^{3} + a\right)} a b^{2}} - \frac{{\left(3 \, \left(-a b^{2}\right)^{\frac{2}{3}} a + \left(-a b^{2}\right)^{\frac{1}{3}} {\left(2 \, a^{2} + b^{2}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{2} b^{3}}}{9 \, d}"," ",0,"-1/9*(9*sin(d*x + c)/b^2 + 2*(3*a*b*(-a/b)^(1/3) + 2*a^2 + b^2)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^2*b^2) + 2*sqrt(3)*(3*(-a*b^2)^(2/3)*a - (-a*b^2)^(1/3)*(2*a^2 + b^2))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b^3) + 3*(3*a*b*sin(d*x + c)^2 + a^2*sin(d*x + c) - b^2*sin(d*x + c) - 3*a*b)/((b*sin(d*x + c)^3 + a)*a*b^2) - (3*(-a*b^2)^(2/3)*a + (-a*b^2)^(1/3)*(2*a^2 + b^2))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b^3))/d","A",0
394,1,228,0,0.255957," ","integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","-\frac{\frac{2 \, {\left(a \left(-\frac{a}{b}\right)^{\frac{1}{3}} + b\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{2} b} + \frac{3 \, {\left(a \sin\left(d x + c\right)^{2} - b \sin\left(d x + c\right) - 2 \, a\right)}}{{\left(b \sin\left(d x + c\right)^{3} + a\right)} a b} - \frac{2 \, \sqrt{3} {\left(\left(-a b^{2}\right)^{\frac{1}{3}} b^{2} - \left(-a b^{2}\right)^{\frac{2}{3}} a\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{2} b^{3}} - \frac{{\left(\left(-a b^{2}\right)^{\frac{1}{3}} b^{2} + \left(-a b^{2}\right)^{\frac{2}{3}} a\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{2} b^{3}}}{9 \, d}"," ",0,"-1/9*(2*(a*(-a/b)^(1/3) + b)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^2*b) + 3*(a*sin(d*x + c)^2 - b*sin(d*x + c) - 2*a)/((b*sin(d*x + c)^3 + a)*a*b) - 2*sqrt(3)*((-a*b^2)^(1/3)*b^2 - (-a*b^2)^(2/3)*a)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b^3) - ((-a*b^2)^(1/3)*b^2 + (-a*b^2)^(2/3)*a)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b^3))/d","A",0
395,1,169,0,0.240311," ","integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","-\frac{\frac{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{2}} - \frac{2 \, \sqrt{3} \left(-a b^{2}\right)^{\frac{1}{3}} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{2} b} - \frac{\left(-a b^{2}\right)^{\frac{1}{3}} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{2} b} - \frac{3 \, {\left(b \sin\left(d x + c\right) + a\right)}}{{\left(b \sin\left(d x + c\right)^{3} + a\right)} a b}}{9 \, d}"," ",0,"-1/9*(2*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/a^2 - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b) - (-a*b^2)^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b) - 3*(b*sin(d*x + c) + a)/((b*sin(d*x + c)^3 + a)*a*b))/d","A",0
396,1,162,0,0.221828," ","integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","-\frac{\frac{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{2}} - \frac{3 \, \sin\left(d x + c\right)}{{\left(b \sin\left(d x + c\right)^{3} + a\right)} a} - \frac{2 \, \sqrt{3} \left(-a b^{2}\right)^{\frac{1}{3}} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{2} b} - \frac{\left(-a b^{2}\right)^{\frac{1}{3}} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{2} b}}{9 \, d}"," ",0,"-1/9*(2*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/a^2 - 3*sin(d*x + c)/((b*sin(d*x + c)^3 + a)*a) - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^2*b) - (-a*b^2)^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^2*b))/d","A",0
397,1,566,0,0.279125," ","integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","-\frac{\frac{12 \, a b \log\left({\left| b \sin\left(d x + c\right)^{3} + a \right|}\right)}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{4 \, {\left(2 \, a^{8} b^{2} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 3 \, a^{6} b^{4} \left(-\frac{a}{b}\right)^{\frac{1}{3}} + a^{2} b^{8} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 4 \, a^{7} b^{3} + 9 \, a^{5} b^{5} - 6 \, a^{3} b^{7} + a b^{9}\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}} + \frac{4 \, {\left({\left(2 \, \sqrt{3} a^{3} + \sqrt{3} a b^{2}\right)} \left(-a b^{2}\right)^{\frac{2}{3}} + {\left(4 \, \sqrt{3} a^{2} b^{2} - \sqrt{3} b^{4}\right)} \left(-a b^{2}\right)^{\frac{1}{3}}\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac{2 \, {\left({\left(2 \, a^{3} + a b^{2}\right)} \left(-a b^{2}\right)^{\frac{2}{3}} - {\left(4 \, a^{2} b^{2} - b^{4}\right)} \left(-a b^{2}\right)^{\frac{1}{3}}\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac{9 \, \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a^{2} - 2 \, a b + b^{2}} + \frac{9 \, \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a^{2} + 2 \, a b + b^{2}} - \frac{6 \, {\left(2 \, a^{2} b^{2} \sin\left(d x + c\right)^{3} + a^{3} b \sin\left(d x + c\right)^{2} - a b^{3} \sin\left(d x + c\right)^{2} - a^{2} b^{2} \sin\left(d x + c\right) + b^{4} \sin\left(d x + c\right) + 3 \, a^{3} b - a b^{3}\right)}}{{\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} {\left(b \sin\left(d x + c\right)^{3} + a\right)}}}{18 \, d}"," ",0,"-1/18*(12*a*b*log(abs(b*sin(d*x + c)^3 + a))/(a^4 - 2*a^2*b^2 + b^4) + 4*(2*a^8*b^2*(-a/b)^(1/3) - 3*a^6*b^4*(-a/b)^(1/3) + a^2*b^8*(-a/b)^(1/3) - 4*a^7*b^3 + 9*a^5*b^5 - 6*a^3*b^7 + a*b^9)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9) + 4*((2*sqrt(3)*a^3 + sqrt(3)*a*b^2)*(-a*b^2)^(2/3) + (4*sqrt(3)*a^2*b^2 - sqrt(3)*b^4)*(-a*b^2)^(1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^6*b - 2*a^4*b^3 + a^2*b^5) - 2*((2*a^3 + a*b^2)*(-a*b^2)^(2/3) - (4*a^2*b^2 - b^4)*(-a*b^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^6*b - 2*a^4*b^3 + a^2*b^5) - 9*log(abs(sin(d*x + c) + 1))/(a^2 - 2*a*b + b^2) + 9*log(abs(sin(d*x + c) - 1))/(a^2 + 2*a*b + b^2) - 6*(2*a^2*b^2*sin(d*x + c)^3 + a^3*b*sin(d*x + c)^2 - a*b^3*sin(d*x + c)^2 - a^2*b^2*sin(d*x + c) + b^4*sin(d*x + c) + 3*a^3*b - a*b^3)/((a^5 - 2*a^3*b^2 + a*b^4)*(b*sin(d*x + c)^3 + a)))/d","A",0
398,1,790,0,0.329456," ","integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","\frac{\frac{8 \, {\left(15 \, a^{10} b^{4} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 42 \, a^{8} b^{6} \left(-\frac{a}{b}\right)^{\frac{1}{3}} + 36 \, a^{6} b^{8} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 6 \, a^{4} b^{10} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 3 \, a^{2} b^{12} \left(-\frac{a}{b}\right)^{\frac{1}{3}} - 8 \, a^{11} b^{3} + 13 \, a^{9} b^{5} + 10 \, a^{7} b^{7} - 28 \, a^{5} b^{9} + 14 \, a^{3} b^{11} - a b^{13}\right)} \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(d x + c\right) \right|}\right)}{a^{15} b - 6 \, a^{13} b^{3} + 15 \, a^{11} b^{5} - 20 \, a^{9} b^{7} + 15 \, a^{7} b^{9} - 6 \, a^{5} b^{11} + a^{3} b^{13}} + \frac{8 \, {\left(3 \, {\left(5 \, \sqrt{3} a^{3} b + \sqrt{3} a b^{3}\right)} \left(-a b^{2}\right)^{\frac{2}{3}} + {\left(8 \, \sqrt{3} a^{4} b + 11 \, \sqrt{3} a^{2} b^{3} - \sqrt{3} b^{5}\right)} \left(-a b^{2}\right)^{\frac{1}{3}}\right)} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(d x + c\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} - \frac{4 \, {\left(3 \, {\left(5 \, a^{3} b + a b^{3}\right)} \left(-a b^{2}\right)^{\frac{2}{3}} - {\left(8 \, a^{4} b + 11 \, a^{2} b^{3} - b^{5}\right)} \left(-a b^{2}\right)^{\frac{1}{3}}\right)} \log\left(\sin\left(d x + c\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(d x + c\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} + \frac{24 \, {\left(a^{3} b + 5 \, a b^{3}\right)} \log\left({\left| b \sin\left(d x + c\right)^{3} + a \right|}\right)}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{9 \, {\left(a - 7 \, b\right)} \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{9 \, {\left(a + 7 \, b\right)} \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac{6 \, {\left(3 \, a^{3} b \sin\left(d x + c\right)^{4} + 9 \, a b^{3} \sin\left(d x + c\right)^{4} - 10 \, a^{2} b^{2} \sin\left(d x + c\right)^{3} - 2 \, b^{4} \sin\left(d x + c\right)^{3} + 2 \, a^{3} b \sin\left(d x + c\right)^{2} - 2 \, a b^{3} \sin\left(d x + c\right)^{2} + 3 \, a^{4} \sin\left(d x + c\right) + 7 \, a^{2} b^{2} \sin\left(d x + c\right) + 2 \, b^{4} \sin\left(d x + c\right) - 8 \, a^{3} b - 4 \, a b^{3}\right)}}{{\left(b \sin\left(d x + c\right)^{5} - b \sin\left(d x + c\right)^{3} + a \sin\left(d x + c\right)^{2} - a\right)} {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)}}}{36 \, d}"," ",0,"1/36*(8*(15*a^10*b^4*(-a/b)^(1/3) - 42*a^8*b^6*(-a/b)^(1/3) + 36*a^6*b^8*(-a/b)^(1/3) - 6*a^4*b^10*(-a/b)^(1/3) - 3*a^2*b^12*(-a/b)^(1/3) - 8*a^11*b^3 + 13*a^9*b^5 + 10*a^7*b^7 - 28*a^5*b^9 + 14*a^3*b^11 - a*b^13)*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^15*b - 6*a^13*b^3 + 15*a^11*b^5 - 20*a^9*b^7 + 15*a^7*b^9 - 6*a^5*b^11 + a^3*b^13) + 8*(3*(5*sqrt(3)*a^3*b + sqrt(3)*a*b^3)*(-a*b^2)^(2/3) + (8*sqrt(3)*a^4*b + 11*sqrt(3)*a^2*b^3 - sqrt(3)*b^5)*(-a*b^2)^(1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6) - 4*(3*(5*a^3*b + a*b^3)*(-a*b^2)^(2/3) - (8*a^4*b + 11*a^2*b^3 - b^5)*(-a*b^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6) + 24*(a^3*b + 5*a*b^3)*log(abs(b*sin(d*x + c)^3 + a))/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) + 9*(a - 7*b)*log(abs(sin(d*x + c) + 1))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 9*(a + 7*b)*log(abs(sin(d*x + c) - 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 6*(3*a^3*b*sin(d*x + c)^4 + 9*a*b^3*sin(d*x + c)^4 - 10*a^2*b^2*sin(d*x + c)^3 - 2*b^4*sin(d*x + c)^3 + 2*a^3*b*sin(d*x + c)^2 - 2*a*b^3*sin(d*x + c)^2 + 3*a^4*sin(d*x + c) + 7*a^2*b^2*sin(d*x + c) + 2*b^4*sin(d*x + c) - 8*a^3*b - 4*a*b^3)/((b*sin(d*x + c)^5 - b*sin(d*x + c)^3 + a*sin(d*x + c)^2 - a)*(a^5 - 2*a^3*b^2 + a*b^4)))/d","A",0
399,-1,0,0,0.000000," ","integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
400,0,0,0,0.000000," ","integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","\int \frac{\cos\left(d x + c\right)^{2}}{{\left(b \sin\left(d x + c\right)^{3} + a\right)}^{2}}\,{d x}"," ",0,"integrate(cos(d*x + c)^2/(b*sin(d*x + c)^3 + a)^2, x)","F",0
401,0,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(d x + c\right)^{3} + a\right)}^{2}}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^3 + a)^(-2), x)","F",0
402,0,0,0,0.000000," ","integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","\int \frac{\sec\left(d x + c\right)^{2}}{{\left(b \sin\left(d x + c\right)^{3} + a\right)}^{2}}\,{d x}"," ",0,"integrate(sec(d*x + c)^2/(b*sin(d*x + c)^3 + a)^2, x)","F",0
403,0,0,0,0.000000," ","integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""giac"")","\int \frac{\sec\left(d x + c\right)^{4}}{{\left(b \sin\left(d x + c\right)^{3} + a\right)}^{2}}\,{d x}"," ",0,"integrate(sec(d*x + c)^4/(b*sin(d*x + c)^3 + a)^2, x)","F",0
404,1,360,0,0.915379," ","integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{8 \, {\left(b^{2} \sin\left(d x + c\right)^{3} - 9 \, b^{2} \sin\left(d x + c\right)\right)}}{b^{3}} - \frac{6 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} - \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b^{4}} - \frac{6 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} - \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b^{4}} + \frac{3 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} + \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b^{4}} - \frac{3 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} + \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b^{4}}}{24 \, d}"," ",0,"1/24*(8*(b^2*sin(d*x + c)^3 - 9*b^2*sin(d*x + c))/b^3 - 6*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^4) - 6*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^4) + 3*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^4) - 3*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^4))/d","B",0
405,1,311,0,0.885816," ","integrate(cos(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{8 \, \sin\left(d x + c\right)}{b} - \frac{2 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} {\left(a b + b^{2}\right)} - 2 \, \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b^{3}} - \frac{2 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} {\left(a b + b^{2}\right)} - 2 \, \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b^{3}} - \frac{\sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} {\left(a b + b^{2}\right)} + 2 \, \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b^{3}} + \frac{\sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} {\left(a b + b^{2}\right)} + 2 \, \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b^{3}}}{8 \, d}"," ",0,"-1/8*(8*sin(d*x + c)/b - 2*sqrt(2)*((-a*b^3)^(1/4)*(a*b + b^2) - 2*(-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^3) - 2*sqrt(2)*((-a*b^3)^(1/4)*(a*b + b^2) - 2*(-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^3) - sqrt(2)*((-a*b^3)^(1/4)*(a*b + b^2) + 2*(-a*b^3)^(3/4))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^3) + sqrt(2)*((-a*b^3)^(1/4)*(a*b + b^2) + 2*(-a*b^3)^(3/4))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^3))/d","B",0
406,1,280,0,0.771003," ","integrate(cos(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{2 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} b^{2} - \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b^{3}} + \frac{2 \, \sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} b^{2} - \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b^{3}} + \frac{\sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} b^{2} + \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b^{3}} - \frac{\sqrt{2} {\left(\left(-a b^{3}\right)^{\frac{1}{4}} b^{2} + \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b^{3}}}{8 \, d}"," ",0,"1/8*(2*sqrt(2)*((-a*b^3)^(1/4)*b^2 - (-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^3) + 2*sqrt(2)*((-a*b^3)^(1/4)*b^2 - (-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^3) + sqrt(2)*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^3) - sqrt(2)*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^3))/d","B",0
407,1,224,0,0.746522," ","integrate(cos(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{2 \, \sqrt{2} \left(-a b^{3}\right)^{\frac{1}{4}} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b} + \frac{2 \, \sqrt{2} \left(-a b^{3}\right)^{\frac{1}{4}} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{a b} + \frac{\sqrt{2} \left(-a b^{3}\right)^{\frac{1}{4}} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b} - \frac{\sqrt{2} \left(-a b^{3}\right)^{\frac{1}{4}} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a b}}{8 \, d}"," ",0,"1/8*(2*sqrt(2)*(-a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b) + 2*sqrt(2)*(-a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b) + sqrt(2)*(-a*b^3)^(1/4)*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b) - sqrt(2)*(-a*b^3)^(1/4)*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b))/d","B",0
408,1,370,0,0.794872," ","integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{4 \, {\left(\left(-a b^{3}\right)^{\frac{1}{4}} b^{2} + \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{\sqrt{2} a^{2} b^{2} - \sqrt{2} a b^{3}} + \frac{4 \, {\left(\left(-a b^{3}\right)^{\frac{1}{4}} b^{2} + \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{\sqrt{2} a^{2} b^{2} - \sqrt{2} a b^{3}} + \frac{{\left(\sqrt{2} \left(-a b^{3}\right)^{\frac{1}{4}} b^{2} - \sqrt{2} \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a^{2} b^{2} - a b^{3}} - \frac{{\left(\sqrt{2} \left(-a b^{3}\right)^{\frac{1}{4}} b^{2} - \sqrt{2} \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a^{2} b^{2} - a b^{3}} - \frac{4 \, \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a - b} + \frac{4 \, \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a - b}}{8 \, d}"," ",0,"-1/8*(4*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^2*b^2 - sqrt(2)*a*b^3) + 4*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^2*b^2 - sqrt(2)*a*b^3) + (sqrt(2)*(-a*b^3)^(1/4)*b^2 - sqrt(2)*(-a*b^3)^(3/4))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a^2*b^2 - a*b^3) - (sqrt(2)*(-a*b^3)^(1/4)*b^2 - sqrt(2)*(-a*b^3)^(3/4))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a^2*b^2 - a*b^3) - 4*log(abs(sin(d*x + c) + 1))/(a - b) + 4*log(abs(sin(d*x + c) - 1))/(a - b))/d","B",0
409,1,475,0,0.799977," ","integrate(sec(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{4 \, {\left(\left(-a b^{3}\right)^{\frac{1}{4}} {\left(a b + b^{2}\right)} + 2 \, \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{\sqrt{2} a^{3} b - 2 \, \sqrt{2} a^{2} b^{2} + \sqrt{2} a b^{3}} + \frac{4 \, {\left(\left(-a b^{3}\right)^{\frac{1}{4}} {\left(a b + b^{2}\right)} + 2 \, \left(-a b^{3}\right)^{\frac{3}{4}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{\sqrt{2} a^{3} b - 2 \, \sqrt{2} a^{2} b^{2} + \sqrt{2} a b^{3}} - \frac{{\left(2 \, \sqrt{2} \left(-a b^{3}\right)^{\frac{3}{4}} - \left(-a b^{3}\right)^{\frac{1}{4}} {\left(\sqrt{2} a b + \sqrt{2} b^{2}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}} + \frac{{\left(2 \, \sqrt{2} \left(-a b^{3}\right)^{\frac{3}{4}} - \left(-a b^{3}\right)^{\frac{1}{4}} {\left(\sqrt{2} a b + \sqrt{2} b^{2}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}} + \frac{2 \, {\left(a - 5 \, b\right)} \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a^{2} - 2 \, a b + b^{2}} - \frac{2 \, {\left(a - 5 \, b\right)} \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a^{2} - 2 \, a b + b^{2}} - \frac{4 \, \sin\left(d x + c\right)}{{\left(\sin\left(d x + c\right)^{2} - 1\right)} {\left(a - b\right)}}}{8 \, d}"," ",0,"1/8*(4*((-a*b^3)^(1/4)*(a*b + b^2) + 2*(-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^3*b - 2*sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 4*((-a*b^3)^(1/4)*(a*b + b^2) + 2*(-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^3*b - 2*sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) - (2*sqrt(2)*(-a*b^3)^(3/4) - (-a*b^3)^(1/4)*(sqrt(2)*a*b + sqrt(2)*b^2))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a^3*b - 2*a^2*b^2 + a*b^3) + (2*sqrt(2)*(-a*b^3)^(3/4) - (-a*b^3)^(1/4)*(sqrt(2)*a*b + sqrt(2)*b^2))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a^3*b - 2*a^2*b^2 + a*b^3) + 2*(a - 5*b)*log(abs(sin(d*x + c) + 1))/(a^2 - 2*a*b + b^2) - 2*(a - 5*b)*log(abs(sin(d*x + c) - 1))/(a^2 - 2*a*b + b^2) - 4*sin(d*x + c)/((sin(d*x + c)^2 - 1)*(a - b)))/d","B",0
410,1,630,0,0.827911," ","integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{8 \, {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} + \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} + 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{\sqrt{2} a^{4} b - 3 \, \sqrt{2} a^{3} b^{2} + 3 \, \sqrt{2} a^{2} b^{3} - \sqrt{2} a b^{4}} + \frac{8 \, {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} + \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} - 2 \, \sin\left(d x + c\right)\right)}}{2 \, \left(-\frac{a}{b}\right)^{\frac{1}{4}}}\right)}{\sqrt{2} a^{4} b - 3 \, \sqrt{2} a^{3} b^{2} + 3 \, \sqrt{2} a^{2} b^{3} - \sqrt{2} a b^{4}} - \frac{4 \, {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} - \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} + \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{\sqrt{2} a^{4} b - 3 \, \sqrt{2} a^{3} b^{2} + 3 \, \sqrt{2} a^{2} b^{3} - \sqrt{2} a b^{4}} + \frac{4 \, {\left(\left(-a b^{3}\right)^{\frac{3}{4}} {\left(a + 3 \, b\right)} - \left(-a b^{3}\right)^{\frac{1}{4}} {\left(3 \, a b^{2} + b^{3}\right)}\right)} \log\left(\sin\left(d x + c\right)^{2} - \sqrt{2} \left(-\frac{a}{b}\right)^{\frac{1}{4}} \sin\left(d x + c\right) + \sqrt{-\frac{a}{b}}\right)}{\sqrt{2} a^{4} b - 3 \, \sqrt{2} a^{3} b^{2} + 3 \, \sqrt{2} a^{2} b^{3} - \sqrt{2} a b^{4}} - \frac{{\left(3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right)} \log\left({\left| \sin\left(d x + c\right) + 1 \right|}\right)}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{{\left(3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right)} \log\left({\left| \sin\left(d x + c\right) - 1 \right|}\right)}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{2 \, {\left(3 \, a \sin\left(d x + c\right)^{3} - 11 \, b \sin\left(d x + c\right)^{3} - 5 \, a \sin\left(d x + c\right) + 13 \, b \sin\left(d x + c\right)\right)}}{{\left(a^{2} - 2 \, a b + b^{2}\right)} {\left(\sin\left(d x + c\right)^{2} - 1\right)}^{2}}}{16 \, d}"," ",0,"-1/16*(8*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) + 8*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) - 4*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) + 4*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) - (3*a^2 - 6*a*b + 35*b^2)*log(abs(sin(d*x + c) + 1))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + (3*a^2 - 6*a*b + 35*b^2)*log(abs(sin(d*x + c) - 1))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 2*(3*a*sin(d*x + c)^3 - 11*b*sin(d*x + c)^3 - 5*a*sin(d*x + c) + 13*b*sin(d*x + c))/((a^2 - 2*a*b + b^2)*(sin(d*x + c)^2 - 1)^2))/d","B",0
411,1,896,0,1.159326," ","integrate(cos(d*x+c)^10/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{24 \, {\left(15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b - 62 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} - 16 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{5} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} - 24 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b + 46 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} + 40 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} + 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b^{2} + \sqrt{a^{2} b^{4} - {\left(a b^{2} - b^{3}\right)} a b^{2}}}{a b^{2} - b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{5} b^{3} - 12 \, a^{4} b^{4} + 14 \, a^{3} b^{5} - 4 \, a^{2} b^{6} - a b^{7}} + \frac{24 \, {\left(15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b - 62 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} - 16 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{5} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} + 24 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 46 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 40 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b^{2} - \sqrt{a^{2} b^{4} - {\left(a b^{2} - b^{3}\right)} a b^{2}}}{a b^{2} - b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{5} b^{3} - 12 \, a^{4} b^{4} + 14 \, a^{3} b^{5} - 4 \, a^{2} b^{6} - a b^{7}} - \frac{9 \, {\left(d x + c\right)} {\left(24 \, a + 35 \, b\right)}}{b^{2}} - \frac{24 \, a \tan\left(d x + c\right)^{5} + 123 \, b \tan\left(d x + c\right)^{5} + 48 \, a \tan\left(d x + c\right)^{3} + 280 \, b \tan\left(d x + c\right)^{3} + 24 \, a \tan\left(d x + c\right) + 165 \, b \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)}^{3} b^{2}}}{48 \, d}"," ",0,"1/48*(24*(15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 62*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 - 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^5 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 - 24*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 46*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 40*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 + sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^3 - 12*a^4*b^4 + 14*a^3*b^5 - 4*a^2*b^6 - a*b^7) + 24*(15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 62*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 - 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^5 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4 + 24*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 46*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 40*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 - sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^3 - 12*a^4*b^4 + 14*a^3*b^5 - 4*a^2*b^6 - a*b^7) - 9*(d*x + c)*(24*a + 35*b)/b^2 - (24*a*tan(d*x + c)^5 + 123*b*tan(d*x + c)^5 + 48*a*tan(d*x + c)^3 + 280*b*tan(d*x + c)^3 + 24*a*tan(d*x + c) + 165*b*tan(d*x + c))/((tan(d*x + c)^2 + 1)^3*b^2))/d","B",0
412,1,836,0,1.100524," ","integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{4 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} + 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b - 34 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{4} - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} + 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b + 28 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} + 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b^{2} + \sqrt{a^{2} b^{4} - {\left(a b^{2} - b^{3}\right)} a b^{2}}}{a b^{2} - b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}} + \frac{4 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} + 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b - 34 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{4} + 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 28 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b^{2} - \sqrt{a^{2} b^{4} - {\left(a b^{2} - b^{3}\right)} a b^{2}}}{a b^{2} - b^{3}}}}\right)\right)} {\left| -a + b \right|}}{3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}} - \frac{{\left(d x + c\right)} {\left(8 \, a + 35 \, b\right)}}{b^{2}} - \frac{11 \, \tan\left(d x + c\right)^{3} + 13 \, \tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)}^{2} b}}{8 \, d}"," ",0,"1/8*(4*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4 + 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b - 34*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4 - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 + 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 28*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 + sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6) + 4*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4 + 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 34*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4 + 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 28*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 - sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6) - (d*x + c)*(8*a + 35*b)/b^2 - (11*tan(d*x + c)^3 + 13*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*b))/d","B",0
413,1,995,0,0.997406," ","integrate(cos(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{5 \, {\left(d x + c\right)}}{b} + \frac{{\left(2 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b - 15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left| -a + b \right|} {\left| b \right|} + {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 7 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b + \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} {\left| b \right|}} - \frac{{\left(2 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} + {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b - 15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left| -a + b \right|} {\left| b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 7 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b - \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} {\left| b \right|}} + \frac{\tan\left(d x + c\right)}{{\left(\tan\left(d x + c\right)^{2} + 1\right)} b}}{2 \, d}"," ",0,"-1/2*(5*(d*x + c)/b + (2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)) - (2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) + (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b - sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)) + tan(d*x + c)/((tan(d*x + c)^2 + 1)*b))/d","B",0
414,1,906,0,0.972810," ","integrate(cos(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{2 \, {\left(d x + c\right)}}{b} + \frac{{\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 7 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left| -a + b \right|} {\left| b \right|} + {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b + \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} {\left| b \right|}} - \frac{{\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} b^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{2} - 7 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{4}\right)} {\left| -a + b \right|} {\left| b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a b - \sqrt{a^{2} b^{2} - {\left(a b - b^{2}\right)} a b}}{a b - b^{2}}}}\right)\right)}}{{\left(3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right)} {\left| b \right|}}}{2 \, d}"," ",0,"-1/2*(2*(d*x + c)/b + ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)) - ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b - sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)))/d","B",0
415,1,559,0,0.933721," ","integrate(cos(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{{\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{3} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} + 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b + \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(d x + c\right)}{\sqrt{\frac{4 \, a + \sqrt{-16 \, {\left(a - b\right)} a + 16 \, a^{2}}}{a - b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}} + \frac{{\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{3} + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{2}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{2 \, \tan\left(d x + c\right)}{\sqrt{\frac{4 \, a - \sqrt{-16 \, {\left(a - b\right)} a + 16 \, a^{2}}}{a - b}}}\right)\right)} {\left| a - b \right|}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}}}{2 \, d}"," ",0,"1/2*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 + 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*tan(d*x + c)/sqrt((4*a + sqrt(-16*(a - b)*a + 16*a^2))/(a - b))))*abs(a - b)/(3*a^5*b - 12*a^4*b^2 + 14*a^3*b^3 - 4*a^2*b^4 - a*b^5) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^3 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*tan(d*x + c)/sqrt((4*a - sqrt(-16*(a - b)*a + 16*a^2))/(a - b))))*abs(a - b)/(3*a^5*b - 12*a^4*b^2 + 14*a^3*b^3 - 4*a^2*b^4 - a*b^5))/d","B",0
416,1,1211,0,1.002114," ","integrate(sec(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","-\frac{\frac{{\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} - 9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b + 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} + 10 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 5 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5} - 2 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a - b\right)}^{2} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b - 12 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 14 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} - 4 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} - a b + \sqrt{{\left(a^{2} - a b\right)}^{2} - {\left(a^{2} - a b\right)} {\left(a^{2} - 2 \, a b + b^{2}\right)}}}{a^{2} - 2 \, a b + b^{2}}}}\right)\right)}}{3 \, a^{8} - 21 \, a^{7} b + 59 \, a^{6} b^{2} - 85 \, a^{5} b^{3} + 65 \, a^{4} b^{4} - 23 \, a^{3} b^{5} + a^{2} b^{6} + a b^{7}} - \frac{{\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} - 9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b + 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} + 10 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 5 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5} - 2 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b - 6 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{2} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{3}\right)} {\left(a - b\right)}^{2} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b - 12 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{2} + 14 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{3} - 4 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{5}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{2} - a b - \sqrt{{\left(a^{2} - a b\right)}^{2} - {\left(a^{2} - a b\right)} {\left(a^{2} - 2 \, a b + b^{2}\right)}}}{a^{2} - 2 \, a b + b^{2}}}}\right)\right)}}{3 \, a^{8} - 21 \, a^{7} b + 59 \, a^{6} b^{2} - 85 \, a^{5} b^{3} + 65 \, a^{4} b^{4} - 23 \, a^{3} b^{5} + a^{2} b^{6} + a b^{7}} - \frac{2 \, \tan\left(d x + c\right)}{a - b}}{2 \, d}"," ",0,"-1/2*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b + sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b + b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8 - 21*a^7*b + 59*a^6*b^2 - 85*a^5*b^3 + 65*a^4*b^4 - 23*a^3*b^5 + a^2*b^6 + a*b^7) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b - sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b + b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8 - 21*a^7*b + 59*a^6*b^2 - 85*a^5*b^3 + 65*a^4*b^4 - 23*a^3*b^5 + a^2*b^6 + a*b^7) - 2*tan(d*x + c)/(a - b))/d","B",0
417,1,2183,0,1.293418," ","integrate(sec(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{2 \, {\left(a^{2} \tan\left(d x + c\right)^{3} - 2 \, a b \tan\left(d x + c\right)^{3} + b^{2} \tan\left(d x + c\right)^{3} + 3 \, a^{2} \tan\left(d x + c\right) - 12 \, a b \tan\left(d x + c\right) + 9 \, b^{2} \tan\left(d x + c\right)\right)}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{3 \, {\left({\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b + 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 19 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)}^{2} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{7} b - 15 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b^{2} + 23 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{3} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{4} - 23 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{5} + 19 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{6} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{7} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{8}\right)} {\left| a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3} \right|} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b - 69 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{2} + 216 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{3} - 352 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{4} + 306 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{5} - 114 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{6} - 16 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{7} + 24 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{8} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{9} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{10}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3} + \sqrt{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)}^{2} - {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)}}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}}}}\right)\right)}}{{\left(3 \, a^{11} - 30 \, a^{10} b + 131 \, a^{9} b^{2} - 328 \, a^{8} b^{3} + 518 \, a^{7} b^{4} - 532 \, a^{6} b^{5} + 350 \, a^{5} b^{6} - 136 \, a^{4} b^{7} + 23 \, a^{3} b^{8} + 2 \, a^{2} b^{9} - a b^{10}\right)} {\left| a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3} \right|}} + \frac{3 \, {\left({\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b + 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{2} - 19 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{3} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{4}\right)} {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)}^{2} {\left| -a + b \right|} + {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{7} b - 15 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b^{2} + 23 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{3} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{4} - 23 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{5} + 19 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{6} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{7} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{8}\right)} {\left| a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3} \right|} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b - 69 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{2} + 216 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{3} - 352 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{4} + 306 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{5} - 114 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{6} - 16 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{7} + 24 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{8} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{9} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{10}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3} - \sqrt{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)}^{2} - {\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} {\left(a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right)}}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}}}}\right)\right)}}{{\left(3 \, a^{11} - 30 \, a^{10} b + 131 \, a^{9} b^{2} - 328 \, a^{8} b^{3} + 518 \, a^{7} b^{4} - 532 \, a^{6} b^{5} + 350 \, a^{5} b^{6} - 136 \, a^{4} b^{7} + 23 \, a^{3} b^{8} + 2 \, a^{2} b^{9} - a b^{10}\right)} {\left| a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3} \right|}}}{6 \, d}"," ",0,"1/6*(2*(a^2*tan(d*x + c)^3 - 2*a*b*tan(d*x + c)^3 + b^2*tan(d*x + c)^3 + 3*a^2*tan(d*x + c) - 12*a*b*tan(d*x + c) + 9*b^2*tan(d*x + c))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 3*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^2 + 23*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^4 - 23*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^5 + 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^6 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^7 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^8)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b - 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 216*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 352*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 306*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 114*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^6 - 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + 24*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^8 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^9 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^10)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3 + sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)^2 - (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4))))/((3*a^11 - 30*a^10*b + 131*a^9*b^2 - 328*a^8*b^3 + 518*a^7*b^4 - 532*a^6*b^5 + 350*a^5*b^6 - 136*a^4*b^7 + 23*a^3*b^8 + 2*a^2*b^9 - a*b^10)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)) + 3*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)^2*abs(-a + b) + (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^2 + 23*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^3 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^4 - 23*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^5 + 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^6 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^7 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^8)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 216*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 352*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 306*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 114*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^6 - 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + 24*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^8 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^9 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^10)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3 - sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)^2 - (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4))))/((3*a^11 - 30*a^10*b + 131*a^9*b^2 - 328*a^8*b^3 + 518*a^7*b^4 - 532*a^6*b^5 + 350*a^5*b^6 - 136*a^4*b^7 + 23*a^3*b^8 + 2*a^2*b^9 - a*b^10)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)))/d","B",0
418,1,3106,0,1.668180," ","integrate(sec(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""giac"")","\frac{\frac{2 \, {\left(3 \, a^{4} \tan\left(d x + c\right)^{5} - 12 \, a^{3} b \tan\left(d x + c\right)^{5} + 18 \, a^{2} b^{2} \tan\left(d x + c\right)^{5} - 12 \, a b^{3} \tan\left(d x + c\right)^{5} + 3 \, b^{4} \tan\left(d x + c\right)^{5} + 10 \, a^{4} \tan\left(d x + c\right)^{3} - 50 \, a^{3} b \tan\left(d x + c\right)^{3} + 90 \, a^{2} b^{2} \tan\left(d x + c\right)^{3} - 70 \, a b^{3} \tan\left(d x + c\right)^{3} + 20 \, b^{4} \tan\left(d x + c\right)^{3} + 15 \, a^{4} \tan\left(d x + c\right) - 75 \, a^{3} b \tan\left(d x + c\right) + 195 \, a^{2} b^{2} \tan\left(d x + c\right) - 225 \, a b^{3} \tan\left(d x + c\right) + 90 \, b^{4} \tan\left(d x + c\right)\right)}}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac{15 \, {\left(4 \, {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 7 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5}\right)} {\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)}^{2} {\left| -a + b \right|} - {\left(9 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{9} b^{2} - 69 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{8} b^{3} + 216 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{7} b^{4} - 352 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{6} b^{5} + 306 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{5} b^{6} - 114 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{4} b^{7} - 16 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{3} b^{8} + 24 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a^{2} b^{9} - 3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} a b^{10} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} b^{11}\right)} {\left| a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5} \right|} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{14} b - 18 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{13} b^{2} - 19 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{12} b^{3} + 508 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{11} b^{4} - 2221 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} b^{5} + 5314 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b^{6} - 8139 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{7} + 8328 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{8} - 5631 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{9} + 2322 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{10} - 417 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{11} - 68 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{12} + 41 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{13} - 2 \, \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{14} - \sqrt{a^{2} - a b + \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{15}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5} + \sqrt{{\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)}^{2} - {\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)} {\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)}}}{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)\right)}}{{\left(3 \, a^{14} - 39 \, a^{13} b + 230 \, a^{12} b^{2} - 814 \, a^{11} b^{3} + 1925 \, a^{10} b^{4} - 3201 \, a^{9} b^{5} + 3828 \, a^{8} b^{6} - 3300 \, a^{7} b^{7} + 2013 \, a^{6} b^{8} - 825 \, a^{5} b^{9} + 198 \, a^{4} b^{10} - 14 \, a^{3} b^{11} - 5 \, a^{2} b^{12} + a b^{13}\right)} {\left| a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5} \right|}} - \frac{15 \, {\left(4 \, {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{2} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{3} - 7 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{4} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{5}\right)} {\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)}^{2} {\left| -a + b \right|} + {\left(9 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{9} b^{2} - 69 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{8} b^{3} + 216 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{7} b^{4} - 352 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{6} b^{5} + 306 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{5} b^{6} - 114 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{4} b^{7} - 16 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{3} b^{8} + 24 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a^{2} b^{9} - 3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} a b^{10} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} b^{11}\right)} {\left| a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5} \right|} {\left| -a + b \right|} - {\left(3 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{14} b - 18 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{13} b^{2} - 19 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{12} b^{3} + 508 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{11} b^{4} - 2221 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{10} b^{5} + 5314 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{9} b^{6} - 8139 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{8} b^{7} + 8328 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{7} b^{8} - 5631 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{6} b^{9} + 2322 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{5} b^{10} - 417 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{4} b^{11} - 68 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{3} b^{12} + 41 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a^{2} b^{13} - 2 \, \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} a b^{14} - \sqrt{a^{2} - a b - \sqrt{a b} {\left(a - b\right)}} \sqrt{a b} b^{15}\right)} {\left| -a + b \right|}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor + \arctan\left(\frac{\tan\left(d x + c\right)}{\sqrt{\frac{a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5} - \sqrt{{\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)}^{2} - {\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)} {\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)}}}{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)\right)}}{{\left(3 \, a^{14} - 39 \, a^{13} b + 230 \, a^{12} b^{2} - 814 \, a^{11} b^{3} + 1925 \, a^{10} b^{4} - 3201 \, a^{9} b^{5} + 3828 \, a^{8} b^{6} - 3300 \, a^{7} b^{7} + 2013 \, a^{6} b^{8} - 825 \, a^{5} b^{9} + 198 \, a^{4} b^{10} - 14 \, a^{3} b^{11} - 5 \, a^{2} b^{12} + a b^{13}\right)} {\left| a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5} \right|}}}{30 \, d}"," ",0,"1/30*(2*(3*a^4*tan(d*x + c)^5 - 12*a^3*b*tan(d*x + c)^5 + 18*a^2*b^2*tan(d*x + c)^5 - 12*a*b^3*tan(d*x + c)^5 + 3*b^4*tan(d*x + c)^5 + 10*a^4*tan(d*x + c)^3 - 50*a^3*b*tan(d*x + c)^3 + 90*a^2*b^2*tan(d*x + c)^3 - 70*a*b^3*tan(d*x + c)^3 + 20*b^4*tan(d*x + c)^3 + 15*a^4*tan(d*x + c) - 75*a^3*b*tan(d*x + c) + 195*a^2*b^2*tan(d*x + c) - 225*a*b^3*tan(d*x + c) + 90*b^4*tan(d*x + c))/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5) + 15*(4*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^5)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)^2*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^9*b^2 - 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b^3 + 216*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b^4 - 352*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^5 + 306*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^6 - 114*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^7 - 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^8 + 24*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^9 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^10 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^11)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b - 18*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^13*b^2 - 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 508*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 2221*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 5314*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 - 8139*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 + 8328*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8 - 5631*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^9 + 2322*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^10 - 417*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^11 - 68*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^12 + 41*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^13 - 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^14 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^15)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5 + sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)^2 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*(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)))/(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))))/((3*a^14 - 39*a^13*b + 230*a^12*b^2 - 814*a^11*b^3 + 1925*a^10*b^4 - 3201*a^9*b^5 + 3828*a^8*b^6 - 3300*a^7*b^7 + 2013*a^6*b^8 - 825*a^5*b^9 + 198*a^4*b^10 - 14*a^3*b^11 - 5*a^2*b^12 + a*b^13)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)) - 15*(4*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^5)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)^2*abs(-a + b) + (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^9*b^2 - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^8*b^3 + 216*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b^4 - 352*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^5 + 306*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^6 - 114*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^7 - 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^8 + 24*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^9 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^10 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^11)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b - 18*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^13*b^2 - 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 508*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 2221*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 5314*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 - 8139*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 + 8328*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8 - 5631*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^9 + 2322*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^10 - 417*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^11 - 68*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^12 + 41*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^13 - 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^14 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^15)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5 - sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)^2 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*(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)))/(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))))/((3*a^14 - 39*a^13*b + 230*a^12*b^2 - 814*a^11*b^3 + 1925*a^10*b^4 - 3201*a^9*b^5 + 3828*a^8*b^6 - 3300*a^7*b^7 + 2013*a^6*b^8 - 825*a^5*b^9 + 198*a^4*b^10 - 14*a^3*b^11 - 5*a^2*b^12 + a*b^13)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)))/d","B",0
419,0,0,0,0.000000," ","integrate(cos(f*x+e)^m*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \cos\left(f x + e\right)^{m}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^m, x)","F",0
420,0,0,0,0.000000," ","integrate(cos(f*x+e)^5*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \cos\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^5, x)","F",0
421,0,0,0,0.000000," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \cos\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^3, x)","F",0
422,0,0,0,0.000000," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \cos\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e), x)","F",0
423,0,0,0,0.000000," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \sec\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*sec(f*x + e), x)","F",0
424,0,0,0,0.000000," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \sec\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*sec(f*x + e)^3, x)","F",0
425,0,0,0,0.000000," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \cos\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^4, x)","F",0
426,0,0,0,0.000000," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \cos\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^2, x)","F",0
427,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p, x)","F",0
428,0,0,0,0.000000," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \sec\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*sec(f*x + e)^2, x)","F",0
429,0,0,0,0.000000," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{4} + a\right)}^{p} \sec\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^4 + a)^p*sec(f*x + e)^4, x)","F",0
430,0,0,0,0.000000," ","integrate(cos(f*x+e)^m*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{m}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^m, x)","F",0
431,0,0,0,0.000000," ","integrate(cos(f*x+e)^5*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{5}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^5, x)","F",0
432,0,0,0,0.000000," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^3, x)","F",0
433,0,0,0,0.000000," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e), x)","F",0
434,0,0,0,0.000000," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e), x)","F",0
435,0,0,0,0.000000," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^3, x)","F",0
436,0,0,0,0.000000," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^4, x)","F",0
437,0,0,0,0.000000," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^2, x)","F",0
438,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p, x)","F",0
439,0,0,0,0.000000," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^2, x)","F",0
440,0,0,0,0.000000," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^4, x)","F",0
441,1,603,0,7.065607," ","integrate(tan(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{30 \, a^{3} \log\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac{60 \, a^{3} \log\left({\left| -\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1 \right|}\right)}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{147 \, a^{3} + \frac{1002 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{120 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{2925 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{960 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{240 \, a b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{4780 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{3600 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{2400 \, a b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{640 \, b^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{2925 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{4}}{{\left(\cos\left(d x + c\right) + 1\right)}^{4}} + \frac{960 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}^{4}}{{\left(\cos\left(d x + c\right) + 1\right)}^{4}} + \frac{240 \, a b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{4}}{{\left(\cos\left(d x + c\right) + 1\right)}^{4}} + \frac{1002 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{5}}{{\left(\cos\left(d x + c\right) + 1\right)}^{5}} + \frac{120 \, a^{2} b {\left(\cos\left(d x + c\right) - 1\right)}^{5}}{{\left(\cos\left(d x + c\right) + 1\right)}^{5}} + \frac{147 \, a^{3} {\left(\cos\left(d x + c\right) - 1\right)}^{6}}{{\left(\cos\left(d x + c\right) + 1\right)}^{6}}}{{\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}^{6}}}{60 \, d}"," ",0,"-1/60*(30*a^3*log(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 60*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + (147*a^3 + 1002*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 120*a^2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2925*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 960*a^2*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 240*a*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 4780*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 3600*a^2*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 2400*a*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 640*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 2925*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 960*a^2*b*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 240*a*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 1002*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 120*a^2*b*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 147*a^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^6))/d","B",0
442,1,393,0,2.746364," ","integrate(tan(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{6 \, a^{2} \log\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac{12 \, a^{2} \log\left({\left| -\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1 \right|}\right)}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{25 \, a^{2} + \frac{124 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{24 \, a b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{246 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{144 \, a b {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{48 \, b^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}} + \frac{124 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{24 \, a b {\left(\cos\left(d x + c\right) - 1\right)}^{3}}{{\left(\cos\left(d x + c\right) + 1\right)}^{3}} + \frac{25 \, a^{2} {\left(\cos\left(d x + c\right) - 1\right)}^{4}}{{\left(\cos\left(d x + c\right) + 1\right)}^{4}}}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}^{4}}}{12 \, d}"," ",0,"1/12*(6*a^2*log(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 12*a^2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + (25*a^2 + 124*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 24*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 246*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 144*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 48*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 124*a^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 24*a*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 25*a^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^4))/d","B",0
443,1,234,0,0.807490," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{a \log\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \, a \log\left({\left| -\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1 \right|}\right)}{a^{2} + 2 \, a b + b^{2}} + \frac{3 \, a + \frac{10 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{3 \, a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} {\left(\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} + 1\right)}^{2}}}{2 \, d}"," ",0,"-1/2*(a*log(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a^2 + 2*a*b + b^2) - 2*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/(a^2 + 2*a*b + b^2) + (3*a + 10*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 3*a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/((a^2 + 2*a*b + b^2)*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2))/d","B",0
444,1,110,0,0.231841," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{\log\left(a - \frac{2 \, a {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} - \frac{4 \, b {\left(\cos\left(d x + c\right) - 1\right)}}{\cos\left(d x + c\right) + 1} + \frac{a {\left(\cos\left(d x + c\right) - 1\right)}^{2}}{{\left(\cos\left(d x + c\right) + 1\right)}^{2}}\right)}{a + b} - \frac{2 \, \log\left({\left| -\frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1} - 1 \right|}\right)}{a + b}}{2 \, d}"," ",0,"1/2*(log(a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a + b) - 2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/(a + b))/d","B",0
445,1,38,0,0.173553," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{\log\left(\sin\left(d x + c\right)^{2}\right)}{a} - \frac{\log\left({\left| b \sin\left(d x + c\right)^{2} + a \right|}\right)}{a}}{2 \, d}"," ",0,"1/2*(log(sin(d*x + c)^2)/a - log(abs(b*sin(d*x + c)^2 + a))/a)/d","A",0
446,1,108,0,0.227798," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}}{a} + \frac{4 \, {\left(a + b\right)} \log\left({\left| -a {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)} + 2 \, a + 4 \, b \right|}\right)}{a^{2}}}{8 \, d}"," ",0,"1/8*(((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))/a + 4*(a + b)*log(abs(-a*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 2*a + 4*b))/a^2)/d","A",0
447,1,205,0,0.275988," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{a {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)}^{2} + 12 \, a {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)} + 8 \, b {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)}}{a^{2}} + \frac{32 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \log\left({\left| -a {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)} + 2 \, a + 4 \, b \right|}\right)}{a^{3}}}{64 \, d}"," ",0,"-1/64*((a*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2 + 12*a*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 8*b*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/a^2 + 32*(a^2 + 2*a*b + b^2)*log(abs(-a*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 2*a + 4*b))/a^3)/d","B",0
448,1,353,0,0.323158," ","integrate(cot(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{a^{2} {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)}^{3} + 12 \, a^{2} {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)}^{2} + 6 \, a b {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)}^{2} + 84 \, a^{2} {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)} + 120 \, a b {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)} + 48 \, b^{2} {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)}}{a^{3}} + \frac{192 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \log\left({\left| -a {\left(\frac{\cos\left(d x + c\right) + 1}{\cos\left(d x + c\right) - 1} + \frac{\cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) + 1}\right)} + 2 \, a + 4 \, b \right|}\right)}{a^{4}}}{384 \, d}"," ",0,"1/384*((a^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^3 + 12*a^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2 + 6*a*b*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1))^2 + 84*a^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 120*a*b*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 48*b^2*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/a^3 + 192*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(abs(-a*((cos(d*x + c) + 1)/(cos(d*x + c) - 1) + (cos(d*x + c) - 1)/(cos(d*x + c) + 1)) + 2*a + 4*b))/a^4)/d","B",0
449,1,472,0,10.508506," ","integrate(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{105 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a^{4}}{{\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \sqrt{a^{2} + a b}} + \frac{15 \, a^{6} \tan\left(d x + c\right)^{7} + 90 \, a^{5} b \tan\left(d x + c\right)^{7} + 225 \, a^{4} b^{2} \tan\left(d x + c\right)^{7} + 300 \, a^{3} b^{3} \tan\left(d x + c\right)^{7} + 225 \, a^{2} b^{4} \tan\left(d x + c\right)^{7} + 90 \, a b^{5} \tan\left(d x + c\right)^{7} + 15 \, b^{6} \tan\left(d x + c\right)^{7} - 21 \, a^{6} \tan\left(d x + c\right)^{5} - 105 \, a^{5} b \tan\left(d x + c\right)^{5} - 210 \, a^{4} b^{2} \tan\left(d x + c\right)^{5} - 210 \, a^{3} b^{3} \tan\left(d x + c\right)^{5} - 105 \, a^{2} b^{4} \tan\left(d x + c\right)^{5} - 21 \, a b^{5} \tan\left(d x + c\right)^{5} + 35 \, a^{6} \tan\left(d x + c\right)^{3} + 140 \, a^{5} b \tan\left(d x + c\right)^{3} + 210 \, a^{4} b^{2} \tan\left(d x + c\right)^{3} + 140 \, a^{3} b^{3} \tan\left(d x + c\right)^{3} + 35 \, a^{2} b^{4} \tan\left(d x + c\right)^{3} - 105 \, a^{6} \tan\left(d x + c\right) - 315 \, a^{5} b \tan\left(d x + c\right) - 315 \, a^{4} b^{2} \tan\left(d x + c\right) - 105 \, a^{3} b^{3} \tan\left(d x + c\right)}{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}}}{105 \, d}"," ",0,"1/105*(105*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a^4/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt(a^2 + a*b)) + (15*a^6*tan(d*x + c)^7 + 90*a^5*b*tan(d*x + c)^7 + 225*a^4*b^2*tan(d*x + c)^7 + 300*a^3*b^3*tan(d*x + c)^7 + 225*a^2*b^4*tan(d*x + c)^7 + 90*a*b^5*tan(d*x + c)^7 + 15*b^6*tan(d*x + c)^7 - 21*a^6*tan(d*x + c)^5 - 105*a^5*b*tan(d*x + c)^5 - 210*a^4*b^2*tan(d*x + c)^5 - 210*a^3*b^3*tan(d*x + c)^5 - 105*a^2*b^4*tan(d*x + c)^5 - 21*a*b^5*tan(d*x + c)^5 + 35*a^6*tan(d*x + c)^3 + 140*a^5*b*tan(d*x + c)^3 + 210*a^4*b^2*tan(d*x + c)^3 + 140*a^3*b^3*tan(d*x + c)^3 + 35*a^2*b^4*tan(d*x + c)^3 - 105*a^6*tan(d*x + c) - 315*a^5*b*tan(d*x + c) - 315*a^4*b^2*tan(d*x + c) - 105*a^3*b^3*tan(d*x + c))/(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))/d","B",0
450,1,296,0,4.321402," ","integrate(tan(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{15 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a^{3}}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{a^{2} + a b}} - \frac{3 \, a^{4} \tan\left(d x + c\right)^{5} + 12 \, a^{3} b \tan\left(d x + c\right)^{5} + 18 \, a^{2} b^{2} \tan\left(d x + c\right)^{5} + 12 \, a b^{3} \tan\left(d x + c\right)^{5} + 3 \, b^{4} \tan\left(d x + c\right)^{5} - 5 \, a^{4} \tan\left(d x + c\right)^{3} - 15 \, a^{3} b \tan\left(d x + c\right)^{3} - 15 \, a^{2} b^{2} \tan\left(d x + c\right)^{3} - 5 \, a b^{3} \tan\left(d x + c\right)^{3} + 15 \, a^{4} \tan\left(d x + c\right) + 30 \, a^{3} b \tan\left(d x + c\right) + 15 \, a^{2} b^{2} \tan\left(d x + c\right)}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}}}{15 \, d}"," ",0,"-1/15*(15*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a^3/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a^2 + a*b)) - (3*a^4*tan(d*x + c)^5 + 12*a^3*b*tan(d*x + c)^5 + 18*a^2*b^2*tan(d*x + c)^5 + 12*a*b^3*tan(d*x + c)^5 + 3*b^4*tan(d*x + c)^5 - 5*a^4*tan(d*x + c)^3 - 15*a^3*b*tan(d*x + c)^3 - 15*a^2*b^2*tan(d*x + c)^3 - 5*a*b^3*tan(d*x + c)^3 + 15*a^4*tan(d*x + c) + 30*a^3*b*tan(d*x + c) + 15*a^2*b^2*tan(d*x + c))/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5))/d","B",0
451,1,164,0,1.354494," ","integrate(tan(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{3 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a^{2}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a^{2} + a b}} + \frac{a^{2} \tan\left(d x + c\right)^{3} + 2 \, a b \tan\left(d x + c\right)^{3} + b^{2} \tan\left(d x + c\right)^{3} - 3 \, a^{2} \tan\left(d x + c\right) - 3 \, a b \tan\left(d x + c\right)}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}{3 \, d}"," ",0,"1/3*(3*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a^2/((a^2 + 2*a*b + b^2)*sqrt(a^2 + a*b)) + (a^2*tan(d*x + c)^3 + 2*a*b*tan(d*x + c)^3 + b^2*tan(d*x + c)^3 - 3*a^2*tan(d*x + c) - 3*a*b*tan(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3))/d","B",0
452,1,86,0,0.487637," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} a}{\sqrt{a^{2} + a b} {\left(a + b\right)}} - \frac{\tan\left(d x + c\right)}{a + b}}{d}"," ",0,"-((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*a/(sqrt(a^2 + a*b)*(a + b)) - tan(d*x + c)/(a + b))/d","A",0
453,1,85,0,0.202983," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{{\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(a + b\right)}}{\sqrt{a^{2} + a b} a} + \frac{1}{a \tan\left(d x + c\right)}}{d}"," ",0,"-((pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(a + b)/(sqrt(a^2 + a*b)*a) + 1/(a*tan(d*x + c)))/d","A",0
454,1,120,0,0.218182," ","integrate(cot(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{3 \, {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)} {\left(a^{2} + 2 \, a b + b^{2}\right)}}{\sqrt{a^{2} + a b} a^{2}} + \frac{3 \, a \tan\left(d x + c\right)^{2} + 3 \, b \tan\left(d x + c\right)^{2} - a}{a^{2} \tan\left(d x + c\right)^{3}}}{3 \, d}"," ",0,"1/3*(3*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(a^2 + 2*a*b + b^2)/(sqrt(a^2 + a*b)*a^2) + (3*a*tan(d*x + c)^2 + 3*b*tan(d*x + c)^2 - a)/(a^2*tan(d*x + c)^3))/d","A",0
455,1,171,0,0.261130," ","integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","-\frac{\frac{15 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{\sqrt{a^{2} + a b} a^{3}} + \frac{15 \, a^{2} \tan\left(d x + c\right)^{4} + 30 \, a b \tan\left(d x + c\right)^{4} + 15 \, b^{2} \tan\left(d x + c\right)^{4} - 5 \, a^{2} \tan\left(d x + c\right)^{2} - 5 \, a b \tan\left(d x + c\right)^{2} + 3 \, a^{2}}{a^{3} \tan\left(d x + c\right)^{5}}}{15 \, d}"," ",0,"-1/15*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*a^3) + (15*a^2*tan(d*x + c)^4 + 30*a*b*tan(d*x + c)^4 + 15*b^2*tan(d*x + c)^4 - 5*a^2*tan(d*x + c)^2 - 5*a*b*tan(d*x + c)^2 + 3*a^2)/(a^3*tan(d*x + c)^5))/d","B",0
456,1,238,0,0.334081," ","integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""giac"")","\frac{\frac{105 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} {\left(\pi \left \lfloor \frac{d x + c}{\pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(2 \, a + 2 \, b\right) + \arctan\left(\frac{a \tan\left(d x + c\right) + b \tan\left(d x + c\right)}{\sqrt{a^{2} + a b}}\right)\right)}}{\sqrt{a^{2} + a b} a^{4}} + \frac{105 \, a^{3} \tan\left(d x + c\right)^{6} + 315 \, a^{2} b \tan\left(d x + c\right)^{6} + 315 \, a b^{2} \tan\left(d x + c\right)^{6} + 105 \, b^{3} \tan\left(d x + c\right)^{6} - 35 \, a^{3} \tan\left(d x + c\right)^{4} - 70 \, a^{2} b \tan\left(d x + c\right)^{4} - 35 \, a b^{2} \tan\left(d x + c\right)^{4} + 21 \, a^{3} \tan\left(d x + c\right)^{2} + 21 \, a^{2} b \tan\left(d x + c\right)^{2} - 15 \, a^{3}}{a^{4} \tan\left(d x + c\right)^{7}}}{105 \, d}"," ",0,"1/105*(105*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))/(sqrt(a^2 + a*b)*a^4) + (105*a^3*tan(d*x + c)^6 + 315*a^2*b*tan(d*x + c)^6 + 315*a*b^2*tan(d*x + c)^6 + 105*b^3*tan(d*x + c)^6 - 35*a^3*tan(d*x + c)^4 - 70*a^2*b*tan(d*x + c)^4 - 35*a*b^2*tan(d*x + c)^4 + 21*a^3*tan(d*x + c)^2 + 21*a^2*b*tan(d*x + c)^2 - 15*a^3)/(a^4*tan(d*x + c)^7))/d","B",0
457,-2,0,0,0.000000," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^5,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(1/3*(12*tan((f*x+exp(1))/2)^2*sign(tan((f*x+exp(1))/2)^4-1)-3*tan((f*x+exp(1))/2)^4*sign(tan((f*x+exp(1))/2)^4-1)-5*sign(tan((f*x+exp(1))/2)^4-1))/(tan((f*x+exp(1))/2)^2-1)^3+sign(tan((f*x+exp(1))/2)^4-1)/(tan((f*x+exp(1))/2)^2+1))","F(-2)",0
458,-2,0,0,0.000000," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^3,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*2*sqrt(a)*sign(tan((f*x+exp(1))/2)^4-1)/(tan((f*x+exp(1))/2)^4-1)","F(-2)",0
459,-2,0,0,0.000000," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*2*sqrt(a)/2*sign(tan((f*x+exp(1))/2)^4-1)/(tan((f*x+exp(1))/2)^2+1)","F(-2)",0
460,1,55,0,0.136914," ","integrate(cot(f*x+e)*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\frac{a {\left(\frac{\arctan\left(\frac{\sqrt{-a \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a}} + \frac{\sqrt{-a \sin\left(f x + e\right)^{2} + a}}{a}\right)}}{f}"," ",0,"a*(arctan(sqrt(-a*sin(f*x + e)^2 + a)/sqrt(-a))/sqrt(-a) + sqrt(-a*sin(f*x + e)^2 + a)/a)/f","A",0
461,-2,0,0,0.000000," ","integrate(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(-1/16*tan((f*x+exp(1))/2)^2*sign(tan((f*x+exp(1))/2)^4-1)+1/16*(14*tan((f*x+exp(1))/2)^2*sign(tan((f*x+exp(1))/2)^4-1)-3*tan((f*x+exp(1))/2)^4*sign(tan((f*x+exp(1))/2)^4-1)+sign(tan((f*x+exp(1))/2)^4-1))/(tan((f*x+exp(1))/2)^4+tan((f*x+exp(1))/2)^2)+3/8*sign(tan((f*x+exp(1))/2)^4-1)*ln(tan((f*x+exp(1))/2)^2))","F(-2)",0
462,-2,0,0,0.000000," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^6,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(-1/8*(-7*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^3*sign(tan((f*x+exp(1))/2)^4-1)+36*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))*sign(tan((f*x+exp(1))/2)^4-1))/((tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^2-4)^2-15/32*sign(tan((f*x+exp(1))/2)^4-1)*ln(abs(tan((f*x+exp(1))/2)+2+1/tan((f*x+exp(1))/2)))+15/32*sign(tan((f*x+exp(1))/2)^4-1)*ln(abs(tan((f*x+exp(1))/2)-2+1/tan((f*x+exp(1))/2)))-sign(tan((f*x+exp(1))/2)^4-1)/(-tan((f*x+exp(1))/2)-1/tan((f*x+exp(1))/2)))","F(-2)",0
463,-2,0,0,0.000000," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^4,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(-1/2*(3*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^2*sign(tan((f*x+exp(1))/2)^4-1)-8*sign(tan((f*x+exp(1))/2)^4-1))/((tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^3-4*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2)))+3/8*sign(tan((f*x+exp(1))/2)^4-1)*ln(abs(tan((f*x+exp(1))/2)+2+1/tan((f*x+exp(1))/2)))-3/8*sign(tan((f*x+exp(1))/2)^4-1)*ln(abs(tan((f*x+exp(1))/2)-2+1/tan((f*x+exp(1))/2))))","F(-2)",0
464,-2,0,0,0.000000," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^2,x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(-1/4*sign(tan((f*x+exp(1))/2)^4-1)*ln(abs(tan((f*x+exp(1))/2)+2+1/tan((f*x+exp(1))/2)))+1/4*sign(tan((f*x+exp(1))/2)^4-1)*ln(abs(tan((f*x+exp(1))/2)-2+1/tan((f*x+exp(1))/2)))-sign(tan((f*x+exp(1))/2)^4-1)/(-tan((f*x+exp(1))/2)-1/tan((f*x+exp(1))/2)))","F(-2)",0
465,-2,0,0,0.000000," ","integrate(cot(f*x+e)^2*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(1/4*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))*sign(tan((f*x+exp(1))/2)^4-1)+sign(tan((f*x+exp(1))/2)^4-1)/(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2)))","F(-2)",0
466,-2,0,0,0.000000," ","integrate(cot(f*x+e)^4*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(-sign(tan((f*x+exp(1))/2)^4-1)/(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))+1/4096*(256/3*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^3*sign(tan((f*x+exp(1))/2)^4-1)-2048*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))*sign(tan((f*x+exp(1))/2)^4-1)))","F(-2)",0
467,-2,0,0,0.000000," ","integrate(cot(f*x+e)^6*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*sqrt(a)*(sign(tan((f*x+exp(1))/2)^4-1)/(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))+1/1073741824*(-67108864*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^3*sign(tan((f*x+exp(1))/2)^4-1)+16777216/5*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^5*sign(tan((f*x+exp(1))/2)^4-1)+805306368*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))*sign(tan((f*x+exp(1))/2)^4-1)))","F(-2)",0
468,-2,0,0,0.000000," ","integrate(tan(f*x+e)^5/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)/15*(80*tan((f*x+exp(1))/2)^4-40*tan((f*x+exp(1))/2)^2+8)/(tan((f*x+exp(1))/2)^2-1)^5/sign(tan((f*x+exp(1))/2)^4-1)","F(-2)",0
469,-2,0,0,0.000000," ","integrate(tan(f*x+e)^3/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)/3*(6*tan((f*x+exp(1))/2)^2-2)/(tan((f*x+exp(1))/2)^2-1)^3/sign(tan((f*x+exp(1))/2)^4-1)","F(-2)",0
470,-2,0,0,0.000000," ","integrate(tan(f*x+e)/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*2/sqrt(a)/2/(tan((f*x+exp(1))/2)^2-1)/sign(tan((f*x+exp(1))/2)^4-1)","F(-2)",0
471,1,32,0,0.151393," ","integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{\sqrt{-a \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} f}"," ",0,"arctan(sqrt(-a*sin(f*x + e)^2 + a)/sqrt(-a))/(sqrt(-a)*f)","A",0
472,-2,0,0,0.000000," ","integrate(cot(f*x+e)^3/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)*(-1/16*tan((f*x+exp(1))/2)^2/sign(tan((f*x+exp(1))/2)^4-1)+1/16*(-2*tan((f*x+exp(1))/2)^2+1)/tan((f*x+exp(1))/2)^2/sign(tan((f*x+exp(1))/2)^4-1)+1/8*ln(tan((f*x+exp(1))/2)^2)/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
473,-2,0,0,0.000000," ","integrate(tan(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)*(-1/8*(-3*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^3+20*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2)))/((tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^2-4)^2/sign(tan((f*x+exp(1))/2)^4-1)-3/32*ln(abs(tan((f*x+exp(1))/2)+2+1/tan((f*x+exp(1))/2)))/sign(tan((f*x+exp(1))/2)^4-1)+3/32*ln(abs(tan((f*x+exp(1))/2)-2+1/tan((f*x+exp(1))/2)))/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
474,-2,0,0,0.000000," ","integrate(tan(f*x+e)^2/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)*(-1/2*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))/((tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^2-4)/sign(tan((f*x+exp(1))/2)^4-1)+1/8*ln(abs(tan((f*x+exp(1))/2)+2+1/tan((f*x+exp(1))/2)))/sign(tan((f*x+exp(1))/2)^4-1)-1/8*ln(abs(tan((f*x+exp(1))/2)-2+1/tan((f*x+exp(1))/2)))/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
475,-2,0,0,0.000000," ","integrate(cot(f*x+e)^2/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)*(1/4*tan((f*x+exp(1))/2)/sign(tan((f*x+exp(1))/2)^4-1)+1/4/tan((f*x+exp(1))/2)/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
476,-2,0,0,0.000000," ","integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)*(1/4096*(256/3*tan((f*x+exp(1))/2)^3-768*tan((f*x+exp(1))/2))/sign(tan((f*x+exp(1))/2)^4-1)+1/48*(-9*tan((f*x+exp(1))/2)^2+1)/tan((f*x+exp(1))/2)^3/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
477,-2,0,0,0.000000," ","integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f/sqrt(a)*(1/960*(150*tan((f*x+exp(1))/2)^4-25*tan((f*x+exp(1))/2)^2+3)/tan((f*x+exp(1))/2)^5/sign(tan((f*x+exp(1))/2)^4-1)+1/1073741824*(16777216/5*tan((f*x+exp(1))/2)^5-83886080/3*tan((f*x+exp(1))/2)^3+167772160*tan((f*x+exp(1))/2))/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
478,1,151,0,0.767787," ","integrate(tan(f*x+e)^5/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\frac{7 \, {\left(3 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}} - 10 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} a + 15 \, \sqrt{a \tan\left(f x + e\right)^{2} + a} a^{2}\right)}}{a^{2}} + \frac{3 \, {\left(5 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{7}{2}} - 21 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}} a + 35 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{a \tan\left(f x + e\right)^{2} + a} a^{3}\right)}}{a^{3}}}{105 \, a^{2} f}"," ",0,"1/105*(7*(3*(a*tan(f*x + e)^2 + a)^(5/2) - 10*(a*tan(f*x + e)^2 + a)^(3/2)*a + 15*sqrt(a*tan(f*x + e)^2 + a)*a^2)/a^2 + 3*(5*(a*tan(f*x + e)^2 + a)^(7/2) - 21*(a*tan(f*x + e)^2 + a)^(5/2)*a + 35*(a*tan(f*x + e)^2 + a)^(3/2)*a^2 - 35*sqrt(a*tan(f*x + e)^2 + a)*a^3)/a^3)/(a^2*f)","B",0
479,1,108,0,0.671941," ","integrate(tan(f*x+e)^3/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\frac{5 \, {\left({\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} - 3 \, \sqrt{a \tan\left(f x + e\right)^{2} + a} a\right)}}{a} + \frac{3 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}} - 10 \, {\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} a + 15 \, \sqrt{a \tan\left(f x + e\right)^{2} + a} a^{2}}{a^{2}}}{15 \, a^{2} f}"," ",0,"1/15*(5*((a*tan(f*x + e)^2 + a)^(3/2) - 3*sqrt(a*tan(f*x + e)^2 + a)*a)/a + (3*(a*tan(f*x + e)^2 + a)^(5/2) - 10*(a*tan(f*x + e)^2 + a)^(3/2)*a + 15*sqrt(a*tan(f*x + e)^2 + a)*a^2)/a^2)/(a^2*f)","B",0
480,1,64,0,0.552534," ","integrate(tan(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{3 \, \sqrt{a \tan\left(f x + e\right)^{2} + a} + \frac{{\left(a \tan\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} - 3 \, \sqrt{a \tan\left(f x + e\right)^{2} + a} a}{a}}{3 \, a^{2} f}"," ",0,"1/3*(3*sqrt(a*tan(f*x + e)^2 + a) + ((a*tan(f*x + e)^2 + a)^(3/2) - 3*sqrt(a*tan(f*x + e)^2 + a)*a)/a)/(a^2*f)","B",0
481,1,59,0,0.160942," ","integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{\sqrt{-a \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a f} + \frac{1}{\sqrt{-a \sin\left(f x + e\right)^{2} + a} a f}"," ",0,"arctan(sqrt(-a*sin(f*x + e)^2 + a)/sqrt(-a))/(sqrt(-a)*a*f) + 1/(sqrt(-a*sin(f*x + e)^2 + a)*a*f)","A",0
482,-2,0,0,0.000000," ","integrate(cot(f*x+e)^3/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*(1/16*(2*sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a))/a^2/tan((f*x+exp(1))/2)^2/sign(tan((f*x+exp(1))/2)^4-1)-1/8*ln(tan((f*x+exp(1))/2)^2)/sqrt(a)/a/sign(tan((f*x+exp(1))/2)^4-1)-1/16/sqrt(a)/a*tan((f*x+exp(1))/2)^2/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
483,-2,0,0,0.000000," ","integrate(tan(f*x+e)^2/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)-2/f/32768*(4096*sqrt(a)*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^3+16384*sqrt(a)*(tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2)))/a^2/((tan((f*x+exp(1))/2)+1/tan((f*x+exp(1))/2))^2-4)^2/sign(tan((f*x+exp(1))/2)^4-1)","F(-2)",0
484,-2,0,0,0.000000," ","integrate(cot(f*x+e)^2/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*(1/4/sqrt(a)/a*tan((f*x+exp(1))/2)/sign(tan((f*x+exp(1))/2)^4-1)+1/4/sqrt(a)/a/tan((f*x+exp(1))/2)/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
485,-2,0,0,0.000000," ","integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*(1/48*(3*sqrt(a)*tan((f*x+exp(1))/2)^2+sqrt(a))/a^2/tan((f*x+exp(1))/2)^3/sign(tan((f*x+exp(1))/2)^4-1)+1/4096*(256/3*sqrt(a)*a^4*tan((f*x+exp(1))/2)^3+256*sqrt(a)*a^4*tan((f*x+exp(1))/2))/a^6/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
486,-2,0,0,0.000000," ","integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*(1/1073741824*(16777216/5*sqrt(a)*a^8*tan((f*x+exp(1))/2)^5-16777216/3*sqrt(a)*a^8*tan((f*x+exp(1))/2)^3-33554432*sqrt(a)*a^8*tan((f*x+exp(1))/2))/a^10/sign(tan((f*x+exp(1))/2)^4-1)+1/960*(-30*sqrt(a)*tan((f*x+exp(1))/2)^4-5*sqrt(a)*tan((f*x+exp(1))/2)^2+3*sqrt(a))/a^2/tan((f*x+exp(1))/2)^5/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
487,-2,0,0,0.000000," ","integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*(1/26880*(525*sqrt(a)*tan((f*x+exp(1))/2)^6+35*sqrt(a)*tan((f*x+exp(1))/2)^4-63*sqrt(a)*tan((f*x+exp(1))/2)^2+15*sqrt(a))/a^2/tan((f*x+exp(1))/2)^7/sign(tan((f*x+exp(1))/2)^4-1)+1/72057594037927936*(281474976710656/7*sqrt(a)*a^12*tan((f*x+exp(1))/2)^7-844424930131968/5*sqrt(a)*a^12*tan((f*x+exp(1))/2)^5+281474976710656/3*sqrt(a)*a^12*tan((f*x+exp(1))/2)^3+1407374883553280*sqrt(a)*a^12*tan((f*x+exp(1))/2))/a^14/sign(tan((f*x+exp(1))/2)^4-1))","F(-2)",0
488,1,2790,0,3.631971," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^5,x, algorithm=""giac"")","-\frac{\frac{{\left(8 \, a^{2} + 24 \, a b + 15 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a + b\right)} \sqrt{-a - b}} - \frac{16 \, {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b - \sqrt{a} b\right)}}{{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} + a + 4 \, b} - \frac{2 \, {\left(8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a^{2} + 16 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a b + 7 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} b^{2} - 56 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{5}{2}} - 80 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{3}{2}} b - 17 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} \sqrt{a} b^{2} - 120 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{3} - 464 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{2} b - 425 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b^{2} - 60 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{3} + 136 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{7}{2}} + 144 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{5}{2}} b - 425 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b^{2} - 468 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{3} + 344 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{4} + 1520 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{3} b + 2093 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b^{2} + 712 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{3} - 240 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{4} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{9}{2}} + 592 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{7}{2}} b + 2165 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b^{2} + 2808 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{3} + 1232 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{4} - 232 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{5} - 1072 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{4} b - 1675 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b^{2} - 652 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{3} + 624 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{4} + 448 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{5} - 104 \, a^{\frac{11}{2}} - 656 \, a^{\frac{9}{2}} b - 1723 \, a^{\frac{7}{2}} b^{2} - 2340 \, a^{\frac{5}{2}} b^{3} - 1616 \, a^{\frac{3}{2}} b^{4} - 448 \, \sqrt{a} b^{5}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{4} {\left(a + b\right)}}}{4 \, f}"," ",0,"-1/4*((8*a^2 + 24*a*b + 15*b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a + b)*sqrt(-a - b)) - 16*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b - sqrt(a)*b)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b) - 2*(8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a^2 + 16*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a*b + 7*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*b^2 - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(5/2) - 80*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(3/2)*b - 17*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*sqrt(a)*b^2 - 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 - 464*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b - 425*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b^2 - 60*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 136*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) + 144*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*b - 425*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b^2 - 468*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^3 + 344*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^4 + 1520*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^3*b + 2093*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b^2 + 712*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^3 - 240*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^4 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2) + 592*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*b + 2165*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b^2 + 2808*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^3 + 1232*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^4 - 232*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^5 - 1072*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*b - 1675*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b^2 - 652*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^3 + 624*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^4 + 448*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^5 - 104*a^(11/2) - 656*a^(9/2)*b - 1723*a^(7/2)*b^2 - 2340*a^(5/2)*b^3 - 1616*a^(3/2)*b^4 - 448*sqrt(a)*b^5)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^4*(a + b)))/f","B",0
489,1,1011,0,1.036163," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^3,x, algorithm=""giac"")","\frac{\frac{{\left(2 \, a + 3 \, b\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{\sqrt{-a - b}} - \frac{4 \, {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b - \sqrt{a} b\right)}}{{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} + a + 4 \, b} - \frac{2 \, {\left(2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a + {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} - {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{2} - 2 \, a^{\frac{5}{2}} - 5 \, a^{\frac{3}{2}} b - 4 \, \sqrt{a} b^{2}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{2}}}{f}"," ",0,"((2*a + 3*b)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/sqrt(-a - b) - 4*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b - sqrt(a)*b)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b) - 2*(2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a + (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2) + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2 - (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^2 - 2*a^(5/2) - 5*a^(3/2)*b - 4*sqrt(a)*b^2)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^2)/f","B",0
490,1,329,0,0.320545," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e),x, algorithm=""giac"")","-\frac{2 \, {\left(\frac{{\left(a + b\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{\sqrt{-a - b}} - \frac{2 \, {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b - \sqrt{a} b\right)}}{{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} + a + 4 \, b}\right)}}{f}"," ",0,"-2*((a + b)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/sqrt(-a - b) - 2*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b - sqrt(a)*b)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b))/f","B",0
491,1,49,0,0.129511," ","integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\frac{\frac{a \arctan\left(\frac{\sqrt{b \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a}} + \sqrt{b \sin\left(f x + e\right)^{2} + a}}{f}"," ",0,"(a*arctan(sqrt(b*sin(f*x + e)^2 + a)/sqrt(-a))/sqrt(-a) + sqrt(b*sin(f*x + e)^2 + a))/f","A",0
492,-2,0,0,0.000000," ","integrate(cot(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to divide, perhaps due to rounding error%%%{4096,[8,8]%%%}+%%%{%%%{16384,[1]%%%},[8,7]%%%}+%%%{%%%{24576,[2]%%%},[8,6]%%%}+%%%{%%%{16384,[3]%%%},[8,5]%%%}+%%%{%%%{4096,[4]%%%},[8,4]%%%}+%%%{%%{[-16384,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,8]%%%}+%%%{%%{[%%%{-65536,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,7]%%%}+%%%{%%{[%%%{-98304,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6]%%%}+%%%{%%{[%%%{-65536,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,5]%%%}+%%%{%%{[%%%{-16384,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,4]%%%}+%%%{32768,[6,9]%%%}+%%%{%%%{147456,[1]%%%},[6,8]%%%}+%%%{%%%{262144,[2]%%%},[6,7]%%%}+%%%{%%%{229376,[3]%%%},[6,6]%%%}+%%%{%%%{98304,[4]%%%},[6,5]%%%}+%%%{%%%{16384,[5]%%%},[6,4]%%%}+%%%{%%{[-65536,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9]%%%}+%%%{%%{[%%%{-245760,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,8]%%%}+%%%{%%{[%%%{-327680,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7]%%%}+%%%{%%{[%%%{-163840,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6]%%%}+%%%{%%{[%%%{16384,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,4]%%%}+%%%{65536,[4,10]%%%}+%%%{%%%{229376,[1]%%%},[4,9]%%%}+%%%{%%%{221184,[2]%%%},[4,8]%%%}+%%%{%%%{-98304,[3]%%%},[4,7]%%%}+%%%{%%%{-311296,[4]%%%},[4,6]%%%}+%%%{%%%{-196608,[5]%%%},[4,5]%%%}+%%%{%%%{-40960,[6]%%%},[4,4]%%%}+%%%{%%{[%%%{131072,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9]%%%}+%%%{%%{[%%%{540672,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,8]%%%}+%%%{%%{[%%%{851968,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,7]%%%}+%%%{%%{[%%%{622592,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6]%%%}+%%%{%%{[%%%{196608,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,5]%%%}+%%%{%%{[%%%{16384,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,4]%%%}+%%%{%%%{-131072,[1]%%%},[2,10]%%%}+%%%{%%%{-557056,[2]%%%},[2,9]%%%}+%%%{%%%{-901120,[3]%%%},[2,8]%%%}+%%%{%%%{-655360,[4]%%%},[2,7]%%%}+%%%{%%%{-163840,[5]%%%},[2,6]%%%}+%%%{%%%{32768,[6]%%%},[2,5]%%%}+%%%{%%%{16384,[7]%%%},[2,4]%%%}+%%%{%%{[%%%{-65536,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9]%%%}+%%%{%%{[%%%{-278528,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,8]%%%}+%%%{%%{[%%%{-458752,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7]%%%}+%%%{%%{[%%%{-360448,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6]%%%}+%%%{%%{[%%%{-131072,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,5]%%%}+%%%{%%{[%%%{-16384,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,4]%%%}+%%%{%%%{65536,[2]%%%},[0,10]%%%}+%%%{%%%{294912,[3]%%%},[0,9]%%%}+%%%{%%%{528384,[4]%%%},[0,8]%%%}+%%%{%%%{475136,[5]%%%},[0,7]%%%}+%%%{%%%{221184,[6]%%%},[0,6]%%%}+%%%{%%%{49152,[7]%%%},[0,5]%%%}+%%%{%%%{4096,[8]%%%},[0,4]%%%} / %%%{%%%{1,[1]%%%},[8,0]%%%}+%%%{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0]%%%}+%%%{%%%{8,[1]%%%},[6,1]%%%}+%%%{%%%{4,[2]%%%},[6,0]%%%}+%%%{%%{poly1[%%%{-16,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1]%%%}+%%%{%%{poly1[%%%{4,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0]%%%}+%%%{%%%{16,[1]%%%},[4,2]%%%}+%%%{%%%{-8,[2]%%%},[4,1]%%%}+%%%{%%%{-10,[3]%%%},[4,0]%%%}+%%%{%%{poly1[%%%{32,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1]%%%}+%%%{%%{poly1[%%%{4,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0]%%%}+%%%{%%%{-32,[2]%%%},[2,2]%%%}+%%%{%%%{-8,[3]%%%},[2,1]%%%}+%%%{%%%{4,[4]%%%},[2,0]%%%}+%%%{%%{poly1[%%%{-16,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1]%%%}+%%%{%%{poly1[%%%{-4,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0]%%%}+%%%{%%%{16,[3]%%%},[0,2]%%%}+%%%{%%%{8,[4]%%%},[0,1]%%%}+%%%{%%%{1,[5]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
493,-2,0,0,0.000000," ","integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Evaluation time: 0.85Unable to divide, perhaps due to rounding error%%%{1,[4,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0]%%%}+%%%{8,[2,1]%%%}+%%%{%%%{6,[1]%%%},[2,0]%%%}+%%%{%%{[-16,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1]%%%}+%%%{%%{[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0]%%%}+%%%{16,[0,2]%%%}+%%%{%%%{8,[1]%%%},[0,1]%%%}+%%%{%%%{1,[2]%%%},[0,0]%%%} / %%%{%%%{1,[1]%%%},[4,0]%%%}+%%%{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0]%%%}+%%%{%%%{8,[1]%%%},[2,1]%%%}+%%%{%%%{6,[2]%%%},[2,0]%%%}+%%%{%%{poly1[%%%{-16,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1]%%%}+%%%{%%{poly1[%%%{-4,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0]%%%}+%%%{%%%{16,[1]%%%},[0,2]%%%}+%%%{%%%{8,[2]%%%},[0,1]%%%}+%%%{%%%{1,[3]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
494,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^4,x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \tan\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*tan(f*x + e)^4, x)","F",0
495,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^2,x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \tan\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*tan(f*x + e)^2, x)","F",0
496,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a), x)","F",0
497,0,0,0,0.000000," ","integrate(cot(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \cot\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*cot(f*x + e)^2, x)","F",0
498,0,0,0,0.000000," ","integrate(cot(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{b \sin\left(f x + e\right)^{2} + a} \cot\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate(sqrt(b*sin(f*x + e)^2 + a)*cot(f*x + e)^4, x)","F",0
499,1,3781,0,5.515800," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^5,x, algorithm=""giac"")","-\frac{\frac{3 \, {\left(8 \, a^{2} + 40 \, a b + 35 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{\sqrt{-a - b}} - \frac{16 \, {\left(6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b + 9 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{2} + 18 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b + 39 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{2} + 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b + 66 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{2} + 88 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{3} - 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b + 6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{2} + 72 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{3} - 18 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b - 75 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{2} - 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{3} + 144 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{4} - 6 \, a^{\frac{7}{2}} b - 45 \, a^{\frac{5}{2}} b^{2} - 136 \, a^{\frac{3}{2}} b^{3} - 144 \, \sqrt{a} b^{4}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} + a + 4 \, b\right)}^{3}} - \frac{6 \, {\left(8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a^{2} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a b + 11 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} b^{2} - 56 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{5}{2}} - 104 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{3}{2}} b - 13 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} \sqrt{a} b^{2} - 120 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{3} - 520 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{2} b - 581 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b^{2} - 76 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{3} + 136 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{7}{2}} + 248 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{5}{2}} b - 357 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b^{2} - 644 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{3} + 344 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{4} + 1736 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{3} b + 2841 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b^{2} + 1320 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{3} - 304 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{4} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{9}{2}} + 584 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{7}{2}} b + 2465 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b^{2} + 3736 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{3} + 1936 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{4} - 232 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{5} - 1240 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{4} b - 2271 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b^{2} - 1244 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{3} + 688 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{4} + 704 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{5} - 104 \, a^{\frac{11}{2}} - 728 \, a^{\frac{9}{2}} b - 2095 \, a^{\frac{7}{2}} b^{2} - 3092 \, a^{\frac{5}{2}} b^{3} - 2320 \, a^{\frac{3}{2}} b^{4} - 704 \, \sqrt{a} b^{5}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{4}}}{12 \, f}"," ",0,"-1/12*(3*(8*a^2 + 40*a*b + 35*b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/sqrt(-a - b) - 16*(6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b + 9*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^2 + 18*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b + 39*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^2 + 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b + 66*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^2 + 88*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^3 - 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b + 6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^2 + 72*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^3 - 18*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b - 75*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^2 - 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^3 + 144*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^4 - 6*a^(7/2)*b - 45*a^(5/2)*b^2 - 136*a^(3/2)*b^3 - 144*sqrt(a)*b^4)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b)^3 - 6*(8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a*b + 11*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*b^2 - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(5/2) - 104*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(3/2)*b - 13*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*sqrt(a)*b^2 - 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 - 520*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b - 581*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b^2 - 76*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 136*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) + 248*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*b - 357*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b^2 - 644*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^3 + 344*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^4 + 1736*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^3*b + 2841*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b^2 + 1320*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^3 - 304*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^4 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2) + 584*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*b + 2465*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b^2 + 3736*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^3 + 1936*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^4 - 232*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^5 - 1240*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*b - 2271*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b^2 - 1244*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^3 + 688*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^4 + 704*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^5 - 104*a^(11/2) - 728*a^(9/2)*b - 2095*a^(7/2)*b^2 - 3092*a^(5/2)*b^3 - 2320*a^(3/2)*b^4 - 704*sqrt(a)*b^5)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^4)/f","B",0
500,1,2185,0,2.377180," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^3,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(2 \, a^{2} + 7 \, a b + 5 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{\sqrt{-a - b}} - \frac{6 \, {\left(2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} + 3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b + {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} + 7 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} - 3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b + 3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{2} + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{3} - 2 \, a^{\frac{7}{2}} - 7 \, a^{\frac{5}{2}} b - 9 \, a^{\frac{3}{2}} b^{2} - 4 \, \sqrt{a} b^{3}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{2}} - \frac{8 \, {\left(3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b + 3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{2} + 9 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b + 15 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{2} + 6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b + 30 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{2} + 32 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{3} - 6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b + 6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{2} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{3} - 9 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b - 33 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{2} + 48 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{4} - 3 \, a^{\frac{7}{2}} b - 21 \, a^{\frac{5}{2}} b^{2} - 56 \, a^{\frac{3}{2}} b^{3} - 48 \, \sqrt{a} b^{4}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} + a + 4 \, b\right)}^{3}}}{3 \, f}"," ",0,"1/3*(3*(2*a^2 + 7*a*b + 5*b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/sqrt(-a - b) - 6*(2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2 + 3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) + 7*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b + 3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2 + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^3 - 2*a^(7/2) - 7*a^(5/2)*b - 9*a^(3/2)*b^2 - 4*sqrt(a)*b^3)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^2 - 8*(3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b + 3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^2 + 9*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b + 15*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^2 + 6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b + 30*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^2 + 32*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^3 - 6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b + 6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^3 - 9*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b - 33*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^2 + 48*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^4 - 3*a^(7/2)*b - 21*a^(5/2)*b^2 - 56*a^(3/2)*b^3 - 48*sqrt(a)*b^4)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b)^3)/f","B",0
501,1,1338,0,0.881197," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e),x, algorithm=""giac"")","-\frac{2 \, {\left(\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{\sqrt{-a - b}} - \frac{2 \, {\left(6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b + 3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{2} + 18 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b + 21 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{2} + 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b + 54 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{2} + 40 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{3} - 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b + 18 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{2} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{3} - 18 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b - 57 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{2} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{3} + 48 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{4} - 6 \, a^{\frac{7}{2}} b - 39 \, a^{\frac{5}{2}} b^{2} - 88 \, a^{\frac{3}{2}} b^{3} - 48 \, \sqrt{a} b^{4}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} + a + 4 \, b\right)}^{3}}\right)}}{3 \, f}"," ",0,"-2/3*(3*(a^2 + 2*a*b + b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/sqrt(-a - b) - 2*(6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b + 3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^2 + 18*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b + 21*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^2 + 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b + 54*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^2 + 40*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^3 - 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b + 18*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^3 - 18*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b - 57*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^3 + 48*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^4 - 6*a^(7/2)*b - 39*a^(5/2)*b^2 - 88*a^(3/2)*b^3 - 48*sqrt(a)*b^4)/((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) + a + 4*b)^3)/f","B",0
502,1,71,0,0.136453," ","integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\frac{3 \, a^{2} \arctan\left(\frac{\sqrt{b \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a}} + {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} + 3 \, \sqrt{b \sin\left(f x + e\right)^{2} + a} a}{3 \, f}"," ",0,"1/3*(3*a^2*arctan(sqrt(b*sin(f*x + e)^2 + a)/sqrt(-a))/sqrt(-a) + (b*sin(f*x + e)^2 + a)^(3/2) + 3*sqrt(b*sin(f*x + e)^2 + a)*a)/f","A",0
503,-2,0,0,0.000000," ","integrate(cot(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Evaluation time: 1.8Unable to divide, perhaps due to rounding error%%%{65536,[8,11]%%%}+%%%{%%%{393216,[1]%%%},[8,10]%%%}+%%%{%%%{983040,[2]%%%},[8,9]%%%}+%%%{%%%{1310720,[3]%%%},[8,8]%%%}+%%%{%%%{983040,[4]%%%},[8,7]%%%}+%%%{%%%{393216,[5]%%%},[8,6]%%%}+%%%{%%%{65536,[6]%%%},[8,5]%%%}+%%%{%%{[-524288,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11]%%%}+%%%{%%{[%%%{-3145728,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,10]%%%}+%%%{%%{[%%%{-7864320,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,9]%%%}+%%%{%%{[%%%{-10485760,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,8]%%%}+%%%{%%{[%%%{-7864320,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,7]%%%}+%%%{%%{[%%%{-3145728,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6]%%%}+%%%{%%{[%%%{-524288,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,5]%%%}+%%%{1048576,[6,12]%%%}+%%%{%%%{8126464,[1]%%%},[6,11]%%%}+%%%{%%%{26738688,[2]%%%},[6,10]%%%}+%%%{%%%{48496640,[3]%%%},[6,9]%%%}+%%%{%%%{52428800,[4]%%%},[6,8]%%%}+%%%{%%%{33816576,[5]%%%},[6,7]%%%}+%%%{%%%{12058624,[6]%%%},[6,6]%%%}+%%%{%%%{1835008,[7]%%%},[6,5]%%%}+%%%{%%{[-6291456,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12]%%%}+%%%{%%{[%%%{-41418752,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11]%%%}+%%%{%%{[%%%{-116391936,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,10]%%%}+%%%{%%{[%%%{-180879360,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9]%%%}+%%%{%%{[%%%{-167772160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,8]%%%}+%%%{%%{[%%%{-92798976,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7]%%%}+%%%{%%{[%%%{-28311552,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6]%%%}+%%%{%%{[%%%{-3670016,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,5]%%%}+%%%{6291456,[4,13]%%%}+%%%{%%%{53477376,[1]%%%},[4,12]%%%}+%%%{%%%{193331200,[2]%%%},[4,11]%%%}+%%%{%%%{389283840,[3]%%%},[4,10]%%%}+%%%{%%%{477757440,[4]%%%},[4,9]%%%}+%%%{%%%{365428736,[5]%%%},[4,8]%%%}+%%%{%%%{169476096,[6]%%%},[4,7]%%%}+%%%{%%%{43253760,[7]%%%},[4,6]%%%}+%%%{%%%{4587520,[8]%%%},[4,5]%%%}+%%%{%%{[-25165824,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13]%%%}+%%%{%%{[%%%{-171966464,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12]%%%}+%%%{%%{[%%%{-506986496,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11]%%%}+%%%{%%{[%%%{-839909376,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,10]%%%}+%%%{%%{[%%%{-851968000,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9]%%%}+%%%{%%{[%%%{-538968064,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,8]%%%}+%%%{%%{[%%%{-206045184,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,7]%%%}+%%%{%%{[%%%{-42991616,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6]%%%}+%%%{%%{[%%%{-3670016,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,5]%%%}+%%%{16777216,[2,14]%%%}+%%%{%%%{138412032,[1]%%%},[2,13]%%%}+%%%{%%%{493879296,[2]%%%},[2,12]%%%}+%%%{%%%{997982208,[3]%%%},[2,11]%%%}+%%%{%%%{1253572608,[4]%%%},[2,10]%%%}+%%%{%%%{1008992256,[5]%%%},[2,9]%%%}+%%%{%%%{515899392,[6]%%%},[2,8]%%%}+%%%{%%%{159645696,[7]%%%},[2,7]%%%}+%%%{%%%{26738688,[8]%%%},[2,6]%%%}+%%%{%%%{1835008,[9]%%%},[2,5]%%%}+%%%{%%{[-33554432,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,14]%%%}+%%%{%%{[%%%{-226492416,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13]%%%}+%%%{%%{[%%%{-660602880,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12]%%%}+%%%{%%{[%%%{-1086849024,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11]%%%}+%%%{%%{[%%%{-1104150528,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,10]%%%}+%%%{%%{[%%%{-712507392,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9]%%%}+%%%{%%{[%%%{-289406976,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,8]%%%}+%%%{%%{[%%%{-70778880,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7]%%%}+%%%{%%{[%%%{-9437184,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6]%%%}+%%%{%%{[%%%{-524288,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,5]%%%}+%%%{16777216,[0,15]%%%}+%%%{%%%{117440512,[1]%%%},[0,14]%%%}+%%%{%%%{358612992,[2]%%%},[0,13]%%%}+%%%{%%%{625999872,[3]%%%},[0,12]%%%}+%%%{%%%{687931392,[4]%%%},[0,11]%%%}+%%%{%%%{494272512,[5]%%%},[0,10]%%%}+%%%{%%%{233766912,[6]%%%},[0,9]%%%}+%%%{%%%{71565312,[7]%%%},[0,8]%%%}+%%%{%%%{13565952,[8]%%%},[0,7]%%%}+%%%{%%%{1441792,[9]%%%},[0,6]%%%}+%%%{%%%{65536,[10]%%%},[0,5]%%%} / %%%{%%%{1,[2]%%%},[8,0]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0]%%%}+%%%{%%%{16,[2]%%%},[6,1]%%%}+%%%{%%%{28,[3]%%%},[6,0]%%%}+%%%{%%{[%%%{-96,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1]%%%}+%%%{%%{poly1[%%%{-56,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0]%%%}+%%%{%%%{96,[2]%%%},[4,2]%%%}+%%%{%%%{240,[3]%%%},[4,1]%%%}+%%%{%%%{70,[4]%%%},[4,0]%%%}+%%%{%%{poly1[%%%{-384,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2]%%%}+%%%{%%{[%%%{-320,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1]%%%}+%%%{%%{poly1[%%%{-56,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0]%%%}+%%%{%%%{256,[2]%%%},[2,3]%%%}+%%%{%%%{576,[3]%%%},[2,2]%%%}+%%%{%%%{240,[4]%%%},[2,1]%%%}+%%%{%%%{28,[5]%%%},[2,0]%%%}+%%%{%%{[%%%{-512,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,3]%%%}+%%%{%%{poly1[%%%{-384,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2]%%%}+%%%{%%{[%%%{-96,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1]%%%}+%%%{%%{poly1[%%%{-8,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0]%%%}+%%%{%%%{256,[2]%%%},[0,4]%%%}+%%%{%%%{256,[3]%%%},[0,3]%%%}+%%%{%%%{96,[4]%%%},[0,2]%%%}+%%%{%%%{16,[5]%%%},[0,1]%%%}+%%%{%%%{1,[6]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
504,-2,0,0,0.000000," ","integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (4*pi/t_nostep/2)>(-4*pi/t_nostep/2)Evaluation time: 2.41Unable to divide, perhaps due to rounding error%%%{524288,[8,12]%%%}+%%%{%%%{3670016,[1]%%%},[8,11]%%%}+%%%{%%%{11010048,[2]%%%},[8,10]%%%}+%%%{%%%{18350080,[3]%%%},[8,9]%%%}+%%%{%%%{18350080,[4]%%%},[8,8]%%%}+%%%{%%%{11010048,[5]%%%},[8,7]%%%}+%%%{%%%{3670016,[6]%%%},[8,6]%%%}+%%%{%%%{524288,[7]%%%},[8,5]%%%}+%%%{%%{[-4194304,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,12]%%%}+%%%{%%{[%%%{-29360128,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,11]%%%}+%%%{%%{[%%%{-88080384,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,10]%%%}+%%%{%%{[%%%{-146800640,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,9]%%%}+%%%{%%{[%%%{-146800640,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,8]%%%}+%%%{%%{[%%%{-88080384,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,7]%%%}+%%%{%%{[%%%{-29360128,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6]%%%}+%%%{%%{[%%%{-4194304,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,5]%%%}+%%%{8388608,[6,13]%%%}+%%%{%%%{73400320,[1]%%%},[6,12]%%%}+%%%{%%%{278921216,[2]%%%},[6,11]%%%}+%%%{%%%{601882624,[3]%%%},[6,10]%%%}+%%%{%%%{807403520,[4]%%%},[6,9]%%%}+%%%{%%%{689963008,[5]%%%},[6,8]%%%}+%%%{%%%{367001600,[6]%%%},[6,7]%%%}+%%%{%%%{111149056,[7]%%%},[6,6]%%%}+%%%{%%%{14680064,[8]%%%},[6,5]%%%}+%%%{%%{[-50331648,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,13]%%%}+%%%{%%{[%%%{-381681664,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,12]%%%}+%%%{%%{[%%%{-1262485504,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,11]%%%}+%%%{%%{[%%%{-2378170368,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,10]%%%}+%%%{%%{[%%%{-2789212160,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,9]%%%}+%%%{%%{[%%%{-2084569088,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,8]%%%}+%%%{%%{[%%%{-968884224,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,7]%%%}+%%%{%%{[%%%{-255852544,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6]%%%}+%%%{%%{[%%%{-29360128,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,5]%%%}+%%%{50331648,[4,14]%%%}+%%%{%%%{478150656,[1]%%%},[4,13]%%%}+%%%{%%%{1974468608,[2]%%%},[4,12]%%%}+%%%{%%%{4660920320,[3]%%%},[4,11]%%%}+%%%{%%%{6936330240,[4]%%%},[4,10]%%%}+%%%{%%%{6745489408,[5]%%%},[4,9]%%%}+%%%{%%%{4279238656,[6]%%%},[4,8]%%%}+%%%{%%%{1701838848,[7]%%%},[4,7]%%%}+%%%{%%%{382730240,[8]%%%},[4,6]%%%}+%%%{%%%{36700160,[9]%%%},[4,5]%%%}+%%%{%%{[-201326592,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,14]%%%}+%%%{%%{[%%%{-1577058304,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,13]%%%}+%%%{%%{[%%%{-5431623680,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,12]%%%}+%%%{%%{[%%%{-10775166976,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,11]%%%}+%%%{%%{[%%%{-13535019008,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,10]%%%}+%%%{%%{[%%%{-11127488512,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,9]%%%}+%%%{%%{[%%%{-5960105984,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,8]%%%}+%%%{%%{[%%%{-1992294400,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,7]%%%}+%%%{%%{[%%%{-373293056,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6]%%%}+%%%{%%{[%%%{-29360128,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,5]%%%}+%%%{134217728,[2,15]%%%}+%%%{%%%{1241513984,[1]%%%},[2,14]%%%}+%%%{%%%{5058330624,[2]%%%},[2,13]%%%}+%%%{%%%{11934892032,[3]%%%},[2,12]%%%}+%%%{%%%{18012438528,[4]%%%},[2,11]%%%}+%%%{%%%{18100518912,[5]%%%},[2,10]%%%}+%%%{%%%{12199133184,[6]%%%},[2,9]%%%}+%%%{%%%{5404360704,[7]%%%},[2,8]%%%}+%%%{%%%{1491075072,[8]%%%},[2,7]%%%}+%%%{%%%{228589568,[9]%%%},[2,6]%%%}+%%%{%%%{14680064,[10]%%%},[2,5]%%%}+%%%{%%{[-268435456,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,15]%%%}+%%%{%%{[%%%{-2080374784,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,14]%%%}+%%%{%%{[%%%{-7096762368,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,13]%%%}+%%%{%%{[%%%{-13979615232,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,12]%%%}+%%%{%%{[%%%{-17527996416,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,11]%%%}+%%%{%%{[%%%{-14533263360,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,10]%%%}+%%%{%%{[%%%{-8015314944,[6]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,9]%%%}+%%%{%%{[%%%{-2881486848,[7]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,8]%%%}+%%%{%%{[%%%{-641728512,[8]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,7]%%%}+%%%{%%{[%%%{-79691776,[9]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,6]%%%}+%%%{%%{[%%%{-4194304,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,5]%%%}+%%%{134217728,[0,16]%%%}+%%%{%%%{1073741824,[1]%%%},[0,15]%%%}+%%%{%%%{3808428032,[2]%%%},[0,14]%%%}+%%%{%%%{7876902912,[3]%%%},[0,13]%%%}+%%%{%%%{10511450112,[4]%%%},[0,12]%%%}+%%%{%%%{9457631232,[5]%%%},[0,11]%%%}+%%%{%%%{5824315392,[6]%%%},[0,10]%%%}+%%%{%%%{2442657792,[7]%%%},[0,9]%%%}+%%%{%%%{681050112,[8]%%%},[0,8]%%%}+%%%{%%%{120061952,[9]%%%},[0,7]%%%}+%%%{%%%{12058624,[10]%%%},[0,6]%%%}+%%%{%%%{524288,[11]%%%},[0,5]%%%} / %%%{%%%{1,[2]%%%},[8,0]%%%}+%%%{%%{poly1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,0]%%%}+%%%{%%%{16,[2]%%%},[6,1]%%%}+%%%{%%%{28,[3]%%%},[6,0]%%%}+%%%{%%{[%%%{-96,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1]%%%}+%%%{%%{poly1[%%%{-56,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,0]%%%}+%%%{%%%{96,[2]%%%},[4,2]%%%}+%%%{%%%{240,[3]%%%},[4,1]%%%}+%%%{%%%{70,[4]%%%},[4,0]%%%}+%%%{%%{poly1[%%%{-384,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2]%%%}+%%%{%%{[%%%{-320,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1]%%%}+%%%{%%{poly1[%%%{-56,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0]%%%}+%%%{%%%{256,[2]%%%},[2,3]%%%}+%%%{%%%{576,[3]%%%},[2,2]%%%}+%%%{%%%{240,[4]%%%},[2,1]%%%}+%%%{%%%{28,[5]%%%},[2,0]%%%}+%%%{%%{[%%%{-512,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,3]%%%}+%%%{%%{poly1[%%%{-384,[3]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2]%%%}+%%%{%%{[%%%{-96,[4]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1]%%%}+%%%{%%{poly1[%%%{-8,[5]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,0]%%%}+%%%{%%%{256,[2]%%%},[0,4]%%%}+%%%{%%%{256,[3]%%%},[0,3]%%%}+%%%{%%%{96,[4]%%%},[0,2]%%%}+%%%{%%%{16,[5]%%%},[0,1]%%%}+%%%{%%%{1,[6]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
505,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^4,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \tan\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*tan(f*x + e)^4, x)","F",0
506,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^2,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \tan\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*tan(f*x + e)^2, x)","F",0
507,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
508,0,0,0,0.000000," ","integrate(cot(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \cot\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*cot(f*x + e)^2, x)","F",0
509,0,0,0,0.000000," ","integrate(cot(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} \cot\left(f x + e\right)^{4}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(3/2)*cot(f*x + e)^4, x)","F",0
510,1,2580,0,3.136349," ","integrate(tan(f*x+e)^5/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","-\frac{\frac{{\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a - b}} - \frac{2 \, {\left(8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a^{2} + 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a b + 3 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} b^{2} - 56 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{5}{2}} - 56 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{3}{2}} b - 21 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} \sqrt{a} b^{2} - 120 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{3} - 408 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{2} b - 269 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b^{2} - 44 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{3} + 136 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{7}{2}} + 40 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{5}{2}} b - 493 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b^{2} - 292 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{3} + 344 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{4} + 1304 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{3} b + 1345 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b^{2} + 104 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{3} - 176 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{4} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{9}{2}} + 600 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{7}{2}} b + 1865 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b^{2} + 1880 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{3} + 528 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{4} - 232 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{5} - 904 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{4} b - 1079 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b^{2} - 60 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{3} + 560 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{4} + 192 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{5} - 104 \, a^{\frac{11}{2}} - 584 \, a^{\frac{9}{2}} b - 1351 \, a^{\frac{7}{2}} b^{2} - 1588 \, a^{\frac{5}{2}} b^{3} - 912 \, a^{\frac{3}{2}} b^{4} - 192 \, \sqrt{a} b^{5}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{4} {\left(a^{2} + 2 \, a b + b^{2}\right)}}}{4 \, f}"," ",0,"-1/4*((8*a^2 + 8*a*b + 3*b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a^2 + 2*a*b + b^2)*sqrt(-a - b)) - 2*(8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a^2 + 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a*b + 3*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*b^2 - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(5/2) - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(3/2)*b - 21*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*sqrt(a)*b^2 - 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 - 408*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b - 269*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b^2 - 44*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 136*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) + 40*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*b - 493*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b^2 - 292*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^3 + 344*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^4 + 1304*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^3*b + 1345*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b^2 + 104*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^3 - 176*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^4 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2) + 600*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*b + 1865*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b^2 + 1880*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^3 + 528*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^4 - 232*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^5 - 904*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*b - 1079*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b^2 - 60*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^3 + 560*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^4 + 192*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^5 - 104*a^(11/2) - 584*a^(9/2)*b - 1351*a^(7/2)*b^2 - 1588*a^(5/2)*b^3 - 912*a^(3/2)*b^4 - 192*sqrt(a)*b^5)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^4*(a^2 + 2*a*b + b^2)))/f","B",0
511,1,793,0,1.286717," ","integrate(tan(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\frac{\frac{{\left(2 \, a + b\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a + b\right)} \sqrt{-a - b}} - \frac{2 \, {\left(2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a + {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} - {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{2} - 2 \, a^{\frac{5}{2}} - 5 \, a^{\frac{3}{2}} b - 4 \, \sqrt{a} b^{2}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{2} {\left(a + b\right)}}}{f}"," ",0,"((2*a + b)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a + b)*sqrt(-a - b)) - 2*(2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a + (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2) + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2 - (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^2 - 2*a^(5/2) - 5*a^(3/2)*b - 4*sqrt(a)*b^2)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^2*(a + b)))/f","B",0
512,1,98,0,0.575700," ","integrate(tan(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","-\frac{2 \, \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{\sqrt{-a - b} f}"," ",0,"-2*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/(sqrt(-a - b)*f)","B",0
513,1,31,0,0.144588," ","integrate(cot(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{\sqrt{b \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} f}"," ",0,"arctan(sqrt(b*sin(f*x + e)^2 + a)/sqrt(-a))/(sqrt(-a)*f)","A",0
514,-2,0,0,0.000000," ","integrate(cot(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[45,77]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-97,-38]Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Evaluation time: 0.65Unable to divide, perhaps due to rounding error%%%{1,[4,0]%%%}+%%%{%%%{-2,[1]%%%},[2,0]%%%}+%%%{%%%{1,[2]%%%},[0,0]%%%} / %%%{%%%{1,[1]%%%},[4,0]%%%}+%%%{%%%{-2,[2]%%%},[2,0]%%%}+%%%{%%%{1,[3]%%%},[0,0]%%%} Error: Bad Argument Value","F(-2)",0
515,1,898,0,1.029228," ","integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","-\frac{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} {\left(\frac{\tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a} - \frac{13 \, a + 6 \, b}{a^{2}}\right)} - \frac{8 \, {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a^{2}} - \frac{4 \, {\left(8 \, a^{\frac{5}{2}} + 8 \, a^{\frac{3}{2}} b + 3 \, \sqrt{a} b^{2}\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{3}} + \frac{4 \, {\left(6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} + 16 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b + 6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{2} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} - 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} - 20 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b - 10 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{2} - 7 \, a^{\frac{7}{2}} - 4 \, a^{\frac{5}{2}} b\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - a\right)}^{2} a^{2}}}{64 \, f}"," ",0,"-1/64*(sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)*(tan(1/2*f*x + 1/2*e)^2/a - (13*a + 6*b)/a^2) - 8*(8*a^2 + 8*a*b + 3*b^2)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 4*(8*a^(5/2) + 8*a^(3/2)*b + 3*sqrt(a)*b^2)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^3 + 4*(6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2 + 16*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + 6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 20*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b - 10*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2 - 7*a^(7/2) - 4*a^(5/2)*b)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^2*a^2))/f","B",0
516,0,0,0,0.000000," ","integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\tan\left(f x + e\right)^{4}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(tan(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
517,0,0,0,0.000000," ","integrate(tan(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\tan\left(f x + e\right)^{2}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(tan(f*x + e)^2/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
518,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(1/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
519,0,0,0,0.000000," ","integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\cot\left(f x + e\right)^{2}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(cot(f*x + e)^2/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
520,0,0,0,0.000000," ","integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""giac"")","\int \frac{\cot\left(f x + e\right)^{4}}{\sqrt{b \sin\left(f x + e\right)^{2} + a}}\,{d x}"," ",0,"integrate(cot(f*x + e)^4/sqrt(b*sin(f*x + e)^2 + a), x)","F",0
521,1,2776,0,4.029415," ","integrate(tan(f*x+e)^5/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","-\frac{\frac{4 \, {\left(\frac{{\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{7} b + 7 \, a^{6} b^{2} + 21 \, a^{5} b^{3} + 35 \, a^{4} b^{4} + 35 \, a^{3} b^{5} + 21 \, a^{2} b^{6} + 7 \, a b^{7} + b^{8}} + \frac{a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}}{a^{7} b + 7 \, a^{6} b^{2} + 21 \, a^{5} b^{3} + 35 \, a^{4} b^{4} + 35 \, a^{3} b^{5} + 21 \, a^{2} b^{6} + 7 \, a b^{7} + b^{8}}\right)}}{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}} + \frac{{\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{-a - b}} - \frac{2 \, {\left(8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a^{2} - {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} b^{2} - 56 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{5}{2}} - 32 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{3}{2}} b - 25 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} \sqrt{a} b^{2} - 120 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{3} - 352 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{2} b - 113 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b^{2} - 28 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{3} + 136 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{7}{2}} - 64 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{5}{2}} b - 561 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b^{2} - 116 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{3} + 344 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{4} + 1088 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{3} b + 597 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b^{2} - 504 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{3} - 112 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{4} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{9}{2}} + 608 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{7}{2}} b + 1565 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b^{2} + 952 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{3} - 176 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{4} - 232 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{5} - 736 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{4} b - 483 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b^{2} + 532 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{3} + 496 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{4} - 64 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{5} - 104 \, a^{\frac{11}{2}} - 512 \, a^{\frac{9}{2}} b - 979 \, a^{\frac{7}{2}} b^{2} - 836 \, a^{\frac{5}{2}} b^{3} - 208 \, a^{\frac{3}{2}} b^{4} + 64 \, \sqrt{a} b^{5}\right)}}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{4}}}{4 \, f}"," ",0,"-1/4*(4*((a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*tan(1/2*f*x + 1/2*e)^2/(a^7*b + 7*a^6*b^2 + 21*a^5*b^3 + 35*a^4*b^4 + 35*a^3*b^5 + 21*a^2*b^6 + 7*a*b^7 + b^8) + (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)/(a^7*b + 7*a^6*b^2 + 21*a^5*b^3 + 35*a^4*b^4 + 35*a^3*b^5 + 21*a^2*b^6 + 7*a*b^7 + b^8))/sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) + (8*a^2 - 8*a*b - b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a - b)) - 2*(8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a^2 - (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*b^2 - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(5/2) - 32*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(3/2)*b - 25*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*sqrt(a)*b^2 - 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 - 352*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b - 113*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b^2 - 28*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 136*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) - 64*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*b - 561*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b^2 - 116*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^3 + 344*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^4 + 1088*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^3*b + 597*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b^2 - 504*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^3 - 112*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^4 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2) + 608*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*b + 1565*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b^2 + 952*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^3 - 176*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^4 - 232*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^5 - 736*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*b - 483*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b^2 + 532*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^3 + 496*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^4 - 64*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^5 - 104*a^(11/2) - 512*a^(9/2)*b - 979*a^(7/2)*b^2 - 836*a^(5/2)*b^3 - 208*a^(3/2)*b^4 + 64*sqrt(a)*b^5)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^4))/f","B",0
522,1,1011,0,1.614100," ","integrate(tan(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\frac{\frac{{\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{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}} + \frac{a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}}{a^{5} b + 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 10 \, a^{2} b^{4} + 5 \, a b^{5} + b^{6}}}{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}} + \frac{{\left(2 \, a - b\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a - b}} - \frac{2 \, {\left(2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a + {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} - {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{2} - 2 \, a^{\frac{5}{2}} - 5 \, a^{\frac{3}{2}} b - 4 \, \sqrt{a} b^{2}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{2} {\left(a^{2} + 2 \, a b + b^{2}\right)}}}{f}"," ",0,"(((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(1/2*f*x + 1/2*e)^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) + (a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)/(a^5*b + 5*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 + 5*a*b^5 + b^6))/sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) + (2*a - b)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a^2 + 2*a*b + b^2)*sqrt(-a - b)) - 2*(2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a + (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2) + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2 - (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^2 - 2*a^(5/2) - 5*a^(3/2)*b - 4*sqrt(a)*b^2)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^2*(a^2 + 2*a*b + b^2)))/f","B",0
523,1,250,0,0.890672," ","integrate(tan(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","-\frac{\frac{\frac{{\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}} + \frac{a^{2} b + 2 \, a b^{2} + b^{3}}{a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}}}{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}} + \frac{2 \, \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a + b\right)} \sqrt{-a - b}}}{f}"," ",0,"-(((a^2*b + 2*a*b^2 + b^3)*tan(1/2*f*x + 1/2*e)^2/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4) + (a^2*b + 2*a*b^2 + b^3)/(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4))/sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) + 2*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a + b)*sqrt(-a - b)))/f","B",0
524,1,57,0,0.155459," ","integrate(cot(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{\sqrt{b \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a f} + \frac{1}{\sqrt{b \sin\left(f x + e\right)^{2} + a} a f}"," ",0,"arctan(sqrt(b*sin(f*x + e)^2 + a)/sqrt(-a))/(sqrt(-a)*a*f) + 1/(sqrt(b*sin(f*x + e)^2 + a)*a*f)","A",0
525,1,525,0,1.007657," ","integrate(cot(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","-\frac{\frac{{\left(\frac{{\left(a^{5} b + a^{4} b^{2}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{6} b + a^{5} b^{2}} + \frac{2 \, {\left(5 \, a^{5} b + 11 \, a^{4} b^{2} + 6 \, a^{3} b^{3}\right)}}{a^{6} b + a^{5} b^{2}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{9 \, a^{5} b + 17 \, a^{4} b^{2} + 8 \, a^{3} b^{3}}{a^{6} b + a^{5} b^{2}}}{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}} + \frac{2 \, {\left(2 \, a^{\frac{3}{2}} + 3 \, \sqrt{a} b\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{3}} - \frac{2 \, {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{\frac{3}{2}} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} b + a^{2}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} - a^{\frac{3}{2}}\right)} a^{2}}}{8 \, f}"," ",0,"-1/8*((((a^5*b + a^4*b^2)*tan(1/2*f*x + 1/2*e)^2/(a^6*b + a^5*b^2) + 2*(5*a^5*b + 11*a^4*b^2 + 6*a^3*b^3)/(a^6*b + a^5*b^2))*tan(1/2*f*x + 1/2*e)^2 + (9*a^5*b + 17*a^4*b^2 + 8*a^3*b^3)/(a^6*b + a^5*b^2))/sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) + 2*(2*a^(3/2) + 3*sqrt(a)*b)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^3 - 2*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^(3/2) + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a)*b + a^2)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a) - a^(3/2))*a^2))/f","B",0
526,1,1150,0,1.452960," ","integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","-\frac{\frac{{\left({\left(\frac{{\left(a^{8} b + a^{7} b^{2}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{9} b + a^{8} b^{2}} - \frac{11 \, a^{8} b + 21 \, a^{7} b^{2} + 10 \, a^{6} b^{3}}{a^{9} b + a^{8} b^{2}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{89 \, a^{8} b + 297 \, a^{7} b^{2} + 328 \, a^{6} b^{3} + 120 \, a^{5} b^{4}}{a^{9} b + a^{8} b^{2}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{77 \, a^{8} b + 219 \, a^{7} b^{2} + 206 \, a^{6} b^{3} + 64 \, a^{5} b^{4}}{a^{9} b + a^{8} b^{2}}}{\sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}} - \frac{8 \, {\left(8 \, a^{2} + 24 \, a b + 15 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a^{3}} - \frac{4 \, {\left(8 \, a^{\frac{5}{2}} + 24 \, a^{\frac{3}{2}} b + 15 \, \sqrt{a} b^{2}\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{4}} + \frac{4 \, {\left(6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} + 20 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b + 14 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{2} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b - 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} - 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b - 18 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{2} - 7 \, a^{\frac{7}{2}} - 8 \, a^{\frac{5}{2}} b\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - a\right)}^{2} a^{3}}}{64 \, f}"," ",0,"-1/64*(((((a^8*b + a^7*b^2)*tan(1/2*f*x + 1/2*e)^2/(a^9*b + a^8*b^2) - (11*a^8*b + 21*a^7*b^2 + 10*a^6*b^3)/(a^9*b + a^8*b^2))*tan(1/2*f*x + 1/2*e)^2 - (89*a^8*b + 297*a^7*b^2 + 328*a^6*b^3 + 120*a^5*b^4)/(a^9*b + a^8*b^2))*tan(1/2*f*x + 1/2*e)^2 - (77*a^8*b + 219*a^7*b^2 + 206*a^6*b^3 + 64*a^5*b^4)/(a^9*b + a^8*b^2))/sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - 8*(8*a^2 + 24*a*b + 15*b^2)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a^3) - 4*(8*a^(5/2) + 24*a^(3/2)*b + 15*sqrt(a)*b^2)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^4 + 4*(6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2 + 20*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + 14*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b - 18*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2 - 7*a^(7/2) - 8*a^(5/2)*b)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^2*a^3))/f","B",0
527,0,0,0,0.000000," ","integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\tan\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(tan(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
528,0,0,0,0.000000," ","integrate(tan(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\tan\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(tan(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
529,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(-3/2), x)","F",0
530,0,0,0,0.000000," ","integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\cot\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(cot(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
531,0,0,0,0.000000," ","integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""giac"")","\int \frac{\cot\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}}}\,{d x}"," ",0,"integrate(cot(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(3/2), x)","F",0
532,1,3826,0,5.035179," ","integrate(tan(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{\frac{3 \, {\left(8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \sqrt{-a - b}} + \frac{4 \, {\left({\left({\left(\frac{{\left(4 \, a^{17} b^{2} + 51 \, a^{16} b^{3} + 294 \, a^{15} b^{4} + 1001 \, a^{14} b^{5} + 2184 \, a^{13} b^{6} + 3003 \, a^{12} b^{7} + 2002 \, a^{11} b^{8} - 1287 \, a^{10} b^{9} - 5148 \, a^{9} b^{10} - 7007 \, a^{8} b^{11} - 6006 \, a^{7} b^{12} - 3549 \, a^{6} b^{13} - 1456 \, a^{5} b^{14} - 399 \, a^{4} b^{15} - 66 \, a^{3} b^{16} - 5 \, a^{2} b^{17}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{18} b^{2} + 18 \, a^{17} b^{3} + 153 \, a^{16} b^{4} + 816 \, a^{15} b^{5} + 3060 \, a^{14} b^{6} + 8568 \, a^{13} b^{7} + 18564 \, a^{12} b^{8} + 31824 \, a^{11} b^{9} + 43758 \, a^{10} b^{10} + 48620 \, a^{9} b^{11} + 43758 \, a^{8} b^{12} + 31824 \, a^{7} b^{13} + 18564 \, a^{6} b^{14} + 8568 \, a^{5} b^{15} + 3060 \, a^{4} b^{16} + 816 \, a^{3} b^{17} + 153 \, a^{2} b^{18} + 18 \, a b^{19} + b^{20}} + \frac{3 \, {\left(4 \, a^{17} b^{2} + 55 \, a^{16} b^{3} + 342 \, a^{15} b^{4} + 1253 \, a^{14} b^{5} + 2912 \, a^{13} b^{6} + 4095 \, a^{12} b^{7} + 2002 \, a^{11} b^{8} - 5291 \, a^{10} b^{9} - 15444 \, a^{9} b^{10} - 22451 \, a^{8} b^{11} - 22022 \, a^{7} b^{12} - 15561 \, a^{6} b^{13} - 8008 \, a^{5} b^{14} - 2947 \, a^{4} b^{15} - 738 \, a^{3} b^{16} - 113 \, a^{2} b^{17} - 8 \, a b^{18}\right)}}{a^{18} b^{2} + 18 \, a^{17} b^{3} + 153 \, a^{16} b^{4} + 816 \, a^{15} b^{5} + 3060 \, a^{14} b^{6} + 8568 \, a^{13} b^{7} + 18564 \, a^{12} b^{8} + 31824 \, a^{11} b^{9} + 43758 \, a^{10} b^{10} + 48620 \, a^{9} b^{11} + 43758 \, a^{8} b^{12} + 31824 \, a^{7} b^{13} + 18564 \, a^{6} b^{14} + 8568 \, a^{5} b^{15} + 3060 \, a^{4} b^{16} + 816 \, a^{3} b^{17} + 153 \, a^{2} b^{18} + 18 \, a b^{19} + b^{20}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{3 \, {\left(4 \, a^{17} b^{2} + 55 \, a^{16} b^{3} + 342 \, a^{15} b^{4} + 1253 \, a^{14} b^{5} + 2912 \, a^{13} b^{6} + 4095 \, a^{12} b^{7} + 2002 \, a^{11} b^{8} - 5291 \, a^{10} b^{9} - 15444 \, a^{9} b^{10} - 22451 \, a^{8} b^{11} - 22022 \, a^{7} b^{12} - 15561 \, a^{6} b^{13} - 8008 \, a^{5} b^{14} - 2947 \, a^{4} b^{15} - 738 \, a^{3} b^{16} - 113 \, a^{2} b^{17} - 8 \, a b^{18}\right)}}{a^{18} b^{2} + 18 \, a^{17} b^{3} + 153 \, a^{16} b^{4} + 816 \, a^{15} b^{5} + 3060 \, a^{14} b^{6} + 8568 \, a^{13} b^{7} + 18564 \, a^{12} b^{8} + 31824 \, a^{11} b^{9} + 43758 \, a^{10} b^{10} + 48620 \, a^{9} b^{11} + 43758 \, a^{8} b^{12} + 31824 \, a^{7} b^{13} + 18564 \, a^{6} b^{14} + 8568 \, a^{5} b^{15} + 3060 \, a^{4} b^{16} + 816 \, a^{3} b^{17} + 153 \, a^{2} b^{18} + 18 \, a b^{19} + b^{20}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{4 \, a^{17} b^{2} + 51 \, a^{16} b^{3} + 294 \, a^{15} b^{4} + 1001 \, a^{14} b^{5} + 2184 \, a^{13} b^{6} + 3003 \, a^{12} b^{7} + 2002 \, a^{11} b^{8} - 1287 \, a^{10} b^{9} - 5148 \, a^{9} b^{10} - 7007 \, a^{8} b^{11} - 6006 \, a^{7} b^{12} - 3549 \, a^{6} b^{13} - 1456 \, a^{5} b^{14} - 399 \, a^{4} b^{15} - 66 \, a^{3} b^{16} - 5 \, a^{2} b^{17}}{a^{18} b^{2} + 18 \, a^{17} b^{3} + 153 \, a^{16} b^{4} + 816 \, a^{15} b^{5} + 3060 \, a^{14} b^{6} + 8568 \, a^{13} b^{7} + 18564 \, a^{12} b^{8} + 31824 \, a^{11} b^{9} + 43758 \, a^{10} b^{10} + 48620 \, a^{9} b^{11} + 43758 \, a^{8} b^{12} + 31824 \, a^{7} b^{13} + 18564 \, a^{6} b^{14} + 8568 \, a^{5} b^{15} + 3060 \, a^{4} b^{16} + 816 \, a^{3} b^{17} + 153 \, a^{2} b^{18} + 18 \, a b^{19} + b^{20}}\right)}}{{\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a\right)}^{\frac{3}{2}}} - \frac{6 \, {\left(8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a^{2} - 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} a b - 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{7} b^{2} - 56 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{5}{2}} - 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} a^{\frac{3}{2}} b - 29 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{6} \sqrt{a} b^{2} - 120 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{3} - 296 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a^{2} b + 43 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} a b^{2} - 12 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{5} b^{3} + 136 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{7}{2}} - 168 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{5}{2}} b - 629 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} a^{\frac{3}{2}} b^{2} + 60 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{4} \sqrt{a} b^{3} + 344 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{4} + 872 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{3} b - 151 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} b^{2} - 1112 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b^{3} - 48 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{4} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{9}{2}} + 616 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{7}{2}} b + 1265 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} b^{2} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b^{3} - 880 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b^{4} - 232 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{5} - 568 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{4} b + 113 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} b^{2} + 1124 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b^{3} + 432 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{4} - 320 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{5} - 104 \, a^{\frac{11}{2}} - 440 \, a^{\frac{9}{2}} b - 607 \, a^{\frac{7}{2}} b^{2} - 84 \, a^{\frac{5}{2}} b^{3} + 496 \, a^{\frac{3}{2}} b^{4} + 320 \, \sqrt{a} b^{5}\right)}}{{\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{4}}}{12 \, f}"," ",0,"-1/12*(3*(8*a^2 - 24*a*b + 3*b^2)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt(-a - b)) + 4*((((4*a^17*b^2 + 51*a^16*b^3 + 294*a^15*b^4 + 1001*a^14*b^5 + 2184*a^13*b^6 + 3003*a^12*b^7 + 2002*a^11*b^8 - 1287*a^10*b^9 - 5148*a^9*b^10 - 7007*a^8*b^11 - 6006*a^7*b^12 - 3549*a^6*b^13 - 1456*a^5*b^14 - 399*a^4*b^15 - 66*a^3*b^16 - 5*a^2*b^17)*tan(1/2*f*x + 1/2*e)^2/(a^18*b^2 + 18*a^17*b^3 + 153*a^16*b^4 + 816*a^15*b^5 + 3060*a^14*b^6 + 8568*a^13*b^7 + 18564*a^12*b^8 + 31824*a^11*b^9 + 43758*a^10*b^10 + 48620*a^9*b^11 + 43758*a^8*b^12 + 31824*a^7*b^13 + 18564*a^6*b^14 + 8568*a^5*b^15 + 3060*a^4*b^16 + 816*a^3*b^17 + 153*a^2*b^18 + 18*a*b^19 + b^20) + 3*(4*a^17*b^2 + 55*a^16*b^3 + 342*a^15*b^4 + 1253*a^14*b^5 + 2912*a^13*b^6 + 4095*a^12*b^7 + 2002*a^11*b^8 - 5291*a^10*b^9 - 15444*a^9*b^10 - 22451*a^8*b^11 - 22022*a^7*b^12 - 15561*a^6*b^13 - 8008*a^5*b^14 - 2947*a^4*b^15 - 738*a^3*b^16 - 113*a^2*b^17 - 8*a*b^18)/(a^18*b^2 + 18*a^17*b^3 + 153*a^16*b^4 + 816*a^15*b^5 + 3060*a^14*b^6 + 8568*a^13*b^7 + 18564*a^12*b^8 + 31824*a^11*b^9 + 43758*a^10*b^10 + 48620*a^9*b^11 + 43758*a^8*b^12 + 31824*a^7*b^13 + 18564*a^6*b^14 + 8568*a^5*b^15 + 3060*a^4*b^16 + 816*a^3*b^17 + 153*a^2*b^18 + 18*a*b^19 + b^20))*tan(1/2*f*x + 1/2*e)^2 + 3*(4*a^17*b^2 + 55*a^16*b^3 + 342*a^15*b^4 + 1253*a^14*b^5 + 2912*a^13*b^6 + 4095*a^12*b^7 + 2002*a^11*b^8 - 5291*a^10*b^9 - 15444*a^9*b^10 - 22451*a^8*b^11 - 22022*a^7*b^12 - 15561*a^6*b^13 - 8008*a^5*b^14 - 2947*a^4*b^15 - 738*a^3*b^16 - 113*a^2*b^17 - 8*a*b^18)/(a^18*b^2 + 18*a^17*b^3 + 153*a^16*b^4 + 816*a^15*b^5 + 3060*a^14*b^6 + 8568*a^13*b^7 + 18564*a^12*b^8 + 31824*a^11*b^9 + 43758*a^10*b^10 + 48620*a^9*b^11 + 43758*a^8*b^12 + 31824*a^7*b^13 + 18564*a^6*b^14 + 8568*a^5*b^15 + 3060*a^4*b^16 + 816*a^3*b^17 + 153*a^2*b^18 + 18*a*b^19 + b^20))*tan(1/2*f*x + 1/2*e)^2 + (4*a^17*b^2 + 51*a^16*b^3 + 294*a^15*b^4 + 1001*a^14*b^5 + 2184*a^13*b^6 + 3003*a^12*b^7 + 2002*a^11*b^8 - 1287*a^10*b^9 - 5148*a^9*b^10 - 7007*a^8*b^11 - 6006*a^7*b^12 - 3549*a^6*b^13 - 1456*a^5*b^14 - 399*a^4*b^15 - 66*a^3*b^16 - 5*a^2*b^17)/(a^18*b^2 + 18*a^17*b^3 + 153*a^16*b^4 + 816*a^15*b^5 + 3060*a^14*b^6 + 8568*a^13*b^7 + 18564*a^12*b^8 + 31824*a^11*b^9 + 43758*a^10*b^10 + 48620*a^9*b^11 + 43758*a^8*b^12 + 31824*a^7*b^13 + 18564*a^6*b^14 + 8568*a^5*b^15 + 3060*a^4*b^16 + 816*a^3*b^17 + 153*a^2*b^18 + 18*a*b^19 + b^20))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) - 6*(8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a^2 - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*a*b - 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^7*b^2 - 56*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(5/2) - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*a^(3/2)*b - 29*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^6*sqrt(a)*b^2 - 120*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^3 - 296*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a^2*b + 43*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*a*b^2 - 12*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^5*b^3 + 136*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(7/2) - 168*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(5/2)*b - 629*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*a^(3/2)*b^2 + 60*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^4*sqrt(a)*b^3 + 344*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^4 + 872*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^3*b - 151*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2*b^2 - 1112*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b^3 - 48*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^4 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(9/2) + 616*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(7/2)*b + 1265*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2)*b^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b^3 - 880*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b^4 - 232*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^5 - 568*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^4*b + 113*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3*b^2 + 1124*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b^3 + 432*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^4 - 320*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^5 - 104*a^(11/2) - 440*a^(9/2)*b - 607*a^(7/2)*b^2 - 84*a^(5/2)*b^3 + 496*a^(3/2)*b^4 + 320*sqrt(a)*b^5)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^4))/f","B",0
533,1,1792,0,2.293085," ","integrate(tan(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\frac{\frac{3 \, {\left(2 \, a - 3 \, b\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{-a - b}} + \frac{2 \, {\left({\left({\left(\frac{{\left(2 \, a^{13} b^{2} + 21 \, a^{12} b^{3} + 99 \, a^{11} b^{4} + 275 \, a^{10} b^{5} + 495 \, a^{9} b^{6} + 594 \, a^{8} b^{7} + 462 \, a^{7} b^{8} + 198 \, a^{6} b^{9} - 55 \, a^{4} b^{11} - 33 \, a^{3} b^{12} - 9 \, a^{2} b^{13} - a b^{14}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{14} b^{2} + 14 \, a^{13} b^{3} + 91 \, a^{12} b^{4} + 364 \, a^{11} b^{5} + 1001 \, a^{10} b^{6} + 2002 \, a^{9} b^{7} + 3003 \, a^{8} b^{8} + 3432 \, a^{7} b^{9} + 3003 \, a^{6} b^{10} + 2002 \, a^{5} b^{11} + 1001 \, a^{4} b^{12} + 364 \, a^{3} b^{13} + 91 \, a^{2} b^{14} + 14 \, a b^{15} + b^{16}} + \frac{3 \, {\left(2 \, a^{13} b^{2} + 23 \, a^{12} b^{3} + 119 \, a^{11} b^{4} + 363 \, a^{10} b^{5} + 715 \, a^{9} b^{6} + 924 \, a^{8} b^{7} + 726 \, a^{7} b^{8} + 198 \, a^{6} b^{9} - 264 \, a^{5} b^{10} - 385 \, a^{4} b^{11} - 253 \, a^{3} b^{12} - 97 \, a^{2} b^{13} - 21 \, a b^{14} - 2 \, b^{15}\right)}}{a^{14} b^{2} + 14 \, a^{13} b^{3} + 91 \, a^{12} b^{4} + 364 \, a^{11} b^{5} + 1001 \, a^{10} b^{6} + 2002 \, a^{9} b^{7} + 3003 \, a^{8} b^{8} + 3432 \, a^{7} b^{9} + 3003 \, a^{6} b^{10} + 2002 \, a^{5} b^{11} + 1001 \, a^{4} b^{12} + 364 \, a^{3} b^{13} + 91 \, a^{2} b^{14} + 14 \, a b^{15} + b^{16}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{3 \, {\left(2 \, a^{13} b^{2} + 23 \, a^{12} b^{3} + 119 \, a^{11} b^{4} + 363 \, a^{10} b^{5} + 715 \, a^{9} b^{6} + 924 \, a^{8} b^{7} + 726 \, a^{7} b^{8} + 198 \, a^{6} b^{9} - 264 \, a^{5} b^{10} - 385 \, a^{4} b^{11} - 253 \, a^{3} b^{12} - 97 \, a^{2} b^{13} - 21 \, a b^{14} - 2 \, b^{15}\right)}}{a^{14} b^{2} + 14 \, a^{13} b^{3} + 91 \, a^{12} b^{4} + 364 \, a^{11} b^{5} + 1001 \, a^{10} b^{6} + 2002 \, a^{9} b^{7} + 3003 \, a^{8} b^{8} + 3432 \, a^{7} b^{9} + 3003 \, a^{6} b^{10} + 2002 \, a^{5} b^{11} + 1001 \, a^{4} b^{12} + 364 \, a^{3} b^{13} + 91 \, a^{2} b^{14} + 14 \, a b^{15} + b^{16}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{2 \, a^{13} b^{2} + 21 \, a^{12} b^{3} + 99 \, a^{11} b^{4} + 275 \, a^{10} b^{5} + 495 \, a^{9} b^{6} + 594 \, a^{8} b^{7} + 462 \, a^{7} b^{8} + 198 \, a^{6} b^{9} - 55 \, a^{4} b^{11} - 33 \, a^{3} b^{12} - 9 \, a^{2} b^{13} - a b^{14}}{a^{14} b^{2} + 14 \, a^{13} b^{3} + 91 \, a^{12} b^{4} + 364 \, a^{11} b^{5} + 1001 \, a^{10} b^{6} + 2002 \, a^{9} b^{7} + 3003 \, a^{8} b^{8} + 3432 \, a^{7} b^{9} + 3003 \, a^{6} b^{10} + 2002 \, a^{5} b^{11} + 1001 \, a^{4} b^{12} + 364 \, a^{3} b^{13} + 91 \, a^{2} b^{14} + 14 \, a b^{15} + b^{16}}\right)}}{{\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a\right)}^{\frac{3}{2}}} - \frac{6 \, {\left(2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a + {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} b - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} - {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b + 4 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} b^{2} - 2 \, a^{\frac{5}{2}} - 5 \, a^{\frac{3}{2}} b - 4 \, \sqrt{a} b^{2}\right)}}{{\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} - 3 \, a - 4 \, b\right)}^{2}}}{3 \, f}"," ",0,"1/3*(3*(2*a - 3*b)*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a - b)) + 2*((((2*a^13*b^2 + 21*a^12*b^3 + 99*a^11*b^4 + 275*a^10*b^5 + 495*a^9*b^6 + 594*a^8*b^7 + 462*a^7*b^8 + 198*a^6*b^9 - 55*a^4*b^11 - 33*a^3*b^12 - 9*a^2*b^13 - a*b^14)*tan(1/2*f*x + 1/2*e)^2/(a^14*b^2 + 14*a^13*b^3 + 91*a^12*b^4 + 364*a^11*b^5 + 1001*a^10*b^6 + 2002*a^9*b^7 + 3003*a^8*b^8 + 3432*a^7*b^9 + 3003*a^6*b^10 + 2002*a^5*b^11 + 1001*a^4*b^12 + 364*a^3*b^13 + 91*a^2*b^14 + 14*a*b^15 + b^16) + 3*(2*a^13*b^2 + 23*a^12*b^3 + 119*a^11*b^4 + 363*a^10*b^5 + 715*a^9*b^6 + 924*a^8*b^7 + 726*a^7*b^8 + 198*a^6*b^9 - 264*a^5*b^10 - 385*a^4*b^11 - 253*a^3*b^12 - 97*a^2*b^13 - 21*a*b^14 - 2*b^15)/(a^14*b^2 + 14*a^13*b^3 + 91*a^12*b^4 + 364*a^11*b^5 + 1001*a^10*b^6 + 2002*a^9*b^7 + 3003*a^8*b^8 + 3432*a^7*b^9 + 3003*a^6*b^10 + 2002*a^5*b^11 + 1001*a^4*b^12 + 364*a^3*b^13 + 91*a^2*b^14 + 14*a*b^15 + b^16))*tan(1/2*f*x + 1/2*e)^2 + 3*(2*a^13*b^2 + 23*a^12*b^3 + 119*a^11*b^4 + 363*a^10*b^5 + 715*a^9*b^6 + 924*a^8*b^7 + 726*a^7*b^8 + 198*a^6*b^9 - 264*a^5*b^10 - 385*a^4*b^11 - 253*a^3*b^12 - 97*a^2*b^13 - 21*a*b^14 - 2*b^15)/(a^14*b^2 + 14*a^13*b^3 + 91*a^12*b^4 + 364*a^11*b^5 + 1001*a^10*b^6 + 2002*a^9*b^7 + 3003*a^8*b^8 + 3432*a^7*b^9 + 3003*a^6*b^10 + 2002*a^5*b^11 + 1001*a^4*b^12 + 364*a^3*b^13 + 91*a^2*b^14 + 14*a*b^15 + b^16))*tan(1/2*f*x + 1/2*e)^2 + (2*a^13*b^2 + 21*a^12*b^3 + 99*a^11*b^4 + 275*a^10*b^5 + 495*a^9*b^6 + 594*a^8*b^7 + 462*a^7*b^8 + 198*a^6*b^9 - 55*a^4*b^11 - 33*a^3*b^12 - 9*a^2*b^13 - a*b^14)/(a^14*b^2 + 14*a^13*b^3 + 91*a^12*b^4 + 364*a^11*b^5 + 1001*a^10*b^6 + 2002*a^9*b^7 + 3003*a^8*b^8 + 3432*a^7*b^9 + 3003*a^6*b^10 + 2002*a^5*b^11 + 1001*a^4*b^12 + 364*a^3*b^13 + 91*a^2*b^14 + 14*a*b^15 + b^16))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) - 6*(2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a + (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2) + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a)*b - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2 - (sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b + 4*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*b^2 - 2*a^(5/2) - 5*a^(3/2)*b - 4*sqrt(a)*b^2)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a) - 3*a - 4*b)^2))/f","B",0
534,1,842,0,1.216811," ","integrate(tan(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{\frac{{\left({\left(\frac{{\left(4 \, a^{9} b^{2} + 33 \, a^{8} b^{3} + 120 \, a^{7} b^{4} + 252 \, a^{6} b^{5} + 336 \, a^{5} b^{6} + 294 \, a^{4} b^{7} + 168 \, a^{3} b^{8} + 60 \, a^{2} b^{9} + 12 \, a b^{10} + b^{11}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{10} b^{2} + 10 \, a^{9} b^{3} + 45 \, a^{8} b^{4} + 120 \, a^{7} b^{5} + 210 \, a^{6} b^{6} + 252 \, a^{5} b^{7} + 210 \, a^{4} b^{8} + 120 \, a^{3} b^{9} + 45 \, a^{2} b^{10} + 10 \, a b^{11} + b^{12}} + \frac{3 \, {\left(4 \, a^{9} b^{2} + 37 \, a^{8} b^{3} + 152 \, a^{7} b^{4} + 364 \, a^{6} b^{5} + 560 \, a^{5} b^{6} + 574 \, a^{4} b^{7} + 392 \, a^{3} b^{8} + 172 \, a^{2} b^{9} + 44 \, a b^{10} + 5 \, b^{11}\right)}}{a^{10} b^{2} + 10 \, a^{9} b^{3} + 45 \, a^{8} b^{4} + 120 \, a^{7} b^{5} + 210 \, a^{6} b^{6} + 252 \, a^{5} b^{7} + 210 \, a^{4} b^{8} + 120 \, a^{3} b^{9} + 45 \, a^{2} b^{10} + 10 \, a b^{11} + b^{12}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{3 \, {\left(4 \, a^{9} b^{2} + 37 \, a^{8} b^{3} + 152 \, a^{7} b^{4} + 364 \, a^{6} b^{5} + 560 \, a^{5} b^{6} + 574 \, a^{4} b^{7} + 392 \, a^{3} b^{8} + 172 \, a^{2} b^{9} + 44 \, a b^{10} + 5 \, b^{11}\right)}}{a^{10} b^{2} + 10 \, a^{9} b^{3} + 45 \, a^{8} b^{4} + 120 \, a^{7} b^{5} + 210 \, a^{6} b^{6} + 252 \, a^{5} b^{7} + 210 \, a^{4} b^{8} + 120 \, a^{3} b^{9} + 45 \, a^{2} b^{10} + 10 \, a b^{11} + b^{12}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{4 \, a^{9} b^{2} + 33 \, a^{8} b^{3} + 120 \, a^{7} b^{4} + 252 \, a^{6} b^{5} + 336 \, a^{5} b^{6} + 294 \, a^{4} b^{7} + 168 \, a^{3} b^{8} + 60 \, a^{2} b^{9} + 12 \, a b^{10} + b^{11}}{a^{10} b^{2} + 10 \, a^{9} b^{3} + 45 \, a^{8} b^{4} + 120 \, a^{7} b^{5} + 210 \, a^{6} b^{6} + 252 \, a^{5} b^{7} + 210 \, a^{4} b^{8} + 120 \, a^{3} b^{9} + 45 \, a^{2} b^{10} + 10 \, a b^{11} + b^{12}}}{{\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a\right)}^{\frac{3}{2}}} + \frac{6 \, \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a} - \sqrt{a}}{2 \, \sqrt{-a - b}}\right)}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a - b}}}{3 \, f}"," ",0,"-1/3*(((((4*a^9*b^2 + 33*a^8*b^3 + 120*a^7*b^4 + 252*a^6*b^5 + 336*a^5*b^6 + 294*a^4*b^7 + 168*a^3*b^8 + 60*a^2*b^9 + 12*a*b^10 + b^11)*tan(1/2*f*x + 1/2*e)^2/(a^10*b^2 + 10*a^9*b^3 + 45*a^8*b^4 + 120*a^7*b^5 + 210*a^6*b^6 + 252*a^5*b^7 + 210*a^4*b^8 + 120*a^3*b^9 + 45*a^2*b^10 + 10*a*b^11 + b^12) + 3*(4*a^9*b^2 + 37*a^8*b^3 + 152*a^7*b^4 + 364*a^6*b^5 + 560*a^5*b^6 + 574*a^4*b^7 + 392*a^3*b^8 + 172*a^2*b^9 + 44*a*b^10 + 5*b^11)/(a^10*b^2 + 10*a^9*b^3 + 45*a^8*b^4 + 120*a^7*b^5 + 210*a^6*b^6 + 252*a^5*b^7 + 210*a^4*b^8 + 120*a^3*b^9 + 45*a^2*b^10 + 10*a*b^11 + b^12))*tan(1/2*f*x + 1/2*e)^2 + 3*(4*a^9*b^2 + 37*a^8*b^3 + 152*a^7*b^4 + 364*a^6*b^5 + 560*a^5*b^6 + 574*a^4*b^7 + 392*a^3*b^8 + 172*a^2*b^9 + 44*a*b^10 + 5*b^11)/(a^10*b^2 + 10*a^9*b^3 + 45*a^8*b^4 + 120*a^7*b^5 + 210*a^6*b^6 + 252*a^5*b^7 + 210*a^4*b^8 + 120*a^3*b^9 + 45*a^2*b^10 + 10*a*b^11 + b^12))*tan(1/2*f*x + 1/2*e)^2 + (4*a^9*b^2 + 33*a^8*b^3 + 120*a^7*b^4 + 252*a^6*b^5 + 336*a^5*b^6 + 294*a^4*b^7 + 168*a^3*b^8 + 60*a^2*b^9 + 12*a*b^10 + b^11)/(a^10*b^2 + 10*a^9*b^3 + 45*a^8*b^4 + 120*a^7*b^5 + 210*a^6*b^6 + 252*a^5*b^7 + 210*a^4*b^8 + 120*a^3*b^9 + 45*a^2*b^10 + 10*a*b^11 + b^12))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) + 6*arctan(-1/2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a) - sqrt(a))/sqrt(-a - b))/((a^2 + 2*a*b + b^2)*sqrt(-a - b)))/f","B",0
535,1,74,0,0.183092," ","integrate(cot(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{\sqrt{b \sin\left(f x + e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a^{2} f} + \frac{3 \, b \sin\left(f x + e\right)^{2} + 4 \, a}{3 \, {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{3}{2}} a^{2} f}"," ",0,"arctan(sqrt(b*sin(f*x + e)^2 + a)/sqrt(-a))/(sqrt(-a)*a^2*f) + 1/3*(3*b*sin(f*x + e)^2 + 4*a)/((b*sin(f*x + e)^2 + a)^(3/2)*a^2*f)","A",0
536,1,751,0,1.355130," ","integrate(cot(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{\frac{{\left({\left({\left(\frac{3 \, {\left(a^{12} b^{2} + 2 \, a^{11} b^{3} + a^{10} b^{4}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{13} b^{2} + 2 \, a^{12} b^{3} + a^{11} b^{4}} + \frac{4 \, {\left(11 \, a^{12} b^{2} + 42 \, a^{11} b^{3} + 51 \, a^{10} b^{4} + 20 \, a^{9} b^{5}\right)}}{a^{13} b^{2} + 2 \, a^{12} b^{3} + a^{11} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{6 \, {\left(19 \, a^{12} b^{2} + 90 \, a^{11} b^{3} + 163 \, a^{10} b^{4} + 132 \, a^{9} b^{5} + 40 \, a^{8} b^{6}\right)}}{a^{13} b^{2} + 2 \, a^{12} b^{3} + a^{11} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{12 \, {\left(9 \, a^{12} b^{2} + 42 \, a^{11} b^{3} + 73 \, a^{10} b^{4} + 56 \, a^{9} b^{5} + 16 \, a^{8} b^{6}\right)}}{a^{13} b^{2} + 2 \, a^{12} b^{3} + a^{11} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + \frac{7 \, {\left(5 \, a^{12} b^{2} + 18 \, a^{11} b^{3} + 21 \, a^{10} b^{4} + 8 \, a^{9} b^{5}\right)}}{a^{13} b^{2} + 2 \, a^{12} b^{3} + a^{11} b^{4}}}{{\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a\right)}^{\frac{3}{2}}} + \frac{6 \, {\left(2 \, a^{\frac{3}{2}} + 5 \, \sqrt{a} b\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{4}} - \frac{6 \, {\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{\frac{3}{2}} + 2 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} \sqrt{a} b + a^{2}\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} \sqrt{a} - a^{\frac{3}{2}}\right)} a^{3}}}{24 \, f}"," ",0,"-1/24*(((((3*(a^12*b^2 + 2*a^11*b^3 + a^10*b^4)*tan(1/2*f*x + 1/2*e)^2/(a^13*b^2 + 2*a^12*b^3 + a^11*b^4) + 4*(11*a^12*b^2 + 42*a^11*b^3 + 51*a^10*b^4 + 20*a^9*b^5)/(a^13*b^2 + 2*a^12*b^3 + a^11*b^4))*tan(1/2*f*x + 1/2*e)^2 + 6*(19*a^12*b^2 + 90*a^11*b^3 + 163*a^10*b^4 + 132*a^9*b^5 + 40*a^8*b^6)/(a^13*b^2 + 2*a^12*b^3 + a^11*b^4))*tan(1/2*f*x + 1/2*e)^2 + 12*(9*a^12*b^2 + 42*a^11*b^3 + 73*a^10*b^4 + 56*a^9*b^5 + 16*a^8*b^6)/(a^13*b^2 + 2*a^12*b^3 + a^11*b^4))*tan(1/2*f*x + 1/2*e)^2 + 7*(5*a^12*b^2 + 18*a^11*b^3 + 21*a^10*b^4 + 8*a^9*b^5)/(a^13*b^2 + 2*a^12*b^3 + a^11*b^4))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) + 6*(2*a^(3/2) + 5*sqrt(a)*b)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^4 - 6*((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^(3/2) + 2*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*sqrt(a)*b + a^2)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*sqrt(a) - a^(3/2))*a^3))/f","B",0
537,1,1411,0,1.993316," ","integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","-\frac{\frac{{\left({\left({\left(3 \, {\left(\frac{{\left(a^{17} b^{2} + 2 \, a^{16} b^{3} + a^{15} b^{4}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2}}{a^{18} b^{2} + 2 \, a^{17} b^{3} + a^{16} b^{4}} - \frac{9 \, a^{17} b^{2} + 32 \, a^{16} b^{3} + 37 \, a^{15} b^{4} + 14 \, a^{14} b^{5}}{a^{18} b^{2} + 2 \, a^{17} b^{3} + a^{16} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{2 \, {\left(197 \, a^{17} b^{2} + 1106 \, a^{16} b^{3} + 2181 \, a^{15} b^{4} + 1832 \, a^{14} b^{5} + 560 \, a^{13} b^{6}\right)}}{a^{18} b^{2} + 2 \, a^{17} b^{3} + a^{16} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{6 \, {\left(165 \, a^{17} b^{2} + 1072 \, a^{16} b^{3} + 2761 \, a^{15} b^{4} + 3526 \, a^{14} b^{5} + 2232 \, a^{13} b^{6} + 560 \, a^{12} b^{7}\right)}}{a^{18} b^{2} + 2 \, a^{17} b^{3} + a^{16} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{3 \, {\left(307 \, a^{17} b^{2} + 1958 \, a^{16} b^{3} + 4835 \, a^{15} b^{4} + 5792 \, a^{14} b^{5} + 3376 \, a^{13} b^{6} + 768 \, a^{12} b^{7}\right)}}{a^{18} b^{2} + 2 \, a^{17} b^{3} + a^{16} b^{4}}\right)} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \frac{295 \, a^{17} b^{2} + 1552 \, a^{16} b^{3} + 2859 \, a^{15} b^{4} + 2242 \, a^{14} b^{5} + 640 \, a^{13} b^{6}}{a^{18} b^{2} + 2 \, a^{17} b^{3} + a^{16} b^{4}}}{{\left(a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a\right)}^{\frac{3}{2}}} - \frac{24 \, {\left(8 \, a^{2} + 40 \, a b + 35 \, b^{2}\right)} \arctan\left(-\frac{\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} a^{4}} - \frac{12 \, {\left(8 \, a^{\frac{5}{2}} + 40 \, a^{\frac{3}{2}} b + 35 \, \sqrt{a} b^{2}\right)} \log\left({\left| -{\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a - a^{\frac{3}{2}} - 2 \, \sqrt{a} b \right|}\right)}{a^{5}} + \frac{12 \, {\left(6 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a^{2} + 24 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} a b + 22 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{3} b^{2} + 5 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{5}{2}} + 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} a^{\frac{3}{2}} b - 8 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{3} - 28 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a^{2} b - 26 \, {\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)} a b^{2} - 7 \, a^{\frac{7}{2}} - 12 \, a^{\frac{5}{2}} b\right)}}{{\left({\left(\sqrt{a} \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} - \sqrt{a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{4} + 2 \, a \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + 4 \, b \tan\left(\frac{1}{2} \, f x + \frac{1}{2} \, e\right)^{2} + a}\right)}^{2} - a\right)}^{2} a^{4}}}{192 \, f}"," ",0,"-1/192*(((((3*((a^17*b^2 + 2*a^16*b^3 + a^15*b^4)*tan(1/2*f*x + 1/2*e)^2/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4) - (9*a^17*b^2 + 32*a^16*b^3 + 37*a^15*b^4 + 14*a^14*b^5)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(1/2*f*x + 1/2*e)^2 - 2*(197*a^17*b^2 + 1106*a^16*b^3 + 2181*a^15*b^4 + 1832*a^14*b^5 + 560*a^13*b^6)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(1/2*f*x + 1/2*e)^2 - 6*(165*a^17*b^2 + 1072*a^16*b^3 + 2761*a^15*b^4 + 3526*a^14*b^5 + 2232*a^13*b^6 + 560*a^12*b^7)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(1/2*f*x + 1/2*e)^2 - 3*(307*a^17*b^2 + 1958*a^16*b^3 + 4835*a^15*b^4 + 5792*a^14*b^5 + 3376*a^13*b^6 + 768*a^12*b^7)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))*tan(1/2*f*x + 1/2*e)^2 - (295*a^17*b^2 + 1552*a^16*b^3 + 2859*a^15*b^4 + 2242*a^14*b^5 + 640*a^13*b^6)/(a^18*b^2 + 2*a^17*b^3 + a^16*b^4))/(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a)^(3/2) - 24*(8*a^2 + 40*a*b + 35*b^2)*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))/(sqrt(-a)*a^4) - 12*(8*a^(5/2) + 40*a^(3/2)*b + 35*sqrt(a)*b^2)*log(abs(-(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a - a^(3/2) - 2*sqrt(a)*b))/a^5 + 12*(6*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a^2 + 24*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*a*b + 22*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^3*b^2 + 5*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(5/2) + 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2*a^(3/2)*b - 8*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^3 - 28*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a^2*b - 26*(sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))*a*b^2 - 7*a^(7/2) - 12*a^(5/2)*b)/(((sqrt(a)*tan(1/2*f*x + 1/2*e)^2 - sqrt(a*tan(1/2*f*x + 1/2*e)^4 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 4*b*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^2*a^4))/f","B",0
538,0,0,0,0.000000," ","integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\tan\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(tan(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
539,0,0,0,0.000000," ","integrate(tan(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\tan\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(tan(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
540,0,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{1}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^(-5/2), x)","F",0
541,0,0,0,0.000000," ","integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\cot\left(f x + e\right)^{2}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(cot(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
542,0,0,0,0.000000," ","integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""giac"")","\int \frac{\cot\left(f x + e\right)^{4}}{{\left(b \sin\left(f x + e\right)^{2} + a\right)}^{\frac{5}{2}}}\,{d x}"," ",0,"integrate(cot(f*x + e)^4/(b*sin(f*x + e)^2 + a)^(5/2), x)","F",0
543,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)^p*(d*tan(f*x+e))^m,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \left(d \tan\left(f x + e\right)\right)^{m}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*(d*tan(f*x + e))^m, x)","F",0
544,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c)^3,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \tan\left(d x + c\right)^{3}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c)^3, x)","F",0
545,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \tan\left(d x + c\right)\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c), x)","F",0
546,0,0,0,0.000000," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \cot\left(d x + c\right)\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c), x)","F",0
547,0,0,0,0.000000," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \cot\left(d x + c\right)^{3}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c)^3, x)","F",0
548,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c)^4,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \tan\left(d x + c\right)^{4}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c)^4, x)","F",0
549,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c)^2,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \tan\left(d x + c\right)^{2}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c)^2, x)","F",0
550,0,0,0,0.000000," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \cot\left(d x + c\right)^{2}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c)^2, x)","F",0
551,0,0,0,0.000000," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{2} + a\right)}^{p} \cot\left(d x + c\right)^{4}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c)^4, x)","F",0
552,1,144,0,0.165872," ","integrate(cot(x)^3/(a+b*sin(x)^3),x, algorithm=""giac"")","\frac{b \left(-\frac{a}{b}\right)^{\frac{1}{3}} \log\left({\left| -\left(-\frac{a}{b}\right)^{\frac{1}{3}} + \sin\left(x\right) \right|}\right)}{3 \, a^{2}} + \frac{\log\left({\left| b \sin\left(x\right)^{3} + a \right|}\right)}{3 \, a} - \frac{\log\left({\left| \sin\left(x\right) \right|}\right)}{a} - \frac{\sqrt{3} \left(-a b^{2}\right)^{\frac{1}{3}} \arctan\left(\frac{\sqrt{3} {\left(\left(-\frac{a}{b}\right)^{\frac{1}{3}} + 2 \, \sin\left(x\right)\right)}}{3 \, \left(-\frac{a}{b}\right)^{\frac{1}{3}}}\right)}{3 \, a^{2}} - \frac{\left(-a b^{2}\right)^{\frac{1}{3}} \log\left(\sin\left(x\right)^{2} + \left(-\frac{a}{b}\right)^{\frac{1}{3}} \sin\left(x\right) + \left(-\frac{a}{b}\right)^{\frac{2}{3}}\right)}{6 \, a^{2}} - \frac{1}{2 \, a \sin\left(x\right)^{2}}"," ",0,"1/3*b*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(x)))/a^2 + 1/3*log(abs(b*sin(x)^3 + a))/a - log(abs(sin(x)))/a - 1/3*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(x))/(-a/b)^(1/3))/a^2 - 1/6*(-a*b^2)^(1/3)*log(sin(x)^2 + (-a/b)^(1/3)*sin(x) + (-a/b)^(2/3))/a^2 - 1/2/(a*sin(x)^2)","A",0
553,1,38,0,0.146240," ","integrate(cot(x)*(a+b*sin(x)^3)^(1/2),x, algorithm=""giac"")","\frac{2 \, a \arctan\left(\frac{\sqrt{b \sin\left(x\right)^{3} + a}}{\sqrt{-a}}\right)}{3 \, \sqrt{-a}} + \frac{2}{3} \, \sqrt{b \sin\left(x\right)^{3} + a}"," ",0,"2/3*a*arctan(sqrt(b*sin(x)^3 + a)/sqrt(-a))/sqrt(-a) + 2/3*sqrt(b*sin(x)^3 + a)","A",0
554,1,24,0,0.119510," ","integrate(cot(x)/(a+b*sin(x)^3)^(1/2),x, algorithm=""giac"")","\frac{2 \, \arctan\left(\frac{\sqrt{b \sin\left(x\right)^{3} + a}}{\sqrt{-a}}\right)}{3 \, \sqrt{-a}}"," ",0,"2/3*arctan(sqrt(b*sin(x)^3 + a)/sqrt(-a))/sqrt(-a)","A",0
555,-1,0,0,0.000000," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
556,0,0,0,0.000000," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\tan\left(d x + c\right)^{3}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(tan(d*x + c)^3/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
557,0,0,0,0.000000," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\tan\left(d x + c\right)}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(tan(d*x + c)/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
558,1,31,0,0.617153," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\frac{\arctan\left(\frac{\sqrt{b \sin\left(d x + c\right)^{4} + a}}{\sqrt{-a}}\right)}{2 \, \sqrt{-a} d}"," ",0,"1/2*arctan(sqrt(b*sin(d*x + c)^4 + a)/sqrt(-a))/(sqrt(-a)*d)","A",0
559,-2,0,0,0.000000," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Evaluation time: 3.09Not invertible Error: Bad Argument Value","F(-2)",0
560,0,0,0,0.000000," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\cot\left(d x + c\right)^{5}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(cot(d*x + c)^5/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
561,0,0,0,0.000000," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\tan\left(d x + c\right)^{2}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(tan(d*x + c)^2/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
562,0,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(1/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
563,0,0,0,0.000000," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""giac"")","\int \frac{\cot\left(d x + c\right)^{2}}{\sqrt{b \sin\left(d x + c\right)^{4} + a}}\,{d x}"," ",0,"integrate(cot(d*x + c)^2/sqrt(b*sin(d*x + c)^4 + a), x)","F",0
564,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \tan\left(d x + c\right)^{m}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c)^m, x)","F",0
565,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^3,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \tan\left(d x + c\right)^{3}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c)^3, x)","F",0
566,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c),x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \tan\left(d x + c\right)\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c), x)","F",0
567,0,0,0,0.000000," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \cot\left(d x + c\right)\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*cot(d*x + c), x)","F",0
568,0,0,0,0.000000," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \cot\left(d x + c\right)^{3}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*cot(d*x + c)^3, x)","F",0
569,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^4,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \tan\left(d x + c\right)^{4}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c)^4, x)","F",0
570,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^2,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \tan\left(d x + c\right)^{2}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c)^2, x)","F",0
571,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p, x)","F",0
572,0,0,0,0.000000," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \cot\left(d x + c\right)^{2}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*cot(d*x + c)^2, x)","F",0
573,0,0,0,0.000000," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^4)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{4} + a\right)}^{p} \cot\left(d x + c\right)^{4}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^4 + a)^p*cot(d*x + c)^4, x)","F",0
574,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^3*tan(d*x+c)^m,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{3} \tan\left(d x + c\right)^{m}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^3*tan(d*x + c)^m, x)","F",0
575,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^2*tan(d*x+c)^m,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{2} \tan\left(d x + c\right)^{m}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^2*tan(d*x + c)^m, x)","F",0
576,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)*tan(d*x+c)^m,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)} \tan\left(d x + c\right)^{m}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)*tan(d*x + c)^m, x)","F",0
577,0,0,0,0.000000," ","integrate(tan(d*x+c)^m/(a+b*sin(d*x+c)^n),x, algorithm=""giac"")","\int \frac{\tan\left(d x + c\right)^{m}}{b \sin\left(d x + c\right)^{n} + a}\,{d x}"," ",0,"integrate(tan(d*x + c)^m/(b*sin(d*x + c)^n + a), x)","F",0
578,0,0,0,0.000000," ","integrate(tan(d*x+c)^m/(a+b*sin(d*x+c)^n)^2,x, algorithm=""giac"")","\int \frac{\tan\left(d x + c\right)^{m}}{{\left(b \sin\left(d x + c\right)^{n} + a\right)}^{2}}\,{d x}"," ",0,"integrate(tan(d*x + c)^m/(b*sin(d*x + c)^n + a)^2, x)","F",0
579,1,46,0,0.150804," ","integrate(cot(x)*(a+b*sin(x)^n)^(1/2),x, algorithm=""giac"")","\frac{2 \, {\left(\frac{a b \arctan\left(\frac{\sqrt{b \sin\left(x\right)^{n} + a}}{\sqrt{-a}}\right)}{\sqrt{-a}} + \sqrt{b \sin\left(x\right)^{n} + a} b\right)}}{b n}"," ",0,"2*(a*b*arctan(sqrt(b*sin(x)^n + a)/sqrt(-a))/sqrt(-a) + sqrt(b*sin(x)^n + a)*b)/(b*n)","A",0
580,1,27,0,0.134259," ","integrate(cot(x)/(a+b*sin(x)^n)^(1/2),x, algorithm=""giac"")","\frac{2 \, \arctan\left(\frac{\sqrt{b \sin\left(x\right)^{n} + a}}{\sqrt{-a}}\right)}{\sqrt{-a} n}"," ",0,"2*arctan(sqrt(b*sin(x)^n + a)/sqrt(-a))/(sqrt(-a)*n)","A",0
581,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{m}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^m, x)","F",0
582,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^3,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{3}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^3, x)","F",0
583,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c), x)","F",0
584,0,0,0,0.000000," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*cot(d*x + c), x)","F",0
585,0,0,0,0.000000," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)^{3}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*cot(d*x + c)^3, x)","F",0
586,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^4,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{4}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^4, x)","F",0
587,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^2,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{2}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^2, x)","F",0
588,0,0,0,0.000000," ","integrate((a+b*sin(d*x+c)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p, x)","F",0
589,0,0,0,0.000000," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)^{2}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*cot(d*x + c)^2, x)","F",0
590,0,0,0,0.000000," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^n)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)^{4}\,{d x}"," ",0,"integrate((b*sin(d*x + c)^n + a)^p*cot(d*x + c)^4, x)","F",0
591,0,0,0,0.000000," ","integrate((a+b*sin(f*x+e)^2)/(g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm=""giac"")","\int \frac{b \sin\left(f x + e\right)^{2} + a}{\left(g \cos\left(f x + e\right)\right)^{\frac{5}{2}} \sqrt{d \sin\left(f x + e\right)}}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)/((g*cos(f*x + e))^(5/2)*sqrt(d*sin(f*x + e))), x)","F",0
592,0,0,0,0.000000," ","integrate((c*cos(f*x+e))^m*(d*sin(f*x+e))^n*(a+b*sin(f*x+e)^2)^p,x, algorithm=""giac"")","\int {\left(b \sin\left(f x + e\right)^{2} + a\right)}^{p} \left(c \cos\left(f x + e\right)\right)^{m} \left(d \sin\left(f x + e\right)\right)^{n}\,{d x}"," ",0,"integrate((b*sin(f*x + e)^2 + a)^p*(c*cos(f*x + e))^m*(d*sin(f*x + e))^n, x)","F",0
593,0,0,0,0.000000," ","integrate((a+(c*cos(f*x+e)+b*sin(f*x+e))^2)^(1/2),x, algorithm=""giac"")","\int \sqrt{{\left(c \cos\left(f x + e\right) + b \sin\left(f x + e\right)\right)}^{2} + a}\,{d x}"," ",0,"integrate(sqrt((c*cos(f*x + e) + b*sin(f*x + e))^2 + a), x)","F",0
594,0,0,0,0.000000," ","integrate(1/(a+(c*cos(f*x+e)+b*sin(f*x+e))^2)^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{{\left(c \cos\left(f x + e\right) + b \sin\left(f x + e\right)\right)}^{2} + a}}\,{d x}"," ",0,"integrate(1/sqrt((c*cos(f*x + e) + b*sin(f*x + e))^2 + a), x)","F",0
