1,1,49,78,3.708124,"\text{Not used}","int(sin(x)^4/(tan(x) + 1i),x)","-\frac{x\,1{}\mathrm{i}}{16}+\frac{\frac{{\mathrm{tan}\left(x\right)}^4\,1{}\mathrm{i}}{16}+\frac{7\,{\mathrm{tan}\left(x\right)}^3}{16}+\frac{{\mathrm{tan}\left(x\right)}^2\,13{}\mathrm{i}}{48}+\frac{11\,\mathrm{tan}\left(x\right)}{48}+\frac{1}{6}{}\mathrm{i}}{{\left(\mathrm{tan}\left(x\right)+1{}\mathrm{i}\right)}^3\,{\left(1+\mathrm{tan}\left(x\right)\,1{}\mathrm{i}\right)}^2}","Not used",1,"((11*tan(x))/48 + (tan(x)^2*13i)/48 + (7*tan(x)^3)/16 + (tan(x)^4*1i)/16 + 1i/6)/((tan(x) + 1i)^3*(tan(x)*1i + 1)^2) - (x*1i)/16","B"
2,1,59,29,3.865209,"\text{Not used}","int(sin(x)^3/(tan(x) + 1i),x)","-\frac{4\,\left(-{\mathrm{tan}\left(\frac{x}{2}\right)}^4\,15{}\mathrm{i}+6\,{\mathrm{tan}\left(\frac{x}{2}\right)}^3+{\mathrm{tan}\left(\frac{x}{2}\right)}^2\,2{}\mathrm{i}+2\,\mathrm{tan}\left(\frac{x}{2}\right)+1{}\mathrm{i}\right)}{15\,{\left(-1+\mathrm{tan}\left(\frac{x}{2}\right)\,1{}\mathrm{i}\right)}^5\,{\left(1+\mathrm{tan}\left(\frac{x}{2}\right)\,1{}\mathrm{i}\right)}^3}","Not used",1,"-(4*(2*tan(x/2) + tan(x/2)^2*2i + 6*tan(x/2)^3 - tan(x/2)^4*15i + 1i))/(15*(tan(x/2)*1i - 1)^5*(tan(x/2)*1i + 1)^3)","B"
3,1,35,50,3.745320,"\text{Not used}","int(sin(x)^2/(tan(x) + 1i),x)","-\frac{x\,1{}\mathrm{i}}{8}+\frac{\frac{{\mathrm{tan}\left(x\right)}^2}{8}-\frac{\mathrm{tan}\left(x\right)\,3{}\mathrm{i}}{8}+\frac{1}{4}}{{\left(\mathrm{tan}\left(x\right)+1{}\mathrm{i}\right)}^2\,\left(1+\mathrm{tan}\left(x\right)\,1{}\mathrm{i}\right)}","Not used",1,"(tan(x)^2/8 - (tan(x)*3i)/8 + 1/4)/((tan(x) + 1i)^2*(tan(x)*1i + 1)) - (x*1i)/8","B"
4,1,39,19,3.828149,"\text{Not used}","int(sin(x)/(tan(x) + 1i),x)","-\frac{2\,\left(3\,{\mathrm{tan}\left(\frac{x}{2}\right)}^2+\mathrm{tan}\left(\frac{x}{2}\right)\,2{}\mathrm{i}-1\right)}{3\,\left(1+\mathrm{tan}\left(\frac{x}{2}\right)\,1{}\mathrm{i}\right)\,{\left(\mathrm{tan}\left(\frac{x}{2}\right)+1{}\mathrm{i}\right)}^3}","Not used",1,"-(2*(tan(x/2)*2i + 3*tan(x/2)^2 - 1))/(3*(tan(x/2)*1i + 1)*(tan(x/2) + 1i)^3)","B"
5,1,20,16,3.799987,"\text{Not used}","int(1/(sin(x)*(tan(x) + 1i)),x)","-\ln\left(\mathrm{tan}\left(\frac{x}{2}\right)\right)\,1{}\mathrm{i}+\frac{2}{\mathrm{tan}\left(\frac{x}{2}\right)+1{}\mathrm{i}}","Not used",1,"2/(tan(x/2) + 1i) - log(tan(x/2))*1i","B"
6,1,19,18,3.738065,"\text{Not used}","int(1/(sin(x)^2*(tan(x) + 1i)),x)","\mathrm{atan}\left(2\,\mathrm{tan}\left(x\right)+1{}\mathrm{i}\right)\,2{}\mathrm{i}+\frac{1{}\mathrm{i}}{\mathrm{tan}\left(x\right)}","Not used",1,"atan(2*tan(x) + 1i)*2i + 1i/tan(x)","B"
7,1,41,24,3.729267,"\text{Not used}","int(1/(sin(x)^3*(tan(x) + 1i)),x)","-\frac{\mathrm{tan}\left(\frac{x}{2}\right)}{2}+\frac{\ln\left(\mathrm{tan}\left(\frac{x}{2}\right)\right)\,1{}\mathrm{i}}{2}-\frac{2\,\mathrm{tan}\left(\frac{x}{2}\right)-\frac{1}{2}{}\mathrm{i}}{4\,{\mathrm{tan}\left(\frac{x}{2}\right)}^2}-\frac{{\mathrm{tan}\left(\frac{x}{2}\right)}^2\,1{}\mathrm{i}}{8}","Not used",1,"(log(tan(x/2))*1i)/2 - tan(x/2)/2 - (2*tan(x/2) - 1i/2)/(4*tan(x/2)^2) - (tan(x/2)^2*1i)/8","B"
8,1,13,19,3.624173,"\text{Not used}","int(1/(sin(x)^4*(tan(x) + 1i)),x)","\frac{{\mathrm{cot}\left(x\right)}^2\,\left(-3+\mathrm{cot}\left(x\right)\,2{}\mathrm{i}\right)}{6}","Not used",1,"(cot(x)^2*(cot(x)*2i - 3))/6","B"
9,1,57,40,3.696443,"\text{Not used}","int(1/(sin(x)^5*(tan(x) + 1i)),x)","-\frac{\mathrm{tan}\left(\frac{x}{2}\right)}{8}+\frac{\ln\left(\mathrm{tan}\left(\frac{x}{2}\right)\right)\,1{}\mathrm{i}}{8}-\frac{2\,{\mathrm{tan}\left(\frac{x}{2}\right)}^3+\frac{2\,\mathrm{tan}\left(\frac{x}{2}\right)}{3}-\frac{1}{4}{}\mathrm{i}}{16\,{\mathrm{tan}\left(\frac{x}{2}\right)}^4}-\frac{{\mathrm{tan}\left(\frac{x}{2}\right)}^3}{24}-\frac{{\mathrm{tan}\left(\frac{x}{2}\right)}^4\,1{}\mathrm{i}}{64}","Not used",1,"(log(tan(x/2))*1i)/8 - tan(x/2)/8 - ((2*tan(x/2))/3 + 2*tan(x/2)^3 - 1i/4)/(16*tan(x/2)^4) - tan(x/2)^3/24 - (tan(x/2)^4*1i)/64","B"
10,1,27,37,3.601000,"\text{Not used}","int(1/(sin(x)^6*(tan(x) + 1i)),x)","\frac{{\mathrm{cot}\left(x\right)}^5\,1{}\mathrm{i}}{5}-\frac{{\mathrm{cot}\left(x\right)}^4}{4}+\frac{{\mathrm{cot}\left(x\right)}^3\,1{}\mathrm{i}}{3}-\frac{{\mathrm{cot}\left(x\right)}^2}{2}","Not used",1,"(cot(x)^3*1i)/3 - cot(x)^2/2 - cot(x)^4/4 + (cot(x)^5*1i)/5","B"
11,1,121,101,3.804270,"\text{Not used}","int(sin(c + d*x)^5*(a + b*tan(c + d*x)),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{d}+\frac{2\,a\,{\cos\left(c+d\,x\right)}^3}{3\,d}-\frac{a\,{\cos\left(c+d\,x\right)}^5}{5\,d}-\frac{a\,\cos\left(c+d\,x\right)}{d}-\frac{23\,b\,\sin\left(c+d\,x\right)}{15\,d}+\frac{11\,b\,{\cos\left(c+d\,x\right)}^2\,\sin\left(c+d\,x\right)}{15\,d}-\frac{b\,{\cos\left(c+d\,x\right)}^4\,\sin\left(c+d\,x\right)}{5\,d}","Not used",1,"(2*b*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (2*a*cos(c + d*x)^3)/(3*d) - (a*cos(c + d*x)^5)/(5*d) - (a*cos(c + d*x))/d - (23*b*sin(c + d*x))/(15*d) + (11*b*cos(c + d*x)^2*sin(c + d*x))/(15*d) - (b*cos(c + d*x)^4*sin(c + d*x))/(5*d)","B"
12,1,155,83,3.786649,"\text{Not used}","int(sin(c + d*x)^4*(a + b*tan(c + d*x)),x)","\frac{3\,a\,x}{8}+\frac{b\,\ln\left({\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}{2\,d}+\frac{3\,b}{4\,d\,\left({\mathrm{tan}\left(c+d\,x\right)}^4+2\,{\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}-\frac{5\,a\,{\mathrm{tan}\left(c+d\,x\right)}^3}{8\,d\,\left({\mathrm{tan}\left(c+d\,x\right)}^4+2\,{\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}+\frac{b\,{\mathrm{tan}\left(c+d\,x\right)}^2}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^4+2\,{\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}-\frac{3\,a\,\mathrm{tan}\left(c+d\,x\right)}{8\,d\,\left({\mathrm{tan}\left(c+d\,x\right)}^4+2\,{\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}","Not used",1,"(3*a*x)/8 + (b*log(tan(c + d*x)^2 + 1))/(2*d) + (3*b)/(4*d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1)) - (5*a*tan(c + d*x)^3)/(8*d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1)) + (b*tan(c + d*x)^2)/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1)) - (3*a*tan(c + d*x))/(8*d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1))","B"
13,1,87,69,3.828404,"\text{Not used}","int(sin(c + d*x)^3*(a + b*tan(c + d*x)),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{d}+\frac{a\,{\cos\left(c+d\,x\right)}^3}{3\,d}-\frac{a\,\cos\left(c+d\,x\right)}{d}-\frac{4\,b\,\sin\left(c+d\,x\right)}{3\,d}+\frac{b\,{\cos\left(c+d\,x\right)}^2\,\sin\left(c+d\,x\right)}{3\,d}","Not used",1,"(2*b*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (a*cos(c + d*x)^3)/(3*d) - (a*cos(c + d*x))/d - (4*b*sin(c + d*x))/(3*d) + (b*cos(c + d*x)^2*sin(c + d*x))/(3*d)","B"
14,1,50,49,3.815172,"\text{Not used}","int(sin(c + d*x)^2*(a + b*tan(c + d*x)),x)","\frac{\frac{b\,{\cos\left(c+d\,x\right)}^2}{2}-\frac{a\,\sin\left(c+d\,x\right)\,\cos\left(c+d\,x\right)}{2}+\frac{b\,\ln\left({\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}{2}+\frac{a\,d\,x}{2}}{d}","Not used",1,"((b*log(tan(c + d*x)^2 + 1))/2 + (b*cos(c + d*x)^2)/2 - (a*cos(c + d*x)*sin(c + d*x))/2 + (a*d*x)/2)/d","B"
15,1,53,37,3.820330,"\text{Not used}","int(sin(c + d*x)*(a + b*tan(c + d*x)),x)","\frac{2\,b\,\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{d}-\frac{2\,a+2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+1\right)}","Not used",1,"(2*b*atanh(tan(c/2 + (d*x)/2)))/d - (2*a + 2*b*tan(c/2 + (d*x)/2))/(d*(tan(c/2 + (d*x)/2)^2 + 1))","B"
16,1,86,26,3.745430,"\text{Not used}","int((a + b*tan(c + d*x))/sin(c + d*x),x)","\frac{a\,\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{d}-\frac{2\,b\,\mathrm{atanh}\left(\frac{b\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)-a\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{a\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)-b\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{d}","Not used",1,"(a*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (2*b*atanh((b*cos(c/2 + (d*x)/2) - a*sin(c/2 + (d*x)/2))/(a*cos(c/2 + (d*x)/2) - b*sin(c/2 + (d*x)/2))))/d","B"
17,1,25,25,3.633597,"\text{Not used}","int((a + b*tan(c + d*x))/sin(c + d*x)^2,x)","\frac{b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}-\frac{a\,\mathrm{cot}\left(c+d\,x\right)}{d}","Not used",1,"(b*log(tan(c + d*x)))/d - (a*cot(c + d*x))/d","B"
18,1,149,60,3.715936,"\text{Not used}","int((a + b*tan(c + d*x))/sin(c + d*x)^3,x)","\frac{a\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}{8\,d}-\frac{\frac{a}{2}+2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{4\,d\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}-\frac{b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{2\,d}-\frac{2\,b\,\mathrm{atanh}\left(\frac{4\,b^2}{2\,a\,b-4\,b^2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}-\frac{2\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{2\,a\,b-4\,b^2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{d}+\frac{a\,\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{2\,d}","Not used",1,"(a*tan(c/2 + (d*x)/2)^2)/(8*d) - (a/2 + 2*b*tan(c/2 + (d*x)/2))/(4*d*tan(c/2 + (d*x)/2)^2) - (b*tan(c/2 + (d*x)/2))/(2*d) - (2*b*atanh((4*b^2)/(2*a*b - 4*b^2*tan(c/2 + (d*x)/2)) - (2*a*b*tan(c/2 + (d*x)/2))/(2*a*b - 4*b^2*tan(c/2 + (d*x)/2))))/d + (a*log(tan(c/2 + (d*x)/2)))/(2*d)","B"
19,1,49,57,3.671725,"\text{Not used}","int((a + b*tan(c + d*x))/sin(c + d*x)^4,x)","\frac{b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}-\frac{a\,{\mathrm{tan}\left(c+d\,x\right)}^2+\frac{b\,\mathrm{tan}\left(c+d\,x\right)}{2}+\frac{a}{3}}{d\,{\mathrm{tan}\left(c+d\,x\right)}^3}","Not used",1,"(b*log(tan(c + d*x)))/d - (a/3 + (b*tan(c + d*x))/2 + a*tan(c + d*x)^2)/(d*tan(c + d*x)^3)","B"
20,1,211,98,3.846081,"\text{Not used}","int((a + b*tan(c + d*x))/sin(c + d*x)^5,x)","\frac{a\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}{8\,d}-\frac{5\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{8\,d}-\frac{2\,b\,\mathrm{atanh}\left(\frac{4\,b^2}{\frac{3\,a\,b}{2}-4\,b^2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}-\frac{3\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{2\,\left(\frac{3\,a\,b}{2}-4\,b^2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}\right)}{d}+\frac{a\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4}{64\,d}-\frac{b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3}{24\,d}+\frac{3\,a\,\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{8\,d}-\frac{10\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3+2\,a\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+\frac{2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{3}+\frac{a}{4}}{16\,d\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4}","Not used",1,"(a*tan(c/2 + (d*x)/2)^2)/(8*d) - (5*b*tan(c/2 + (d*x)/2))/(8*d) - (2*b*atanh((4*b^2)/((3*a*b)/2 - 4*b^2*tan(c/2 + (d*x)/2)) - (3*a*b*tan(c/2 + (d*x)/2))/(2*((3*a*b)/2 - 4*b^2*tan(c/2 + (d*x)/2)))))/d + (a*tan(c/2 + (d*x)/2)^4)/(64*d) - (b*tan(c/2 + (d*x)/2)^3)/(24*d) + (3*a*log(tan(c/2 + (d*x)/2)))/(8*d) - (a/4 + (2*b*tan(c/2 + (d*x)/2))/3 + 2*a*tan(c/2 + (d*x)/2)^2 + 10*b*tan(c/2 + (d*x)/2)^3)/(16*d*tan(c/2 + (d*x)/2)^4)","B"
21,1,70,87,3.929899,"\text{Not used}","int((a + b*tan(c + d*x))/sin(c + d*x)^6,x)","\frac{b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}-\frac{a\,{\mathrm{tan}\left(c+d\,x\right)}^4+b\,{\mathrm{tan}\left(c+d\,x\right)}^3+\frac{2\,a\,{\mathrm{tan}\left(c+d\,x\right)}^2}{3}+\frac{b\,\mathrm{tan}\left(c+d\,x\right)}{4}+\frac{a}{5}}{d\,{\mathrm{tan}\left(c+d\,x\right)}^5}","Not used",1,"(b*log(tan(c + d*x)))/d - (a/5 + (b*tan(c + d*x))/4 + (2*a*tan(c + d*x)^2)/3 + a*tan(c + d*x)^4 + b*tan(c + d*x)^3)/(d*tan(c + d*x)^5)","B"
22,1,127,113,3.820481,"\text{Not used}","int(sin(c + d*x)^4*(a + b*tan(c + d*x))^2,x)","x\,\left(\frac{3\,a^2}{8}-\frac{15\,b^2}{8}\right)+\frac{b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}+\frac{\left(\frac{9\,b^2}{8}-\frac{5\,a^2}{8}\right)\,{\mathrm{tan}\left(c+d\,x\right)}^3+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^2+\left(\frac{7\,b^2}{8}-\frac{3\,a^2}{8}\right)\,\mathrm{tan}\left(c+d\,x\right)+\frac{3\,a\,b}{2}}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^4+2\,{\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}+\frac{a\,b\,\ln\left({\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}{d}","Not used",1,"x*((3*a^2)/8 - (15*b^2)/8) + (b^2*tan(c + d*x))/d + ((3*a*b)/2 - tan(c + d*x)*((3*a^2)/8 - (7*b^2)/8) - tan(c + d*x)^3*((5*a^2)/8 - (9*b^2)/8) + 2*a*b*tan(c + d*x)^2)/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1)) + (a*b*log(tan(c + d*x)^2 + 1))/d","B"
23,1,174,122,6.706257,"\text{Not used}","int(sin(c + d*x)^3*(a + b*tan(c + d*x))^2,x)","\frac{4\,a\,b\,\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{d}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{8\,a^2}{3}-\frac{32\,b^2}{3}\right)-4\,a^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+\frac{4\,a^2}{3}-\frac{16\,b^2}{3}+\frac{28\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3}{3}-\frac{28\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5}{3}-4\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7+4\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d\,\left(-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8-2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+1\right)}","Not used",1,"(4*a*b*atanh(tan(c/2 + (d*x)/2)))/d - (tan(c/2 + (d*x)/2)^2*((8*a^2)/3 - (32*b^2)/3) - 4*a^2*tan(c/2 + (d*x)/2)^4 + (4*a^2)/3 - (16*b^2)/3 + (28*a*b*tan(c/2 + (d*x)/2)^3)/3 - (28*a*b*tan(c/2 + (d*x)/2)^5)/3 - 4*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2))/(d*(2*tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2)^6 - tan(c/2 + (d*x)/2)^8 + 1))","B"
24,1,75,76,3.688020,"\text{Not used}","int(sin(c + d*x)^2*(a + b*tan(c + d*x))^2,x)","\frac{{\cos\left(c+d\,x\right)}^2\,\left(a\,b-\mathrm{tan}\left(c+d\,x\right)\,\left(\frac{a^2}{2}-\frac{b^2}{2}\right)\right)+b^2\,\mathrm{tan}\left(c+d\,x\right)+a\,b\,\ln\left({\mathrm{tan}\left(c+d\,x\right)}^2+1\right)+d\,x\,\left(\frac{a^2}{2}-\frac{3\,b^2}{2}\right)}{d}","Not used",1,"(cos(c + d*x)^2*(a*b - tan(c + d*x)*(a^2/2 - b^2/2)) + b^2*tan(c + d*x) + a*b*log(tan(c + d*x)^2 + 1) + d*x*(a^2/2 - (3*b^2)/2))/d","B"
25,1,93,68,4.042048,"\text{Not used}","int(sin(c + d*x)*(a + b*tan(c + d*x))^2,x)","\frac{4\,a\,b\,\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{d}-\frac{2\,a^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2-2\,a^2+4\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3-4\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)+4\,b^2}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4-1\right)}","Not used",1,"(4*a*b*atanh(tan(c/2 + (d*x)/2)))/d - (2*a^2*tan(c/2 + (d*x)/2)^2 - 2*a^2 + 4*b^2 + 4*a*b*tan(c/2 + (d*x)/2)^3 - 4*a*b*tan(c/2 + (d*x)/2))/(d*(tan(c/2 + (d*x)/2)^4 - 1))","B"
26,1,125,43,3.721166,"\text{Not used}","int((a + b*tan(c + d*x))^2/sin(c + d*x),x)","\frac{a^2\,\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{d}-\frac{2\,b^2}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2-1\right)}-\frac{4\,a\,b\,\mathrm{atanh}\left(\frac{16\,a^2\,b^2}{8\,a^3\,b-16\,a^2\,b^2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}-\frac{8\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{8\,a^3\,b-16\,a^2\,b^2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{d}","Not used",1,"(a^2*log(tan(c/2 + (d*x)/2)))/d - (2*b^2)/(d*(tan(c/2 + (d*x)/2)^2 - 1)) - (4*a*b*atanh((16*a^2*b^2)/(8*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2)) - (8*a^3*b*tan(c/2 + (d*x)/2))/(8*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2))))/d","B"
27,1,44,42,3.640530,"\text{Not used}","int((a + b*tan(c + d*x))^2/sin(c + d*x)^2,x)","\frac{b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}-\frac{a^2}{d\,\mathrm{tan}\left(c+d\,x\right)}+\frac{2\,a\,b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}","Not used",1,"(b^2*tan(c + d*x))/d - a^2/(d*tan(c + d*x)) + (2*a*b*log(tan(c + d*x)))/d","B"
28,1,292,95,3.844641,"\text{Not used}","int((a + b*tan(c + d*x))^2/sin(c + d*x)^3,x)","\frac{a^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}{8\,d}+\frac{\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(\frac{a^2}{2}+b^2\right)}{d}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{a^2}{2}+8\,b^2\right)-\frac{a^2}{2}+4\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3-4\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d\,\left(4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2-4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\right)}+\frac{4\,a\,b\,\mathrm{atanh}\left(\frac{8\,a\,b^3\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{4\,a^3\,b-16\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^2+8\,a\,b^3}-\frac{16\,a^2\,b^2}{4\,a^3\,b-16\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^2+8\,a\,b^3}+\frac{4\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{4\,a^3\,b-16\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^2+8\,a\,b^3}\right)}{d}-\frac{a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d}","Not used",1,"(a^2*tan(c/2 + (d*x)/2)^2)/(8*d) + (log(tan(c/2 + (d*x)/2))*(a^2/2 + b^2))/d + (tan(c/2 + (d*x)/2)^2*(a^2/2 + 8*b^2) - a^2/2 + 4*a*b*tan(c/2 + (d*x)/2)^3 - 4*a*b*tan(c/2 + (d*x)/2))/(d*(4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^4)) + (4*a*b*atanh((8*a*b^3*tan(c/2 + (d*x)/2))/(8*a*b^3 + 4*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2)) - (16*a^2*b^2)/(8*a*b^3 + 4*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2)) + (4*a^3*b*tan(c/2 + (d*x)/2))/(8*a*b^3 + 4*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2))))/d - (a*b*tan(c/2 + (d*x)/2))/d","B"
29,1,72,79,3.787064,"\text{Not used}","int((a + b*tan(c + d*x))^2/sin(c + d*x)^4,x)","\frac{b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^2+b^2\right)+\frac{a^2}{3}+a\,b\,\mathrm{tan}\left(c+d\,x\right)}{d\,{\mathrm{tan}\left(c+d\,x\right)}^3}+\frac{2\,a\,b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}","Not used",1,"(b^2*tan(c + d*x))/d - (tan(c + d*x)^2*(a^2 + b^2) + a^2/3 + a*b*tan(c + d*x))/(d*tan(c + d*x)^3) + (2*a*b*log(tan(c + d*x)))/d","B"
30,1,378,165,3.908540,"\text{Not used}","int((a + b*tan(c + d*x))^2/sin(c + d*x)^5,x)","\frac{\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(\frac{3\,a^2}{8}+\frac{3\,b^2}{2}\right)}{d}+\frac{a^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4}{64\,d}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{7\,a^2}{4}+2\,b^2\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(2\,a^2+34\,b^2\right)+\frac{a^2}{4}+\frac{56\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3}{3}-20\,a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5+\frac{4\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{3}}{d\,\left(16\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4-16\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\right)}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{a^2}{8}+\frac{b^2}{8}\right)}{d}-\frac{a\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3}{12\,d}+\frac{4\,a\,b\,\mathrm{atanh}\left(\frac{12\,a\,b^3\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{3\,a^3\,b-16\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^2+12\,a\,b^3}-\frac{16\,a^2\,b^2}{3\,a^3\,b-16\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^2+12\,a\,b^3}+\frac{3\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{3\,a^3\,b-16\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^2+12\,a\,b^3}\right)}{d}-\frac{5\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{4\,d}","Not used",1,"(log(tan(c/2 + (d*x)/2))*((3*a^2)/8 + (3*b^2)/2))/d + (a^2*tan(c/2 + (d*x)/2)^4)/(64*d) - (tan(c/2 + (d*x)/2)^2*((7*a^2)/4 + 2*b^2) - tan(c/2 + (d*x)/2)^4*(2*a^2 + 34*b^2) + a^2/4 + (56*a*b*tan(c/2 + (d*x)/2)^3)/3 - 20*a*b*tan(c/2 + (d*x)/2)^5 + (4*a*b*tan(c/2 + (d*x)/2))/3)/(d*(16*tan(c/2 + (d*x)/2)^4 - 16*tan(c/2 + (d*x)/2)^6)) + (tan(c/2 + (d*x)/2)^2*(a^2/8 + b^2/8))/d - (a*b*tan(c/2 + (d*x)/2)^3)/(12*d) + (4*a*b*atanh((12*a*b^3*tan(c/2 + (d*x)/2))/(12*a*b^3 + 3*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2)) - (16*a^2*b^2)/(12*a*b^3 + 3*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2)) + (3*a^3*b*tan(c/2 + (d*x)/2))/(12*a*b^3 + 3*a^3*b - 16*a^2*b^2*tan(c/2 + (d*x)/2))))/d - (5*a*b*tan(c/2 + (d*x)/2))/(4*d)","B"
31,1,107,122,4.060203,"\text{Not used}","int((a + b*tan(c + d*x))^2/sin(c + d*x)^6,x)","\frac{b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^2+2\,b^2\right)+\frac{a^2}{5}+{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(\frac{2\,a^2}{3}+\frac{b^2}{3}\right)+\frac{a\,b\,\mathrm{tan}\left(c+d\,x\right)}{2}+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3}{d\,{\mathrm{tan}\left(c+d\,x\right)}^5}+\frac{2\,a\,b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}","Not used",1,"(b^2*tan(c + d*x))/d - (tan(c + d*x)^4*(a^2 + 2*b^2) + a^2/5 + tan(c + d*x)^2*((2*a^2)/3 + b^2/3) + (a*b*tan(c + d*x))/2 + 2*a*b*tan(c + d*x)^3)/(d*tan(c + d*x)^5) + (2*a*b*log(tan(c + d*x)))/d","B"
32,1,291,205,6.436069,"\text{Not used}","int(sin(c + d*x)^3*(a + b*tan(c + d*x))^3,x)","\frac{\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(6\,a^2\,b-5\,b^3\right)}{d}-\frac{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(6\,a^2\,b-5\,b^3\right)+4\,a^3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6-16\,a\,b^2-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(16\,a\,b^2-\frac{4\,a^3}{3}\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(32\,a\,b^2-\frac{20\,a^3}{3}\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^9\,\left(6\,a^2\,b-5\,b^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(8\,a^2\,b-\frac{20\,b^3}{3}\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(8\,a^2\,b-\frac{20\,b^3}{3}\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(28\,a^2\,b-\frac{22\,b^3}{3}\right)+\frac{4\,a^3}{3}}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{10}+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8-2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6-2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+1\right)}","Not used",1,"(atanh(tan(c/2 + (d*x)/2))*(6*a^2*b - 5*b^3))/d - (tan(c/2 + (d*x)/2)*(6*a^2*b - 5*b^3) + 4*a^3*tan(c/2 + (d*x)/2)^6 - 16*a*b^2 - tan(c/2 + (d*x)/2)^2*(16*a*b^2 - (4*a^3)/3) + tan(c/2 + (d*x)/2)^4*(32*a*b^2 - (20*a^3)/3) + tan(c/2 + (d*x)/2)^9*(6*a^2*b - 5*b^3) + tan(c/2 + (d*x)/2)^3*(8*a^2*b - (20*b^3)/3) + tan(c/2 + (d*x)/2)^7*(8*a^2*b - (20*b^3)/3) - tan(c/2 + (d*x)/2)^5*(28*a^2*b - (22*b^3)/3) + (4*a^3)/3)/(d*(tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2)^4 - 2*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1))","B"
33,1,151,103,3.695473,"\text{Not used}","int(sin(c + d*x)^2*(a + b*tan(c + d*x))^3,x)","\frac{b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{2\,d}+\frac{{\cos\left(c+d\,x\right)}^2\,\left(\frac{3\,a^2\,b}{2}-\frac{b^3}{2}+\mathrm{tan}\left(c+d\,x\right)\,\left(\frac{3\,a\,b^2}{2}-\frac{a^3}{2}\right)\right)}{d}+\frac{\ln\left({\mathrm{tan}\left(c+d\,x\right)}^2+1\right)\,\left(\frac{3\,a^2\,b}{2}-b^3\right)}{d}+\frac{3\,a\,b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}-\frac{a\,\mathrm{atan}\left(\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(a-3\,b\right)\,\left(a+3\,b\right)}{2\,\left(\frac{9\,a\,b^2}{2}-\frac{a^3}{2}\right)}\right)\,\left(a-3\,b\right)\,\left(a+3\,b\right)}{2\,d}","Not used",1,"(b^3*tan(c + d*x)^2)/(2*d) + (cos(c + d*x)^2*((3*a^2*b)/2 - b^3/2 + tan(c + d*x)*((3*a*b^2)/2 - a^3/2)))/d + (log(tan(c + d*x)^2 + 1)*((3*a^2*b)/2 - b^3))/d + (3*a*b^2*tan(c + d*x))/d - (a*atan((a*tan(c + d*x)*(a - 3*b)*(a + 3*b))/(2*((9*a*b^2)/2 - a^3/2)))*(a - 3*b)*(a + 3*b))/(2*d)","B"
34,1,193,133,5.799139,"\text{Not used}","int(sin(c + d*x)*(a + b*tan(c + d*x))^3,x)","\frac{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(6\,a^2\,b-3\,b^3\right)+2\,a^3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4-12\,a\,b^2+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(12\,a\,b^2-4\,a^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(6\,a^2\,b-3\,b^3\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(12\,a^2\,b-2\,b^3\right)+2\,a^3}{d\,\left(-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2-1\right)}+\frac{\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(6\,a^2\,b-3\,b^3\right)}{d}","Not used",1,"(tan(c/2 + (d*x)/2)*(6*a^2*b - 3*b^3) + 2*a^3*tan(c/2 + (d*x)/2)^4 - 12*a*b^2 + tan(c/2 + (d*x)/2)^2*(12*a*b^2 - 4*a^3) + tan(c/2 + (d*x)/2)^5*(6*a^2*b - 3*b^3) - tan(c/2 + (d*x)/2)^3*(12*a^2*b - 2*b^3) + 2*a^3)/(d*(tan(c/2 + (d*x)/2)^2 + tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)^6 - 1)) + (atanh(tan(c/2 + (d*x)/2))*(6*a^2*b - 3*b^3))/d","B"
35,1,278,86,4.209416,"\text{Not used}","int((a + b*tan(c + d*x))^3/sin(c + d*x),x)","\frac{2\,\left(\frac{a^3\,\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{2}-\frac{b^3\,\mathrm{atan}\left(\frac{2\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^3-6\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b+\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^3}{2{}\mathrm{i}\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^3-6{}\mathrm{i}\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b+1{}\mathrm{i}\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^3}\right)\,1{}\mathrm{i}}{2}+a^2\,b\,\mathrm{atan}\left(\frac{2\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^3-6\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b+\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^3}{2{}\mathrm{i}\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^3-6{}\mathrm{i}\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b+1{}\mathrm{i}\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^3}\right)\,3{}\mathrm{i}\right)}{d}+\frac{\frac{\sin\left(c+d\,x\right)\,b^3}{2}+3\,a\,\cos\left(c+d\,x\right)\,b^2}{d\,\left(\frac{\cos\left(2\,c+2\,d\,x\right)}{2}+\frac{1}{2}\right)}","Not used",1,"(2*((a^3*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 - (b^3*atan((b^3*cos(c/2 + (d*x)/2) + 2*a^3*sin(c/2 + (d*x)/2) - 6*a^2*b*cos(c/2 + (d*x)/2))/(a^3*cos(c/2 + (d*x)/2)*2i + b^3*sin(c/2 + (d*x)/2)*1i - a^2*b*sin(c/2 + (d*x)/2)*6i))*1i)/2 + a^2*b*atan((b^3*cos(c/2 + (d*x)/2) + 2*a^3*sin(c/2 + (d*x)/2) - 6*a^2*b*cos(c/2 + (d*x)/2))/(a^3*cos(c/2 + (d*x)/2)*2i + b^3*sin(c/2 + (d*x)/2)*1i - a^2*b*sin(c/2 + (d*x)/2)*6i))*3i))/d + ((b^3*sin(c + d*x))/2 + 3*a*b^2*cos(c + d*x))/(d*(cos(2*c + 2*d*x)/2 + 1/2))","B"
36,1,62,64,3.661695,"\text{Not used}","int((a + b*tan(c + d*x))^3/sin(c + d*x)^2,x)","\frac{b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{2\,d}-\frac{a^3\,\mathrm{cot}\left(c+d\,x\right)}{d}+\frac{3\,a^2\,b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}+\frac{3\,a\,b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}","Not used",1,"(b^3*tan(c + d*x)^2)/(2*d) - (a^3*cot(c + d*x))/d + (3*a^2*b*log(tan(c + d*x)))/d + (3*a*b^2*tan(c + d*x))/d","B"
37,1,581,141,3.965688,"\text{Not used}","int((a + b*tan(c + d*x))^3/sin(c + d*x)^3,x)","\frac{a^3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}{8\,d}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(\frac{a^3}{2}+24\,a\,b^2\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(a^3+24\,a\,b^2\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(6\,a^2\,b-4\,b^3\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(12\,a^2\,b+4\,b^3\right)+\frac{a^3}{2}+6\,a^2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d\,\left(4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6-8\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\right)}+\frac{\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(\frac{a^3}{2}+3\,a\,b^2\right)}{d}-\frac{\mathrm{atan}\left(-\frac{\left(3\,a^2\,b+\frac{b^3}{2}\right)\,\left(6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{b^3}{2}\right)+6\,a^2\,b+b^3-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^3+6\,a\,b^2\right)\right)\,1{}\mathrm{i}-\left(3\,a^2\,b+\frac{b^3}{2}\right)\,\left(6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{b^3}{2}\right)-6\,a^2\,b-b^3+\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^3+6\,a\,b^2\right)\right)\,1{}\mathrm{i}}{2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(36\,a^4\,b^2+12\,a^2\,b^4+b^6\right)-\left(3\,a^2\,b+\frac{b^3}{2}\right)\,\left(6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{b^3}{2}\right)+6\,a^2\,b+b^3-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^3+6\,a\,b^2\right)\right)-\left(3\,a^2\,b+\frac{b^3}{2}\right)\,\left(6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{b^3}{2}\right)-6\,a^2\,b-b^3+\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^3+6\,a\,b^2\right)\right)+6\,a\,b^5+6\,a^5\,b+37\,a^3\,b^3}\right)\,\left(a^2\,b\,6{}\mathrm{i}+b^3\,1{}\mathrm{i}\right)}{d}-\frac{3\,a^2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{2\,d}","Not used",1,"(a^3*tan(c/2 + (d*x)/2)^2)/(8*d) - (tan(c/2 + (d*x)/2)^4*(24*a*b^2 + a^3/2) - tan(c/2 + (d*x)/2)^2*(24*a*b^2 + a^3) + tan(c/2 + (d*x)/2)^5*(6*a^2*b - 4*b^3) - tan(c/2 + (d*x)/2)^3*(12*a^2*b + 4*b^3) + a^3/2 + 6*a^2*b*tan(c/2 + (d*x)/2))/(d*(4*tan(c/2 + (d*x)/2)^2 - 8*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6)) + (log(tan(c/2 + (d*x)/2))*(3*a*b^2 + a^3/2))/d - (atan(-((3*a^2*b + b^3/2)*(6*tan(c/2 + (d*x)/2)*(3*a^2*b + b^3/2) + 6*a^2*b + b^3 - tan(c/2 + (d*x)/2)*(6*a*b^2 + a^3))*1i - (3*a^2*b + b^3/2)*(6*tan(c/2 + (d*x)/2)*(3*a^2*b + b^3/2) - 6*a^2*b - b^3 + tan(c/2 + (d*x)/2)*(6*a*b^2 + a^3))*1i)/(2*tan(c/2 + (d*x)/2)*(b^6 + 12*a^2*b^4 + 36*a^4*b^2) - (3*a^2*b + b^3/2)*(6*tan(c/2 + (d*x)/2)*(3*a^2*b + b^3/2) + 6*a^2*b + b^3 - tan(c/2 + (d*x)/2)*(6*a*b^2 + a^3)) - (3*a^2*b + b^3/2)*(6*tan(c/2 + (d*x)/2)*(3*a^2*b + b^3/2) - 6*a^2*b - b^3 + tan(c/2 + (d*x)/2)*(6*a*b^2 + a^3)) + 6*a*b^5 + 6*a^5*b + 37*a^3*b^3))*(a^2*b*6i + b^3*1i))/d - (3*a^2*b*tan(c/2 + (d*x)/2))/(2*d)","B"
38,1,103,113,3.719448,"\text{Not used}","int((a + b*tan(c + d*x))^3/sin(c + d*x)^4,x)","\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)\,\left(3\,a^2\,b+b^3\right)}{d}-\frac{{\mathrm{cot}\left(c+d\,x\right)}^3\,\left(\frac{a^3}{3}+{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^3+3\,a\,b^2\right)+\frac{3\,a^2\,b\,\mathrm{tan}\left(c+d\,x\right)}{2}\right)}{d}+\frac{b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{2\,d}+\frac{3\,a\,b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}","Not used",1,"(log(tan(c + d*x))*(3*a^2*b + b^3))/d - (cot(c + d*x)^3*(a^3/3 + tan(c + d*x)^2*(3*a*b^2 + a^3) + (3*a^2*b*tan(c + d*x))/2))/d + (b^3*tan(c + d*x)^2)/(2*d) + (3*a*b^2*tan(c + d*x))/d","B"
39,1,698,229,4.038364,"\text{Not used}","int((a + b*tan(c + d*x))^3/sin(c + d*x)^5,x)","\frac{a^3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4}{64\,d}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{a^3}{8}+\frac{3\,a\,b^2}{8}\right)}{d}-\frac{\mathrm{atan}\left(\frac{\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)\,\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^3}{4}+9\,a\,b^2\right)+6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)-6\,a^2\,b-3\,b^3\right)\,1{}\mathrm{i}-\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)\,\left(6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^3}{4}+9\,a\,b^2\right)+6\,a^2\,b+3\,b^3\right)\,1{}\mathrm{i}}{2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(36\,a^4\,b^2+36\,a^2\,b^4+9\,b^6\right)+27\,a\,b^5+\frac{9\,a^5\,b}{2}-\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)\,\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^3}{4}+9\,a\,b^2\right)+6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)-6\,a^2\,b-3\,b^3\right)-\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)\,\left(6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(3\,a^2\,b+\frac{3\,b^3}{2}\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^3}{4}+9\,a\,b^2\right)+6\,a^2\,b+3\,b^3\right)+\frac{225\,a^3\,b^3}{4}}\right)\,\left(a^2\,b\,6{}\mathrm{i}+b^3\,3{}\mathrm{i}\right)}{d}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{3\,a^3}{2}+6\,a\,b^2\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\,\left(2\,a^3+102\,a\,b^2\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(\frac{15\,a^3}{4}+108\,a\,b^2\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(26\,a^2\,b+8\,b^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(30\,a^2\,b-8\,b^3\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(58\,a^2\,b+32\,b^3\right)+\frac{a^3}{4}+2\,a^2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d\,\left(16\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8-32\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+16\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\right)}-\frac{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{15\,a^2\,b}{8}+\frac{b^3}{2}\right)}{d}+\frac{3\,a\,\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(a^2+12\,b^2\right)}{8\,d}-\frac{a^2\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3}{8\,d}","Not used",1,"(a^3*tan(c/2 + (d*x)/2)^4)/(64*d) + (tan(c/2 + (d*x)/2)^2*((3*a*b^2)/8 + a^3/8))/d - (atan(((3*a^2*b + (3*b^3)/2)*(tan(c/2 + (d*x)/2)*(9*a*b^2 + (3*a^3)/4) + 6*tan(c/2 + (d*x)/2)*(3*a^2*b + (3*b^3)/2) - 6*a^2*b - 3*b^3)*1i - (3*a^2*b + (3*b^3)/2)*(6*tan(c/2 + (d*x)/2)*(3*a^2*b + (3*b^3)/2) - tan(c/2 + (d*x)/2)*(9*a*b^2 + (3*a^3)/4) + 6*a^2*b + 3*b^3)*1i)/(2*tan(c/2 + (d*x)/2)*(9*b^6 + 36*a^2*b^4 + 36*a^4*b^2) + 27*a*b^5 + (9*a^5*b)/2 - (3*a^2*b + (3*b^3)/2)*(tan(c/2 + (d*x)/2)*(9*a*b^2 + (3*a^3)/4) + 6*tan(c/2 + (d*x)/2)*(3*a^2*b + (3*b^3)/2) - 6*a^2*b - 3*b^3) - (3*a^2*b + (3*b^3)/2)*(6*tan(c/2 + (d*x)/2)*(3*a^2*b + (3*b^3)/2) - tan(c/2 + (d*x)/2)*(9*a*b^2 + (3*a^3)/4) + 6*a^2*b + 3*b^3) + (225*a^3*b^3)/4))*(a^2*b*6i + b^3*3i))/d - (tan(c/2 + (d*x)/2)^2*(6*a*b^2 + (3*a^3)/2) + tan(c/2 + (d*x)/2)^6*(102*a*b^2 + 2*a^3) - tan(c/2 + (d*x)/2)^4*(108*a*b^2 + (15*a^3)/4) + tan(c/2 + (d*x)/2)^3*(26*a^2*b + 8*b^3) + tan(c/2 + (d*x)/2)^7*(30*a^2*b - 8*b^3) - tan(c/2 + (d*x)/2)^5*(58*a^2*b + 32*b^3) + a^3/4 + 2*a^2*b*tan(c/2 + (d*x)/2))/(d*(16*tan(c/2 + (d*x)/2)^4 - 32*tan(c/2 + (d*x)/2)^6 + 16*tan(c/2 + (d*x)/2)^8)) - (tan(c/2 + (d*x)/2)*((15*a^2*b)/8 + b^3/2))/d + (3*a*log(tan(c/2 + (d*x)/2))*(a^2 + 12*b^2))/(8*d) - (a^2*b*tan(c/2 + (d*x)/2)^3)/(8*d)","B"
40,1,146,167,3.876917,"\text{Not used}","int((a + b*tan(c + d*x))^3/sin(c + d*x)^6,x)","\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)\,\left(3\,a^2\,b+2\,b^3\right)}{d}-\frac{{\mathrm{cot}\left(c+d\,x\right)}^5\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(\frac{2\,a^3}{3}+a\,b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(3\,a^2\,b+\frac{b^3}{2}\right)+\frac{a^3}{5}+{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^3+6\,a\,b^2\right)+\frac{3\,a^2\,b\,\mathrm{tan}\left(c+d\,x\right)}{4}\right)}{d}+\frac{b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{2\,d}+\frac{3\,a\,b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}","Not used",1,"(log(tan(c + d*x))*(3*a^2*b + 2*b^3))/d - (cot(c + d*x)^5*(tan(c + d*x)^2*(a*b^2 + (2*a^3)/3) + tan(c + d*x)^3*(3*a^2*b + b^3/2) + a^3/5 + tan(c + d*x)^4*(6*a*b^2 + a^3) + (3*a^2*b*tan(c + d*x))/4))/d + (b^3*tan(c + d*x)^2)/(2*d) + (3*a*b^2*tan(c + d*x))/d","B"
41,1,319,275,7.213283,"\text{Not used}","int(sin(c + d*x)^3*(a + b*tan(c + d*x))^4,x)","-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(8\,a^4-96\,a^2\,b^2+32\,b^4\right)+4\,a^4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8+\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(20\,a\,b^3-8\,a^3\,b\right)-\frac{4\,a^4}{3}-\frac{32\,b^4}{3}+32\,a^2\,b^2-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\,\left(\frac{32\,a^4}{3}-64\,a^2\,b^2\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(\frac{20\,a\,b^3}{3}-\frac{8\,a^3\,b}{3}\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{11}\,\left(20\,a\,b^3-8\,a^3\,b\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^9\,\left(\frac{20\,a\,b^3}{3}-\frac{8\,a^3\,b}{3}\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(56\,a\,b^3-48\,a^3\,b\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(56\,a\,b^3-48\,a^3\,b\right)}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{12}-3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8+3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4-1\right)}-\frac{\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(20\,a\,b^3-8\,a^3\,b\right)}{d}","Not used",1,"- (tan(c/2 + (d*x)/2)^4*(8*a^4 + 32*b^4 - 96*a^2*b^2) + 4*a^4*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)*(20*a*b^3 - 8*a^3*b) - (4*a^4)/3 - (32*b^4)/3 + 32*a^2*b^2 - tan(c/2 + (d*x)/2)^6*((32*a^4)/3 - 64*a^2*b^2) + tan(c/2 + (d*x)/2)^3*((20*a*b^3)/3 - (8*a^3*b)/3) - tan(c/2 + (d*x)/2)^11*(20*a*b^3 - 8*a^3*b) - tan(c/2 + (d*x)/2)^9*((20*a*b^3)/3 - (8*a^3*b)/3) - tan(c/2 + (d*x)/2)^5*(56*a*b^3 - 48*a^3*b) + tan(c/2 + (d*x)/2)^7*(56*a*b^3 - 48*a^3*b))/(d*(3*tan(c/2 + (d*x)/2)^4 - 3*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^12 - 1)) - (atanh(tan(c/2 + (d*x)/2))*(20*a*b^3 - 8*a^3*b))/d","B"
42,1,161,139,3.793857,"\text{Not used}","int(sin(c + d*x)^2*(a + b*tan(c + d*x))^4,x)","x\,\left(\frac{a^4}{2}-9\,a^2\,b^2+\frac{5\,b^4}{2}\right)-\frac{\ln\left({\mathrm{tan}\left(c+d\,x\right)}^2+1\right)\,\left(4\,a\,b^3-2\,a^3\,b\right)}{d}-\frac{{\cos\left(c+d\,x\right)}^2\,\left(\mathrm{tan}\left(c+d\,x\right)\,\left(\frac{a^4}{2}-3\,a^2\,b^2+\frac{b^4}{2}\right)+2\,a\,b^3-2\,a^3\,b\right)}{d}+\frac{b^4\,{\mathrm{tan}\left(c+d\,x\right)}^3}{3\,d}-\frac{\mathrm{tan}\left(c+d\,x\right)\,\left(2\,b^4-6\,a^2\,b^2\right)}{d}+\frac{2\,a\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{d}","Not used",1,"x*(a^4/2 + (5*b^4)/2 - 9*a^2*b^2) - (log(tan(c + d*x)^2 + 1)*(4*a*b^3 - 2*a^3*b))/d - (cos(c + d*x)^2*(tan(c + d*x)*(a^4/2 + b^4/2 - 3*a^2*b^2) + 2*a*b^3 - 2*a^3*b))/d + (b^4*tan(c + d*x)^3)/(3*d) - (tan(c + d*x)*(2*b^4 - 6*a^2*b^2))/d + (2*a*b^3*tan(c + d*x)^2)/d","B"
43,1,268,180,7.285173,"\text{Not used}","int(sin(c + d*x)*(a + b*tan(c + d*x))^4,x)","-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(6\,a^4-48\,a^2\,b^2+\frac{32\,b^4}{3}\right)+2\,a^4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(12\,a\,b^3-8\,a^3\,b\right)-2\,a^4-\frac{16\,b^4}{3}+24\,a^2\,b^2-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(6\,a^4-24\,a^2\,b^2\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(12\,a\,b^3-8\,a^3\,b\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(20\,a\,b^3-24\,a^3\,b\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(20\,a\,b^3-24\,a^3\,b\right)}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8-2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2-1\right)}-\frac{\mathrm{atanh}\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(12\,a\,b^3-8\,a^3\,b\right)}{d}","Not used",1,"- (tan(c/2 + (d*x)/2)^2*(6*a^4 + (32*b^4)/3 - 48*a^2*b^2) + 2*a^4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)*(12*a*b^3 - 8*a^3*b) - 2*a^4 - (16*b^4)/3 + 24*a^2*b^2 - tan(c/2 + (d*x)/2)^4*(6*a^4 - 24*a^2*b^2) - tan(c/2 + (d*x)/2)^7*(12*a*b^3 - 8*a^3*b) - tan(c/2 + (d*x)/2)^3*(20*a*b^3 - 24*a^3*b) + tan(c/2 + (d*x)/2)^5*(20*a*b^3 - 24*a^3*b))/(d*(2*tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 - 1)) - (atanh(tan(c/2 + (d*x)/2))*(12*a*b^3 - 8*a^3*b))/d","B"
44,1,496,118,4.143902,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x),x)","\frac{a^4\,\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)}{d}-\frac{12\,a^2\,b^2-\frac{4\,b^4}{3}+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(4\,b^4-24\,a^2\,b^2\right)+12\,a^2\,b^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+4\,a\,b^3\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)-4\,a\,b^3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6-3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2-1\right)}-\frac{a\,b\,\mathrm{atan}\left(\frac{a\,b\,\left(2\,a^2-b^2\right)\,\left(4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)-12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2-b^2\right)\right)\,2{}\mathrm{i}+a\,b\,\left(2\,a^2-b^2\right)\,\left(4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)+12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2-b^2\right)\right)\,2{}\mathrm{i}}{2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(64\,a^6\,b^2-64\,a^4\,b^4+16\,a^2\,b^6\right)+16\,a^7\,b-8\,a^5\,b^3+2\,a\,b\,\left(2\,a^2-b^2\right)\,\left(4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)-12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2-b^2\right)\right)-2\,a\,b\,\left(2\,a^2-b^2\right)\,\left(4\,a\,b^3-8\,a^3\,b+2\,a^4\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)+12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2-b^2\right)\right)}\right)\,\left(2\,a^2-b^2\right)\,4{}\mathrm{i}}{d}","Not used",1,"(a^4*log(tan(c/2 + (d*x)/2)))/d - (12*a^2*b^2 - (4*b^4)/3 + tan(c/2 + (d*x)/2)^2*(4*b^4 - 24*a^2*b^2) + 12*a^2*b^2*tan(c/2 + (d*x)/2)^4 + 4*a*b^3*tan(c/2 + (d*x)/2) - 4*a*b^3*tan(c/2 + (d*x)/2)^5)/(d*(3*tan(c/2 + (d*x)/2)^2 - 3*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 - 1)) - (a*b*atan((a*b*(2*a^2 - b^2)*(4*a*b^3 - 8*a^3*b + 2*a^4*tan(c/2 + (d*x)/2) - 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2))*2i + a*b*(2*a^2 - b^2)*(4*a*b^3 - 8*a^3*b + 2*a^4*tan(c/2 + (d*x)/2) + 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2))*2i)/(2*tan(c/2 + (d*x)/2)*(16*a^2*b^6 - 64*a^4*b^4 + 64*a^6*b^2) + 16*a^7*b - 8*a^5*b^3 + 2*a*b*(2*a^2 - b^2)*(4*a*b^3 - 8*a^3*b + 2*a^4*tan(c/2 + (d*x)/2) - 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2)) - 2*a*b*(2*a^2 - b^2)*(4*a*b^3 - 8*a^3*b + 2*a^4*tan(c/2 + (d*x)/2) + 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 - b^2))))*(2*a^2 - b^2)*4i)/d","B"
45,1,81,83,3.662552,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x)^2,x)","\frac{b^4\,{\mathrm{tan}\left(c+d\,x\right)}^3}{3\,d}-\frac{a^4\,\mathrm{cot}\left(c+d\,x\right)}{d}+\frac{6\,a^2\,b^2\,\mathrm{tan}\left(c+d\,x\right)}{d}+\frac{2\,a\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{d}+\frac{4\,a^3\,b\,\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)}{d}","Not used",1,"(b^4*tan(c + d*x)^3)/(3*d) - (a^4*cot(c + d*x))/d + (6*a^2*b^2*tan(c + d*x))/d + (2*a*b^3*tan(c + d*x)^2)/d + (4*a^3*b*log(tan(c + d*x)))/d","B"
46,1,670,161,4.068819,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x)^3,x)","\frac{a^4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}{8\,d}-\frac{\frac{a^4}{2}-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{3\,a^4}{2}+48\,a^2\,b^2+\frac{8\,b^4}{3}\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\,\left(\frac{a^4}{2}+48\,a^2\,b^2+8\,b^4\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(\frac{3\,a^4}{2}+96\,a^2\,b^2\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(16\,a\,b^3-8\,a^3\,b\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(24\,a^3\,b+16\,a\,b^3\right)+8\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)+24\,a^3\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5}{d\,\left(-4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8+12\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6-12\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\right)}+\frac{\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(\frac{a^4}{2}+6\,a^2\,b^2\right)}{d}-\frac{2\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{d}-\frac{a\,b\,\mathrm{atan}\left(\frac{a\,b\,\left(2\,a^2+b^2\right)\,\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^4+12\,a^2\,b^2\right)-4\,a\,b^3-8\,a^3\,b+12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2+b^2\right)\right)\,2{}\mathrm{i}-a\,b\,\left(2\,a^2+b^2\right)\,\left(4\,a\,b^3-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^4+12\,a^2\,b^2\right)+8\,a^3\,b+12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2+b^2\right)\right)\,2{}\mathrm{i}}{2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(64\,a^6\,b^2+64\,a^4\,b^4+16\,a^2\,b^6\right)+8\,a^7\,b+48\,a^3\,b^5+100\,a^5\,b^3-2\,a\,b\,\left(2\,a^2+b^2\right)\,\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^4+12\,a^2\,b^2\right)-4\,a\,b^3-8\,a^3\,b+12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2+b^2\right)\right)-2\,a\,b\,\left(2\,a^2+b^2\right)\,\left(4\,a\,b^3-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^4+12\,a^2\,b^2\right)+8\,a^3\,b+12\,a\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^2+b^2\right)\right)}\right)\,\left(2\,a^2+b^2\right)\,4{}\mathrm{i}}{d}","Not used",1,"(a^4*tan(c/2 + (d*x)/2)^2)/(8*d) - (a^4/2 - tan(c/2 + (d*x)/2)^2*((3*a^4)/2 + (8*b^4)/3 + 48*a^2*b^2) - tan(c/2 + (d*x)/2)^6*(a^4/2 + 8*b^4 + 48*a^2*b^2) + tan(c/2 + (d*x)/2)^4*((3*a^4)/2 + 96*a^2*b^2) + tan(c/2 + (d*x)/2)^7*(16*a*b^3 - 8*a^3*b) - tan(c/2 + (d*x)/2)^3*(16*a*b^3 + 24*a^3*b) + 8*a^3*b*tan(c/2 + (d*x)/2) + 24*a^3*b*tan(c/2 + (d*x)/2)^5)/(d*(4*tan(c/2 + (d*x)/2)^2 - 12*tan(c/2 + (d*x)/2)^4 + 12*tan(c/2 + (d*x)/2)^6 - 4*tan(c/2 + (d*x)/2)^8)) + (log(tan(c/2 + (d*x)/2))*(a^4/2 + 6*a^2*b^2))/d - (2*a^3*b*tan(c/2 + (d*x)/2))/d - (a*b*atan((a*b*(2*a^2 + b^2)*(tan(c/2 + (d*x)/2)*(a^4 + 12*a^2*b^2) - 4*a*b^3 - 8*a^3*b + 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 + b^2))*2i - a*b*(2*a^2 + b^2)*(4*a*b^3 - tan(c/2 + (d*x)/2)*(a^4 + 12*a^2*b^2) + 8*a^3*b + 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 + b^2))*2i)/(2*tan(c/2 + (d*x)/2)*(16*a^2*b^6 + 64*a^4*b^4 + 64*a^6*b^2) + 8*a^7*b + 48*a^3*b^5 + 100*a^5*b^3 - 2*a*b*(2*a^2 + b^2)*(tan(c/2 + (d*x)/2)*(a^4 + 12*a^2*b^2) - 4*a*b^3 - 8*a^3*b + 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 + b^2)) - 2*a*b*(2*a^2 + b^2)*(4*a*b^3 - tan(c/2 + (d*x)/2)*(a^4 + 12*a^2*b^2) + 8*a^3*b + 12*a*b*tan(c/2 + (d*x)/2)*(2*a^2 + b^2))))*(2*a^2 + b^2)*4i)/d","B"
47,1,132,137,3.761361,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x)^4,x)","\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)\,\left(4\,a^3\,b+4\,a\,b^3\right)}{d}-\frac{{\mathrm{cot}\left(c+d\,x\right)}^3\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^4+6\,a^2\,b^2\right)+\frac{a^4}{3}+2\,a^3\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{d}+\frac{\mathrm{tan}\left(c+d\,x\right)\,\left(6\,a^2\,b^2+b^4\right)}{d}+\frac{b^4\,{\mathrm{tan}\left(c+d\,x\right)}^3}{3\,d}+\frac{2\,a\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{d}","Not used",1,"(log(tan(c + d*x))*(4*a*b^3 + 4*a^3*b))/d - (cot(c + d*x)^3*(tan(c + d*x)^2*(a^4 + 6*a^2*b^2) + a^4/3 + 2*a^3*b*tan(c + d*x)))/d + (tan(c + d*x)*(b^4 + 6*a^2*b^2))/d + (b^4*tan(c + d*x)^3)/(3*d) + (2*a*b^3*tan(c + d*x)^2)/d","B"
48,1,857,274,4.171067,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x)^5,x)","\frac{a^4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4}{64\,d}-\frac{\mathrm{atan}\left(-\frac{\left(4\,a^3\,b+6\,a\,b^3\right)\,\left(12\,a\,b^3+8\,a^3\,b-6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+6\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right)\right)\,1{}\mathrm{i}+\left(4\,a^3\,b+6\,a\,b^3\right)\,\left(12\,a\,b^3+8\,a^3\,b+6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+6\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right)\right)\,1{}\mathrm{i}}{2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(64\,a^6\,b^2+192\,a^4\,b^4+144\,a^2\,b^6\right)+\left(4\,a^3\,b+6\,a\,b^3\right)\,\left(12\,a\,b^3+8\,a^3\,b-6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+6\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right)\right)-\left(4\,a^3\,b+6\,a\,b^3\right)\,\left(12\,a\,b^3+8\,a^3\,b+6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+6\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right)\right)+24\,a\,b^7+6\,a^7\,b+232\,a^3\,b^5+153\,a^5\,b^3}\right)\,\left(a^3\,b\,8{}\mathrm{i}+a\,b^3\,12{}\mathrm{i}\right)}{d}-\frac{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(2\,a^3\,b+\frac{a\,b\,\left(a^2+4\,b^2\right)}{2}\right)}{d}+\frac{\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(\frac{3\,a^4}{8}+9\,a^2\,b^2+b^4\right)}{d}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{a^4}{8}+\frac{3\,a^2\,b^2}{4}\right)}{d}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\,\left(\frac{23\,a^4}{4}+420\,a^2\,b^2+64\,b^4\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(\frac{21\,a^4}{4}+228\,a^2\,b^2+\frac{128\,b^4}{3}\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8\,\left(2\,a^4+204\,a^2\,b^2+64\,b^4\right)+\frac{a^4}{4}+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{5\,a^4}{4}+12\,a^2\,b^2\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(32\,a^3\,b+32\,a\,b^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^9\,\left(32\,a\,b^3-40\,a^3\,b\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(112\,a^3\,b+160\,a\,b^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(\frac{352\,a^3\,b}{3}+96\,a\,b^3\right)+\frac{8\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{3}}{d\,\left(-16\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{10}+48\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8-48\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+16\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\right)}-\frac{a^3\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3}{6\,d}","Not used",1,"(a^4*tan(c/2 + (d*x)/2)^4)/(64*d) - (atan(-((6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2))*1i + (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2))*1i)/(2*tan(c/2 + (d*x)/2)*(144*a^2*b^6 + 192*a^4*b^4 + 64*a^6*b^2) + (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2)) - (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2)) + 24*a*b^7 + 6*a^7*b + 232*a^3*b^5 + 153*a^5*b^3))*(a*b^3*12i + a^3*b*8i))/d - (tan(c/2 + (d*x)/2)*(2*a^3*b + (a*b*(a^2 + 4*b^2))/2))/d + (log(tan(c/2 + (d*x)/2))*((3*a^4)/8 + b^4 + 9*a^2*b^2))/d + (tan(c/2 + (d*x)/2)^2*(a^4/8 + (3*a^2*b^2)/4))/d - (tan(c/2 + (d*x)/2)^6*((23*a^4)/4 + 64*b^4 + 420*a^2*b^2) - tan(c/2 + (d*x)/2)^4*((21*a^4)/4 + (128*b^4)/3 + 228*a^2*b^2) - tan(c/2 + (d*x)/2)^8*(2*a^4 + 64*b^4 + 204*a^2*b^2) + a^4/4 + tan(c/2 + (d*x)/2)^2*((5*a^4)/4 + 12*a^2*b^2) + tan(c/2 + (d*x)/2)^3*(32*a*b^3 + 32*a^3*b) + tan(c/2 + (d*x)/2)^9*(32*a*b^3 - 40*a^3*b) - tan(c/2 + (d*x)/2)^5*(160*a*b^3 + 112*a^3*b) + tan(c/2 + (d*x)/2)^7*(96*a*b^3 + (352*a^3*b)/3) + (8*a^3*b*tan(c/2 + (d*x)/2))/3)/(d*(16*tan(c/2 + (d*x)/2)^4 - 48*tan(c/2 + (d*x)/2)^6 + 48*tan(c/2 + (d*x)/2)^8 - 16*tan(c/2 + (d*x)/2)^10)) - (a^3*b*tan(c/2 + (d*x)/2)^3)/(6*d)","B"
49,1,181,194,3.827144,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x)^6,x)","\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)\right)\,\left(4\,a^3\,b+8\,a\,b^3\right)}{d}-\frac{{\mathrm{cot}\left(c+d\,x\right)}^5\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(\frac{2\,a^4}{3}+2\,a^2\,b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(4\,a^3\,b+2\,a\,b^3\right)+\frac{a^4}{5}+{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^4+12\,a^2\,b^2+b^4\right)+a^3\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{d}+\frac{b^4\,{\mathrm{tan}\left(c+d\,x\right)}^3}{3\,d}+\frac{\mathrm{tan}\left(c+d\,x\right)\,\left(6\,a^2\,b^2+2\,b^4\right)}{d}+\frac{2\,a\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^2}{d}","Not used",1,"(log(tan(c + d*x))*(8*a*b^3 + 4*a^3*b))/d - (cot(c + d*x)^5*(tan(c + d*x)^2*((2*a^4)/3 + 2*a^2*b^2) + tan(c + d*x)^3*(2*a*b^3 + 4*a^3*b) + a^4/5 + tan(c + d*x)^4*(a^4 + b^4 + 12*a^2*b^2) + a^3*b*tan(c + d*x)))/d + (b^4*tan(c + d*x)^3)/(3*d) + (tan(c + d*x)*(2*b^4 + 6*a^2*b^2))/d + (2*a*b^3*tan(c + d*x)^2)/d","B"
50,1,990,402,4.034712,"\text{Not used}","int((a + b*tan(c + d*x))^4/sin(c + d*x)^7,x)","\frac{a^4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6}{384\,d}-\frac{\mathrm{atan}\left(-\frac{\left(4\,a^3\,b+10\,a\,b^3\right)\,\left(20\,a\,b^3+8\,a^3\,b-6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+10\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{5\,a^4}{8}+\frac{45\,a^2\,b^2}{2}+5\,b^4\right)\right)\,1{}\mathrm{i}+\left(4\,a^3\,b+10\,a\,b^3\right)\,\left(20\,a\,b^3+8\,a^3\,b+6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+10\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{5\,a^4}{8}+\frac{45\,a^2\,b^2}{2}+5\,b^4\right)\right)\,1{}\mathrm{i}}{2\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(64\,a^6\,b^2+320\,a^4\,b^4+400\,a^2\,b^6\right)+\left(4\,a^3\,b+10\,a\,b^3\right)\,\left(20\,a\,b^3+8\,a^3\,b-6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+10\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{5\,a^4}{8}+\frac{45\,a^2\,b^2}{2}+5\,b^4\right)\right)-\left(4\,a^3\,b+10\,a\,b^3\right)\,\left(20\,a\,b^3+8\,a^3\,b+6\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(4\,a^3\,b+10\,a\,b^3\right)-\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{5\,a^4}{8}+\frac{45\,a^2\,b^2}{2}+5\,b^4\right)\right)+100\,a\,b^7+5\,a^7\,b+490\,a^3\,b^5+\frac{385\,a^5\,b^3}{2}}\right)\,\left(a^3\,b\,8{}\mathrm{i}+a\,b^3\,20{}\mathrm{i}\right)}{d}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(\frac{a^2\,\left(a^2+12\,b^2\right)}{128}+\frac{a^4}{64}\right)}{d}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(\frac{a^2\,\left(a^2+12\,b^2\right)}{16}+\frac{7\,a^4}{128}+\frac{b^4}{8}+\frac{3\,a^2\,b^2}{4}\right)}{d}-\frac{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(\frac{11\,a^3\,b}{4}+\frac{9\,a\,b^3}{2}\right)}{d}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(\frac{7\,a^4}{2}+78\,a^2\,b^2+8\,b^4\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{10}\,\left(\frac{15\,a^4}{2}+864\,a^2\,b^2+392\,b^4\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\,\left(\frac{109\,a^4}{6}+1038\,a^2\,b^2+\frac{968\,b^4}{3}\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8\,\left(21\,a^4+1818\,a^2\,b^2+536\,b^4\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(a^4+6\,a^2\,b^2\right)+\frac{a^4}{6}-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{11}\,\left(176\,a^3\,b+32\,a\,b^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(\frac{208\,a^3\,b}{15}+\frac{32\,a\,b^3}{3}\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(\frac{624\,a^3\,b}{5}+256\,a\,b^3\right)-{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(\frac{2368\,a^3\,b}{5}+1088\,a\,b^3\right)+{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^9\,\left(\frac{1528\,a^3\,b}{3}+\frac{2560\,a\,b^3}{3}\right)+\frac{8\,a^3\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{5}}{d\,\left(-64\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{12}+192\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{10}-192\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8+64\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\right)}-\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(\frac{7\,a^3\,b}{24}+\frac{a\,b^3}{6}\right)}{d}+\frac{\ln\left(\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\right)\,\left(\frac{5\,a^4}{16}+\frac{45\,a^2\,b^2}{4}+\frac{5\,b^4}{2}\right)}{d}-\frac{a^3\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5}{40\,d}","Not used",1,"(a^4*tan(c/2 + (d*x)/2)^6)/(384*d) - (atan(-((10*a*b^3 + 4*a^3*b)*(20*a*b^3 + 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2))*1i + (10*a*b^3 + 4*a^3*b)*(20*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2))*1i)/(2*tan(c/2 + (d*x)/2)*(400*a^2*b^6 + 320*a^4*b^4 + 64*a^6*b^2) + (10*a*b^3 + 4*a^3*b)*(20*a*b^3 + 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2)) - (10*a*b^3 + 4*a^3*b)*(20*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(10*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((5*a^4)/8 + 5*b^4 + (45*a^2*b^2)/2)) + 100*a*b^7 + 5*a^7*b + 490*a^3*b^5 + (385*a^5*b^3)/2))*(a*b^3*20i + a^3*b*8i))/d + (tan(c/2 + (d*x)/2)^4*((a^2*(a^2 + 12*b^2))/128 + a^4/64))/d + (tan(c/2 + (d*x)/2)^2*((a^2*(a^2 + 12*b^2))/16 + (7*a^4)/128 + b^4/8 + (3*a^2*b^2)/4))/d - (tan(c/2 + (d*x)/2)*((9*a*b^3)/2 + (11*a^3*b)/4))/d - (tan(c/2 + (d*x)/2)^4*((7*a^4)/2 + 8*b^4 + 78*a^2*b^2) - tan(c/2 + (d*x)/2)^10*((15*a^4)/2 + 392*b^4 + 864*a^2*b^2) - tan(c/2 + (d*x)/2)^6*((109*a^4)/6 + (968*b^4)/3 + 1038*a^2*b^2) + tan(c/2 + (d*x)/2)^8*(21*a^4 + 536*b^4 + 1818*a^2*b^2) + tan(c/2 + (d*x)/2)^2*(a^4 + 6*a^2*b^2) + a^4/6 - tan(c/2 + (d*x)/2)^11*(32*a*b^3 + 176*a^3*b) + tan(c/2 + (d*x)/2)^3*((32*a*b^3)/3 + (208*a^3*b)/15) + tan(c/2 + (d*x)/2)^5*(256*a*b^3 + (624*a^3*b)/5) - tan(c/2 + (d*x)/2)^7*(1088*a*b^3 + (2368*a^3*b)/5) + tan(c/2 + (d*x)/2)^9*((2560*a*b^3)/3 + (1528*a^3*b)/3) + (8*a^3*b*tan(c/2 + (d*x)/2))/5)/(d*(64*tan(c/2 + (d*x)/2)^6 - 192*tan(c/2 + (d*x)/2)^8 + 192*tan(c/2 + (d*x)/2)^10 - 64*tan(c/2 + (d*x)/2)^12)) - (tan(c/2 + (d*x)/2)^3*((a*b^3)/6 + (7*a^3*b)/24))/d + (log(tan(c/2 + (d*x)/2))*((5*a^4)/16 + (5*b^4)/2 + (45*a^2*b^2)/4))/d - (a^3*b*tan(c/2 + (d*x)/2)^5)/(40*d)","B"
51,1,683,274,6.830027,"\text{Not used}","int(sin(c + d*x)^5/(a + b*tan(c + d*x)),x)","\frac{\frac{2\,\left(-8\,a^5+9\,a^3\,b^2+2\,a\,b^4\right)}{15\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{8\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(4\,a^4\,b+a^2\,b^3\right)}{3\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6\,\left(3\,a^3\,b^2+a\,b^4\right)}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac{4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2\,\left(-4\,a^5+3\,a^3\,b^2+a\,b^4\right)}{3\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5\,\left(89\,a^4\,b+68\,a^2\,b^3+24\,b^5\right)}{15\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}-\frac{4\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4\,\left(8\,a^5+a\,b^4\right)}{3\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{2\,a^3\,b^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac{8\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^7\,\left(4\,a^4+a^2\,b^2\right)}{3\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{2\,a^4\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac{2\,a^4\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^9}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^{10}+5\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^8+10\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+10\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+5\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+1\right)}+\frac{2\,a^5\,b\,\mathrm{atanh}\left(\frac{2\,a^6\,b+2\,b^7+6\,a^2\,b^5+6\,a^4\,b^3-2\,a\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}{2\,{\left(a^2+b^2\right)}^{7/2}}\right)}{d\,{\left(a^2+b^2\right)}^{7/2}}","Not used",1,"((2*(2*a*b^4 - 8*a^5 + 9*a^3*b^2))/(15*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (8*tan(c/2 + (d*x)/2)^3*(4*a^4*b + a^2*b^3))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^6*(a*b^4 + 3*a^3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*tan(c/2 + (d*x)/2)^2*(a*b^4 - 4*a^5 + 3*a^3*b^2))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^5*(89*a^4*b + 24*b^5 + 68*a^2*b^3))/(15*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*tan(c/2 + (d*x)/2)^4*(a*b^4 + 8*a^5))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*a^3*b^2*tan(c/2 + (d*x)/2)^8)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (8*b*tan(c/2 + (d*x)/2)^7*(4*a^4 + a^2*b^2))/(3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*a^4*b*tan(c/2 + (d*x)/2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*a^4*b*tan(c/2 + (d*x)/2)^9)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (2*a^5*b*atanh((2*a^6*b + 2*b^7 + 6*a^2*b^5 + 6*a^4*b^3 - 2*a*tan(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(2*(a^2 + b^2)^(7/2))))/(d*(a^2 + b^2)^(7/2))","B"
52,1,313,158,4.269229,"\text{Not used}","int(sin(c + d*x)^4/(a + b*tan(c + d*x)),x)","\frac{a^4\,b\,\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}{d\,{\left(a^2+b^2\right)}^3}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(a\,b-a^2\,3{}\mathrm{i}\right)}{16\,d\,\left(-a^3-a^2\,b\,3{}\mathrm{i}+3\,a\,b^2+b^3\,1{}\mathrm{i}\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(-3\,a^2+a\,b\,1{}\mathrm{i}\right)}{16\,d\,\left(-a^3\,1{}\mathrm{i}-3\,a^2\,b+a\,b^2\,3{}\mathrm{i}+b^3\right)}-\frac{\frac{3\,a^2\,b+b^3}{4\,\left(a^4+2\,a^2\,b^2+b^4\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(5\,a^3+a\,b^2\right)}{8\,\left(a^4+2\,a^2\,b^2+b^4\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(2\,a^2\,b+b^3\right)}{2\,\left(a^4+2\,a^2\,b^2+b^4\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(3\,a^2-b^2\right)}{8\,\left(a^4+2\,a^2\,b^2+b^4\right)}}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^4+2\,{\mathrm{tan}\left(c+d\,x\right)}^2+1\right)}","Not used",1,"(a^4*b*log(a + b*tan(c + d*x)))/(d*(a^2 + b^2)^3) - (log(tan(c + d*x) - 1i)*(a*b - a^2*3i))/(16*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) - (log(tan(c + d*x) + 1i)*(a*b*1i - 3*a^2))/(16*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) - ((3*a^2*b + b^3)/(4*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)^3*(a*b^2 + 5*a^3))/(8*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)^2*(2*a^2*b + b^3))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (a*tan(c + d*x)*(3*a^2 - b^2))/(8*(a^4 + b^4 + 2*a^2*b^2)))/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1))","B"
53,1,324,168,6.496746,"\text{Not used}","int(sin(c + d*x)^3/(a + b*tan(c + d*x)),x)","\frac{\frac{\frac{2\,a\,b^2}{3}-\frac{4\,a^3}{3}}{a^4+2\,a^2\,b^2+b^4}-\frac{4\,a^3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2}{a^4+2\,a^2\,b^2+b^4}+\frac{{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^3\,\left(\frac{20\,a^2\,b}{3}+\frac{8\,b^3}{3}\right)}{a^4+2\,a^2\,b^2+b^4}+\frac{2\,a^2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{a^4+2\,a^2\,b^2+b^4}+\frac{2\,a\,b^2\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4}{a^4+2\,a^2\,b^2+b^4}+\frac{2\,a^2\,b\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^5}{a^4+2\,a^2\,b^2+b^4}}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^6+3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^4+3\,{\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+1\right)}+\frac{2\,a^3\,b\,\mathrm{atanh}\left(\frac{a^4\,b+b^5+2\,a^2\,b^3-a\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^4+2\,a^2\,b^2+b^4\right)}{{\left(a^2+b^2\right)}^{5/2}}\right)}{d\,{\left(a^2+b^2\right)}^{5/2}}","Not used",1,"(((2*a*b^2)/3 - (4*a^3)/3)/(a^4 + b^4 + 2*a^2*b^2) - (4*a^3*tan(c/2 + (d*x)/2)^2)/(a^4 + b^4 + 2*a^2*b^2) + (tan(c/2 + (d*x)/2)^3*((20*a^2*b)/3 + (8*b^3)/3))/(a^4 + b^4 + 2*a^2*b^2) + (2*a^2*b*tan(c/2 + (d*x)/2))/(a^4 + b^4 + 2*a^2*b^2) + (2*a*b^2*tan(c/2 + (d*x)/2)^4)/(a^4 + b^4 + 2*a^2*b^2) + (2*a^2*b*tan(c/2 + (d*x)/2)^5)/(a^4 + b^4 + 2*a^2*b^2))/(d*(3*tan(c/2 + (d*x)/2)^2 + 3*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 + 1)) + (2*a^3*b*atanh((a^4*b + b^5 + 2*a^2*b^3 - a*tan(c/2 + (d*x)/2)*(a^4 + b^4 + 2*a^2*b^2))/(a^2 + b^2)^(5/2)))/(d*(a^2 + b^2)^(5/2))","B"
54,1,147,94,3.865193,"\text{Not used}","int(sin(c + d*x)^2/(a + b*tan(c + d*x)),x)","\frac{a^2\,b\,\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}{d\,{\left(a^2+b^2\right)}^2}-\frac{a\,\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)}{4\,d\,\left(-a^2\,1{}\mathrm{i}+2\,a\,b+b^2\,1{}\mathrm{i}\right)}-\frac{{\cos\left(c+d\,x\right)}^2\,\left(\frac{b}{2\,\left(a^2+b^2\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)}{2\,\left(a^2+b^2\right)}\right)}{d}-\frac{a\,\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,1{}\mathrm{i}}{4\,d\,\left(-a^2+a\,b\,2{}\mathrm{i}+b^2\right)}","Not used",1,"(a^2*b*log(a + b*tan(c + d*x)))/(d*(a^2 + b^2)^2) - (a*log(tan(c + d*x) + 1i)*1i)/(4*d*(a*b*2i - a^2 + b^2)) - (a*log(tan(c + d*x) - 1i))/(4*d*(2*a*b - a^2*1i + b^2*1i)) - (cos(c + d*x)^2*(b/(2*(a^2 + b^2)) + (a*tan(c + d*x))/(2*(a^2 + b^2))))/d","B"
55,1,110,90,3.913266,"\text{Not used}","int(sin(c + d*x)/(a + b*tan(c + d*x)),x)","\frac{2\,a\,b\,\mathrm{atanh}\left(\frac{a^2\,b+b^3-a\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^2+b^2\right)}{{\left(a^2+b^2\right)}^{3/2}}\right)}{d\,{\left(a^2+b^2\right)}^{3/2}}-\frac{\frac{2\,a}{a^2+b^2}-\frac{2\,b\,\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{a^2+b^2}}{d\,\left({\mathrm{tan}\left(\frac{c}{2}+\frac{d\,x}{2}\right)}^2+1\right)}","Not used",1,"(2*a*b*atanh((a^2*b + b^3 - a*tan(c/2 + (d*x)/2)*(a^2 + b^2))/(a^2 + b^2)^(3/2)))/(d*(a^2 + b^2)^(3/2)) - ((2*a)/(a^2 + b^2) - (2*b*tan(c/2 + (d*x)/2))/(a^2 + b^2))/(d*(tan(c/2 + (d*x)/2)^2 + 1))","B"
56,1,174,66,4.091822,"\text{Not used}","int(1/(sin(c + d*x)*(a + b*tan(c + d*x))),x)","\frac{\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{a\,d}-\frac{2\,b\,\mathrm{atanh}\left(\frac{\sqrt{a^2+b^2}\,\left(1{}\mathrm{i}\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2+2{}\mathrm{i}\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a\,b+4{}\mathrm{i}\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^2\right)}{b^3\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,4{}\mathrm{i}+a\,b^2\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,1{}\mathrm{i}+a^2\,b\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,3{}\mathrm{i}+a\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\left(a^2+b^2\right)\,1{}\mathrm{i}}\right)}{a\,d\,\sqrt{a^2+b^2}}","Not used",1,"log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))/(a*d) - (2*b*atanh(((a^2 + b^2)^(1/2)*(a^2*sin(c/2 + (d*x)/2)*1i + b^2*sin(c/2 + (d*x)/2)*4i + a*b*cos(c/2 + (d*x)/2)*2i))/(b^3*sin(c/2 + (d*x)/2)*4i + a*b^2*cos(c/2 + (d*x)/2)*1i + a^2*b*sin(c/2 + (d*x)/2)*3i + a*cos(c/2 + (d*x)/2)*(a^2 + b^2)*1i)))/(a*d*(a^2 + b^2)^(1/2))","B"
57,1,39,50,3.736921,"\text{Not used}","int(1/(sin(c + d*x)^2*(a + b*tan(c + d*x))),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{2\,b\,\mathrm{tan}\left(c+d\,x\right)}{a}+1\right)}{a^2\,d}-\frac{\mathrm{cot}\left(c+d\,x\right)}{a\,d}","Not used",1,"(2*b*atanh((2*b*tan(c + d*x))/a + 1))/(a^2*d) - cot(c + d*x)/(a*d)","B"
58,1,764,122,4.447580,"\text{Not used}","int(1/(sin(c + d*x)^3*(a + b*tan(c + d*x))),x)","\frac{b^2\,\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{2\,\left(\frac{a^3\,d}{2}-\frac{a^3\,d\,\cos\left(2\,c+2\,d\,x\right)}{2}\right)}-\frac{a^2\,\left(\frac{\cos\left(c+d\,x\right)}{2}-\frac{\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)}{4}+\frac{\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)\,\cos\left(2\,c+2\,d\,x\right)}{4}\right)}{\frac{a^3\,d}{2}-\frac{a^3\,d\,\cos\left(2\,c+2\,d\,x\right)}{2}}+\frac{a\,b\,\sin\left(c+d\,x\right)}{\frac{a^3\,d}{2}-\frac{a^3\,d\,\cos\left(2\,c+2\,d\,x\right)}{2}}-\frac{b^2\,\ln\left(\frac{\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)}\right)\,\cos\left(2\,c+2\,d\,x\right)}{2\,\left(\frac{a^3\,d}{2}-\frac{a^3\,d\,\cos\left(2\,c+2\,d\,x\right)}{2}\right)}+\frac{b\,\mathrm{atan}\left(\frac{a^4\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,1{}\mathrm{i}+b^4\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,8{}\mathrm{i}+a\,b^3\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,4{}\mathrm{i}+a^3\,b\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,3{}\mathrm{i}+a^2\,b^2\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,8{}\mathrm{i}}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^5+4\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^4\,b+5\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^3\,b^2+12\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^3+4\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a\,b^4+8\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^5}\right)\,\sqrt{a^2+b^2}\,1{}\mathrm{i}}{\frac{a^3\,d}{2}-\frac{a^3\,d\,\cos\left(2\,c+2\,d\,x\right)}{2}}-\frac{b\,\cos\left(2\,c+2\,d\,x\right)\,\mathrm{atan}\left(\frac{a^4\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,1{}\mathrm{i}+b^4\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,8{}\mathrm{i}+a\,b^3\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,4{}\mathrm{i}+a^3\,b\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,3{}\mathrm{i}+a^2\,b^2\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,\sqrt{a^2+b^2}\,8{}\mathrm{i}}{\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^5+4\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^4\,b+5\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^3\,b^2+12\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a^2\,b^3+4\,\cos\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,a\,b^4+8\,\sin\left(\frac{c}{2}+\frac{d\,x}{2}\right)\,b^5}\right)\,\sqrt{a^2+b^2}\,1{}\mathrm{i}}{\frac{a^3\,d}{2}-\frac{a^3\,d\,\cos\left(2\,c+2\,d\,x\right)}{2}}","Not used",1,"(b^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(2*((a^3*d)/2 - (a^3*d*cos(2*c + 2*d*x))/2)) - (a^2*(cos(c + d*x)/2 - log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))/4 + (log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x))/4))/((a^3*d)/2 - (a^3*d*cos(2*c + 2*d*x))/2) + (a*b*sin(c + d*x))/((a^3*d)/2 - (a^3*d*cos(2*c + 2*d*x))/2) - (b^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x))/(2*((a^3*d)/2 - (a^3*d*cos(2*c + 2*d*x))/2)) + (b*atan((a^4*sin(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*1i + b^4*sin(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*8i + a*b^3*cos(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*4i + a^3*b*cos(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*3i + a^2*b^2*sin(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*8i)/(a^5*cos(c/2 + (d*x)/2) + 8*b^5*sin(c/2 + (d*x)/2) + 4*a*b^4*cos(c/2 + (d*x)/2) + 4*a^4*b*sin(c/2 + (d*x)/2) + 5*a^3*b^2*cos(c/2 + (d*x)/2) + 12*a^2*b^3*sin(c/2 + (d*x)/2)))*(a^2 + b^2)^(1/2)*1i)/((a^3*d)/2 - (a^3*d*cos(2*c + 2*d*x))/2) - (b*cos(2*c + 2*d*x)*atan((a^4*sin(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*1i + b^4*sin(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*8i + a*b^3*cos(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*4i + a^3*b*cos(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*3i + a^2*b^2*sin(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2)*8i)/(a^5*cos(c/2 + (d*x)/2) + 8*b^5*sin(c/2 + (d*x)/2) + 4*a*b^4*cos(c/2 + (d*x)/2) + 4*a^4*b*sin(c/2 + (d*x)/2) + 5*a^3*b^2*cos(c/2 + (d*x)/2) + 12*a^2*b^3*sin(c/2 + (d*x)/2)))*(a^2 + b^2)^(1/2)*1i)/((a^3*d)/2 - (a^3*d*cos(2*c + 2*d*x))/2)","B"
59,1,102,108,3.842648,"\text{Not used}","int(1/(sin(c + d*x)^4*(a + b*tan(c + d*x))),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{b\,\left(a^2+b^2\right)\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{a\,\left(a^2\,b+b^3\right)}\right)\,\left(a^2+b^2\right)}{a^4\,d}-\frac{\frac{1}{3\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^2+b^2\right)}{a^3}-\frac{b\,\mathrm{tan}\left(c+d\,x\right)}{2\,a^2}}{d\,{\mathrm{tan}\left(c+d\,x\right)}^3}","Not used",1,"(2*b*atanh((b*(a^2 + b^2)*(a + 2*b*tan(c + d*x)))/(a*(a^2*b + b^3)))*(a^2 + b^2))/(a^4*d) - (1/(3*a) + (tan(c + d*x)^2*(a^2 + b^2))/a^3 - (b*tan(c + d*x))/(2*a^2))/(d*tan(c + d*x)^3)","B"
60,1,167,169,4.329225,"\text{Not used}","int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{b\,{\left(a^2+b^2\right)}^2\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{a\,\left(a^4\,b+2\,a^2\,b^3+b^5\right)}\right)\,{\left(a^2+b^2\right)}^2}{a^6\,d}-\frac{\frac{1}{5\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(2\,a^2+b^2\right)}{3\,a^3}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^4+2\,a^2\,b^2+b^4\right)}{a^5}-\frac{b\,\mathrm{tan}\left(c+d\,x\right)}{4\,a^2}-\frac{b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(2\,a^2+b^2\right)}{2\,a^4}}{d\,{\mathrm{tan}\left(c+d\,x\right)}^5}","Not used",1,"(2*b*atanh((b*(a^2 + b^2)^2*(a + 2*b*tan(c + d*x)))/(a*(a^4*b + b^5 + 2*a^2*b^3)))*(a^2 + b^2)^2)/(a^6*d) - (1/(5*a) + (tan(c + d*x)^2*(2*a^2 + b^2))/(3*a^3) + (tan(c + d*x)^4*(a^4 + b^4 + 2*a^2*b^2))/a^5 - (b*tan(c + d*x))/(4*a^2) - (b*tan(c + d*x)^3*(2*a^2 + b^2))/(2*a^4))/(d*tan(c + d*x)^5)","B"
61,1,757,297,5.543125,"\text{Not used}","int(sin(c + d*x)^6/(a + b*tan(c + d*x))^2,x)","\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{2\,a\,b}{{\left(a^2+b^2\right)}^2}-\frac{12\,a\,b^3}{{\left(a^2+b^2\right)}^3}+\frac{18\,a\,b^5}{{\left(a^2+b^2\right)}^4}-\frac{8\,a\,b^7}{{\left(a^2+b^2\right)}^5}\right)}{d}+\frac{\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(-5\,a^5+9\,a^3\,b^2+2\,a\,b^4\right)}{6\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(-11\,a^5+6\,a^3\,b^2+a\,b^4\right)}{16\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^6\,\left(-27\,a^6\,b+43\,a^4\,b^3+7\,a^2\,b^5+b^7\right)}{16\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{\mathrm{tan}\left(c+d\,x\right)\,\left(-15\,a^5+38\,a^3\,b^2+5\,a\,b^4\right)}{48\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{a\,\left(-17\,a^5\,b+8\,a^3\,b^3+a\,b^5\right)}{6\,\left(a^2+b^2\right)\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(41\,a^6\,b-31\,a^4\,b^3+a^2\,b^5+b^7\right)}{6\,\left(a^2+b^2\right)\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(125\,a^6\,b-69\,a^4\,b^3-a^2\,b^5+b^7\right)}{16\,\left(a^2+b^2\right)\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}}{d\,\left(b\,{\mathrm{tan}\left(c+d\,x\right)}^7+a\,{\mathrm{tan}\left(c+d\,x\right)}^6+3\,b\,{\mathrm{tan}\left(c+d\,x\right)}^5+3\,a\,{\mathrm{tan}\left(c+d\,x\right)}^4+3\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3+3\,a\,{\mathrm{tan}\left(c+d\,x\right)}^2+b\,\mathrm{tan}\left(c+d\,x\right)+a\right)}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(a^3\,5{}\mathrm{i}-7\,a^2\,b+a\,b^2\,5{}\mathrm{i}+b^3\right)}{32\,d\,\left(a^5-a^4\,b\,5{}\mathrm{i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm{i}+5\,a\,b^4-b^5\,1{}\mathrm{i}\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(a^3\,5{}\mathrm{i}+7\,a^2\,b+a\,b^2\,5{}\mathrm{i}-b^3\right)}{32\,d\,\left(a^5+a^4\,b\,5{}\mathrm{i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm{i}+5\,a\,b^4+b^5\,1{}\mathrm{i}\right)}","Not used",1,"(log(a + b*tan(c + d*x))*((2*a*b)/(a^2 + b^2)^2 - (12*a*b^3)/(a^2 + b^2)^3 + (18*a*b^5)/(a^2 + b^2)^4 - (8*a*b^7)/(a^2 + b^2)^5))/d + ((tan(c + d*x)^3*(2*a*b^4 - 5*a^5 + 9*a^3*b^2))/(6*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^5*(a*b^4 - 11*a^5 + 6*a^3*b^2))/(16*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^6*(b^7 - 27*a^6*b + 7*a^2*b^5 + 43*a^4*b^3))/(16*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)*(5*a*b^4 - 15*a^5 + 38*a^3*b^2))/(48*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*(a*b^5 - 17*a^5*b + 8*a^3*b^3))/(6*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^4*(41*a^6*b + b^7 + a^2*b^5 - 31*a^4*b^3))/(6*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^2*(125*a^6*b + b^7 - a^2*b^5 - 69*a^4*b^3))/(16*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a + b*tan(c + d*x) + 3*a*tan(c + d*x)^2 + 3*a*tan(c + d*x)^4 + a*tan(c + d*x)^6 + 3*b*tan(c + d*x)^3 + 3*b*tan(c + d*x)^5 + b*tan(c + d*x)^7)) + (log(tan(c + d*x) + 1i)*(a*b^2*5i - 7*a^2*b + a^3*5i + b^3))/(32*d*(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)) - (log(tan(c + d*x) - 1i)*(a*b^2*5i + 7*a^2*b + a^3*5i - b^3))/(32*d*(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2))","B"
62,1,481,217,4.819928,"\text{Not used}","int(sin(c + d*x)^4/(a + b*tan(c + d*x))^2,x)","\frac{\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(a\,b^2-5\,a^3\right)}{8\,\left(a^4+2\,a^2\,b^2+b^4\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(-13\,a^4\,b+12\,a^2\,b^3+b^5\right)}{8\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{3\,\mathrm{tan}\left(c+d\,x\right)\,\left(a\,b^2-a^3\right)}{8\,\left(a^4+2\,a^2\,b^2+b^4\right)}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(35\,a^4\,b-12\,a^2\,b^3+b^5\right)}{8\,\left(a^2+b^2\right)\,\left(a^4+2\,a^2\,b^2+b^4\right)}+\frac{a\,\left(a\,b^3-5\,a^3\,b\right)}{2\,\left(a^2+b^2\right)\,\left(a^4+2\,a^2\,b^2+b^4\right)}}{d\,\left(b\,{\mathrm{tan}\left(c+d\,x\right)}^5+a\,{\mathrm{tan}\left(c+d\,x\right)}^4+2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3+2\,a\,{\mathrm{tan}\left(c+d\,x\right)}^2+b\,\mathrm{tan}\left(c+d\,x\right)+a\right)}+\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{2\,a\,b}{{\left(a^2+b^2\right)}^2}-\frac{8\,a\,b^3}{{\left(a^2+b^2\right)}^3}+\frac{6\,a\,b^5}{{\left(a^2+b^2\right)}^4}\right)}{d}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(3\,a^2-a\,b\,4{}\mathrm{i}+b^2\right)}{16\,d\,\left(a^4\,1{}\mathrm{i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm{i}+4\,a\,b^3+b^4\,1{}\mathrm{i}\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(3\,a^2+a\,b\,4{}\mathrm{i}+b^2\right)}{16\,d\,\left(a^4\,1{}\mathrm{i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm{i}-4\,a\,b^3+b^4\,1{}\mathrm{i}\right)}","Not used",1,"((tan(c + d*x)^3*(a*b^2 - 5*a^3))/(8*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)^4*(b^5 - 13*a^4*b + 12*a^2*b^3))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (3*tan(c + d*x)*(a*b^2 - a^3))/(8*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c + d*x)^2*(35*a^4*b + b^5 - 12*a^2*b^3))/(8*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) + (a*(a*b^3 - 5*a^3*b))/(2*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a + b*tan(c + d*x) + 2*a*tan(c + d*x)^2 + a*tan(c + d*x)^4 + 2*b*tan(c + d*x)^3 + b*tan(c + d*x)^5)) + (log(a + b*tan(c + d*x))*((2*a*b)/(a^2 + b^2)^2 - (8*a*b^3)/(a^2 + b^2)^3 + (6*a*b^5)/(a^2 + b^2)^4))/d + (log(tan(c + d*x) - 1i)*(3*a^2 - a*b*4i + b^2))/(16*d*(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)) - (log(tan(c + d*x) + 1i)*(a*b*4i + 3*a^2 + b^2))/(16*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i))","B"
63,1,255,148,4.056497,"\text{Not used}","int(sin(c + d*x)^2/(a + b*tan(c + d*x))^2,x)","\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{2\,a\,b}{{\left(a^2+b^2\right)}^2}-\frac{4\,a\,b^3}{{\left(a^2+b^2\right)}^3}\right)}{d}-\frac{\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(3\,a^2\,b-b^3\right)}{2\,\left(a^4+2\,a^2\,b^2+b^4\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)}{2\,\left(a^2+b^2\right)}+\frac{2\,a^2\,b}{{\left(a^2+b^2\right)}^2}}{d\,\left(b\,{\mathrm{tan}\left(c+d\,x\right)}^3+a\,{\mathrm{tan}\left(c+d\,x\right)}^2+b\,\mathrm{tan}\left(c+d\,x\right)+a\right)}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(a+b\,1{}\mathrm{i}\right)}{4\,d\,\left(-a^3\,1{}\mathrm{i}-3\,a^2\,b+a\,b^2\,3{}\mathrm{i}+b^3\right)}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(b+a\,1{}\mathrm{i}\right)}{4\,d\,\left(-a^3-a^2\,b\,3{}\mathrm{i}+3\,a\,b^2+b^3\,1{}\mathrm{i}\right)}","Not used",1,"(log(a + b*tan(c + d*x))*((2*a*b)/(a^2 + b^2)^2 - (4*a*b^3)/(a^2 + b^2)^3))/d - ((tan(c + d*x)^2*(3*a^2*b - b^3))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (a*tan(c + d*x))/(2*(a^2 + b^2)) + (2*a^2*b)/(a^2 + b^2)^2)/(d*(a + b*tan(c + d*x) + a*tan(c + d*x)^2 + b*tan(c + d*x)^3)) + (log(tan(c + d*x) + 1i)*(a + b*1i))/(4*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) + (log(tan(c + d*x) - 1i)*(a*1i + b))/(4*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i))","B"
64,1,79,72,3.841963,"\text{Not used}","int(1/(sin(c + d*x)^2*(a + b*tan(c + d*x))^2),x)","\frac{2\,b\,\ln\left(\frac{a+b\,\mathrm{tan}\left(c+d\,x\right)}{\mathrm{tan}\left(c+d\,x\right)}\right)}{a^3\,d}-\frac{2\,b}{a^2\,d\,\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}-\frac{1}{a\,d\,\mathrm{tan}\left(c+d\,x\right)\,\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}","Not used",1,"(2*b*log((a + b*tan(c + d*x))/tan(c + d*x)))/(a^3*d) - (2*b)/(a^2*d*(a + b*tan(c + d*x))) - 1/(a*d*tan(c + d*x)*(a + b*tan(c + d*x)))","B"
65,1,150,140,3.914626,"\text{Not used}","int(1/(sin(c + d*x)^4*(a + b*tan(c + d*x))^2),x)","\frac{4\,b\,\mathrm{atanh}\left(\frac{2\,b\,\left(a^2+2\,b^2\right)\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{a\,\left(2\,a^2\,b+4\,b^3\right)}\right)\,\left(a^2+2\,b^2\right)}{a^5\,d}-\frac{\frac{1}{3\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^2+2\,b^2\right)}{a^3}-\frac{2\,b\,\mathrm{tan}\left(c+d\,x\right)}{3\,a^2}+\frac{2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(a^2+2\,b^2\right)}{a^4}}{d\,\left(b\,{\mathrm{tan}\left(c+d\,x\right)}^4+a\,{\mathrm{tan}\left(c+d\,x\right)}^3\right)}","Not used",1,"(4*b*atanh((2*b*(a^2 + 2*b^2)*(a + 2*b*tan(c + d*x)))/(a*(2*a^2*b + 4*b^3)))*(a^2 + 2*b^2))/(a^5*d) - (1/(3*a) + (tan(c + d*x)^2*(a^2 + 2*b^2))/a^3 - (2*b*tan(c + d*x))/(3*a^2) + (2*b*tan(c + d*x)^3*(a^2 + 2*b^2))/a^4)/(d*(a*tan(c + d*x)^3 + b*tan(c + d*x)^4))","B"
66,1,237,219,5.014880,"\text{Not used}","int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))^2),x)","\frac{4\,b\,\mathrm{atanh}\left(\frac{2\,b\,\left(a^2+3\,b^2\right)\,\left(a^2+b^2\right)\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{a\,\left(2\,a^4\,b+8\,a^2\,b^3+6\,b^5\right)}\right)\,\left(a^2+3\,b^2\right)\,\left(a^2+b^2\right)}{a^7\,d}-\frac{\frac{1}{5\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^4+4\,a^2\,b^2+3\,b^4\right)}{a^5}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(4\,a^2+3\,b^2\right)}{6\,a^3}-\frac{3\,b\,\mathrm{tan}\left(c+d\,x\right)}{10\,a^2}+\frac{2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(a^4+4\,a^2\,b^2+3\,b^4\right)}{a^6}-\frac{b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(4\,a^2+3\,b^2\right)}{3\,a^4}}{d\,\left(b\,{\mathrm{tan}\left(c+d\,x\right)}^6+a\,{\mathrm{tan}\left(c+d\,x\right)}^5\right)}","Not used",1,"(4*b*atanh((2*b*(a^2 + 3*b^2)*(a^2 + b^2)*(a + 2*b*tan(c + d*x)))/(a*(2*a^4*b + 6*b^5 + 8*a^2*b^3)))*(a^2 + 3*b^2)*(a^2 + b^2))/(a^7*d) - (1/(5*a) + (tan(c + d*x)^4*(a^4 + 3*b^4 + 4*a^2*b^2))/a^5 + (tan(c + d*x)^2*(4*a^2 + 3*b^2))/(6*a^3) - (3*b*tan(c + d*x))/(10*a^2) + (2*b*tan(c + d*x)^5*(a^4 + 3*b^4 + 4*a^2*b^2))/a^6 - (b*tan(c + d*x)^3*(4*a^2 + 3*b^2))/(3*a^4))/(d*(a*tan(c + d*x)^5 + b*tan(c + d*x)^6))","B"
67,1,1068,382,5.842373,"\text{Not used}","int(sin(c + d*x)^6/(a + b*tan(c + d*x))^3,x)","\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{3\,b}{{\left(a^2+b^2\right)}^2}-\frac{34\,b^3}{{\left(a^2+b^2\right)}^3}+\frac{99\,b^5}{{\left(a^2+b^2\right)}^4}-\frac{108\,b^7}{{\left(a^2+b^2\right)}^5}+\frac{40\,b^9}{{\left(a^2+b^2\right)}^6}\right)}{d}-\frac{\frac{{\mathrm{tan}\left(c+d\,x\right)}^6\,\left(31\,a^8\,b-127\,a^6\,b^3+5\,a^4\,b^5+3\,a^2\,b^7\right)}{8\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^7\,\left(43\,a^7\,b^2-215\,a^5\,b^4+65\,a^3\,b^6+3\,a\,b^8\right)}{16\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(33\,a^9+403\,a^7\,b^2-2005\,a^5\,b^4+529\,a^3\,b^6+24\,a\,b^8\right)}{48\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(164\,a^8\,b-515\,a^6\,b^3+65\,a^4\,b^5+27\,a^2\,b^7+3\,b^9\right)}{12\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{a^2\,\left(63\,a^6\,b-161\,a^4\,b^3+17\,a^2\,b^5+b^7\right)}{12\,\left(a^2+b^2\right)\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(40\,a^9+335\,a^7\,b^2-2171\,a^5\,b^4+429\,a^3\,b^6+15\,a\,b^8\right)}{48\,\left(a^2+b^2\right)\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(357\,a^8\,b-987\,a^6\,b^3+125\,a^4\,b^5+31\,a^2\,b^7+2\,b^9\right)}{24\,\left(a^2+b^2\right)\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(15\,a^8+93\,a^6\,b^2-763\,a^4\,b^4+127\,a^2\,b^6+8\,b^8\right)}{48\,\left(a^2+b^2\right)\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(3\,a^2+b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^6\,\left(a^2+3\,b^2\right)+a^2+{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(3\,a^2+3\,b^2\right)+b^2\,{\mathrm{tan}\left(c+d\,x\right)}^8+2\,a\,b\,\mathrm{tan}\left(c+d\,x\right)+6\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3+6\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^5+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^7\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(a^3\,5{}\mathrm{i}-18\,a^2\,b+a\,b^2\,3{}\mathrm{i}\right)}{32\,d\,\left(-a^6+a^5\,b\,6{}\mathrm{i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm{i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm{i}+b^6\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(5\,a^3-a^2\,b\,18{}\mathrm{i}+3\,a\,b^2\right)}{32\,d\,\left(-a^6\,1{}\mathrm{i}+6\,a^5\,b+a^4\,b^2\,15{}\mathrm{i}-20\,a^3\,b^3-a^2\,b^4\,15{}\mathrm{i}+6\,a\,b^5+b^6\,1{}\mathrm{i}\right)}","Not used",1,"(log(a + b*tan(c + d*x))*((3*b)/(a^2 + b^2)^2 - (34*b^3)/(a^2 + b^2)^3 + (99*b^5)/(a^2 + b^2)^4 - (108*b^7)/(a^2 + b^2)^5 + (40*b^9)/(a^2 + b^2)^6))/d - ((tan(c + d*x)^6*(31*a^8*b + 3*a^2*b^7 + 5*a^4*b^5 - 127*a^6*b^3))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^7*(3*a*b^8 + 65*a^3*b^6 - 215*a^5*b^4 + 43*a^7*b^2))/(16*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^5*(24*a*b^8 + 33*a^9 + 529*a^3*b^6 - 2005*a^5*b^4 + 403*a^7*b^2))/(48*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^4*(164*a^8*b + 3*b^9 + 27*a^2*b^7 + 65*a^4*b^5 - 515*a^6*b^3))/(12*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (a^2*(63*a^6*b + b^7 + 17*a^2*b^5 - 161*a^4*b^3))/(12*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)^3*(15*a*b^8 + 40*a^9 + 429*a^3*b^6 - 2171*a^5*b^4 + 335*a^7*b^2))/(48*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)^2*(357*a^8*b + 2*b^9 + 31*a^2*b^7 + 125*a^4*b^5 - 987*a^6*b^3))/(24*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (a*tan(c + d*x)*(15*a^8 + 8*b^8 + 127*a^2*b^6 - 763*a^4*b^4 + 93*a^6*b^2))/(48*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)))/(d*(tan(c + d*x)^2*(3*a^2 + b^2) + tan(c + d*x)^6*(a^2 + 3*b^2) + a^2 + tan(c + d*x)^4*(3*a^2 + 3*b^2) + b^2*tan(c + d*x)^8 + 2*a*b*tan(c + d*x) + 6*a*b*tan(c + d*x)^3 + 6*a*b*tan(c + d*x)^5 + 2*a*b*tan(c + d*x)^7)) - (log(tan(c + d*x) + 1i)*(a*b^2*3i - 18*a^2*b + a^3*5i))/(32*d*(a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)) - (log(tan(c + d*x) - 1i)*(3*a*b^2 - a^2*b*18i + 5*a^3))/(32*d*(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i))","B"
68,1,717,285,5.335442,"\text{Not used}","int(sin(c + d*x)^4/(a + b*tan(c + d*x))^3,x)","\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{3\,b}{{\left(a^2+b^2\right)}^2}-\frac{24\,b^3}{{\left(a^2+b^2\right)}^3}+\frac{45\,b^5}{{\left(a^2+b^2\right)}^4}-\frac{24\,b^7}{{\left(a^2+b^2\right)}^5}\right)}{d}-\frac{\frac{19\,a^6\,b-28\,a^4\,b^3+a^2\,b^5}{4\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(35\,a^6\,b-61\,a^4\,b^3+a^2\,b^5+b^7\right)}{4\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}-\frac{3\,{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(-5\,a^6\,b+12\,a^4\,b^3+a^2\,b^5\right)}{4\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{3\,{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(7\,a^5\,b^2-22\,a^3\,b^4+3\,a\,b^6\right)}{8\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(5\,a^7+49\,a^5\,b^2-133\,a^3\,b^4+15\,a\,b^6\right)}{8\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(3\,a^6+22\,a^4\,b^2-73\,a^2\,b^4+4\,b^6\right)}{8\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(2\,a^2+b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^2+2\,b^2\right)+a^2+b^2\,{\mathrm{tan}\left(c+d\,x\right)}^6+2\,a\,b\,\mathrm{tan}\left(c+d\,x\right)+4\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^5\right)}-\frac{3\,\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(a^2\,1{}\mathrm{i}+3\,b\,a\right)}{16\,d\,\left(a^5+a^4\,b\,5{}\mathrm{i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm{i}+5\,a\,b^4+b^5\,1{}\mathrm{i}\right)}-\frac{3\,\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(3\,a\,b-a^2\,1{}\mathrm{i}\right)}{16\,d\,\left(a^5-a^4\,b\,5{}\mathrm{i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm{i}+5\,a\,b^4-b^5\,1{}\mathrm{i}\right)}","Not used",1,"(log(a + b*tan(c + d*x))*((3*b)/(a^2 + b^2)^2 - (24*b^3)/(a^2 + b^2)^3 + (45*b^5)/(a^2 + b^2)^4 - (24*b^7)/(a^2 + b^2)^5))/d - ((19*a^6*b + a^2*b^5 - 28*a^4*b^3)/(4*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)^2*(35*a^6*b + b^7 + a^2*b^5 - 61*a^4*b^3))/(4*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) - (3*tan(c + d*x)^4*(a^2*b^5 - 5*a^6*b + 12*a^4*b^3))/(4*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (3*tan(c + d*x)^5*(3*a*b^6 - 22*a^3*b^4 + 7*a^5*b^2))/(8*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)^3*(15*a*b^6 + 5*a^7 - 133*a^3*b^4 + 49*a^5*b^2))/(8*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (a*tan(c + d*x)*(3*a^6 + 4*b^6 - 73*a^2*b^4 + 22*a^4*b^2))/(8*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)))/(d*(tan(c + d*x)^2*(2*a^2 + b^2) + tan(c + d*x)^4*(a^2 + 2*b^2) + a^2 + b^2*tan(c + d*x)^6 + 2*a*b*tan(c + d*x) + 4*a*b*tan(c + d*x)^3 + 2*a*b*tan(c + d*x)^5)) - (3*log(tan(c + d*x) - 1i)*(3*a*b + a^2*1i))/(16*d*(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)) - (3*log(tan(c + d*x) + 1i)*(3*a*b - a^2*1i))/(16*d*(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2))","B"
69,1,433,206,4.622633,"\text{Not used}","int(sin(c + d*x)^2/(a + b*tan(c + d*x))^3,x)","\frac{\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(-7\,a^4\,b+6\,a^2\,b^3+b^5\right)}{2\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(7\,a\,b^4-5\,a^3\,b^2\right)}{2\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}-\frac{2\,a^2\,\left(2\,a^2\,b-b^3\right)}{\left(a^2+b^2\right)\,\left(a^4+2\,a^2\,b^2+b^4\right)}-\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(a^4+7\,a^2\,b^2-6\,b^4\right)}{2\,\left(a^2+b^2\right)\,\left(a^4+2\,a^2\,b^2+b^4\right)}}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^2+b^2\right)+a^2+b^2\,{\mathrm{tan}\left(c+d\,x\right)}^4+2\,a\,b\,\mathrm{tan}\left(c+d\,x\right)+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3\right)}+\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{3\,b}{{\left(a^2+b^2\right)}^2}-\frac{14\,b^3}{{\left(a^2+b^2\right)}^3}+\frac{12\,b^5}{{\left(a^2+b^2\right)}^4}\right)}{d}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(-2\,b+a\,1{}\mathrm{i}\right)}{4\,d\,\left(a^4-a^3\,b\,4{}\mathrm{i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm{i}+b^4\right)}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(a-b\,2{}\mathrm{i}\right)}{4\,d\,\left(a^4\,1{}\mathrm{i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm{i}+4\,a\,b^3+b^4\,1{}\mathrm{i}\right)}","Not used",1,"((tan(c + d*x)^2*(b^5 - 7*a^4*b + 6*a^2*b^3))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^3*(7*a*b^4 - 5*a^3*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (2*a^2*(2*a^2*b - b^3))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) - (a*tan(c + d*x)*(a^4 - 6*b^4 + 7*a^2*b^2))/(2*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)))/(d*(tan(c + d*x)^2*(a^2 + b^2) + a^2 + b^2*tan(c + d*x)^4 + 2*a*b*tan(c + d*x) + 2*a*b*tan(c + d*x)^3)) + (log(a + b*tan(c + d*x))*((3*b)/(a^2 + b^2)^2 - (14*b^3)/(a^2 + b^2)^3 + (12*b^5)/(a^2 + b^2)^4))/d + (log(tan(c + d*x) + 1i)*(a*1i - 2*b))/(4*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)) + (log(tan(c + d*x) - 1i)*(a - b*2i))/(4*d*(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i))","B"
70,1,99,95,3.869155,"\text{Not used}","int(1/(sin(c + d*x)^2*(a + b*tan(c + d*x))^3),x)","\frac{6\,b\,\mathrm{atanh}\left(\frac{2\,b\,\mathrm{tan}\left(c+d\,x\right)}{a}+1\right)}{a^4\,d}-\frac{\frac{1}{a}+\frac{3\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^2}{a^3}+\frac{9\,b\,\mathrm{tan}\left(c+d\,x\right)}{2\,a^2}}{d\,\left(a^2\,\mathrm{tan}\left(c+d\,x\right)+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^2+b^2\,{\mathrm{tan}\left(c+d\,x\right)}^3\right)}","Not used",1,"(6*b*atanh((2*b*tan(c + d*x))/a + 1))/(a^4*d) - (1/a + (3*b^2*tan(c + d*x)^2)/a^3 + (9*b*tan(c + d*x))/(2*a^2))/(d*(a^2*tan(c + d*x) + b^2*tan(c + d*x)^3 + 2*a*b*tan(c + d*x)^2))","B"
71,1,200,178,4.367135,"\text{Not used}","int(1/(sin(c + d*x)^4*(a + b*tan(c + d*x))^3),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{b\,\left(3\,a^2+10\,b^2\right)\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{a\,\left(3\,a^2\,b+10\,b^3\right)}\right)\,\left(3\,a^2+10\,b^2\right)}{a^6\,d}-\frac{\frac{1}{3\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(3\,a^2+10\,b^2\right)}{3\,a^3}-\frac{5\,b\,\mathrm{tan}\left(c+d\,x\right)}{6\,a^2}+\frac{b^2\,{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(3\,a^2+10\,b^2\right)}{a^5}+\frac{3\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(3\,a^2+10\,b^2\right)}{2\,a^4}}{d\,\left(a^2\,{\mathrm{tan}\left(c+d\,x\right)}^3+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^4+b^2\,{\mathrm{tan}\left(c+d\,x\right)}^5\right)}","Not used",1,"(2*b*atanh((b*(3*a^2 + 10*b^2)*(a + 2*b*tan(c + d*x)))/(a*(3*a^2*b + 10*b^3)))*(3*a^2 + 10*b^2))/(a^6*d) - (1/(3*a) + (tan(c + d*x)^2*(3*a^2 + 10*b^2))/(3*a^3) - (5*b*tan(c + d*x))/(6*a^2) + (b^2*tan(c + d*x)^4*(3*a^2 + 10*b^2))/a^5 + (3*b*tan(c + d*x)^3*(3*a^2 + 10*b^2))/(2*a^4))/(d*(a^2*tan(c + d*x)^3 + b^2*tan(c + d*x)^5 + 2*a*b*tan(c + d*x)^4))","B"
72,1,297,265,5.320840,"\text{Not used}","int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))^3),x)","\frac{2\,b\,\mathrm{atanh}\left(\frac{b\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(3\,a^4+20\,a^2\,b^2+21\,b^4\right)}{a\,\left(3\,a^4\,b+20\,a^2\,b^3+21\,b^5\right)}\right)\,\left(3\,a^4+20\,a^2\,b^2+21\,b^4\right)}{a^8\,d}-\frac{\frac{1}{5\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(3\,a^4+20\,a^2\,b^2+21\,b^4\right)}{3\,a^5}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(20\,a^2+21\,b^2\right)}{30\,a^3}-\frac{7\,b\,\mathrm{tan}\left(c+d\,x\right)}{20\,a^2}+\frac{b^2\,{\mathrm{tan}\left(c+d\,x\right)}^6\,\left(3\,a^4+20\,a^2\,b^2+21\,b^4\right)}{a^7}+\frac{3\,b\,{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(3\,a^4+20\,a^2\,b^2+21\,b^4\right)}{2\,a^6}-\frac{b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(20\,a^2+21\,b^2\right)}{12\,a^4}}{d\,\left(a^2\,{\mathrm{tan}\left(c+d\,x\right)}^5+2\,a\,b\,{\mathrm{tan}\left(c+d\,x\right)}^6+b^2\,{\mathrm{tan}\left(c+d\,x\right)}^7\right)}","Not used",1,"(2*b*atanh((b*(a + 2*b*tan(c + d*x))*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(a*(3*a^4*b + 21*b^5 + 20*a^2*b^3)))*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(a^8*d) - (1/(5*a) + (tan(c + d*x)^4*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(3*a^5) + (tan(c + d*x)^2*(20*a^2 + 21*b^2))/(30*a^3) - (7*b*tan(c + d*x))/(20*a^2) + (b^2*tan(c + d*x)^6*(3*a^4 + 21*b^4 + 20*a^2*b^2))/a^7 + (3*b*tan(c + d*x)^5*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(2*a^6) - (b*tan(c + d*x)^3*(20*a^2 + 21*b^2))/(12*a^4))/(d*(a^2*tan(c + d*x)^5 + b^2*tan(c + d*x)^7 + 2*a*b*tan(c + d*x)^6))","B"
73,1,962,366,5.662452,"\text{Not used}","int(sin(c + d*x)^4/(a + b*tan(c + d*x))^4,x)","\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{4\,a\,b}{{\left(a^2+b^2\right)}^3}-\frac{48\,a\,b^3}{{\left(a^2+b^2\right)}^4}+\frac{120\,a\,b^5}{{\left(a^2+b^2\right)}^5}-\frac{80\,a\,b^7}{{\left(a^2+b^2\right)}^6}\right)}{d}-\frac{\frac{2\,\left(11\,a^8\,b-38\,a^6\,b^3+11\,a^4\,b^5\right)}{3\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^6\,\left(-29\,a^6\,b^3+185\,a^4\,b^5-103\,a^2\,b^7+3\,b^9\right)}{8\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(71\,a^7\,b^2-411\,a^5\,b^4+165\,a^3\,b^6+7\,a\,b^8\right)}{8\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(331\,a^8\,b-1183\,a^6\,b^3-239\,a^4\,b^5+315\,a^2\,b^7\right)}{24\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(5\,a^9+152\,a^7\,b^2-822\,a^5\,b^4+320\,a^3\,b^6+9\,a\,b^8\right)}{8\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}-\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(-149\,a^8\,b+512\,a^6\,b^3+1006\,a^4\,b^5-600\,a^2\,b^7+15\,b^9\right)}{24\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(3\,a^8+73\,a^6\,b^2-423\,a^4\,b^4+147\,a^2\,b^6\right)}{8\,\left(a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right)}}{d\,\left({\mathrm{tan}\left(c+d\,x\right)}^2\,\left(2\,a^3+3\,a\,b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(3\,a^2\,b+2\,b^3\right)+a^3+{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^3+6\,a\,b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(6\,a^2\,b+b^3\right)+b^3\,{\mathrm{tan}\left(c+d\,x\right)}^7+3\,a\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^6+3\,a^2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(-3\,a^2+a\,b\,14{}\mathrm{i}+3\,b^2\right)}{16\,d\,\left(-a^6\,1{}\mathrm{i}+6\,a^5\,b+a^4\,b^2\,15{}\mathrm{i}-20\,a^3\,b^3-a^2\,b^4\,15{}\mathrm{i}+6\,a\,b^5+b^6\,1{}\mathrm{i}\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(3\,a^2+a\,b\,14{}\mathrm{i}-3\,b^2\right)}{16\,d\,\left(a^6\,1{}\mathrm{i}+6\,a^5\,b-a^4\,b^2\,15{}\mathrm{i}-20\,a^3\,b^3+a^2\,b^4\,15{}\mathrm{i}+6\,a\,b^5-b^6\,1{}\mathrm{i}\right)}","Not used",1,"(log(a + b*tan(c + d*x))*((4*a*b)/(a^2 + b^2)^3 - (48*a*b^3)/(a^2 + b^2)^4 + (120*a*b^5)/(a^2 + b^2)^5 - (80*a*b^7)/(a^2 + b^2)^6))/d - ((2*(11*a^8*b + 11*a^4*b^5 - 38*a^6*b^3))/(3*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) - (tan(c + d*x)^6*(3*b^9 - 103*a^2*b^7 + 185*a^4*b^5 - 29*a^6*b^3))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^5*(7*a*b^8 + 165*a^3*b^6 - 411*a^5*b^4 + 71*a^7*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^2*(331*a^8*b + 315*a^2*b^7 - 239*a^4*b^5 - 1183*a^6*b^3))/(24*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^3*(9*a*b^8 + 5*a^9 + 320*a^3*b^6 - 822*a^5*b^4 + 152*a^7*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) - (tan(c + d*x)^4*(15*b^9 - 149*a^8*b - 600*a^2*b^7 + 1006*a^4*b^5 + 512*a^6*b^3))/(24*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (a*tan(c + d*x)*(3*a^8 + 147*a^2*b^6 - 423*a^4*b^4 + 73*a^6*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)))/(d*(tan(c + d*x)^2*(3*a*b^2 + 2*a^3) + tan(c + d*x)^5*(3*a^2*b + 2*b^3) + a^3 + tan(c + d*x)^4*(6*a*b^2 + a^3) + tan(c + d*x)^3*(6*a^2*b + b^3) + b^3*tan(c + d*x)^7 + 3*a*b^2*tan(c + d*x)^6 + 3*a^2*b*tan(c + d*x))) + (log(tan(c + d*x) - 1i)*(a*b*14i - 3*a^2 + 3*b^2))/(16*d*(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)) - (log(tan(c + d*x) + 1i)*(a*b*14i + 3*a^2 - 3*b^2))/(16*d*(6*a*b^5 + 6*a^5*b + a^6*1i - b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i))","B"
74,1,597,264,5.087672,"\text{Not used}","int(sin(c + d*x)^2/(a + b*tan(c + d*x))^4,x)","\frac{\ln\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(\frac{4\,a\,b}{{\left(a^2+b^2\right)}^3}-\frac{28\,a\,b^3}{{\left(a^2+b^2\right)}^4}+\frac{32\,a\,b^5}{{\left(a^2+b^2\right)}^5}\right)}{d}-\frac{\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(35\,a^4\,b-79\,a^2\,b^3+6\,b^5\right)}{6\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(7\,a^4\,b^3-22\,a^2\,b^5+3\,b^7\right)}{2\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(17\,a^5\,b^2-46\,a^3\,b^4+a\,b^6\right)}{2\,\left(a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right)}+\frac{a^2\,\left(19\,a^4\,b-28\,a^2\,b^3+b^5\right)}{3\,\left(a^2+b^2\right)\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}+\frac{a\,\mathrm{tan}\left(c+d\,x\right)\,\left(a^6+20\,a^4\,b^2-43\,a^2\,b^4+2\,b^6\right)}{2\,\left(a^2+b^2\right)\,\left(a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right)}}{d\,\left(a^3+{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^3+3\,a\,b^2\right)+{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(3\,a^2\,b+b^3\right)+b^3\,{\mathrm{tan}\left(c+d\,x\right)}^5+3\,a\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^4+3\,a^2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}-\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)-\mathrm{i}\right)\,\left(3\,b+a\,1{}\mathrm{i}\right)}{4\,d\,\left(a^5+a^4\,b\,5{}\mathrm{i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm{i}+5\,a\,b^4+b^5\,1{}\mathrm{i}\right)}+\frac{\ln\left(\mathrm{tan}\left(c+d\,x\right)+1{}\mathrm{i}\right)\,\left(-3\,b+a\,1{}\mathrm{i}\right)}{4\,d\,\left(a^5-a^4\,b\,5{}\mathrm{i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm{i}+5\,a\,b^4-b^5\,1{}\mathrm{i}\right)}","Not used",1,"(log(a + b*tan(c + d*x))*((4*a*b)/(a^2 + b^2)^3 - (28*a*b^3)/(a^2 + b^2)^4 + (32*a*b^5)/(a^2 + b^2)^5))/d - ((tan(c + d*x)^2*(35*a^4*b + 6*b^5 - 79*a^2*b^3))/(6*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^4*(3*b^7 - 22*a^2*b^5 + 7*a^4*b^3))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)^3*(a*b^6 - 46*a^3*b^4 + 17*a^5*b^2))/(2*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (a^2*(19*a^4*b + b^5 - 28*a^2*b^3))/(3*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*tan(c + d*x)*(a^6 + 2*b^6 - 43*a^2*b^4 + 20*a^4*b^2))/(2*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + tan(c + d*x)^2*(3*a*b^2 + a^3) + tan(c + d*x)^3*(3*a^2*b + b^3) + b^3*tan(c + d*x)^5 + 3*a*b^2*tan(c + d*x)^4 + 3*a^2*b*tan(c + d*x))) - (log(tan(c + d*x) - 1i)*(a*1i + 3*b))/(4*d*(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)) + (log(tan(c + d*x) + 1i)*(a*1i - 3*b))/(4*d*(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2))","B"
75,1,131,116,3.975776,"\text{Not used}","int(1/(sin(c + d*x)^2*(a + b*tan(c + d*x))^4),x)","\frac{8\,b\,\mathrm{atanh}\left(\frac{2\,b\,\mathrm{tan}\left(c+d\,x\right)}{a}+1\right)}{a^5\,d}-\frac{\frac{1}{a}+\frac{10\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^2}{a^3}+\frac{4\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^3}{a^4}+\frac{22\,b\,\mathrm{tan}\left(c+d\,x\right)}{3\,a^2}}{d\,\left(a^3\,\mathrm{tan}\left(c+d\,x\right)+3\,a^2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^2+3\,a\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^3+b^3\,{\mathrm{tan}\left(c+d\,x\right)}^4\right)}","Not used",1,"(8*b*atanh((2*b*tan(c + d*x))/a + 1))/(a^5*d) - (1/a + (10*b^2*tan(c + d*x)^2)/a^3 + (4*b^3*tan(c + d*x)^3)/a^4 + (22*b*tan(c + d*x))/(3*a^2))/(d*(a^3*tan(c + d*x) + b^3*tan(c + d*x)^4 + 3*a^2*b*tan(c + d*x)^2 + 3*a*b^2*tan(c + d*x)^3))","B"
76,1,232,205,5.057747,"\text{Not used}","int(1/(sin(c + d*x)^4*(a + b*tan(c + d*x))^4),x)","\frac{8\,b\,\mathrm{atanh}\left(\frac{4\,b\,\left(a^2+5\,b^2\right)\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)}{a\,\left(4\,a^2\,b+20\,b^3\right)}\right)\,\left(a^2+5\,b^2\right)}{a^7\,d}-\frac{\frac{1}{3\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(a^2+5\,b^2\right)}{a^3}-\frac{b\,\mathrm{tan}\left(c+d\,x\right)}{a^2}+\frac{22\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(a^2+5\,b^2\right)}{3\,a^4}+\frac{10\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^2+5\,b^2\right)}{a^5}+\frac{4\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(a^2+5\,b^2\right)}{a^6}}{d\,\left(a^3\,{\mathrm{tan}\left(c+d\,x\right)}^3+3\,a^2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^4+3\,a\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^5+b^3\,{\mathrm{tan}\left(c+d\,x\right)}^6\right)}","Not used",1,"(8*b*atanh((4*b*(a^2 + 5*b^2)*(a + 2*b*tan(c + d*x)))/(a*(4*a^2*b + 20*b^3)))*(a^2 + 5*b^2))/(a^7*d) - (1/(3*a) + (tan(c + d*x)^2*(a^2 + 5*b^2))/a^3 - (b*tan(c + d*x))/a^2 + (22*b*tan(c + d*x)^3*(a^2 + 5*b^2))/(3*a^4) + (10*b^2*tan(c + d*x)^4*(a^2 + 5*b^2))/a^5 + (4*b^3*tan(c + d*x)^5*(a^2 + 5*b^2))/a^6)/(d*(a^3*tan(c + d*x)^3 + b^3*tan(c + d*x)^6 + 3*a^2*b*tan(c + d*x)^4 + 3*a*b^2*tan(c + d*x)^5))","B"
77,1,337,300,5.647129,"\text{Not used}","int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))^4),x)","\frac{8\,b\,\mathrm{atanh}\left(\frac{4\,b\,\left(a+2\,b\,\mathrm{tan}\left(c+d\,x\right)\right)\,\left(a^4+10\,a^2\,b^2+14\,b^4\right)}{a\,\left(4\,a^4\,b+40\,a^2\,b^3+56\,b^5\right)}\right)\,\left(a^4+10\,a^2\,b^2+14\,b^4\right)}{a^9\,d}-\frac{\frac{1}{5\,a}+\frac{{\mathrm{tan}\left(c+d\,x\right)}^4\,\left(a^4+10\,a^2\,b^2+14\,b^4\right)}{a^5}+\frac{2\,{\mathrm{tan}\left(c+d\,x\right)}^2\,\left(5\,a^2+7\,b^2\right)}{15\,a^3}-\frac{2\,b\,\mathrm{tan}\left(c+d\,x\right)}{5\,a^2}+\frac{22\,b\,{\mathrm{tan}\left(c+d\,x\right)}^5\,\left(a^4+10\,a^2\,b^2+14\,b^4\right)}{3\,a^6}+\frac{10\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^6\,\left(a^4+10\,a^2\,b^2+14\,b^4\right)}{a^7}+\frac{4\,b^3\,{\mathrm{tan}\left(c+d\,x\right)}^7\,\left(a^4+10\,a^2\,b^2+14\,b^4\right)}{a^8}-\frac{2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^3\,\left(5\,a^2+7\,b^2\right)}{5\,a^4}}{d\,\left(a^3\,{\mathrm{tan}\left(c+d\,x\right)}^5+3\,a^2\,b\,{\mathrm{tan}\left(c+d\,x\right)}^6+3\,a\,b^2\,{\mathrm{tan}\left(c+d\,x\right)}^7+b^3\,{\mathrm{tan}\left(c+d\,x\right)}^8\right)}","Not used",1,"(8*b*atanh((4*b*(a + 2*b*tan(c + d*x))*(a^4 + 14*b^4 + 10*a^2*b^2))/(a*(4*a^4*b + 56*b^5 + 40*a^2*b^3)))*(a^4 + 14*b^4 + 10*a^2*b^2))/(a^9*d) - (1/(5*a) + (tan(c + d*x)^4*(a^4 + 14*b^4 + 10*a^2*b^2))/a^5 + (2*tan(c + d*x)^2*(5*a^2 + 7*b^2))/(15*a^3) - (2*b*tan(c + d*x))/(5*a^2) + (22*b*tan(c + d*x)^5*(a^4 + 14*b^4 + 10*a^2*b^2))/(3*a^6) + (10*b^2*tan(c + d*x)^6*(a^4 + 14*b^4 + 10*a^2*b^2))/a^7 + (4*b^3*tan(c + d*x)^7*(a^4 + 14*b^4 + 10*a^2*b^2))/a^8 - (2*b*tan(c + d*x)^3*(5*a^2 + 7*b^2))/(5*a^4))/(d*(a^3*tan(c + d*x)^5 + b^3*tan(c + d*x)^8 + 3*a^2*b*tan(c + d*x)^6 + 3*a*b^2*tan(c + d*x)^7))","B"
78,1,38,26,3.881336,"\text{Not used}","int(1/(sin(x)*(tan(x) + 1)),x)","\ln\left(\mathrm{tan}\left(\frac{x}{2}\right)\right)-\sqrt{2}\,\mathrm{atanh}\left(\frac{5\,\sqrt{2}\,\mathrm{tan}\left(\frac{x}{2}\right)+2\,\sqrt{2}}{7\,\mathrm{tan}\left(\frac{x}{2}\right)+3}\right)","Not used",1,"log(tan(x/2)) - 2^(1/2)*atanh((5*2^(1/2)*tan(x/2) + 2*2^(1/2))/(7*tan(x/2) + 3))","B"
79,0,-1,229,0.000000,"\text{Not used}","int(sin(c + d*x)^m*(a + b*tan(c + d*x))^3,x)","\int {\sin\left(c+d\,x\right)}^m\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^3 \,d x","Not used",1,"int(sin(c + d*x)^m*(a + b*tan(c + d*x))^3, x)","F"
80,0,-1,179,0.000000,"\text{Not used}","int(sin(c + d*x)^m*(a + b*tan(c + d*x))^2,x)","\int {\sin\left(c+d\,x\right)}^m\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^2 \,d x","Not used",1,"int(sin(c + d*x)^m*(a + b*tan(c + d*x))^2, x)","F"
81,0,-1,109,0.000000,"\text{Not used}","int(sin(c + d*x)^m*(a + b*tan(c + d*x)),x)","\int {\sin\left(c+d\,x\right)}^m\,\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right) \,d x","Not used",1,"int(sin(c + d*x)^m*(a + b*tan(c + d*x)), x)","F"
82,0,-1,765,0.000000,"\text{Not used}","int(sin(c + d*x)^m/(a + b*tan(c + d*x)),x)","\int \frac{{\sin\left(c+d\,x\right)}^m}{a+b\,\mathrm{tan}\left(c+d\,x\right)} \,d x","Not used",1,"int(sin(c + d*x)^m/(a + b*tan(c + d*x)), x)","F"
83,0,-1,24,0.000000,"\text{Not used}","int(sin(c + d*x)^m*(a + b*tan(c + d*x))^n,x)","\int {\sin\left(c+d\,x\right)}^m\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n \,d x","Not used",0,"int(sin(c + d*x)^m*(a + b*tan(c + d*x))^n, x)","F"
84,0,-1,435,0.000000,"\text{Not used}","int(sin(c + d*x)^4*(a + b*tan(c + d*x))^n,x)","\int {\sin\left(c+d\,x\right)}^4\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n \,d x","Not used",1,"int(sin(c + d*x)^4*(a + b*tan(c + d*x))^n, x)","F"
85,0,-1,276,0.000000,"\text{Not used}","int(sin(c + d*x)^2*(a + b*tan(c + d*x))^n,x)","\int {\sin\left(c+d\,x\right)}^2\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n \,d x","Not used",1,"int(sin(c + d*x)^2*(a + b*tan(c + d*x))^n, x)","F"
86,0,-1,48,0.000000,"\text{Not used}","int((a + b*tan(c + d*x))^n/sin(c + d*x)^2,x)","\int \frac{{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n}{{\sin\left(c+d\,x\right)}^2} \,d x","Not used",1,"int((a + b*tan(c + d*x))^n/sin(c + d*x)^2, x)","F"
87,0,-1,140,0.000000,"\text{Not used}","int((a + b*tan(c + d*x))^n/sin(c + d*x)^4,x)","\int \frac{{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n}{{\sin\left(c+d\,x\right)}^4} \,d x","Not used",1,"int((a + b*tan(c + d*x))^n/sin(c + d*x)^4, x)","F"
88,0,-1,24,0.000000,"\text{Not used}","int(sin(c + d*x)^3*(a + b*tan(c + d*x))^n,x)","\int {\sin\left(c+d\,x\right)}^3\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n \,d x","Not used",0,"int(sin(c + d*x)^3*(a + b*tan(c + d*x))^n, x)","F"
89,0,-1,22,0.000000,"\text{Not used}","int(sin(c + d*x)*(a + b*tan(c + d*x))^n,x)","\int \sin\left(c+d\,x\right)\,{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n \,d x","Not used",0,"int(sin(c + d*x)*(a + b*tan(c + d*x))^n, x)","F"
90,0,-1,22,0.000000,"\text{Not used}","int((a + b*tan(c + d*x))^n/sin(c + d*x),x)","\int \frac{{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n}{\sin\left(c+d\,x\right)} \,d x","Not used",0,"int((a + b*tan(c + d*x))^n/sin(c + d*x), x)","F"
91,0,-1,24,0.000000,"\text{Not used}","int((a + b*tan(c + d*x))^n/sin(c + d*x)^3,x)","\int \frac{{\left(a+b\,\mathrm{tan}\left(c+d\,x\right)\right)}^n}{{\sin\left(c+d\,x\right)}^3} \,d x","Not used",0,"int((a + b*tan(c + d*x))^n/sin(c + d*x)^3, x)","F"