Integrand size = 31, antiderivative size = 71 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x) \]
1/12*(d-2*e)/(2+x)-1/18*(d+e)*ln(1-x)+1/48*(d+2*e)*ln(2-x)+1/6*(d-e)*ln(1+ x)-1/144*(19*d-26*e)*ln(2+x)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \left (\frac {12 (d-2 e)}{2+x}+24 (d-e) \log (-1-x)-8 (d+e) \log (1-x)+3 (d+2 e) \log (2-x)+(-19 d+26 e) \log (2+x)\right ) \]
((12*(d - 2*e))/(2 + x) + 24*(d - e)*Log[-1 - x] - 8*(d + e)*Log[1 - x] + 3*(d + 2*e)*Log[2 - x] + (-19*d + 26*e)*Log[2 + x])/144
Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2019, 2462, 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.
\(\displaystyle \int \frac {\left (x^3-2 x^2-x+2\right ) (d+e x)}{\left (x^4-5 x^2+4\right )^2} \, dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int \frac {d+e x}{(x+2)^2 \left (x^3-2 x^2-x+2\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {-d-e}{18 (x-1)}+\frac {d+2 e}{48 (x-2)}+\frac {d-e}{6 (x+1)}+\frac {26 e-19 d}{144 (x+2)}+\frac {2 e-d}{12 (x+2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d-2 e}{12 (x+2)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (x+1)-\frac {1}{144} (19 d-26 e) \log (x+2)\) |
(d - 2*e)/(12*(2 + x)) - ((d + e)*Log[1 - x])/18 + ((d + 2*e)*Log[2 - x])/ 48 + ((d - e)*Log[1 + x])/6 - ((19*d - 26*e)*Log[2 + x])/144
3.1.86.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90
method | result | size |
default | \(\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )-\frac {-\frac {d}{12}+\frac {e}{6}}{x +2}+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )\) | \(64\) |
risch | \(\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )}-\frac {\ln \left (x -1\right ) d}{18}-\frac {\ln \left (x -1\right ) e}{18}+\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}-\frac {19 \ln \left (-x -2\right ) d}{144}+\frac {13 \ln \left (-x -2\right ) e}{72}+\frac {\ln \left (2-x \right ) d}{48}+\frac {\ln \left (2-x \right ) e}{24}\) | \(82\) |
norman | \(\frac {\left (-\frac {d}{12}+\frac {e}{6}\right ) x +\left (\frac {d}{12}-\frac {e}{6}\right ) x^{3}+\left (-\frac {d}{6}+\frac {e}{3}\right ) x^{2}+\frac {d}{6}-\frac {e}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (x +1\right )+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )\) | \(101\) |
parallelrisch | \(\frac {3 \ln \left (x -2\right ) x d +6 \ln \left (x -2\right ) x e -8 \ln \left (x -1\right ) x d -8 \ln \left (x -1\right ) x e +24 \ln \left (x +1\right ) x d -24 \ln \left (x +1\right ) x e -19 \ln \left (x +2\right ) x d +26 \ln \left (x +2\right ) x e +6 \ln \left (x -2\right ) d +12 \ln \left (x -2\right ) e -16 \ln \left (x -1\right ) d -16 \ln \left (x -1\right ) e +48 \ln \left (x +1\right ) d -48 \ln \left (x +1\right ) e -38 \ln \left (x +2\right ) d +52 \ln \left (x +2\right ) e +12 d -24 e}{144 x +288}\) | \(135\) |
(-19/144*d+13/72*e)*ln(x+2)-(-1/12*d+1/6*e)/(x+2)+(1/6*d-1/6*e)*ln(x+1)+(- 1/18*d-1/18*e)*ln(x-1)+(1/48*d+1/24*e)*ln(x-2)
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {{\left ({\left (19 \, d - 26 \, e\right )} x + 38 \, d - 52 \, e\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e\right )} x + 2 \, d - 2 \, e\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e}{144 \, {\left (x + 2\right )}} \]
-1/144*(((19*d - 26*e)*x + 38*d - 52*e)*log(x + 2) - 24*((d - e)*x + 2*d - 2*e)*log(x + 1) + 8*((d + e)*x + 2*d + 2*e)*log(x - 1) - 3*((d + 2*e)*x + 2*d + 4*e)*log(x - 2) - 12*d + 24*e)/(x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (60) = 120\).
Time = 7.10 (sec) , antiderivative size = 1188, normalized size of antiderivative = 16.73 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Too large to display} \]
(d - 2*e)/(12*x + 24) + (d - e)*log(x + (-1534775*d**6 + 8032360*d**5*e - 984027*d**5*(d - e) - 12991180*d**4*e**2 + 11797266*d**4*e*(d - e) + 35671 68*d**4*(d - e)**2 + 1075200*d**3*e**3 - 32721528*d**3*e**2*(d - e) - 8725 248*d**3*e*(d - e)**2 - 247104*d**3*(d - e)**3 + 16959280*d**2*e**4 + 3897 7296*d**2*e**3*(d - e) - 2820096*d**2*e**2*(d - e)**2 - 10357632*d**2*e*(d - e)**3 - 15836800*d*e**5 - 21294960*d*e**4*(d - e) + 15436800*d*e**3*(d - e)**2 + 16277760*d*e**2*(d - e)**3 + 4283840*e**6 + 3876000*e**5*(d - e) - 6865920*e**4*(d - e)**2 - 4078080*e**3*(d - e)**3)/(801262*d**6 - 46622 51*d**5*e + 7296938*d**4*e**2 + 1388616*d**3*e**3 - 12447440*d**2*e**4 + 9 990800*d*e**5 - 2380000*e**6))/6 - (d + e)*log(x + (-1534775*d**6 + 803236 0*d**5*e + 328009*d**5*(d + e) - 12991180*d**4*e**2 - 3932422*d**4*e*(d + e) + 396352*d**4*(d + e)**2 + 1075200*d**3*e**3 + 10907176*d**3*e**2*(d + e) - 969472*d**3*e*(d + e)**2 + 9152*d**3*(d + e)**3 + 16959280*d**2*e**4 - 12992432*d**2*e**3*(d + e) - 313344*d**2*e**2*(d + e)**2 + 383616*d**2*e *(d + e)**3 - 15836800*d*e**5 + 7098320*d*e**4*(d + e) + 1715200*d*e**3*(d + e)**2 - 602880*d*e**2*(d + e)**3 + 4283840*e**6 - 1292000*e**5*(d + e) - 762880*e**4*(d + e)**2 + 151040*e**3*(d + e)**3)/(801262*d**6 - 4662251* d**5*e + 7296938*d**4*e**2 + 1388616*d**3*e**3 - 12447440*d**2*e**4 + 9990 800*d*e**5 - 2380000*e**6))/18 + (d + 2*e)*log(x + (-1534775*d**6 + 803236 0*d**5*e - 984027*d**5*(d + 2*e)/8 - 12991180*d**4*e**2 + 5898633*d**4*...
Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \]
-1/144*(19*d - 26*e)*log(x + 2) + 1/6*(d - e)*log(x + 1) - 1/18*(d + e)*lo g(x - 1) + 1/48*(d + 2*e)*log(x - 2) + 1/12*(d - 2*e)/(x + 2)
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \]
-1/144*(19*d - 26*e)*log(abs(x + 2)) + 1/6*(d - e)*log(abs(x + 1)) - 1/18* (d + e)*log(abs(x - 1)) + 1/48*(d + 2*e)*log(abs(x - 2)) + 1/12*(d - 2*e)/ (x + 2)
Time = 7.95 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {\frac {d}{12}-\frac {e}{6}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}\right ) \]