Integrand size = 31, antiderivative size = 136 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=h x-\frac {(d+f-2 h) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(d+f-2 h) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(2 e-g) \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} (d-f) \log \left (1-x+x^2\right )+\frac {1}{4} (d-f) \log \left (1+x+x^2\right )+\frac {1}{4} g \log \left (1+x^2+x^4\right ) \]
h*x-1/4*(d-f)*ln(x^2-x+1)+1/4*(d-f)*ln(x^2+x+1)+1/4*g*ln(x^4+x^2+1)-1/6*(d +f-2*h)*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/6*(d+f-2*h)*arctan(1/3*(1+2* x)*3^(1/2))*3^(1/2)+1/6*(2*e-g)*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.21 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=\frac {1}{24} \left (24 h x+4 \left (\left (3 i+\sqrt {3}\right ) d+\left (-3 i+\sqrt {3}\right ) f-2 \sqrt {3} h\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+4 \left (\left (-3 i+\sqrt {3}\right ) d+\left (3 i+\sqrt {3}\right ) f-2 \sqrt {3} h\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )-8 \sqrt {3} e \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )+4 \sqrt {3} g \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )+6 g \log \left (1+x^2+x^4\right )\right ) \]
(24*h*x + 4*((3*I + Sqrt[3])*d + (-3*I + Sqrt[3])*f - 2*Sqrt[3]*h)*ArcTan[ ((-I + Sqrt[3])*x)/2] + 4*((-3*I + Sqrt[3])*d + (3*I + Sqrt[3])*f - 2*Sqrt [3]*h)*ArcTan[((I + Sqrt[3])*x)/2] - 8*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2 )] + 4*Sqrt[3]*g*ArcTan[Sqrt[3]/(1 + 2*x^2)] + 6*g*Log[1 + x^2 + x^4])/24
Time = 0.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2202, 1576, 1142, 1083, 217, 1103, 2205, 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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4}{x^4+x^2+1} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{x^4+x^2+1}dx+\int \frac {x \left (g x^2+e\right )}{x^4+x^2+1}dx\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{x^4+x^2+1}dx+\frac {1}{2} \int \frac {g x^2+e}{x^4+x^2+1}dx^2\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{x^4+x^2+1}dx+\frac {1}{2} \left (\frac {1}{2} (2 e-g) \int \frac {1}{x^4+x^2+1}dx^2+\frac {1}{2} g \int \frac {2 x^2+1}{x^4+x^2+1}dx^2\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{x^4+x^2+1}dx+\frac {1}{2} \left (\frac {1}{2} g \int \frac {2 x^2+1}{x^4+x^2+1}dx^2-(2 e-g) \int \frac {1}{-x^4-3}d\left (2 x^2+1\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} g \int \frac {2 x^2+1}{x^4+x^2+1}dx^2+\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g)}{\sqrt {3}}\right )+\int \frac {h x^4+f x^2+d}{x^4+x^2+1}dx\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{x^4+x^2+1}dx+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g)}{\sqrt {3}}+\frac {1}{2} g \log \left (x^4+x^2+1\right )\right )\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \int \left (h+\frac {(f-h) x^2+d-h}{x^4+x^2+1}\right )dx+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g)}{\sqrt {3}}+\frac {1}{2} g \log \left (x^4+x^2+1\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right ) (d+f-2 h)}{2 \sqrt {3}}+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (d+f-2 h)}{2 \sqrt {3}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g)}{\sqrt {3}}+\frac {1}{2} g \log \left (x^4+x^2+1\right )\right )-\frac {1}{4} (d-f) \log \left (x^2-x+1\right )+\frac {1}{4} (d-f) \log \left (x^2+x+1\right )+h x\) |
h*x - ((d + f - 2*h)*ArcTan[(1 - 2*x)/Sqrt[3]])/(2*Sqrt[3]) + ((d + f - 2* h)*ArcTan[(1 + 2*x)/Sqrt[3]])/(2*Sqrt[3]) - ((d - f)*Log[1 - x + x^2])/4 + ((d - f)*Log[1 + x + x^2])/4 + (((2*e - g)*ArcTan[(1 + 2*x^2)/Sqrt[3]])/S qrt[3] + (g*Log[1 + x^2 + x^4])/2)/2
3.1.18.3.1 Defintions of rubi rules used
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[((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[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
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), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(h x +\frac {\left (f -d +g \right ) \ln \left (x^{2}-x +1\right )}{4}+\frac {\left (\frac {d}{2}+e +\frac {f}{2}-\frac {g}{2}-h \right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\left (d -f +g \right ) \ln \left (x^{2}+x +1\right )}{4}+\frac {\left (\frac {d}{2}-e +\frac {f}{2}+\frac {g}{2}-h \right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(99\) |
| risch | \(\text {Expression too large to display}\) | \(140079\) |
h*x+1/4*(f-d+g)*ln(x^2-x+1)+1/3*(1/2*d+e+1/2*f-1/2*g-h)*3^(1/2)*arctan(1/3 *(2*x-1)*3^(1/2))+1/4*(d-f+g)*ln(x^2+x+1)+1/3*(1/2*d-e+1/2*f+1/2*g-h)*arct an(1/3*(1+2*x)*3^(1/2))*3^(1/2)
Time = 1.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g - 2 \, h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g - 2 \, h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + h x + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \]
1/6*sqrt(3)*(d - 2*e + f + g - 2*h)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sq rt(3)*(d + 2*e + f - g - 2*h)*arctan(1/3*sqrt(3)*(2*x - 1)) + h*x + 1/4*(d - f + g)*log(x^2 + x + 1) - 1/4*(d - f - g)*log(x^2 - x + 1)
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g - 2 \, h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g - 2 \, h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + h x + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \]
1/6*sqrt(3)*(d - 2*e + f + g - 2*h)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sq rt(3)*(d + 2*e + f - g - 2*h)*arctan(1/3*sqrt(3)*(2*x - 1)) + h*x + 1/4*(d - f + g)*log(x^2 + x + 1) - 1/4*(d - f - g)*log(x^2 - x + 1)
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g - 2 \, h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g - 2 \, h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + h x + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \]
1/6*sqrt(3)*(d - 2*e + f + g - 2*h)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sq rt(3)*(d + 2*e + f - g - 2*h)*arctan(1/3*sqrt(3)*(2*x - 1)) + h*x + 1/4*(d - f + g)*log(x^2 + x + 1) - 1/4*(d - f - g)*log(x^2 - x + 1)
Time = 11.14 (sec) , antiderivative size = 1209, normalized size of antiderivative = 8.89 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{1+x^2+x^4} \, dx=\text {Too large to display} \]
log(d*f*9i - d*e*6i + d*g*3i - d*h*3i + e*h*6i + f*h*3i - g*h*3i - 3*3^(1/ 2)*d^2 - d^2*x*6i - f^2*x*3i - d^2*3i - f^2*6i + 2*3^(1/2)*d*e + 3*3^(1/2) *d*f - 3^(1/2)*d*g - 4*3^(1/2)*e*f + 3*3^(1/2)*d*h + 2*3^(1/2)*e*h + 2*3^( 1/2)*f*g - 3*3^(1/2)*f*h - 3^(1/2)*g*h + d*f*x*9i - e*f*x*6i + d*h*x*3i + e*h*x*6i + f*g*x*3i - f*h*x*3i - g*h*x*3i + 3*3^(1/2)*f^2*x - 3*3^(1/2)*d* f*x - 2*3^(1/2)*d*g*x - 2*3^(1/2)*e*f*x + 3*3^(1/2)*d*h*x - 2*3^(1/2)*e*h* x + 3^(1/2)*f*g*x - 3*3^(1/2)*f*h*x + 3^(1/2)*g*h*x + 4*3^(1/2)*d*e*x)*(d/ 4 - f/4 + g/4 - (3^(1/2)*d*1i)/12 + (3^(1/2)*e*1i)/6 - (3^(1/2)*f*1i)/12 - (3^(1/2)*g*1i)/12 + (3^(1/2)*h*1i)/6) - log(d*g*3i - d*f*9i - d*e*6i + d* h*3i + e*h*6i - f*h*3i - g*h*3i - 3*3^(1/2)*d^2 - d^2*x*6i - f^2*x*3i + d^ 2*3i + f^2*6i - 2*3^(1/2)*d*e + 3*3^(1/2)*d*f + 3^(1/2)*d*g + 4*3^(1/2)*e* f + 3*3^(1/2)*d*h - 2*3^(1/2)*e*h - 2*3^(1/2)*f*g - 3*3^(1/2)*f*h + 3^(1/2 )*g*h + d*f*x*9i + e*f*x*6i + d*h*x*3i - e*h*x*6i - f*g*x*3i - f*h*x*3i + g*h*x*3i - 3*3^(1/2)*f^2*x + 3*3^(1/2)*d*f*x - 2*3^(1/2)*d*g*x - 2*3^(1/2) *e*f*x - 3*3^(1/2)*d*h*x - 2*3^(1/2)*e*h*x + 3^(1/2)*f*g*x + 3*3^(1/2)*f*h *x + 3^(1/2)*g*h*x + 4*3^(1/2)*d*e*x)*(d/4 - f/4 - g/4 + (3^(1/2)*d*1i)/12 + (3^(1/2)*e*1i)/6 + (3^(1/2)*f*1i)/12 - (3^(1/2)*g*1i)/12 - (3^(1/2)*h*1 i)/6) + log(d*f*9i - d*e*6i + d*g*3i - d*h*3i + e*h*6i + f*h*3i - g*h*3i + 3*3^(1/2)*d^2 - d^2*x*6i - f^2*x*3i - d^2*3i - f^2*6i - 2*3^(1/2)*d*e - 3 *3^(1/2)*d*f + 3^(1/2)*d*g + 4*3^(1/2)*e*f - 3*3^(1/2)*d*h - 2*3^(1/2)*...