1,1,1652,0,1.180736," ","integrate(x/(a+b*sin(x)^2),x, algorithm=""fricas"")","\frac{4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right)}{16 \, {\left(a^{2} + a b\right)}}"," ",0,"1/16*(4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1))/(a^2 + a*b)","B",0
2,1,2452,0,2.608507," ","integrate(x^2/(a+b*sin(x)^2),x, algorithm=""fricas"")","\frac{4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right)}{16 \, {\left(a^{2} + a b\right)}}"," ",0,"1/16*(4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b))/(a^2 + a*b)","C",0
3,1,3252,0,1.590015," ","integrate(x^3/(a+b*sin(x)^2),x, algorithm=""fricas"")","\frac{4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right)}{16 \, {\left(a^{2} + a b\right)}}"," ",0,"1/16*(4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b))/(a^2 + a*b)","C",0
4,1,4000,0,2.032593," ","integrate(x/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{4 \, {\left(a^{2} b + a b^{2}\right)} d x \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3} - {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - {\left(-i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right)}{8 \, {\left({\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} d^{2}\right)}}"," ",0,"-1/8*(4*(a^2*b + a*b^2)*d*x*cos(d*x + c)*sin(d*x + c) + (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - (2*a^2*b + 3*a*b^2 + b^3 - (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - (-I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - (I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - (I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - (-I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - (I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - (-I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) - (-I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - (I*(2*a^2*b + 3*a*b^2 + b^3)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*cos(d*x + c) + 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*cos(d*x + c) - 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*cos(d*x + c) + 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) - (a^3 + 2*a^2*b + a*b^2 - (a^2*b + a*b^2)*cos(d*x + c)^2 + (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + 3*a*b^2 + b^3)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*cos(d*x + c) - 2*I*sin(d*x + c)))/((a^4*b + 2*a^3*b^2 + a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d^2)","B",0
5,1,8,0,0.668946," ","integrate(x*(sin(x)^2)^(1/2),x, algorithm=""fricas"")","-x \cos\left(x\right) + \sin\left(x\right)"," ",0,"-x*cos(x) + sin(x)","A",0
6,1,1652,0,1.087420," ","integrate(x/(a+b*cos(x)^2),x, algorithm=""fricas"")","\frac{4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 4 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right)}{16 \, {\left(a^{2} + a b\right)}}"," ",0,"1/16*(4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 4*b*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1))/(a^2 + a*b)","B",0
7,1,2452,0,2.117341," ","integrate(x^2/(a+b*cos(x)^2),x, algorithm=""fricas"")","\frac{4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 8 \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) - 8 i \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right)}{16 \, {\left(a^{2} + a b\right)}}"," ",0,"1/16*(4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b))/(a^2 + a*b)","C",0
8,1,3252,0,0.836920," ","integrate(x^3/(a+b*cos(x)^2),x, algorithm=""fricas"")","\frac{4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) + 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - 4 i \, b x^{3} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) + 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) - 12 \, b x^{2} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) - 24 i \, b x \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(3, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) - 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) - 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, \frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) - i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right) + 24 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm polylog}\left(4, -\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(x\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(x\right) + 4 \, {\left(b \cos\left(x\right) + i \, b \sin\left(x\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}}}{2 \, b}\right)}{16 \, {\left(a^{2} + a b\right)}}"," ",0,"1/16*(4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x^3*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 12*b*x^2*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 24*I*b*x*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 24*b*sqrt((a^2 + a*b)/b^2)*polylog(4, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b))/(a^2 + a*b)","C",0
9,1,3799,0,1.178058," ","integrate(x/(a+b*cos(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{4 \, {\left(a^{2} b + a b^{2}\right)} d x \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b} + 1\right) - {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b} + 1\right) + {\left(2 \, a^{2} b + a b^{2} + {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} {\rm Li}_2\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b} + 1\right) - {\left(i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) - 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(4 i \, a + 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) - {\left(i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} d x + i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) + {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) - i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, b}{2 \, b}\right) - {\left(-i \, {\left(2 \, a^{2} b + a b^{2}\right)} d x + {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} d x - i \, {\left(2 \, a b^{2} + b^{3}\right)} c\right)} \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}} \log\left(-\frac{{\left(2 \, {\left(2 \, a + b\right)} \cos\left(d x + c\right) - {\left(-4 i \, a - 2 i \, b\right)} \sin\left(d x + c\right) + 4 \, {\left(b \cos\left(d x + c\right) + i \, b \sin\left(d x + c\right)\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, b}{2 \, b}\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} + 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{-\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} - 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} + 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(-i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} - i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, \cos\left(d x + c\right) + 2 i \, \sin\left(d x + c\right)\right) + {\left(a^{3} + a^{2} b + {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(i \, {\left(2 \, a b^{2} + b^{3}\right)} c \cos\left(d x + c\right)^{2} + i \, {\left(2 \, a^{2} b + a b^{2}\right)} c\right)} \sqrt{\frac{a^{2} + a b}{b^{2}}}\right)} \log\left(2 \, \sqrt{\frac{2 \, b \sqrt{\frac{a^{2} + a b}{b^{2}}} - 2 \, a - b}{b}} - 2 \, \cos\left(d x + c\right) - 2 i \, \sin\left(d x + c\right)\right)}{8 \, {\left({\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} d^{2}\right)}}"," ",0,"-1/8*(4*(a^2*b + a*b^2)*d*x*cos(d*x + c)*sin(d*x + c) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) - (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) + (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) + (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) + (2*a^2*b + a*b^2 + (2*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) - 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (4*I*a + 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) - (I*(2*a^2*b + a*b^2)*d*x + (I*(2*a*b^2 + b^3)*d*x + I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(d*x + c) + (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - (-I*(2*a^2*b + a*b^2)*d*x + (-I*(2*a*b^2 + b^3)*d*x - I*(2*a*b^2 + b^3)*c)*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(d*x + c) - (-4*I*a - 2*I*b)*sin(d*x + c) + 4*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*cos(d*x + c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*cos(d*x + c) - 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*cos(d*x + c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (-I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 - I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) + (a^3 + a^2*b + (a^2*b + a*b^2)*cos(d*x + c)^2 - (I*(2*a*b^2 + b^3)*c*cos(d*x + c)^2 + I*(2*a^2*b + a*b^2)*c)*sqrt((a^2 + a*b)/b^2))*log(2*sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*cos(d*x + c) - 2*I*sin(d*x + c)))/((a^4*b + 2*a^3*b^2 + a^2*b^3)*d^2*cos(d*x + c)^2 + (a^5 + 2*a^4*b + a^3*b^2)*d^2)","B",0
