1,1,103,27,0.069000," ","int(tan(x)^4/(a+a*cos(x)),x)","-\frac{1}{3 a \left(\tan \left(\frac{x}{2}\right)-1\right)^{3}}-\frac{1}{a \left(\tan \left(\frac{x}{2}\right)-1\right)^{2}}-\frac{1}{2 a \left(\tan \left(\frac{x}{2}\right)-1\right)}-\frac{\ln \left(\tan \left(\frac{x}{2}\right)-1\right)}{2 a}-\frac{1}{3 a \left(\tan \left(\frac{x}{2}\right)+1\right)^{3}}+\frac{1}{a \left(\tan \left(\frac{x}{2}\right)+1\right)^{2}}-\frac{1}{2 a \left(\tan \left(\frac{x}{2}\right)+1\right)}+\frac{\ln \left(\tan \left(\frac{x}{2}\right)+1\right)}{2 a}"," ",0,"-1/3/a/(tan(1/2*x)-1)^3-1/a/(tan(1/2*x)-1)^2-1/2/a/(tan(1/2*x)-1)-1/2/a*ln(tan(1/2*x)-1)-1/3/a/(tan(1/2*x)+1)^3+1/a/(tan(1/2*x)+1)^2-1/2/a/(tan(1/2*x)+1)+1/2/a*ln(tan(1/2*x)+1)","B"
2,1,18,17,0.051000," ","int(tan(x)^3/(a+a*cos(x)),x)","\frac{\frac{1}{2 \cos \left(x \right)^{2}}-\frac{1}{\cos \left(x \right)}}{a}"," ",0,"1/a*(1/2/cos(x)^2-1/cos(x))","A"
3,1,51,15,0.063000," ","int(tan(x)^2/(a+a*cos(x)),x)","-\frac{1}{a \left(\tan \left(\frac{x}{2}\right)-1\right)}+\frac{\ln \left(\tan \left(\frac{x}{2}\right)-1\right)}{a}-\frac{1}{a \left(\tan \left(\frac{x}{2}\right)+1\right)}-\frac{\ln \left(\tan \left(\frac{x}{2}\right)+1\right)}{a}"," ",0,"-1/a/(tan(1/2*x)-1)+1/a*ln(tan(1/2*x)-1)-1/a/(tan(1/2*x)+1)-1/a*ln(tan(1/2*x)+1)","B"
4,1,19,18,0.048000," ","int(tan(x)/(a+a*cos(x)),x)","-\frac{\ln \left(\cos \left(x \right)\right)}{a}+\frac{\ln \left(\cos \left(x \right)+1\right)}{a}"," ",0,"-ln(cos(x))/a+ln(cos(x)+1)/a","A"
5,1,33,27,0.057000," ","int(cot(x)/(a+a*cos(x)),x)","\frac{\ln \left(-1+\cos \left(x \right)\right)}{4 a}-\frac{1}{2 a \left(\cos \left(x \right)+1\right)}-\frac{\ln \left(\cos \left(x \right)+1\right)}{4 a}"," ",0,"1/4/a*ln(-1+cos(x))-1/2/a/(cos(x)+1)-1/4*ln(cos(x)+1)/a","A"
6,1,29,26,0.046000," ","int(cot(x)^2/(a+a*cos(x)),x)","\frac{\frac{\left(\tan^{3}\left(\frac{x}{2}\right)\right)}{3}-2 \tan \left(\frac{x}{2}\right)-\frac{1}{\tan \left(\frac{x}{2}\right)}}{4 a}"," ",0,"1/4/a*(1/3*tan(1/2*x)^3-2*tan(1/2*x)-1/tan(1/2*x))","A"
7,1,55,38,0.055000," ","int(cot(x)^3/(a+a*cos(x)),x)","\frac{1}{8 a \left(-1+\cos \left(x \right)\right)}-\frac{3 \ln \left(-1+\cos \left(x \right)\right)}{16 a}-\frac{1}{8 a \left(\cos \left(x \right)+1\right)^{2}}+\frac{1}{2 a \left(\cos \left(x \right)+1\right)}+\frac{3 \ln \left(\cos \left(x \right)+1\right)}{16 a}"," ",0,"1/8/a/(-1+cos(x))-3/16/a*ln(-1+cos(x))-1/8/a/(cos(x)+1)^2+1/2/a/(cos(x)+1)+3/16*ln(cos(x)+1)/a","A"
8,1,45,34,0.055000," ","int(cot(x)^4/(a+a*cos(x)),x)","\frac{\frac{\left(\tan^{5}\left(\frac{x}{2}\right)\right)}{5}-\frac{4 \left(\tan^{3}\left(\frac{x}{2}\right)\right)}{3}+6 \tan \left(\frac{x}{2}\right)-\frac{1}{3 \tan \left(\frac{x}{2}\right)^{3}}+\frac{4}{\tan \left(\frac{x}{2}\right)}}{16 a}"," ",0,"1/16/a*(1/5*tan(1/2*x)^5-4/3*tan(1/2*x)^3+6*tan(1/2*x)-1/3/tan(1/2*x)^3+4/tan(1/2*x))","A"
9,1,28,27,0.052000," ","int(tan(3*x)/(1+cos(3*x))^2,x)","-\frac{1}{3 \left(1+\cos \left(3 x \right)\right)}-\frac{\ln \left(\cos \left(3 x \right)\right)}{3}+\frac{\ln \left(1+\cos \left(3 x \right)\right)}{3}"," ",0,"-1/3/(1+cos(3*x))-1/3*ln(cos(3*x))+1/3*ln(1+cos(3*x))","A"
10,1,338,95,0.058000," ","int(tan(x)^4/(a+b*cos(x)),x)","\frac{2 \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}-\frac{4 \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right) b^{2}}{a^{2} \sqrt{\left(a -b \right) \left(a +b \right)}}+\frac{2 \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right) b^{4}}{a^{4} \sqrt{\left(a -b \right) \left(a +b \right)}}-\frac{1}{3 a \left(\tan \left(\frac{x}{2}\right)-1\right)^{3}}-\frac{1}{2 a \left(\tan \left(\frac{x}{2}\right)-1\right)^{2}}-\frac{b}{2 a^{2} \left(\tan \left(\frac{x}{2}\right)-1\right)^{2}}+\frac{1}{a \left(\tan \left(\frac{x}{2}\right)-1\right)}-\frac{b}{2 a^{2} \left(\tan \left(\frac{x}{2}\right)-1\right)}-\frac{b^{2}}{a^{3} \left(\tan \left(\frac{x}{2}\right)-1\right)}-\frac{3 b \ln \left(\tan \left(\frac{x}{2}\right)-1\right)}{2 a^{2}}+\frac{b^{3} \ln \left(\tan \left(\frac{x}{2}\right)-1\right)}{a^{4}}-\frac{1}{3 a \left(\tan \left(\frac{x}{2}\right)+1\right)^{3}}+\frac{1}{2 a \left(\tan \left(\frac{x}{2}\right)+1\right)^{2}}+\frac{b}{2 a^{2} \left(\tan \left(\frac{x}{2}\right)+1\right)^{2}}+\frac{1}{a \left(\tan \left(\frac{x}{2}\right)+1\right)}-\frac{b}{2 a^{2} \left(\tan \left(\frac{x}{2}\right)+1\right)}-\frac{b^{2}}{a^{3} \left(\tan \left(\frac{x}{2}\right)+1\right)}+\frac{3 b \ln \left(\tan \left(\frac{x}{2}\right)+1\right)}{2 a^{2}}-\frac{b^{3} \ln \left(\tan \left(\frac{x}{2}\right)+1\right)}{a^{4}}"," ",0,"2/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))-4/a^2/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))*b^2+2/a^4/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))*b^4-1/3/a/(tan(1/2*x)-1)^3-1/2/a/(tan(1/2*x)-1)^2-1/2/a^2/(tan(1/2*x)-1)^2*b+1/a/(tan(1/2*x)-1)-1/2/a^2/(tan(1/2*x)-1)*b-1/a^3/(tan(1/2*x)-1)*b^2-3/2*b/a^2*ln(tan(1/2*x)-1)+b^3/a^4*ln(tan(1/2*x)-1)-1/3/a/(tan(1/2*x)+1)^3+1/2/a/(tan(1/2*x)+1)^2+1/2/a^2/(tan(1/2*x)+1)^2*b+1/a/(tan(1/2*x)+1)-1/2/a^2/(tan(1/2*x)+1)*b-1/a^3/(tan(1/2*x)+1)*b^2+3/2*b/a^2*ln(tan(1/2*x)+1)-b^3/a^4*ln(tan(1/2*x)+1)","B"
11,1,65,55,0.053000," ","int(tan(x)^3/(a+b*cos(x)),x)","-\frac{\ln \left(a +b \cos \left(x \right)\right)}{a}+\frac{\ln \left(a +b \cos \left(x \right)\right) b^{2}}{a^{3}}-\frac{b}{a^{2} \cos \left(x \right)}+\frac{\ln \left(\cos \left(x \right)\right)}{a}-\frac{\ln \left(\cos \left(x \right)\right) b^{2}}{a^{3}}+\frac{1}{2 a \cos \left(x \right)^{2}}"," ",0,"-ln(a+b*cos(x))/a+1/a^3*ln(a+b*cos(x))*b^2-1/a^2*b/cos(x)+ln(cos(x))/a-1/a^3*ln(cos(x))*b^2+1/2/a/cos(x)^2","A"
12,1,129,51,0.050000," ","int(tan(x)^2/(a+b*cos(x)),x)","-\frac{2 \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}+\frac{2 \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right) b^{2}}{a^{2} \sqrt{\left(a -b \right) \left(a +b \right)}}-\frac{1}{a \left(\tan \left(\frac{x}{2}\right)-1\right)}+\frac{b \ln \left(\tan \left(\frac{x}{2}\right)-1\right)}{a^{2}}-\frac{1}{a \left(\tan \left(\frac{x}{2}\right)+1\right)}-\frac{b \ln \left(\tan \left(\frac{x}{2}\right)+1\right)}{a^{2}}"," ",0,"-2/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))+2/a^2/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))*b^2-1/a/(tan(1/2*x)-1)+b/a^2*ln(tan(1/2*x)-1)-1/a/(tan(1/2*x)+1)-b/a^2*ln(tan(1/2*x)+1)","B"
13,1,21,20,0.039000," ","int(tan(x)/(a+b*cos(x)),x)","-\frac{\ln \left(\cos \left(x \right)\right)}{a}+\frac{\ln \left(a +b \cos \left(x \right)\right)}{a}"," ",0,"-ln(cos(x))/a+ln(a+b*cos(x))/a","A"
14,1,54,50,0.048000," ","int(cot(x)/(a+b*cos(x)),x)","-\frac{a \ln \left(a +b \cos \left(x \right)\right)}{\left(a +b \right) \left(a -b \right)}+\frac{\ln \left(-1+\cos \left(x \right)\right)}{2 a +2 b}+\frac{\ln \left(\cos \left(x \right)+1\right)}{2 a -2 b}"," ",0,"-a/(a+b)/(a-b)*ln(a+b*cos(x))+1/(2*a+2*b)*ln(-1+cos(x))+1/(2*a-2*b)*ln(cos(x)+1)","A"
15,1,78,67,0.056000," ","int(cot(x)^2/(a+b*cos(x)),x)","\frac{\tan \left(\frac{x}{2}\right)}{2 a -2 b}-\frac{2 a^{2} \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right)}{\left(a -b \right) \left(a +b \right) \sqrt{\left(a -b \right) \left(a +b \right)}}-\frac{1}{2 \left(a +b \right) \tan \left(\frac{x}{2}\right)}"," ",0,"1/2/(a-b)*tan(1/2*x)-2/(a-b)/(a+b)*a^2/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))-1/2/(a+b)/tan(1/2*x)","A"
16,1,114,87,0.064000," ","int(cot(x)^3/(a+b*cos(x)),x)","\frac{a^{3} \ln \left(a +b \cos \left(x \right)\right)}{\left(a +b \right)^{2} \left(a -b \right)^{2}}+\frac{1}{\left(4 a +4 b \right) \left(-1+\cos \left(x \right)\right)}-\frac{\ln \left(-1+\cos \left(x \right)\right) a}{2 \left(a +b \right)^{2}}-\frac{\ln \left(-1+\cos \left(x \right)\right) b}{4 \left(a +b \right)^{2}}-\frac{1}{\left(4 a -4 b \right) \left(\cos \left(x \right)+1\right)}-\frac{\ln \left(\cos \left(x \right)+1\right) a}{2 \left(a -b \right)^{2}}+\frac{\ln \left(\cos \left(x \right)+1\right) b}{4 \left(a -b \right)^{2}}"," ",0,"a^3/(a+b)^2/(a-b)^2*ln(a+b*cos(x))+1/(4*a+4*b)/(-1+cos(x))-1/2/(a+b)^2*ln(-1+cos(x))*a-1/4/(a+b)^2*ln(-1+cos(x))*b-1/(4*a-4*b)/(cos(x)+1)-1/2/(a-b)^2*ln(cos(x)+1)*a+1/4/(a-b)^2*ln(cos(x)+1)*b","A"
17,1,153,124,0.059000," ","int(cot(x)^4/(a+b*cos(x)),x)","\frac{a \left(\tan^{3}\left(\frac{x}{2}\right)\right)}{24 \left(a -b \right)^{2}}-\frac{\left(\tan^{3}\left(\frac{x}{2}\right)\right) b}{24 \left(a -b \right)^{2}}-\frac{5 a \tan \left(\frac{x}{2}\right)}{8 \left(a -b \right)^{2}}+\frac{3 \tan \left(\frac{x}{2}\right) b}{8 \left(a -b \right)^{2}}+\frac{2 a^{4} \arctan \left(\frac{\tan \left(\frac{x}{2}\right) \left(a -b \right)}{\sqrt{\left(a -b \right) \left(a +b \right)}}\right)}{\left(a -b \right)^{2} \left(a +b \right)^{2} \sqrt{\left(a -b \right) \left(a +b \right)}}-\frac{1}{24 \left(a +b \right) \tan \left(\frac{x}{2}\right)^{3}}+\frac{5 a}{8 \left(a +b \right)^{2} \tan \left(\frac{x}{2}\right)}+\frac{3 b}{8 \left(a +b \right)^{2} \tan \left(\frac{x}{2}\right)}"," ",0,"1/24/(a-b)^2*a*tan(1/2*x)^3-1/24/(a-b)^2*tan(1/2*x)^3*b-5/8/(a-b)^2*a*tan(1/2*x)+3/8/(a-b)^2*tan(1/2*x)*b+2/(a-b)^2/(a+b)^2*a^4/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*x)*(a-b)/((a-b)*(a+b))^(1/2))-1/24/(a+b)/tan(1/2*x)^3+5/8/(a+b)^2/tan(1/2*x)*a+3/8/(a+b)^2/tan(1/2*x)*b","A"
18,1,81,33,0.493000," ","int(cot(x)/(3-cos(x))^(1/2),x)","-\frac{\sqrt{2}\, \arctanh \left(\frac{-\sqrt{2}\, \cos \left(\frac{x}{2}\right)+2 \sqrt{2}}{\sqrt{2 \left(\sin^{2}\left(\frac{x}{2}\right)\right)+2}}\right)}{4}-\frac{\sqrt{2}\, \arctanh \left(\frac{\left(2+\cos \left(\frac{x}{2}\right)\right) \sqrt{2}}{\sqrt{2 \left(\sin^{2}\left(\frac{x}{2}\right)\right)+2}}\right)}{4}-\frac{\arctanh \left(\frac{2}{\sqrt{2 \left(\sin^{2}\left(\frac{x}{2}\right)\right)+2}}\right)}{2}"," ",0,"-1/4*2^(1/2)*arctanh((-2^(1/2)*cos(1/2*x)+2*2^(1/2))/(2*sin(1/2*x)^2+2)^(1/2))-1/4*2^(1/2)*arctanh(1/(2*sin(1/2*x)^2+2)^(1/2)*(2+cos(1/2*x))*2^(1/2))-1/2*arctanh(2/(2*sin(1/2*x)^2+2)^(1/2))","B"
19,1,30,29,0.035000," ","int((a+b*cos(x))^(1/2)*tan(x),x)","2 \arctanh \left(\frac{\sqrt{a +b \cos \left(x \right)}}{\sqrt{a}}\right) \sqrt{a}-2 \sqrt{a +b \cos \left(x \right)}"," ",0,"2*arctanh((a+b*cos(x))^(1/2)/a^(1/2))*a^(1/2)-2*(a+b*cos(x))^(1/2)","A"
20,1,19,18,0.039000," ","int(tan(x)/(a+b*cos(x))^(1/2),x)","\frac{2 \arctanh \left(\frac{\sqrt{a +b \cos \left(x \right)}}{\sqrt{a}}\right)}{\sqrt{a}}"," ",0,"2*arctanh((a+b*cos(x))^(1/2)/a^(1/2))/a^(1/2)","A"
21,1,546,164,0.398000," ","int((e*tan(d*x+c))^(1/2)/(a+b*cos(d*x+c)),x)","\frac{\sqrt{\frac{-1+\cos \left(d x +c \right)}{\sin \left(d x +c \right)}}\, \sqrt{\frac{-1+\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}\, \sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}\, \sqrt{\frac{e \sin \left(d x +c \right)}{\cos \left(d x +c \right)}}\, \left(1+\cos \left(d x +c \right)\right)^{2} \left(-1+\cos \left(d x +c \right)\right) \left(\sqrt{-a^{2}+b^{2}}\, \EllipticPi \left(\sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}, \frac{a -b}{a -b +\sqrt{-\left(a -b \right) \left(a +b \right)}}, \frac{\sqrt{2}}{2}\right)-\sqrt{-a^{2}+b^{2}}\, \EllipticPi \left(\sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}, -\frac{a -b}{-a +b +\sqrt{-\left(a -b \right) \left(a +b \right)}}, \frac{\sqrt{2}}{2}\right)-a \EllipticPi \left(\sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}, \frac{a -b}{a -b +\sqrt{-\left(a -b \right) \left(a +b \right)}}, \frac{\sqrt{2}}{2}\right)+\EllipticPi \left(\sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}, \frac{a -b}{a -b +\sqrt{-\left(a -b \right) \left(a +b \right)}}, \frac{\sqrt{2}}{2}\right) b -a \EllipticPi \left(\sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}, -\frac{a -b}{-a +b +\sqrt{-\left(a -b \right) \left(a +b \right)}}, \frac{\sqrt{2}}{2}\right)+\EllipticPi \left(\sqrt{\frac{1-\cos \left(d x +c \right)+\sin \left(d x +c \right)}{\sin \left(d x +c \right)}}, -\frac{a -b}{-a +b +\sqrt{-\left(a -b \right) \left(a +b \right)}}, \frac{\sqrt{2}}{2}\right) b \right) \sqrt{2}}{d \sin \left(d x +c \right)^{3} \left(\sqrt{-a^{2}+b^{2}}-a +b \right) \left(\sqrt{-a^{2}+b^{2}}+a -b \right)}"," ",0,"1/d*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(e*sin(d*x+c)/cos(d*x+c))^(1/2)*(1+cos(d*x+c))^2*(-1+cos(d*x+c))*((-a^2+b^2)^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))-(-a^2+b^2)^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))-a*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))+EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),(a-b)/(a-b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b-a*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))+EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),-(a-b)/(-a+b+(-(a-b)*(a+b))^(1/2)),1/2*2^(1/2))*b)/sin(d*x+c)^3*2^(1/2)/((-a^2+b^2)^(1/2)-a+b)/((-a^2+b^2)^(1/2)+a-b)","B"
22,0,0,48,1.331000," ","int((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x)","\int \left(a +b \cos \left(f x +e \right)\right)^{m} \left(g \tan \left(f x +e \right)\right)^{p}\, dx"," ",0,"int((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x)","F"