1,1,545,203,0.565101,"\int \frac{x}{a+b \sin ^2(x)} \, dx","Integrate[x/(a + b*Sin[x]^2),x]","\frac{i \left(\text{Li}_2\left(\frac{\left(2 a+b-2 i \sqrt{-a (a+b)}\right) \left(\sqrt{-a (a+b)} \tan (x)-a\right)}{b \left(a+\sqrt{-a (a+b)} \tan (x)\right)}\right)-\text{Li}_2\left(\frac{\left(2 a+b+2 i \sqrt{-a (a+b)}\right) \left(\sqrt{-a (a+b)} \tan (x)-a\right)}{b \left(a+\sqrt{-a (a+b)} \tan (x)\right)}\right)\right)+4 x \tanh ^{-1}\left(\frac{a \cot (x)}{\sqrt{-a (a+b)}}\right)-2 \cos ^{-1}\left(\frac{2 a}{b}+1\right) \tanh ^{-1}\left(\frac{\sqrt{-a (a+b)} \tan (x)}{a}\right)-\log \left(\frac{2 a \left(-i \sqrt{-a (a+b)}+a+b\right) (1-i \tan (x))}{b \left(\sqrt{-a (a+b)} \tan (x)+a\right)}\right) \left(\cos ^{-1}\left(\frac{2 a}{b}+1\right)+2 i \tanh ^{-1}\left(\frac{\sqrt{-a (a+b)} \tan (x)}{a}\right)\right)-\log \left(\frac{2 a \left(i \sqrt{-a (a+b)}+a+b\right) (1+i \tan (x))}{b \left(\sqrt{-a (a+b)} \tan (x)+a\right)}\right) \left(\cos ^{-1}\left(\frac{2 a}{b}+1\right)-2 i \tanh ^{-1}\left(\frac{\sqrt{-a (a+b)} \tan (x)}{a}\right)\right)+\log \left(\frac{\sqrt{2} e^{-i x} \sqrt{-a (a+b)}}{\sqrt{-b} \sqrt{2 a-b \cos (2 x)+b}}\right) \left(2 i \tanh ^{-1}\left(\frac{\sqrt{-a (a+b)} \tan (x)}{a}\right)-2 i \tanh ^{-1}\left(\frac{a \cot (x)}{\sqrt{-a (a+b)}}\right)+\cos ^{-1}\left(\frac{2 a}{b}+1\right)\right)+\log \left(\frac{\sqrt{2} e^{i x} \sqrt{-a (a+b)}}{\sqrt{-b} \sqrt{2 a-b \cos (2 x)+b}}\right) \left(\cos ^{-1}\left(\frac{2 a}{b}+1\right)+2 i \left(\tanh ^{-1}\left(\frac{a \cot (x)}{\sqrt{-a (a+b)}}\right)-\tanh ^{-1}\left(\frac{\sqrt{-a (a+b)} \tan (x)}{a}\right)\right)\right)}{4 \sqrt{-a (a+b)}}","-\frac{\text{Li}_2\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{Li}_2\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x \log \left(1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left(1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}",1,"(4*x*ArcTanh[(a*Cot[x])/Sqrt[-(a*(a + b))]] - 2*ArcCos[1 + (2*a)/b]*ArcTanh[(Sqrt[-(a*(a + b))]*Tan[x])/a] + (ArcCos[1 + (2*a)/b] - (2*I)*ArcTanh[(a*Cot[x])/Sqrt[-(a*(a + b))]] + (2*I)*ArcTanh[(Sqrt[-(a*(a + b))]*Tan[x])/a])*Log[(Sqrt[2]*Sqrt[-(a*(a + b))])/(Sqrt[-b]*E^(I*x)*Sqrt[2*a + b - b*Cos[2*x]])] + (ArcCos[1 + (2*a)/b] + (2*I)*(ArcTanh[(a*Cot[x])/Sqrt[-(a*(a + b))]] - ArcTanh[(Sqrt[-(a*(a + b))]*Tan[x])/a]))*Log[(Sqrt[2]*Sqrt[-(a*(a + b))]*E^(I*x))/(Sqrt[-b]*Sqrt[2*a + b - b*Cos[2*x]])] - (ArcCos[1 + (2*a)/b] + (2*I)*ArcTanh[(Sqrt[-(a*(a + b))]*Tan[x])/a])*Log[(2*a*(a + b - I*Sqrt[-(a*(a + b))])*(1 - I*Tan[x]))/(b*(a + Sqrt[-(a*(a + b))]*Tan[x]))] - (ArcCos[1 + (2*a)/b] - (2*I)*ArcTanh[(Sqrt[-(a*(a + b))]*Tan[x])/a])*Log[(2*a*(a + b + I*Sqrt[-(a*(a + b))])*(1 + I*Tan[x]))/(b*(a + Sqrt[-(a*(a + b))]*Tan[x]))] + I*(PolyLog[2, ((2*a + b - (2*I)*Sqrt[-(a*(a + b))])*(-a + Sqrt[-(a*(a + b))]*Tan[x]))/(b*(a + Sqrt[-(a*(a + b))]*Tan[x]))] - PolyLog[2, ((2*a + b + (2*I)*Sqrt[-(a*(a + b))])*(-a + Sqrt[-(a*(a + b))]*Tan[x]))/(b*(a + Sqrt[-(a*(a + b))]*Tan[x]))]))/(4*Sqrt[-(a*(a + b))])","B",1
2,1,237,309,2.5834931,"\int \frac{x^2}{a+b \sin ^2(x)} \, dx","Integrate[x^2/(a + b*Sin[x]^2),x]","-\frac{i \left(-2 i x \text{Li}_2\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)+2 i x \text{Li}_2\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)+\text{Li}_3\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)-\text{Li}_3\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)+2 x^2 \log \left(1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)-2 x^2 \log \left(1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)\right)}{4 \sqrt{a} \sqrt{a+b}}","-\frac{x \text{Li}_2\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \text{Li}_3\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{i \text{Li}_3\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x^2 \log \left(1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left(1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}",1,"((-1/4*I)*(2*x^2*Log[1 - (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - 2*x^2*Log[1 - (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] - (2*I)*x*PolyLog[2, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + (2*I)*x*PolyLog[2, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] + PolyLog[3, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - PolyLog[3, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])]))/(Sqrt[a]*Sqrt[a + b])","A",1
3,1,315,411,2.5130763,"\int \frac{x^3}{a+b \sin ^2(x)} \, dx","Integrate[x^3/(a + b*Sin[x]^2),x]","\frac{-6 x^2 \text{Li}_2\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)+6 x^2 \text{Li}_2\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)-6 i x \text{Li}_3\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)+6 i x \text{Li}_3\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)+3 \text{Li}_4\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)-3 \text{Li}_4\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)-4 i x^3 \log \left(1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)+4 i x^3 \log \left(1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{8 \sqrt{a} \sqrt{a+b}}","-\frac{3 x^2 \text{Li}_2\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{3 x^2 \text{Li}_2\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}-\frac{3 i x \text{Li}_3\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{3 i x \text{Li}_3\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{3 \text{Li}_4\left(\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 \sqrt{a} \sqrt{a+b}}-\frac{3 \text{Li}_4\left(\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 \sqrt{a} \sqrt{a+b}}-\frac{i x^3 \log \left(1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^3 \log \left(1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}",1,"((-4*I)*x^3*Log[1 - (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + (4*I)*x^3*Log[1 - (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] - 6*x^2*PolyLog[2, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + 6*x^2*PolyLog[2, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] - (6*I)*x*PolyLog[3, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + (6*I)*x*PolyLog[3, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] + 3*PolyLog[4, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - 3*PolyLog[4, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(8*Sqrt[a]*Sqrt[a + b])","A",1
4,1,855,329,16.1084617,"\int \frac{x}{\left(a+b \sin ^2(c+d x)\right)^2} \, dx","Integrate[x/(a + b*Sin[c + d*x]^2)^2,x]","\frac{\left(-\frac{4 (2 a+b) c \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{a+b}}+2 \log \left(\sec ^2(c+d x)\right)-2 \log \left((a+b) \tan ^2(c+d x)+a\right)-\frac{i (2 a+b) \left(\log (1-i \tan (c+d x)) \log \left(\frac{\sqrt{a}-\sqrt{-a-b} \tan (c+d x)}{\sqrt{a}+i \sqrt{-a-b}}\right)+\text{Li}_2\left(\frac{\sqrt{-a-b} (1-i \tan (c+d x))}{\sqrt{-a-b}-i \sqrt{a}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}+\frac{i (2 a+b) \left(\log (1-i \tan (c+d x)) \log \left(\frac{\sqrt{-a-b} \tan (c+d x)+\sqrt{a}}{\sqrt{a}-i \sqrt{-a-b}}\right)+\text{Li}_2\left(\frac{\sqrt{-a-b} (1-i \tan (c+d x))}{i \sqrt{a}+\sqrt{-a-b}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}-\frac{i (2 a+b) \left(\log (i \tan (c+d x)+1) \log \left(\frac{\sqrt{-a-b} \tan (c+d x)+\sqrt{a}}{\sqrt{a}+i \sqrt{-a-b}}\right)+\text{Li}_2\left(\frac{\sqrt{-a-b} (i \tan (c+d x)+1)}{\sqrt{-a-b}-i \sqrt{a}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}+\frac{i (2 a+b) \left(\log (i \tan (c+d x)+1) \log \left(\frac{\sqrt{a}-\sqrt{-a-b} \tan (c+d x)}{\sqrt{a}-i \sqrt{-a-b}}\right)+\text{Li}_2\left(\frac{\sqrt{-a-b} (i \tan (c+d x)+1)}{i \sqrt{a}+\sqrt{-a-b}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}\right) (2 (2 a+b) d x-b \sin (2 (c+d x))) \left(\sqrt{a}-\sqrt{-a-b} \tan (c+d x)\right) \left(\sqrt{-a-b} \tan (c+d x)+\sqrt{a}\right) \cos ^2(c+d x)}{4 a (a+b) d^2 (-2 a-b+b \cos (2 (c+d x))) ((2 a+b) (2 c-i \log (1-i \tan (c+d x))+i \log (i \tan (c+d x)+1))+b \sin (2 (c+d x)))}+\frac{b c \sin (2 (c+d x))-b (c+d x) \sin (2 (c+d x))}{2 a (a+b) d^2 (-2 a-b+b \cos (2 (c+d x)))}","-\frac{(2 a+b) \text{Li}_2\left(\frac{b e^{2 i (c+d x)}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 a^{3/2} d^2 (a+b)^{3/2}}+\frac{(2 a+b) \text{Li}_2\left(\frac{b e^{2 i (c+d x)}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 a^{3/2} d^2 (a+b)^{3/2}}-\frac{i x (2 a+b) \log \left(1-\frac{b e^{2 i (c+d x)}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{4 a^{3/2} d (a+b)^{3/2}}+\frac{i x (2 a+b) \log \left(1-\frac{b e^{2 i (c+d x)}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{4 a^{3/2} d (a+b)^{3/2}}-\frac{\log (2 a-b \cos (2 c+2 d x)+b)}{4 a d^2 (a+b)}+\frac{b x \sin (2 c+2 d x)}{2 a d (a+b) (2 a-b \cos (2 c+2 d x)+b)}",1,"(b*c*Sin[2*(c + d*x)] - b*(c + d*x)*Sin[2*(c + d*x)])/(2*a*(a + b)*d^2*(-2*a - b + b*Cos[2*(c + d*x)])) + (Cos[c + d*x]^2*((-4*(2*a + b)*c*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*Sqrt[a + b]) + 2*Log[Sec[c + d*x]^2] - 2*Log[a + (a + b)*Tan[c + d*x]^2] - (I*(2*a + b)*(Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[a] - Sqrt[-a - b]*Tan[c + d*x])/(Sqrt[a] + I*Sqrt[-a - b])] + PolyLog[2, (Sqrt[-a - b]*(1 - I*Tan[c + d*x]))/((-I)*Sqrt[a] + Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]) + (I*(2*a + b)*(Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[a] + Sqrt[-a - b]*Tan[c + d*x])/(Sqrt[a] - I*Sqrt[-a - b])] + PolyLog[2, (Sqrt[-a - b]*(1 - I*Tan[c + d*x]))/(I*Sqrt[a] + Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]) - (I*(2*a + b)*(Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[a] + Sqrt[-a - b]*Tan[c + d*x])/(Sqrt[a] + I*Sqrt[-a - b])] + PolyLog[2, (Sqrt[-a - b]*(1 + I*Tan[c + d*x]))/((-I)*Sqrt[a] + Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]) + (I*(2*a + b)*(Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[a] - Sqrt[-a - b]*Tan[c + d*x])/(Sqrt[a] - I*Sqrt[-a - b])] + PolyLog[2, (Sqrt[-a - b]*(1 + I*Tan[c + d*x]))/(I*Sqrt[a] + Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]))*(2*(2*a + b)*d*x - b*Sin[2*(c + d*x)])*(Sqrt[a] - Sqrt[-a - b]*Tan[c + d*x])*(Sqrt[a] + Sqrt[-a - b]*Tan[c + d*x]))/(4*a*(a + b)*d^2*(-2*a - b + b*Cos[2*(c + d*x)])*((2*a + b)*(2*c - I*Log[1 - I*Tan[c + d*x]] + I*Log[1 + I*Tan[c + d*x]]) + b*Sin[2*(c + d*x)]))","B",0
5,1,16,22,0.0150995,"\int x \sqrt{\sin ^2(x)} \, dx","Integrate[x*Sqrt[Sin[x]^2],x]","\sqrt{\sin ^2(x)} (1-x \cot (x))","\sqrt{\sin ^2(x)}-x \sqrt{\sin ^2(x)} \cot (x)",1,"(1 - x*Cot[x])*Sqrt[Sin[x]^2]","A",1
6,1,532,203,0.5621049,"\int \frac{x}{a+b \cos ^2(x)} \, dx","Integrate[x/(a + b*Cos[x]^2),x]","\frac{i \left(\text{Li}_2\left(\frac{\left(2 a+b-2 i \sqrt{-a (a+b)}\right) \left(a+b-\sqrt{-a (a+b)} \tan (x)\right)}{b \left(a+b+\sqrt{-a (a+b)} \tan (x)\right)}\right)-\text{Li}_2\left(\frac{\left(2 a+b+2 i \sqrt{-a (a+b)}\right) \left(a+b-\sqrt{-a (a+b)} \tan (x)\right)}{b \left(a+b+\sqrt{-a (a+b)} \tan (x)\right)}\right)\right)+4 x \tanh ^{-1}\left(\frac{(a+b) \cot (x)}{\sqrt{-a (a+b)}}\right)+2 \cos ^{-1}\left(-\frac{2 a}{b}-1\right) \tanh ^{-1}\left(\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right)-\log \left(\frac{2 (a+b) \left(\sqrt{-a (a+b)}-i a\right) (\tan (x)-i)}{b \left(\sqrt{-a (a+b)} \tan (x)+a+b\right)}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}-1\right)+2 i \tanh ^{-1}\left(\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right)\right)-\log \left(\frac{2 (a+b) \left(\sqrt{-a (a+b)}+i a\right) (\tan (x)+i)}{b \left(\sqrt{-a (a+b)} \tan (x)+a+b\right)}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}-1\right)-2 i \tanh ^{-1}\left(\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right)\right)+\log \left(\frac{\sqrt{2} e^{-i x} \sqrt{-a (a+b)}}{\sqrt{b} \sqrt{2 a+b \cos (2 x)+b}}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}-1\right)-2 i \left(\tanh ^{-1}\left(\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right)+\tanh ^{-1}\left(\frac{(a+b) \cot (x)}{\sqrt{-a (a+b)}}\right)\right)\right)+\log \left(\frac{\sqrt{2} e^{i x} \sqrt{-a (a+b)}}{\sqrt{b} \sqrt{2 a+b \cos (2 x)+b}}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}-1\right)+2 i \left(\tanh ^{-1}\left(\frac{a \tan (x)}{\sqrt{-a (a+b)}}\right)+\tanh ^{-1}\left(\frac{(a+b) \cot (x)}{\sqrt{-a (a+b)}}\right)\right)\right)}{4 \sqrt{-a (a+b)}}","-\frac{\text{Li}_2\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{\text{Li}_2\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x \log \left(1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x \log \left(1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}",1,"(4*x*ArcTanh[((a + b)*Cot[x])/Sqrt[-(a*(a + b))]] + 2*ArcCos[-1 - (2*a)/b]*ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]] + (ArcCos[-1 - (2*a)/b] - (2*I)*(ArcTanh[((a + b)*Cot[x])/Sqrt[-(a*(a + b))]] + ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]]))*Log[(Sqrt[2]*Sqrt[-(a*(a + b))])/(Sqrt[b]*E^(I*x)*Sqrt[2*a + b + b*Cos[2*x]])] + (ArcCos[-1 - (2*a)/b] + (2*I)*(ArcTanh[((a + b)*Cot[x])/Sqrt[-(a*(a + b))]] + ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]]))*Log[(Sqrt[2]*Sqrt[-(a*(a + b))]*E^(I*x))/(Sqrt[b]*Sqrt[2*a + b + b*Cos[2*x]])] - (ArcCos[-1 - (2*a)/b] + (2*I)*ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]])*Log[(2*(a + b)*((-I)*a + Sqrt[-(a*(a + b))])*(-I + Tan[x]))/(b*(a + b + Sqrt[-(a*(a + b))]*Tan[x]))] - (ArcCos[-1 - (2*a)/b] - (2*I)*ArcTanh[(a*Tan[x])/Sqrt[-(a*(a + b))]])*Log[(2*(a + b)*(I*a + Sqrt[-(a*(a + b))])*(I + Tan[x]))/(b*(a + b + Sqrt[-(a*(a + b))]*Tan[x]))] + I*(PolyLog[2, ((2*a + b - (2*I)*Sqrt[-(a*(a + b))])*(a + b - Sqrt[-(a*(a + b))]*Tan[x]))/(b*(a + b + Sqrt[-(a*(a + b))]*Tan[x]))] - PolyLog[2, ((2*a + b + (2*I)*Sqrt[-(a*(a + b))])*(a + b - Sqrt[-(a*(a + b))]*Tan[x]))/(b*(a + b + Sqrt[-(a*(a + b))]*Tan[x]))]))/(4*Sqrt[-(a*(a + b))])","B",1
7,1,239,311,1.9040544,"\int \frac{x^2}{a+b \cos ^2(x)} \, dx","Integrate[x^2/(a + b*Cos[x]^2),x]","-\frac{i \left(-2 i x \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)+2 i x \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)+\text{Li}_3\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)-\text{Li}_3\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)+2 x^2 \log \left(1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)-2 x^2 \log \left(1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)\right)}{4 \sqrt{a} \sqrt{a+b}}","-\frac{x \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \text{Li}_3\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{i \text{Li}_3\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x^2 \log \left(1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left(1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}",1,"((-1/4*I)*(2*x^2*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - 2*x^2*Log[1 + (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] - (2*I)*x*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] + (2*I)*x*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] + PolyLog[3, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] - PolyLog[3, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]))/(Sqrt[a]*Sqrt[a + b])","A",1
8,1,319,415,1.7215054,"\int \frac{x^3}{a+b \cos ^2(x)} \, dx","Integrate[x^3/(a + b*Cos[x]^2),x]","\frac{-6 x^2 \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)+6 x^2 \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)-6 i x \text{Li}_3\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)+6 i x \text{Li}_3\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)+3 \text{Li}_4\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)-3 \text{Li}_4\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)-4 i x^3 \log \left(1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)+4 i x^3 \log \left(1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{8 \sqrt{a} \sqrt{a+b}}","-\frac{3 x^2 \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{3 x^2 \text{Li}_2\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}-\frac{3 i x \text{Li}_3\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{3 i x \text{Li}_3\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{4 \sqrt{a} \sqrt{a+b}}+\frac{3 \text{Li}_4\left(-\frac{b e^{2 i x}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 \sqrt{a} \sqrt{a+b}}-\frac{3 \text{Li}_4\left(-\frac{b e^{2 i x}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 \sqrt{a} \sqrt{a+b}}-\frac{i x^3 \log \left(1+\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^3 \log \left(1+\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{2 \sqrt{a} \sqrt{a+b}}",1,"((-4*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + (4*I)*x^3*Log[1 + (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] - 6*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] + 6*x^2*PolyLog[2, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] - (6*I)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] + (6*I)*x*PolyLog[3, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))] + 3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))] - 3*PolyLog[4, -((b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a]*Sqrt[a + b])","A",1
9,1,825,327,13.5964116,"\int \frac{x}{\left(a+b \cos ^2(c+d x)\right)^2} \, dx","Integrate[x/(a + b*Cos[c + d*x]^2)^2,x]","\frac{\left(-\frac{4 (2 a+b) c \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+b}}\right)}{\sqrt{a} \sqrt{a+b}}+2 \log \left(\sec ^2(c+d x)\right)-2 \log \left(a \tan ^2(c+d x)+a+b\right)-\frac{i (2 a+b) \left(\log (1-i \tan (c+d x)) \log \left(\frac{\sqrt{a} \tan (c+d x)+\sqrt{-a-b}}{\sqrt{-a-b}-i \sqrt{a}}\right)+\text{Li}_2\left(\frac{\sqrt{a} (1-i \tan (c+d x))}{\sqrt{a}+i \sqrt{-a-b}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}+\frac{i (2 a+b) \left(\log (i \tan (c+d x)+1) \log \left(\frac{\sqrt{a} \tan (c+d x)+\sqrt{-a-b}}{i \sqrt{a}+\sqrt{-a-b}}\right)+\text{Li}_2\left(\frac{\sqrt{a} (i \tan (c+d x)+1)}{\sqrt{a}-i \sqrt{-a-b}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}-\frac{i (2 a+b) \left(\log (i \tan (c+d x)+1) \log \left(\frac{\sqrt{-a-b}-\sqrt{a} \tan (c+d x)}{\sqrt{-a-b}-i \sqrt{a}}\right)+\text{Li}_2\left(\frac{\sqrt{a} (i \tan (c+d x)+1)}{\sqrt{a}+i \sqrt{-a-b}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}+\frac{i (2 a+b) \left(\log (1-i \tan (c+d x)) \log \left(\frac{\sqrt{-a-b}-\sqrt{a} \tan (c+d x)}{i \sqrt{a}+\sqrt{-a-b}}\right)+\text{Li}_2\left(\frac{\sqrt{a} (\tan (c+d x)+i)}{i \sqrt{a}+\sqrt{-a-b}}\right)\right)}{\sqrt{a} \sqrt{-a-b}}\right) (2 (2 a+b) d x+b \sin (2 (c+d x))) \left(\sqrt{a} \tan (c+d x)-\sqrt{-a-b}\right) \left(\sqrt{a} \tan (c+d x)+\sqrt{-a-b}\right) \cos ^2(c+d x)}{4 a (a+b) d^2 (2 a+b+b \cos (2 (c+d x))) (b \sin (2 (c+d x))-(2 a+b) (2 c-i \log (1-i \tan (c+d x))+i \log (i \tan (c+d x)+1)))}+\frac{b c \sin (2 (c+d x))-b (c+d x) \sin (2 (c+d x))}{2 a (a+b) d^2 (2 a+b+b \cos (2 (c+d x)))}","-\frac{(2 a+b) \text{Li}_2\left(-\frac{b e^{2 i (c+d x)}}{2 a-2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 a^{3/2} d^2 (a+b)^{3/2}}+\frac{(2 a+b) \text{Li}_2\left(-\frac{b e^{2 i (c+d x)}}{2 a+2 \sqrt{a+b} \sqrt{a}+b}\right)}{8 a^{3/2} d^2 (a+b)^{3/2}}-\frac{i x (2 a+b) \log \left(1+\frac{b e^{2 i (c+d x)}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{4 a^{3/2} d (a+b)^{3/2}}+\frac{i x (2 a+b) \log \left(1+\frac{b e^{2 i (c+d x)}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right)}{4 a^{3/2} d (a+b)^{3/2}}-\frac{\log (2 a+b \cos (2 c+2 d x)+b)}{4 a d^2 (a+b)}-\frac{b x \sin (2 c+2 d x)}{2 a d (a+b) (2 a+b \cos (2 c+2 d x)+b)}",1,"(b*c*Sin[2*(c + d*x)] - b*(c + d*x)*Sin[2*(c + d*x)])/(2*a*(a + b)*d^2*(2*a + b + b*Cos[2*(c + d*x)])) + (Cos[c + d*x]^2*((-4*(2*a + b)*c*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*Sqrt[a + b]) + 2*Log[Sec[c + d*x]^2] - 2*Log[a + b + a*Tan[c + d*x]^2] - (I*(2*a + b)*(Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[-a - b] + Sqrt[a]*Tan[c + d*x])/((-I)*Sqrt[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(1 - I*Tan[c + d*x]))/(Sqrt[a] + I*Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]) + (I*(2*a + b)*(Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[-a - b] + Sqrt[a]*Tan[c + d*x])/(I*Sqrt[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(1 + I*Tan[c + d*x]))/(Sqrt[a] - I*Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]) - (I*(2*a + b)*(Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[-a - b] - Sqrt[a]*Tan[c + d*x])/((-I)*Sqrt[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(1 + I*Tan[c + d*x]))/(Sqrt[a] + I*Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]) + (I*(2*a + b)*(Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[-a - b] - Sqrt[a]*Tan[c + d*x])/(I*Sqrt[a] + Sqrt[-a - b])] + PolyLog[2, (Sqrt[a]*(I + Tan[c + d*x]))/(I*Sqrt[a] + Sqrt[-a - b])]))/(Sqrt[a]*Sqrt[-a - b]))*(2*(2*a + b)*d*x + b*Sin[2*(c + d*x)])*(-Sqrt[-a - b] + Sqrt[a]*Tan[c + d*x])*(Sqrt[-a - b] + Sqrt[a]*Tan[c + d*x]))/(4*a*(a + b)*d^2*(2*a + b + b*Cos[2*(c + d*x)])*(-((2*a + b)*(2*c - I*Log[1 - I*Tan[c + d*x]] + I*Log[1 + I*Tan[c + d*x]])) + b*Sin[2*(c + d*x)]))","B",0