Integrand size = 23, antiderivative size = 51 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=-\frac {1}{6} (d+4 f) \text {arctanh}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f) \text {arctanh}(x)-\frac {1}{6} e \log \left (1-x^2\right )+\frac {1}{6} e \log \left (4-x^2\right ) \]
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.14 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=\frac {1}{12} (-2 (d+e+f) \log (1-x)+(d+2 e+4 f) \log (2-x)+2 (d-e+f) \log (1+x)-(d-2 e+4 f) \log (2+x)) \]
(-2*(d + e + f)*Log[1 - x] + (d + 2*e + 4*f)*Log[2 - x] + 2*(d - e + f)*Lo g[1 + x] - (d - 2*e + 4*f)*Log[2 + x])/12
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2202, 27, 1432, 1081, 1480, 220, 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.
\(\displaystyle \int \frac {d+e x+f x^2}{x^4-5 x^2+4} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {f x^2+d}{x^4-5 x^2+4}dx+\int \frac {e x}{x^4-5 x^2+4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {f x^2+d}{x^4-5 x^2+4}dx+e \int \frac {x}{x^4-5 x^2+4}dx\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle \int \frac {f x^2+d}{x^4-5 x^2+4}dx+\frac {1}{2} e \int \frac {1}{x^4-5 x^2+4}dx^2\) |
\(\Big \downarrow \) 1081 |
\(\displaystyle \int \frac {f x^2+d}{x^4-5 x^2+4}dx+\frac {1}{2} e \int \left (\frac {1}{3 \left (1-x^2\right )}-\frac {1}{3 \left (4-x^2\right )}\right )dx^2\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle \frac {1}{3} (d+4 f) \int \frac {1}{x^2-4}dx-\frac {1}{3} (d+f) \int \frac {1}{x^2-1}dx+\frac {1}{2} e \int \left (\frac {1}{3 \left (1-x^2\right )}-\frac {1}{3 \left (4-x^2\right )}\right )dx^2\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{2} e \int \left (\frac {1}{3 \left (1-x^2\right )}-\frac {1}{3 \left (4-x^2\right )}\right )dx^2-\frac {1}{6} \text {arctanh}\left (\frac {x}{2}\right ) (d+4 f)+\frac {1}{3} \text {arctanh}(x) (d+f)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{6} \text {arctanh}\left (\frac {x}{2}\right ) (d+4 f)+\frac {1}{3} \text {arctanh}(x) (d+f)+\frac {1}{2} e \left (\frac {1}{3} \log \left (4-x^2\right )-\frac {1}{3} \log \left (1-x^2\right )\right )\) |
-1/6*((d + 4*f)*ArcTanh[x/2]) + ((d + f)*ArcTanh[x])/3 + (e*(-1/3*Log[1 - x^2] + Log[4 - x^2]/3))/2
3.1.11.3.1 Defintions of rubi rules used
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))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
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]
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.22
method | result | size |
default | \(\left (-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}\right ) \ln \left (x +2\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{6}-\frac {e}{6}-\frac {f}{6}\right ) \ln \left (x -1\right )+\left (\frac {d}{12}+\frac {e}{6}+\frac {f}{3}\right ) \ln \left (x -2\right )\) | \(62\) |
norman | \(\left (-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}\right ) \ln \left (x +2\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{6}-\frac {e}{6}-\frac {f}{6}\right ) \ln \left (x -1\right )+\left (\frac {d}{12}+\frac {e}{6}+\frac {f}{3}\right ) \ln \left (x -2\right )\) | \(62\) |
parallelrisch | \(\frac {\ln \left (x -2\right ) d}{12}+\frac {\ln \left (x -2\right ) e}{6}+\frac {\ln \left (x -2\right ) f}{3}-\frac {\ln \left (x -1\right ) d}{6}-\frac {\ln \left (x -1\right ) e}{6}-\frac {\ln \left (x -1\right ) f}{6}+\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}+\frac {\ln \left (x +1\right ) f}{6}-\frac {\ln \left (x +2\right ) d}{12}+\frac {\ln \left (x +2\right ) e}{6}-\frac {\ln \left (x +2\right ) f}{3}\) | \(86\) |
risch | \(-\frac {\ln \left (1-x \right ) d}{6}-\frac {\ln \left (1-x \right ) e}{6}-\frac {\ln \left (1-x \right ) f}{6}+\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}+\frac {\ln \left (x +1\right ) f}{6}+\frac {\ln \left (2-x \right ) d}{12}+\frac {\ln \left (2-x \right ) e}{6}+\frac {\ln \left (2-x \right ) f}{3}-\frac {\ln \left (x +2\right ) d}{12}+\frac {\ln \left (x +2\right ) e}{6}-\frac {\ln \left (x +2\right ) f}{3}\) | \(98\) |
(-1/12*d+1/6*e-1/3*f)*ln(x+2)+(1/6*d-1/6*e+1/6*f)*ln(x+1)+(-1/6*d-1/6*e-1/ 6*f)*ln(x-1)+(1/12*d+1/6*e+1/3*f)*ln(x-2)
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=-\frac {1}{12} \, {\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + e + f\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \]
-1/12*(d - 2*e + 4*f)*log(x + 2) + 1/6*(d - e + f)*log(x + 1) - 1/6*(d + e + f)*log(x - 1) + 1/12*(d + 2*e + 4*f)*log(x - 2)
Leaf count of result is larger than twice the leaf count of optimal. 2195 vs. \(2 (44) = 88\).
Time = 95.81 (sec) , antiderivative size = 2195, normalized size of antiderivative = 43.04 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=\text {Too large to display} \]
-(d - 2*e + 4*f)*log(x + (-35*d**5*e + 51*d**5*(d - 2*e + 4*f)/2 - 820*d** 4*e*f + 90*d**4*f*(d - 2*e + 4*f) - 180*d**3*e**3 - 90*d**3*e**2*(d - 2*e + 4*f) - 4100*d**3*e*f**2 + 41*d**3*e*(d - 2*e + 4*f)**2 + 42*d**3*f**2*(d - 2*e + 4*f) - 15*d**3*(d - 2*e + 4*f)**3/2 - 432*d**2*e**2*f*(d - 2*e + 4*f) - 8000*d**2*e*f**3 + 240*d**2*e*f*(d - 2*e + 4*f)**2 - 240*d**2*f**3* (d - 2*e + 4*f) - 12*d**2*f*(d - 2*e + 4*f)**3 + 320*d*e**5 - 96*d*e**4*(d - 2*e + 4*f) + 720*d*e**3*f**2 - 80*d*e**3*(d - 2*e + 4*f)**2 - 1080*d*e* *2*f**2*(d - 2*e + 4*f) + 24*d*e**2*(d - 2*e + 4*f)**3 - 6400*d*e*f**4 + 4 92*d*e*f**2*(d - 2*e + 4*f)**2 - 576*d*f**4*(d - 2*e + 4*f) + 30*d*f**2*(d - 2*e + 4*f)**3 + 512*e**5*f - 128*e**3*f*(d - 2*e + 4*f)**2 - 576*e**2*f **3*(d - 2*e + 4*f) - 1472*e*f**5 + 320*e*f**3*(d - 2*e + 4*f)**2 - 480*f* *5*(d - 2*e + 4*f) + 48*f**3*(d - 2*e + 4*f)**3)/(9*d**6 + 45*d**5*f - 160 *d**4*e**2 - 36*d**4*f**2 - 1312*d**3*e**2*f - 360*d**3*f**3 + 256*d**2*e* *4 - 3840*d**2*e**2*f**2 - 144*d**2*f**4 + 1280*d*e**4*f - 5248*d*e**2*f** 3 + 720*d*f**5 + 1024*e**4*f**2 - 2560*e**2*f**4 + 576*f**6))/12 + (d - e + f)*log(x + (-35*d**5*e - 51*d**5*(d - e + f) - 820*d**4*e*f - 180*d**4*f *(d - e + f) - 180*d**3*e**3 + 180*d**3*e**2*(d - e + f) - 4100*d**3*e*f** 2 + 164*d**3*e*(d - e + f)**2 - 84*d**3*f**2*(d - e + f) + 60*d**3*(d - e + f)**3 + 864*d**2*e**2*f*(d - e + f) - 8000*d**2*e*f**3 + 960*d**2*e*f*(d - e + f)**2 + 480*d**2*f**3*(d - e + f) + 96*d**2*f*(d - e + f)**3 + 3...
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=-\frac {1}{12} \, {\left (d - 2 \, e + 4 \, f\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + e + f\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \]
-1/12*(d - 2*e + 4*f)*log(x + 2) + 1/6*(d - e + f)*log(x + 1) - 1/6*(d + e + f)*log(x - 1) + 1/12*(d + 2*e + 4*f)*log(x - 2)
Time = 0.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=-\frac {1}{12} \, {\left (d - 2 \, e + 4 \, f\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e + f\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, {\left (d + e + f\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{12} \, {\left (d + 2 \, e + 4 \, f\right )} \log \left ({\left | x - 2 \right |}\right ) \]
-1/12*(d - 2*e + 4*f)*log(abs(x + 2)) + 1/6*(d - e + f)*log(abs(x + 1)) - 1/6*(d + e + f)*log(abs(x - 1)) + 1/12*(d + 2*e + 4*f)*log(abs(x - 2))
Time = 7.76 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.24 \[ \int \frac {d+e x+f x^2}{4-5 x^2+x^4} \, dx=\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{6}+\frac {e}{6}+\frac {f}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{12}+\frac {e}{6}+\frac {f}{3}\right )-\ln \left (x+2\right )\,\left (\frac {d}{12}-\frac {e}{6}+\frac {f}{3}\right ) \]