Integrand size = 18, antiderivative size = 161 \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=-\frac {11+6 \sqrt {2}}{49 \left (\sqrt {2}+x\right )}-\frac {2 \left (12-11 \sqrt {2}\right )-\left (57-40 \sqrt {2}\right ) x}{3 \left (193-132 \sqrt {2}\right ) \left (1+x+x^2\right )}+\frac {2 \left (129-85 \sqrt {2}\right ) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3} \left (193-132 \sqrt {2}\right )}+\frac {2}{343} \left (71+61 \sqrt {2}\right ) \log \left (\sqrt {2}+x\right )+\frac {7 \left (1-\sqrt {2}\right ) \log \left (1+x+x^2\right )}{193-132 \sqrt {2}} \] Output:
-1/49*(11+6*2^(1/2))/(2^(1/2)+x)-1/3*(24-22*2^(1/2)-(57-40*2^(1/2))*x)/(19 3-132*2^(1/2))/(x^2+x+1)+2/9*(129-85*2^(1/2))*arctan(1/3*(1+2*x)*3^(1/2))* 3^(1/2)/(193-132*2^(1/2))+2/343*(71+61*2^(1/2))*ln(2^(1/2)+x)+7*(1-2^(1/2) )*ln(x^2+x+1)/(193-132*2^(1/2))
Time = 0.43 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=\frac {809873-566592 \sqrt {2}}{\left (-15707707+11091750 \sqrt {2}\right ) \left (\sqrt {2}+x\right )}+\frac {30129079608055070283236-21304476501766810187526 \sqrt {2}+\left (-62085794360705285564525+43901286207808112899953 \sqrt {2}\right ) x}{3 \left (72097-50952 \sqrt {2}\right )^2 \left (113+72 \sqrt {2}\right ) \left (-193+132 \sqrt {2}\right ) \left (-843+589 \sqrt {2}\right ) \left (-1404491+993054 \sqrt {2}\right ) \left (1+x+x^2\right )}+\frac {2 \left (-1731811420232099418355476610308730834650895+1224575598982423225758721688341580503956966 \sqrt {2}\right ) \arctan \left (\frac {\left (-193+132 \sqrt {2}\right ) (1+2 x)}{\sqrt {3 \left (72097-50952 \sqrt {2}\right )}}\right )}{3 \left (6949256678653070099896757513353533545085-4913866521681490443934087594624264749029 \sqrt {2}\right ) \sqrt {3 \left (72097-50952 \sqrt {2}\right )}}-\frac {14 \left (-116382400935985545095123522239775815+82294784912606974633398219105049322 \sqrt {2}\right ) \log \left (\sqrt {2}+x\right )}{-18303114225793024607028352809840100149+12942256185890213449773808413084826577 \sqrt {2}}+\frac {7 \left (-44187626597573451506575969636009753303+31245370411683238056802161222924926726 \sqrt {2}\right ) \left (\log \left (193-132 \sqrt {2}\right )+\log \left (1+x+x^2\right )\right )}{-6949256678653070099896757513353533545085+4913866521681490443934087594624264749029 \sqrt {2}} \] Input:
Integrate[1/((Sqrt[2] + x)^2*(1 + x + x^2)^2),x]
Output:
(809873 - 566592*Sqrt[2])/((-15707707 + 11091750*Sqrt[2])*(Sqrt[2] + x)) + (30129079608055070283236 - 21304476501766810187526*Sqrt[2] + (-6208579436 0705285564525 + 43901286207808112899953*Sqrt[2])*x)/(3*(72097 - 50952*Sqrt [2])^2*(113 + 72*Sqrt[2])*(-193 + 132*Sqrt[2])*(-843 + 589*Sqrt[2])*(-1404 491 + 993054*Sqrt[2])*(1 + x + x^2)) + (2*(-173181142023209941835547661030 8730834650895 + 1224575598982423225758721688341580503956966*Sqrt[2])*ArcTa n[((-193 + 132*Sqrt[2])*(1 + 2*x))/Sqrt[3*(72097 - 50952*Sqrt[2])]])/(3*(6 949256678653070099896757513353533545085 - 49138665216814904439340875946242 64749029*Sqrt[2])*Sqrt[3*(72097 - 50952*Sqrt[2])]) - (14*(-116382400935985 545095123522239775815 + 82294784912606974633398219105049322*Sqrt[2])*Log[S qrt[2] + x])/(-18303114225793024607028352809840100149 + 129422561858902134 49773808413084826577*Sqrt[2]) + (7*(-4418762659757345150657596963600975330 3 + 31245370411683238056802161222924926726*Sqrt[2])*(Log[193 - 132*Sqrt[2] ] + Log[1 + x + x^2]))/(-6949256678653070099896757513353533545085 + 491386 6521681490443934087594624264749029*Sqrt[2])
Time = 0.41 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1165, 27, 1200, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (x+\sqrt {2}\right )^2 \left (x^2+x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {\int \frac {2 \left (4-\left (1-2 \sqrt {2}\right ) x\right )}{\left (x+\sqrt {2}\right )^2 \left (x^2+x+1\right )}dx}{3 \left (3-\sqrt {2}\right )}+\frac {-\left (\left (1-2 \sqrt {2}\right ) x\right )+\sqrt {2}+1}{3 \left (3-\sqrt {2}\right ) \left (x+\sqrt {2}\right ) \left (x^2+x+1\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {4-\left (1-2 \sqrt {2}\right ) x}{\left (x+\sqrt {2}\right )^2 \left (x^2+x+1\right )}dx}{3 \left (3-\sqrt {2}\right )}+\frac {-\left (\left (1-2 \sqrt {2}\right ) x\right )+\sqrt {2}+1}{3 \left (3-\sqrt {2}\right ) \left (x+\sqrt {2}\right ) \left (x^2+x+1\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {2 \int \left (\frac {3 \left (1-2 \sqrt {2}\right ) x-12 \sqrt {2}+17}{\left (3-\sqrt {2}\right )^2 \left (x^2+x+1\right )}+\frac {3 \left (-1+2 \sqrt {2}\right )}{\left (-3+\sqrt {2}\right )^2 \left (x+\sqrt {2}\right )}-\frac {\sqrt {2}}{\left (-3+\sqrt {2}\right ) \left (x+\sqrt {2}\right )^2}\right )dx}{3 \left (3-\sqrt {2}\right )}+\frac {-\left (\left (1-2 \sqrt {2}\right ) x\right )+\sqrt {2}+1}{3 \left (3-\sqrt {2}\right ) \left (x+\sqrt {2}\right ) \left (x^2+x+1\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {\left (31-18 \sqrt {2}\right ) \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3} \left (3-\sqrt {2}\right )^2}+\frac {3 \left (1-2 \sqrt {2}\right ) \log \left (x^2+x+1\right )}{2 \left (3-\sqrt {2}\right )^2}-\frac {\sqrt {2}}{\left (3-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}-\frac {3 \left (1-2 \sqrt {2}\right ) \log \left (x+\sqrt {2}\right )}{\left (3-\sqrt {2}\right )^2}\right )}{3 \left (3-\sqrt {2}\right )}+\frac {-\left (\left (1-2 \sqrt {2}\right ) x\right )+\sqrt {2}+1}{3 \left (3-\sqrt {2}\right ) \left (x+\sqrt {2}\right ) \left (x^2+x+1\right )}\) |
Input:
Int[1/((Sqrt[2] + x)^2*(1 + x + x^2)^2),x]
Output:
(1 + Sqrt[2] - (1 - 2*Sqrt[2])*x)/(3*(3 - Sqrt[2])*(Sqrt[2] + x)*(1 + x + x^2)) + (2*(-(Sqrt[2]/((3 - Sqrt[2])*(Sqrt[2] + x))) + ((31 - 18*Sqrt[2])* ArcTan[(1 + 2*x)/Sqrt[3]])/(Sqrt[3]*(3 - Sqrt[2])^2) - (3*(1 - 2*Sqrt[2])* Log[Sqrt[2] + x])/(3 - Sqrt[2])^2 + (3*(1 - 2*Sqrt[2])*Log[1 + x + x^2])/( 2*(3 - Sqrt[2])^2)))/(3*(3 - Sqrt[2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Time = 1.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(-\frac {\left (\frac {28 \sqrt {2}}{3}-21\right ) x -\frac {154 \sqrt {2}}{3}-56}{343 \left (x^{2}+x +1\right )}-\frac {\left (183 \sqrt {2}+213\right ) \ln \left (x^{2}+x +1\right )}{1029}-\frac {4 \left (-\frac {89 \sqrt {2}}{2}-\frac {351}{2}\right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3087}-\frac {-3+\sqrt {2}}{\left (-45+29 \sqrt {2}\right ) \left (\sqrt {2}+x \right )}+\frac {\left (-4 \sqrt {2}+2\right ) \ln \left (\sqrt {2}+x \right )}{-45+29 \sqrt {2}}\) | \(113\) |
Input:
int(1/(2^(1/2)+x)^2/(x^2+x+1)^2,x,method=_RETURNVERBOSE)
Output:
-1/343*((28/3*2^(1/2)-21)*x-154/3*2^(1/2)-56)/(x^2+x+1)-1/1029*(183*2^(1/2 )+213)*ln(x^2+x+1)-4/3087*(-89/2*2^(1/2)-351/2)*arctan(1/3*(1+2*x)*3^(1/2) )*3^(1/2)-1/(-45+29*2^(1/2))*(-3+2^(1/2))/(2^(1/2)+x)+1/(-45+29*2^(1/2))*( -4*2^(1/2)+2)*ln(2^(1/2)+x)
Time = 0.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=-\frac {168 \, x^{3} + 2 \, {\left (x^{4} + x^{3} - x^{2} - 2 \, x - 2\right )} \sqrt {20826 \, \sqrt {2} + \frac {139043}{3}} \arctan \left (\frac {1}{107359} \, {\left (89 \, \sqrt {2} {\left (2 \, x + 1\right )} - 702 \, x - 351\right )} \sqrt {20826 \, \sqrt {2} + \frac {139043}{3}}\right ) - 189 \, x^{2} + 3 \, {\left (71 \, x^{4} + 71 \, x^{3} - 71 \, x^{2} + 61 \, \sqrt {2} {\left (x^{4} + x^{3} - x^{2} - 2 \, x - 2\right )} - 142 \, x - 142\right )} \log \left (x^{2} + x + 1\right ) - 6 \, {\left (71 \, x^{4} + 71 \, x^{3} - 71 \, x^{2} + 61 \, \sqrt {2} {\left (x^{4} + x^{3} - x^{2} - 2 \, x - 2\right )} - 142 \, x - 142\right )} \log \left (x + \sqrt {2}\right ) + 7 \, \sqrt {2} {\left (22 \, x^{3} - 37 \, x^{2} - 23 \, x + 11\right )} + 105 \, x + 84}{1029 \, {\left (x^{4} + x^{3} - x^{2} - 2 \, x - 2\right )}} \] Input:
integrate(1/(2^(1/2)+x)^2/(x^2+x+1)^2,x, algorithm="fricas")
Output:
-1/1029*(168*x^3 + 2*(x^4 + x^3 - x^2 - 2*x - 2)*sqrt(20826*sqrt(2) + 1390 43/3)*arctan(1/107359*(89*sqrt(2)*(2*x + 1) - 702*x - 351)*sqrt(20826*sqrt (2) + 139043/3)) - 189*x^2 + 3*(71*x^4 + 71*x^3 - 71*x^2 + 61*sqrt(2)*(x^4 + x^3 - x^2 - 2*x - 2) - 142*x - 142)*log(x^2 + x + 1) - 6*(71*x^4 + 71*x ^3 - 71*x^2 + 61*sqrt(2)*(x^4 + x^3 - x^2 - 2*x - 2) - 142*x - 142)*log(x + sqrt(2)) + 7*sqrt(2)*(22*x^3 - 37*x^2 - 23*x + 11) + 105*x + 84)/(x^4 + x^3 - x^2 - 2*x - 2)
Timed out. \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(2**(1/2)+x)**2/(x**2+x+1)**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (18 \, \sqrt {2} - 31\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right )}{9 \, {\left (29 \, \sqrt {2} - 45\right )}} + \frac {{\left (2 \, \sqrt {2} - 1\right )} \log \left (x^{2} + x + 1\right )}{29 \, \sqrt {2} - 45} - \frac {2 \, {\left (2 \, \sqrt {2} - 1\right )} \log \left (x + \sqrt {2}\right )}{29 \, \sqrt {2} - 45} + \frac {2 \, \sqrt {2} x^{2} - x {\left (5 \, \sqrt {2} - 7\right )} - 1}{3 \, {\left (x^{3} {\left (6 \, \sqrt {2} - 11\right )} - x^{2} {\left (5 \, \sqrt {2} - 1\right )} - x {\left (5 \, \sqrt {2} - 1\right )} - 11 \, \sqrt {2} + 12\right )}} \] Input:
integrate(1/(2^(1/2)+x)^2/(x^2+x+1)^2,x, algorithm="maxima")
Output:
2/9*sqrt(3)*(18*sqrt(2) - 31)*arctan(1/3*sqrt(3)*(2*x + 1))/(29*sqrt(2) - 45) + (2*sqrt(2) - 1)*log(x^2 + x + 1)/(29*sqrt(2) - 45) - 2*(2*sqrt(2) - 1)*log(x + sqrt(2))/(29*sqrt(2) - 45) + 1/3*(2*sqrt(2)*x^2 - x*(5*sqrt(2) - 7) - 1)/(x^3*(6*sqrt(2) - 11) - x^2*(5*sqrt(2) - 1) - x*(5*sqrt(2) - 1) - 11*sqrt(2) + 12)
Exception generated. \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(2^(1/2)+x)^2/(x^2+x+1)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[3884404797504,5496426689544]:[1,0,-2]%%},[2]%%%}+%%%{% %{[448093
Time = 5.70 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=\frac {\left (-\frac {22\,\sqrt {2}}{147}-\frac {8}{49}\right )\,x^2+\left (\frac {13\,\sqrt {2}}{147}-\frac {17}{147}\right )\,x+\frac {2\,\sqrt {2}}{49}+\frac {11}{147}}{x^3+\left (\sqrt {2}+1\right )\,x^2+\left (\sqrt {2}+1\right )\,x+\sqrt {2}}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {11}{147}-\frac {\sqrt {2}\,\left (\frac {59}{147}+\frac {\sqrt {3}\,71{}\mathrm {i}}{441}\right )+\frac {2}{49}-\frac {\sqrt {3}\,124{}\mathrm {i}}{441}}{\sqrt {2}-3}+\frac {\sqrt {3}\,83{}\mathrm {i}}{441}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\frac {2}{49}-\sqrt {2}\,\left (-\frac {59}{147}+\frac {\sqrt {3}\,71{}\mathrm {i}}{441}\right )+\frac {\sqrt {3}\,124{}\mathrm {i}}{441}}{\sqrt {2}-3}-\frac {11}{147}+\frac {\sqrt {3}\,83{}\mathrm {i}}{441}\right )-\ln \left (x+\sqrt {2}\right )\,\left (\frac {\frac {118\,\sqrt {2}}{147}+\frac {4}{49}}{\sqrt {2}-3}-\frac {22}{147}\right ) \] Input:
int(1/((x + 2^(1/2))^2*(x + x^2 + 1)^2),x)
Output:
((2*2^(1/2))/49 - x^2*((22*2^(1/2))/147 + 8/49) + x*((13*2^(1/2))/147 - 17 /147) + 11/147)/(2^(1/2) + x*(2^(1/2) + 1) + x^3 + x^2*(2^(1/2) + 1)) - lo g(x - (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*83i)/441 - (2^(1/2)*((3^(1/2)*71i)/4 41 + 59/147) - (3^(1/2)*124i)/441 + 2/49)/(2^(1/2) - 3) + 11/147) + log(x + (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*83i)/441 + ((3^(1/2)*124i)/441 - 2^(1/2) *((3^(1/2)*71i)/441 - 59/147) + 2/49)/(2^(1/2) - 3) - 11/147) - log(x + 2^ (1/2))*(((118*2^(1/2))/147 + 4/49)/(2^(1/2) - 3) - 22/147)
Time = 0.23 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.80 \[ \int \frac {1}{\left (\sqrt {2}+x\right )^2 \left (1+x+x^2\right )^2} \, dx=\frac {-1260-1323 x +63 x^{2}-356 \sqrt {6}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right )+1098 \sqrt {2}\, \mathrm {log}\left (x^{2}+x +1\right )-2196 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}+x \right )+462 \sqrt {2}\, x^{4}-639 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{4}-639 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{3}+639 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{2}+1278 \,\mathrm {log}\left (x^{2}+x +1\right ) x +1278 \,\mathrm {log}\left (\sqrt {2}+x \right ) x^{4}+1278 \,\mathrm {log}\left (\sqrt {2}+x \right ) x^{3}-1278 \,\mathrm {log}\left (\sqrt {2}+x \right ) x^{2}-2556 \,\mathrm {log}\left (\sqrt {2}+x \right ) x -1155 \sqrt {2}-702 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{2}-1404 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x -2556 \,\mathrm {log}\left (\sqrt {2}+x \right )+504 x^{4}+1278 \,\mathrm {log}\left (x^{2}+x +1\right )+315 \sqrt {2}\, x^{2}+178 \sqrt {6}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{4}+178 \sqrt {6}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{3}-178 \sqrt {6}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{2}-356 \sqrt {6}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x +702 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{4}+702 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{3}-549 \sqrt {2}\, \mathrm {log}\left (x^{2}+x +1\right ) x^{4}-549 \sqrt {2}\, \mathrm {log}\left (x^{2}+x +1\right ) x^{3}+549 \sqrt {2}\, \mathrm {log}\left (x^{2}+x +1\right ) x^{2}+1098 \sqrt {2}\, \mathrm {log}\left (x^{2}+x +1\right ) x +1098 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}+x \right ) x^{4}+1098 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}+x \right ) x^{3}-1098 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}+x \right ) x^{2}-2196 \sqrt {2}\, \mathrm {log}\left (\sqrt {2}+x \right ) x -1404 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right )-441 \sqrt {2}\, x}{3087 x^{4}+3087 x^{3}-3087 x^{2}-6174 x -6174} \] Input:
int(1/(2^(1/2)+x)^2/(x^2+x+1)^2,x)
Output:
(178*sqrt(6)*atan((2*x + 1)/sqrt(3))*x**4 + 178*sqrt(6)*atan((2*x + 1)/sqr t(3))*x**3 - 178*sqrt(6)*atan((2*x + 1)/sqrt(3))*x**2 - 356*sqrt(6)*atan(( 2*x + 1)/sqrt(3))*x - 356*sqrt(6)*atan((2*x + 1)/sqrt(3)) + 702*sqrt(3)*at an((2*x + 1)/sqrt(3))*x**4 + 702*sqrt(3)*atan((2*x + 1)/sqrt(3))*x**3 - 70 2*sqrt(3)*atan((2*x + 1)/sqrt(3))*x**2 - 1404*sqrt(3)*atan((2*x + 1)/sqrt( 3))*x - 1404*sqrt(3)*atan((2*x + 1)/sqrt(3)) - 549*sqrt(2)*log(x**2 + x + 1)*x**4 - 549*sqrt(2)*log(x**2 + x + 1)*x**3 + 549*sqrt(2)*log(x**2 + x + 1)*x**2 + 1098*sqrt(2)*log(x**2 + x + 1)*x + 1098*sqrt(2)*log(x**2 + x + 1 ) + 1098*sqrt(2)*log(sqrt(2) + x)*x**4 + 1098*sqrt(2)*log(sqrt(2) + x)*x** 3 - 1098*sqrt(2)*log(sqrt(2) + x)*x**2 - 2196*sqrt(2)*log(sqrt(2) + x)*x - 2196*sqrt(2)*log(sqrt(2) + x) + 462*sqrt(2)*x**4 + 315*sqrt(2)*x**2 - 441 *sqrt(2)*x - 1155*sqrt(2) - 639*log(x**2 + x + 1)*x**4 - 639*log(x**2 + x + 1)*x**3 + 639*log(x**2 + x + 1)*x**2 + 1278*log(x**2 + x + 1)*x + 1278*l og(x**2 + x + 1) + 1278*log(sqrt(2) + x)*x**4 + 1278*log(sqrt(2) + x)*x**3 - 1278*log(sqrt(2) + x)*x**2 - 2556*log(sqrt(2) + x)*x - 2556*log(sqrt(2) + x) + 504*x**4 + 63*x**2 - 1323*x - 1260)/(3087*(x**4 + x**3 - x**2 - 2* x - 2))