Integrand size = 36, antiderivative size = 249 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{12 \left (1+x^2+x^4\right )^2}+\frac {e-2 g+i+(2 e-g-i) x^2}{12 \left (1+x^2+x^4\right )^2}+\frac {(2 e-g+i) \left (1+2 x^2\right )}{12 \left (1+x^2+x^4\right )}+\frac {x \left (2 d+3 f-h-(7 d-7 f+4 h) x^2\right )}{24 \left (1+x^2+x^4\right )}-\frac {(13 d+2 f+h) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {(13 d+2 f+h) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{48 \sqrt {3}}+\frac {(2 e-g+i) \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{16} (9 d-4 f+3 h) \text {arctanh}\left (\frac {x}{1+x^2}\right ) \] Output:
1/12*x*(d+f-2*h-(d-2*f+h)*x^2)/(x^4+x^2+1)^2+1/12*(e-2*g+i+(2*e-g-i)*x^2)/ (x^4+x^2+1)^2+(2*e-g+i)*(2*x^2+1)/(12*x^4+12*x^2+12)+x*(2*d+3*f-h-(7*d-7*f +4*h)*x^2)/(24*x^4+24*x^2+24)-1/144*(13*d+2*f+h)*arctan(1/3*(1-2*x)*3^(1/2 ))*3^(1/2)+1/144*(13*d+2*f+h)*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/9*(2*e -g+i)*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)+1/16*(9*d-4*f+3*h)*arctanh(x/( x^2+1))
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.31 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \left (\frac {12 \left (e+i+d x+f x-2 h x+2 e x^2-i x^2-d x^3+2 f x^3-h x^3-g \left (2+x^2\right )\right )}{\left (1+x^2+x^4\right )^2}+\frac {6 \left (2 i+2 d x+3 f x-h x+4 i x^2-7 d x^3+7 f x^3-4 h x^3-2 g \left (1+2 x^2\right )+e \left (4+8 x^2\right )\right )}{1+x^2+x^4}-\frac {\left (\left (-47 i+7 \sqrt {3}\right ) d+\left (17 i-7 \sqrt {3}\right ) f+2 \left (-7 i+2 \sqrt {3}\right ) h\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\left (\left (47 i+7 \sqrt {3}\right ) d-\left (17 i+7 \sqrt {3}\right ) f+2 \left (7 i+2 \sqrt {3}\right ) h\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-16 \sqrt {3} (2 e-g+i) \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )\right ) \] Input:
Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^3,x]
Output:
((12*(e + i + d*x + f*x - 2*h*x + 2*e*x^2 - i*x^2 - d*x^3 + 2*f*x^3 - h*x^ 3 - g*(2 + x^2)))/(1 + x^2 + x^4)^2 + (6*(2*i + 2*d*x + 3*f*x - h*x + 4*i* x^2 - 7*d*x^3 + 7*f*x^3 - 4*h*x^3 - 2*g*(1 + 2*x^2) + e*(4 + 8*x^2)))/(1 + x^2 + x^4) - (((-47*I + 7*Sqrt[3])*d + (17*I - 7*Sqrt[3])*f + 2*(-7*I + 2 *Sqrt[3])*h)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[3])/6] - (((47 *I + 7*Sqrt[3])*d - (17*I + 7*Sqrt[3])*f + 2*(7*I + 2*Sqrt[3])*h)*ArcTan[( (I + Sqrt[3])*x)/2])/Sqrt[(1 - I*Sqrt[3])/6] - 16*Sqrt[3]*(2*e - g + i)*Ar cTan[Sqrt[3]/(1 + 2*x^2)])/144
Time = 0.74 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2202, 2194, 2191, 27, 1086, 1083, 217, 2206, 1492, 27, 1483, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (x^4+x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\int \frac {x \left (i x^4+g x^2+e\right )}{\left (x^4+x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{2} \int \frac {i x^4+g x^2+e}{\left (x^4+x^2+1\right )^3}dx^2\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{2} \left (\frac {1}{6} \int \frac {3 (2 e-g+i)}{\left (x^4+x^2+1\right )^2}dx^2+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{2} \left (\frac {1}{2} (2 e-g+i) \int \frac {1}{\left (x^4+x^2+1\right )^2}dx^2+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{2} \left (\frac {1}{2} (2 e-g+i) \left (\frac {2}{3} \int \frac {1}{x^4+x^2+1}dx^2+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right )+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{2} \left (\frac {1}{2} (2 e-g+i) \left (\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}-\frac {4}{3} \int \frac {1}{-x^4-3}d\left (2 x^2+1\right )\right )+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{\left (x^4+x^2+1\right )^3}dx+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{12} \int \frac {-5 (d-2 f+h) x^2+11 d-f+2 h}{\left (x^4+x^2+1\right )^2}dx+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{6} \int \frac {3 \left (5 (4 d-f+h)-(7 d-7 f+4 h) x^2\right )}{x^4+x^2+1}dx+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \int \frac {5 (4 d-f+h)-(7 d-7 f+4 h) x^2}{x^4+x^2+1}dx+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {5 (4 d-f+h)-3 (9 d-4 f+3 h) x}{x^2-x+1}dx+\frac {1}{2} \int \frac {5 (4 d-f+h)+3 (9 d-4 f+3 h) x}{x^2+x+1}dx\right )+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (13 d+2 f+h) \int \frac {1}{x^2-x+1}dx-\frac {3}{2} (9 d-4 f+3 h) \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} (13 d+2 f+h) \int \frac {1}{x^2+x+1}dx+\frac {3}{2} (9 d-4 f+3 h) \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} (13 d+2 f+h) \int \frac {1}{x^2-x+1}dx+\frac {3}{2} (9 d-4 f+3 h) \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{2} \left (\frac {1}{2} (13 d+2 f+h) \int \frac {1}{x^2+x+1}dx+\frac {3}{2} (9 d-4 f+3 h) \int \frac {2 x+1}{x^2+x+1}dx\right )\right )+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {3}{2} (9 d-4 f+3 h) \int \frac {1-2 x}{x^2-x+1}dx-(13 d+2 f+h) \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{2} \left (\frac {3}{2} (9 d-4 f+3 h) \int \frac {2 x+1}{x^2+x+1}dx-(13 d+2 f+h) \int \frac {1}{-(2 x+1)^2-3}d(2 x+1)\right )\right )+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {3}{2} (9 d-4 f+3 h) \int \frac {1-2 x}{x^2-x+1}dx+\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f+h)}{\sqrt {3}}\right )+\frac {1}{2} \left (\frac {3}{2} (9 d-4 f+3 h) \int \frac {2 x+1}{x^2+x+1}dx+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f+h)}{\sqrt {3}}\right )\right )+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right ) (13 d+2 f+h)}{\sqrt {3}}-\frac {3}{2} \log \left (x^2-x+1\right ) (9 d-4 f+3 h)\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (13 d+2 f+h)}{\sqrt {3}}+\frac {3}{2} \log \left (x^2+x+1\right ) (9 d-4 f+3 h)\right )\right )+\frac {x \left (-\left (x^2 (7 d-7 f+4 h)\right )+2 d+3 f-h\right )}{2 \left (x^4+x^2+1\right )}\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {4 \arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x^2+1}{3 \left (x^4+x^2+1\right )}\right ) (2 e-g+i)+\frac {x^2 (2 e-g-i)+e-2 g+i}{6 \left (x^4+x^2+1\right )^2}\right )+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{12 \left (x^4+x^2+1\right )^2}\) |
Input:
Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(1 + x^2 + x^4)^3,x]
Output:
(x*(d + f - 2*h - (d - 2*f + h)*x^2))/(12*(1 + x^2 + x^4)^2) + ((e - 2*g + i + (2*e - g - i)*x^2)/(6*(1 + x^2 + x^4)^2) + ((2*e - g + i)*((1 + 2*x^2 )/(3*(1 + x^2 + x^4)) + (4*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3])))/2)/2 + ((x*(2*d + 3*f - h - (7*d - 7*f + 4*h)*x^2))/(2*(1 + x^2 + x^4)) + (((( 13*d + 2*f + h)*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3] - (3*(9*d - 4*f + 3*h) *Log[1 - x + x^2])/2)/2 + (((13*d + 2*f + h)*ArcTan[(1 + 2*x)/Sqrt[3]])/Sq rt[3] + (3*(9*d - 4*f + 3*h)*Log[1 + x + x^2])/2)/2)/2)/12
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.36 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(\frac {\left (-\frac {7 d}{3}+\frac {7 f}{3}-\frac {4 h}{3}-\frac {4 e}{3}-\frac {g}{3}+\frac {i}{3}\right ) x^{3}+\left (-6 d +4 f -2 h -2 g +2 i \right ) x^{2}+\left (-\frac {20 d}{3}+\frac {13 f}{3}-\frac {5 h}{3}+\frac {e}{3}-\frac {8 g}{3}+\frac {7 i}{3}\right ) x -4 d +\frac {4 f}{3}+2 e -2 g +\frac {4 i}{3}}{16 \left (x^{2}+x +1\right )^{2}}+\frac {\left (27 d -12 f +9 h \right ) \ln \left (x^{2}+x +1\right )}{96}+\frac {\left (\frac {13 d}{2}-16 e +f +8 g +\frac {h}{2}-8 i \right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{72}-\frac {\left (\frac {7 d}{3}-\frac {7 f}{3}+\frac {4 h}{3}-\frac {4 e}{3}-\frac {g}{3}+\frac {i}{3}\right ) x^{3}+\left (-6 d +4 f -2 h +2 g -2 i \right ) x^{2}+\left (\frac {20 d}{3}-\frac {13 f}{3}+\frac {5 h}{3}+\frac {e}{3}-\frac {8 g}{3}+\frac {7 i}{3}\right ) x -4 d +\frac {4 f}{3}-2 e +2 g -\frac {4 i}{3}}{16 \left (x^{2}-x +1\right )^{2}}-\frac {\left (27 d -12 f +9 h \right ) \ln \left (x^{2}-x +1\right )}{96}-\frac {\left (-\frac {13 d}{2}-16 e -f +8 g -\frac {h}{2}-8 i \right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{72}\) | \(292\) |
| risch | \(\text {Expression too large to display}\) | \(463265\) |
Input:
int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x,method=_RETURNVERBOSE)
Output:
1/16*((-7/3*d+7/3*f-4/3*h-4/3*e-1/3*g+1/3*i)*x^3+(-6*d+4*f-2*h-2*g+2*i)*x^ 2+(-20/3*d+13/3*f-5/3*h+1/3*e-8/3*g+7/3*i)*x-4*d+4/3*f+2*e-2*g+4/3*i)/(x^2 +x+1)^2+1/96*(27*d-12*f+9*h)*ln(x^2+x+1)+1/72*(13/2*d-16*e+f+8*g+1/2*h-8*i )*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/16*((7/3*d-7/3*f+4/3*h-4/3*e-1/3*g +1/3*i)*x^3+(-6*d+4*f-2*h+2*g-2*i)*x^2+(20/3*d-13/3*f+5/3*h+1/3*e-8/3*g+7/ 3*i)*x-4*d+4/3*f-2*e+2*g-4/3*i)/(x^2-x+1)^2-1/96*(27*d-12*f+9*h)*ln(x^2-x+ 1)-1/72*(-13/2*d-16*e-f+8*g-1/2*h-8*i)*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (224) = 448\).
Time = 4.78 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.09 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=-\frac {12 \, {\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 48 \, {\left (2 \, e - g + i\right )} x^{6} + 60 \, {\left (d - 2 \, f + h\right )} x^{5} - 72 \, {\left (2 \, e - g + i\right )} x^{4} + 84 \, {\left (d - 2 \, f + h\right )} x^{3} - 48 \, {\left (4 \, e - 2 \, g + i\right )} x^{2} - 2 \, \sqrt {3} {\left ({\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{8} + 2 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{6} + 3 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{4} + 2 \, {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} x^{2} + 13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{8} + 2 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{6} + 3 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{4} + 2 \, {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} x^{2} + 13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (4 \, d + 5 \, f - 5 \, h\right )} x - 9 \, {\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (9 \, d - 4 \, f + 3 \, h\right )} x^{8} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{6} + 3 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{4} + 2 \, {\left (9 \, d - 4 \, f + 3 \, h\right )} x^{2} + 9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - 72 \, e + 72 \, g - 48 \, i}{288 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x, algorithm="fric as")
Output:
-1/288*(12*(7*d - 7*f + 4*h)*x^7 - 48*(2*e - g + i)*x^6 + 60*(d - 2*f + h) *x^5 - 72*(2*e - g + i)*x^4 + 84*(d - 2*f + h)*x^3 - 48*(4*e - 2*g + i)*x^ 2 - 2*sqrt(3)*((13*d - 32*e + 2*f + 16*g + h - 16*i)*x^8 + 2*(13*d - 32*e + 2*f + 16*g + h - 16*i)*x^6 + 3*(13*d - 32*e + 2*f + 16*g + h - 16*i)*x^4 + 2*(13*d - 32*e + 2*f + 16*g + h - 16*i)*x^2 + 13*d - 32*e + 2*f + 16*g + h - 16*i)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*sqrt(3)*((13*d + 32*e + 2*f - 16*g + h + 16*i)*x^8 + 2*(13*d + 32*e + 2*f - 16*g + h + 16*i)*x^6 + 3*( 13*d + 32*e + 2*f - 16*g + h + 16*i)*x^4 + 2*(13*d + 32*e + 2*f - 16*g + h + 16*i)*x^2 + 13*d + 32*e + 2*f - 16*g + h + 16*i)*arctan(1/3*sqrt(3)*(2* x - 1)) - 12*(4*d + 5*f - 5*h)*x - 9*((9*d - 4*f + 3*h)*x^8 + 2*(9*d - 4*f + 3*h)*x^6 + 3*(9*d - 4*f + 3*h)*x^4 + 2*(9*d - 4*f + 3*h)*x^2 + 9*d - 4* f + 3*h)*log(x^2 + x + 1) + 9*((9*d - 4*f + 3*h)*x^8 + 2*(9*d - 4*f + 3*h) *x^6 + 3*(9*d - 4*f + 3*h)*x^4 + 2*(9*d - 4*f + 3*h)*x^2 + 9*d - 4*f + 3*h )*log(x^2 - x + 1) - 72*e + 72*g - 48*i)/(x^8 + 2*x^6 + 3*x^4 + 2*x^2 + 1)
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4+x**2+1)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac {{\left (7 \, d - 7 \, f + 4 \, h\right )} x^{7} - 4 \, {\left (2 \, e - g + i\right )} x^{6} + 5 \, {\left (d - 2 \, f + h\right )} x^{5} - 6 \, {\left (2 \, e - g + i\right )} x^{4} + 7 \, {\left (d - 2 \, f + h\right )} x^{3} - 4 \, {\left (4 \, e - 2 \, g + i\right )} x^{2} - {\left (4 \, d + 5 \, f - 5 \, h\right )} x - 6 \, e + 6 \, g - 4 \, i}{24 \, {\left (x^{8} + 2 \, x^{6} + 3 \, x^{4} + 2 \, x^{2} + 1\right )}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x, algorithm="maxi ma")
Output:
1/144*sqrt(3)*(13*d - 32*e + 2*f + 16*g + h - 16*i)*arctan(1/3*sqrt(3)*(2* x + 1)) + 1/144*sqrt(3)*(13*d + 32*e + 2*f - 16*g + h + 16*i)*arctan(1/3*s qrt(3)*(2*x - 1)) + 1/32*(9*d - 4*f + 3*h)*log(x^2 + x + 1) - 1/32*(9*d - 4*f + 3*h)*log(x^2 - x + 1) - 1/24*((7*d - 7*f + 4*h)*x^7 - 4*(2*e - g + i )*x^6 + 5*(d - 2*f + h)*x^5 - 6*(2*e - g + i)*x^4 + 7*(d - 2*f + h)*x^3 - 4*(4*e - 2*g + i)*x^2 - (4*d + 5*f - 5*h)*x - 6*e + 6*g - 4*i)/(x^8 + 2*x^ 6 + 3*x^4 + 2*x^2 + 1)
Time = 0.16 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=\frac {1}{144} \, \sqrt {3} {\left (13 \, d - 32 \, e + 2 \, f + 16 \, g + h - 16 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{144} \, \sqrt {3} {\left (13 \, d + 32 \, e + 2 \, f - 16 \, g + h + 16 \, i\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{32} \, {\left (9 \, d - 4 \, f + 3 \, h\right )} \log \left (x^{2} - x + 1\right ) - \frac {7 \, d x^{7} - 7 \, f x^{7} + 4 \, h x^{7} - 8 \, e x^{6} + 4 \, g x^{6} - 4 \, i x^{6} + 5 \, d x^{5} - 10 \, f x^{5} + 5 \, h x^{5} - 12 \, e x^{4} + 6 \, g x^{4} - 6 \, i x^{4} + 7 \, d x^{3} - 14 \, f x^{3} + 7 \, h x^{3} - 16 \, e x^{2} + 8 \, g x^{2} - 4 \, i x^{2} - 4 \, d x - 5 \, f x + 5 \, h x - 6 \, e + 6 \, g - 4 \, i}{24 \, {\left (x^{4} + x^{2} + 1\right )}^{2}} \] Input:
integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x, algorithm="giac ")
Output:
1/144*sqrt(3)*(13*d - 32*e + 2*f + 16*g + h - 16*i)*arctan(1/3*sqrt(3)*(2* x + 1)) + 1/144*sqrt(3)*(13*d + 32*e + 2*f - 16*g + h + 16*i)*arctan(1/3*s qrt(3)*(2*x - 1)) + 1/32*(9*d - 4*f + 3*h)*log(x^2 + x + 1) - 1/32*(9*d - 4*f + 3*h)*log(x^2 - x + 1) - 1/24*(7*d*x^7 - 7*f*x^7 + 4*h*x^7 - 8*e*x^6 + 4*g*x^6 - 4*i*x^6 + 5*d*x^5 - 10*f*x^5 + 5*h*x^5 - 12*e*x^4 + 6*g*x^4 - 6*i*x^4 + 7*d*x^3 - 14*f*x^3 + 7*h*x^3 - 16*e*x^2 + 8*g*x^2 - 4*i*x^2 - 4* d*x - 5*f*x + 5*h*x - 6*e + 6*g - 4*i)/(x^4 + x^2 + 1)^2
Time = 27.39 (sec) , antiderivative size = 1963, normalized size of antiderivative = 7.88 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(x^2 + x^4 + 1)^3,x)
Output:
(e/4 - g/4 + i/6 + x*(d/6 + (5*f)/24 - (5*h)/24) - x^7*((7*d)/24 - (7*f)/2 4 + h/6) - x^5*((5*d)/24 - (5*f)/12 + (5*h)/24) - x^3*((7*d)/24 - (7*f)/12 + (7*h)/24) + x^4*(e/2 - g/4 + i/4) + x^2*((2*e)/3 - g/3 + i/6) + x^6*(e/ 3 - g/6 + i/6))/(2*x^2 + 3*x^4 + 2*x^6 + x^8 + 1) - log(960*d*g - 2763*d*f - 1920*d*e + 480*e*f + 1971*d*h - 960*d*i - 480*e*h - 240*f*g - 981*f*h + 240*f*i + 240*g*h - 240*h*i + 3^(1/2)*d^2*1620i + 3^(1/2)*f^2*180i + 3^(1 /2)*h^2*135i - 3807*d^2*x - 612*f^2*x - 378*h^2*x + 2754*d^2 + 684*f^2 + 3 51*h^2 + 3^(1/2)*d*e*1088i - 3^(1/2)*d*f*1125i - 3^(1/2)*d*g*544i - 3^(1/2 )*e*f*608i + 3^(1/2)*d*h*945i + 3^(1/2)*d*i*544i + 3^(1/2)*e*h*416i + 3^(1 /2)*f*g*304i - 3^(1/2)*f*h*315i - 3^(1/2)*f*i*304i - 3^(1/2)*g*h*208i + 3^ (1/2)*h*i*208i - 672*d*e*x + 3069*d*f*x + 336*d*g*x + 672*e*f*x - 2403*d*h *x - 336*d*i*x - 384*e*h*x - 336*f*g*x + 963*f*h*x + 336*f*i*x + 192*g*h*x - 192*h*i*x + 3^(1/2)*d^2*x*567i + 3^(1/2)*f^2*x*252i + 3^(1/2)*h^2*x*108 i - 3^(1/2)*d*f*x*819i + 3^(1/2)*d*g*x*752i + 3^(1/2)*e*f*x*544i + 3^(1/2) *d*h*x*513i - 3^(1/2)*d*i*x*752i - 3^(1/2)*e*h*x*448i - 3^(1/2)*f*g*x*272i - 3^(1/2)*f*h*x*333i + 3^(1/2)*f*i*x*272i + 3^(1/2)*g*h*x*224i - 3^(1/2)* h*i*x*224i - 3^(1/2)*d*e*x*1504i)*((9*d)/32 - f/8 + (3*h)/32 + (3^(1/2)*d* 13i)/288 + (3^(1/2)*e*1i)/9 + (3^(1/2)*f*1i)/144 - (3^(1/2)*g*1i)/18 + (3^ (1/2)*h*1i)/288 + (3^(1/2)*i*1i)/18) - log(1920*d*e - 2763*d*f - 960*d*g - 480*e*f + 1971*d*h + 960*d*i + 480*e*h + 240*f*g - 981*f*h - 240*f*i -...
Time = 0.17 (sec) , antiderivative size = 1637, normalized size of antiderivative = 6.57 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{\left (1+x^2+x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4+x^2+1)^3,x)
Output:
(26*sqrt(3)*atan((2*x - 1)/sqrt(3))*d*x**8 + 52*sqrt(3)*atan((2*x - 1)/sqr t(3))*d*x**6 + 78*sqrt(3)*atan((2*x - 1)/sqrt(3))*d*x**4 + 52*sqrt(3)*atan ((2*x - 1)/sqrt(3))*d*x**2 + 26*sqrt(3)*atan((2*x - 1)/sqrt(3))*d + 64*sqr t(3)*atan((2*x - 1)/sqrt(3))*e*x**8 + 128*sqrt(3)*atan((2*x - 1)/sqrt(3))* e*x**6 + 192*sqrt(3)*atan((2*x - 1)/sqrt(3))*e*x**4 + 128*sqrt(3)*atan((2* x - 1)/sqrt(3))*e*x**2 + 64*sqrt(3)*atan((2*x - 1)/sqrt(3))*e + 4*sqrt(3)* atan((2*x - 1)/sqrt(3))*f*x**8 + 8*sqrt(3)*atan((2*x - 1)/sqrt(3))*f*x**6 + 12*sqrt(3)*atan((2*x - 1)/sqrt(3))*f*x**4 + 8*sqrt(3)*atan((2*x - 1)/sqr t(3))*f*x**2 + 4*sqrt(3)*atan((2*x - 1)/sqrt(3))*f - 32*sqrt(3)*atan((2*x - 1)/sqrt(3))*g*x**8 - 64*sqrt(3)*atan((2*x - 1)/sqrt(3))*g*x**6 - 96*sqrt (3)*atan((2*x - 1)/sqrt(3))*g*x**4 - 64*sqrt(3)*atan((2*x - 1)/sqrt(3))*g* x**2 - 32*sqrt(3)*atan((2*x - 1)/sqrt(3))*g + 2*sqrt(3)*atan((2*x - 1)/sqr t(3))*h*x**8 + 4*sqrt(3)*atan((2*x - 1)/sqrt(3))*h*x**6 + 6*sqrt(3)*atan(( 2*x - 1)/sqrt(3))*h*x**4 + 4*sqrt(3)*atan((2*x - 1)/sqrt(3))*h*x**2 + 2*sq rt(3)*atan((2*x - 1)/sqrt(3))*h + 32*sqrt(3)*atan((2*x - 1)/sqrt(3))*i*x** 8 + 64*sqrt(3)*atan((2*x - 1)/sqrt(3))*i*x**6 + 96*sqrt(3)*atan((2*x - 1)/ sqrt(3))*i*x**4 + 64*sqrt(3)*atan((2*x - 1)/sqrt(3))*i*x**2 + 32*sqrt(3)*a tan((2*x - 1)/sqrt(3))*i + 26*sqrt(3)*atan((2*x + 1)/sqrt(3))*d*x**8 + 52* sqrt(3)*atan((2*x + 1)/sqrt(3))*d*x**6 + 78*sqrt(3)*atan((2*x + 1)/sqrt(3) )*d*x**4 + 52*sqrt(3)*atan((2*x + 1)/sqrt(3))*d*x**2 + 26*sqrt(3)*atan(...