Integrand size = 36, antiderivative size = 117 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {d-e+f-g}{6 (1+x)}-\frac {d-2 e+4 f-8 g}{12 (2+x)}-\frac {1}{36} (d+e+f+g) \log (1-x)+\frac {1}{144} (d+2 e+4 f+8 g) \log (2-x)-\frac {1}{36} (7 d-13 e+19 f-25 g) \log (1+x)+\frac {1}{144} (31 d-50 e+76 f-104 g) \log (2+x) \]
1/6*(-d+e-f+g)/(1+x)+1/12*(-d+2*e-4*f+8*g)/(2+x)-1/36*(d+e+f+g)*ln(1-x)+1/ 144*(d+2*e+4*f+8*g)*ln(2-x)-1/36*(7*d-13*e+19*f-25*g)*ln(1+x)+1/144*(31*d- 50*e+76*f-104*g)*ln(2+x)
Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \left (\frac {12 (-5 d+6 e-8 f+12 g-3 d x+4 e x-6 f x+10 g x)}{2+3 x+x^2}-4 (d+e+f+g) \log (1-x)+(d+2 e+4 f+8 g) \log (2-x)+4 (-7 d+13 e-19 f+25 g) \log (1+x)+(31 d-50 e+76 f-104 g) \log (2+x)\right ) \]
((12*(-5*d + 6*e - 8*f + 12*g - 3*d*x + 4*e*x - 6*f*x + 10*g*x))/(2 + 3*x + x^2) - 4*(d + e + f + g)*Log[1 - x] + (d + 2*e + 4*f + 8*g)*Log[2 - x] + 4*(-7*d + 13*e - 19*f + 25*g)*Log[1 + x] + (31*d - 50*e + 76*f - 104*g)*L og[2 + x])/144
Time = 0.48 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2019, 7279, 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.
| \(\displaystyle \int \frac {\left (x^2-3 x+2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (x^4-5 x^2+4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3}{\left (x^2-3 x+2\right ) \left (x^2+3 x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {31 d-50 e+76 f-104 g}{144 (x+2)}+\frac {d+2 e+4 f+8 g}{144 (x-2)}+\frac {-d-e-f-g}{36 (x-1)}+\frac {-7 d+13 e-19 f+25 g}{36 (x+1)}+\frac {d-e+f-g}{6 (x+1)^2}+\frac {d-2 e+4 f-8 g}{12 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d-2 e+4 f-8 g}{12 (x+2)}-\frac {d-e+f-g}{6 (x+1)}-\frac {1}{36} \log (1-x) (d+e+f+g)+\frac {1}{144} \log (2-x) (d+2 e+4 f+8 g)-\frac {1}{36} \log (x+1) (7 d-13 e+19 f-25 g)+\frac {1}{144} \log (x+2) (31 d-50 e+76 f-104 g)\) |
-1/6*(d - e + f - g)/(1 + x) - (d - 2*e + 4*f - 8*g)/(12*(2 + x)) - ((d + e + f + g)*Log[1 - x])/36 + ((d + 2*e + 4*f + 8*g)*Log[2 - x])/144 - ((7*d - 13*e + 19*f - 25*g)*Log[1 + x])/36 + ((31*d - 50*e + 76*f - 104*g)*Log[ 2 + x])/144
3.1.94.3.1 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_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2 g}{3}}{x +2}+\left (\frac {31 d}{144}-\frac {25 e}{72}+\frac {19 f}{36}-\frac {13 g}{18}\right ) \ln \left (x +2\right )+\left (-\frac {7 d}{36}+\frac {13 e}{36}-\frac {19 f}{36}+\frac {25 g}{36}\right ) \ln \left (x +1\right )-\frac {\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}}{x +1}+\left (-\frac {d}{36}-\frac {e}{36}-\frac {f}{36}-\frac {g}{36}\right ) \ln \left (x -1\right )+\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}\right ) \ln \left (x -2\right )\) | \(114\) |
| norman | \(\frac {\left (-\frac {d}{4}+\frac {e}{3}-\frac {f}{2}+\frac {5 g}{6}\right ) x^{3}+\left (\frac {3 d}{4}-\frac {5 e}{6}+f -\frac {4 g}{3}\right ) x +\left (\frac {d}{3}-\frac {e}{2}+\frac {5 f}{6}-\frac {3 g}{2}\right ) x^{2}-\frac {5 d}{6}+e +2 g -\frac {4 f}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {7 d}{36}+\frac {13 e}{36}-\frac {19 f}{36}+\frac {25 g}{36}\right ) \ln \left (x +1\right )+\left (-\frac {d}{36}-\frac {e}{36}-\frac {f}{36}-\frac {g}{36}\right ) \ln \left (x -1\right )+\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}\right ) \ln \left (x -2\right )+\left (\frac {31 d}{144}-\frac {25 e}{72}+\frac {19 f}{36}-\frac {13 g}{18}\right ) \ln \left (x +2\right )\) | \(145\) |
| risch | \(\frac {\left (-\frac {d}{4}+\frac {e}{3}-\frac {f}{2}+\frac {5 g}{6}\right ) x -\frac {5 d}{12}+\frac {e}{2}-\frac {2 f}{3}+g}{x^{2}+3 x +2}-\frac {\ln \left (x -1\right ) d}{36}-\frac {\ln \left (x -1\right ) e}{36}-\frac {\ln \left (x -1\right ) f}{36}-\frac {\ln \left (x -1\right ) g}{36}-\frac {7 \ln \left (-x -1\right ) d}{36}+\frac {13 \ln \left (-x -1\right ) e}{36}-\frac {19 \ln \left (-x -1\right ) f}{36}+\frac {25 \ln \left (-x -1\right ) g}{36}+\frac {\ln \left (2-x \right ) d}{144}+\frac {\ln \left (2-x \right ) e}{72}+\frac {\ln \left (2-x \right ) f}{36}+\frac {\ln \left (2-x \right ) g}{18}+\frac {31 \ln \left (x +2\right ) d}{144}-\frac {25 \ln \left (x +2\right ) e}{72}+\frac {19 \ln \left (x +2\right ) f}{36}-\frac {13 \ln \left (x +2\right ) g}{18}\) | \(167\) |
| parallelrisch | \(\frac {-96 f +144 g +120 g x -60 d +72 e -36 d x +2 \ln \left (x -2\right ) d +4 \ln \left (x -2\right ) e -8 \ln \left (x -1\right ) d -8 \ln \left (x -1\right ) e +24 \ln \left (x -2\right ) x g -12 \ln \left (x -1\right ) x g +300 \ln \left (x +1\right ) x g -312 \ln \left (x +2\right ) x g +152 \ln \left (x +2\right ) f -152 \ln \left (x +1\right ) f -150 \ln \left (x +2\right ) x e +6 \ln \left (x -2\right ) x e -12 \ln \left (x -1\right ) x d -12 \ln \left (x -1\right ) x e -84 \ln \left (x +1\right ) x d +156 \ln \left (x +1\right ) x e +93 \ln \left (x +2\right ) x d +2 \ln \left (x -2\right ) x^{2} e -4 \ln \left (x -1\right ) x^{2} d -4 \ln \left (x -1\right ) x^{2} e -28 \ln \left (x +1\right ) x^{2} d +52 \ln \left (x +1\right ) x^{2} e +31 \ln \left (x +2\right ) x^{2} d -50 \ln \left (x +2\right ) x^{2} e +62 \ln \left (x +2\right ) d +12 \ln \left (x -2\right ) x f -12 \ln \left (x -1\right ) x f -228 \ln \left (x +1\right ) x f +228 \ln \left (x +2\right ) x f -100 \ln \left (x +2\right ) e -56 \ln \left (x +1\right ) d +104 \ln \left (x +1\right ) e +3 \ln \left (x -2\right ) x d +200 \ln \left (x +1\right ) g -208 \ln \left (x +2\right ) g +8 \ln \left (x -2\right ) x^{2} g -4 \ln \left (x -1\right ) x^{2} g +100 \ln \left (x +1\right ) x^{2} g -104 \ln \left (x +2\right ) x^{2} g +48 e x +16 \ln \left (x -2\right ) g -8 \ln \left (x -1\right ) g +4 \ln \left (x -2\right ) x^{2} f -4 \ln \left (x -1\right ) x^{2} f -76 \ln \left (x +1\right ) x^{2} f +76 \ln \left (x +2\right ) x^{2} f +8 \ln \left (x -2\right ) f -8 \ln \left (x -1\right ) f +\ln \left (x -2\right ) x^{2} d -72 f x}{144 x^{2}+432 x +288}\) | \(441\) |
-(1/12*d-1/6*e+1/3*f-2/3*g)/(x+2)+(31/144*d-25/72*e+19/36*f-13/18*g)*ln(x+ 2)+(-7/36*d+13/36*e-19/36*f+25/36*g)*ln(x+1)-(1/6*d-1/6*e+1/6*f-1/6*g)/(x+ 1)+(-1/36*d-1/36*e-1/36*f-1/36*g)*ln(x-1)+(1/144*d+1/72*e+1/36*f+1/18*g)*l n(x-2)
Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (105) = 210\).
Time = 0.72 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.96 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (3 \, d - 4 \, e + 6 \, f - 10 \, g\right )} x - {\left ({\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} x^{2} + 3 \, {\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} x + 62 \, d - 100 \, e + 152 \, f - 208 \, g\right )} \log \left (x + 2\right ) + 4 \, {\left ({\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} x^{2} + 3 \, {\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} x + 14 \, d - 26 \, e + 38 \, f - 50 \, g\right )} \log \left (x + 1\right ) + 4 \, {\left ({\left (d + e + f + g\right )} x^{2} + 3 \, {\left (d + e + f + g\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g\right )} \log \left (x - 1\right ) - {\left ({\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x^{2} + 3 \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e + 96 \, f - 144 \, g}{144 \, {\left (x^{2} + 3 \, x + 2\right )}} \]
-1/144*(12*(3*d - 4*e + 6*f - 10*g)*x - ((31*d - 50*e + 76*f - 104*g)*x^2 + 3*(31*d - 50*e + 76*f - 104*g)*x + 62*d - 100*e + 152*f - 208*g)*log(x + 2) + 4*((7*d - 13*e + 19*f - 25*g)*x^2 + 3*(7*d - 13*e + 19*f - 25*g)*x + 14*d - 26*e + 38*f - 50*g)*log(x + 1) + 4*((d + e + f + g)*x^2 + 3*(d + e + f + g)*x + 2*d + 2*e + 2*f + 2*g)*log(x - 1) - ((d + 2*e + 4*f + 8*g)*x ^2 + 3*(d + 2*e + 4*f + 8*g)*x + 2*d + 4*e + 8*f + 16*g)*log(x - 2) + 60*d - 72*e + 96*f - 144*g)/(x^2 + 3*x + 2)
Timed out. \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \, {\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} \log \left (x + 2\right ) - \frac {1}{36} \, {\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} \log \left (x + 1\right ) - \frac {1}{36} \, {\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac {1}{144} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) - \frac {{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g}{12 \, {\left (x^{2} + 3 \, x + 2\right )}} \]
1/144*(31*d - 50*e + 76*f - 104*g)*log(x + 2) - 1/36*(7*d - 13*e + 19*f - 25*g)*log(x + 1) - 1/36*(d + e + f + g)*log(x - 1) + 1/144*(d + 2*e + 4*f + 8*g)*log(x - 2) - 1/12*((3*d - 4*e + 6*f - 10*g)*x + 5*d - 6*e + 8*f - 1 2*g)/(x^2 + 3*x + 2)
Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \, {\left (31 \, d - 50 \, e + 76 \, f - 104 \, g\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{36} \, {\left (7 \, d - 13 \, e + 19 \, f - 25 \, g\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{36} \, {\left (d + e + f + g\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{144} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g}{12 \, {\left (x + 2\right )} {\left (x + 1\right )}} \]
1/144*(31*d - 50*e + 76*f - 104*g)*log(abs(x + 2)) - 1/36*(7*d - 13*e + 19 *f - 25*g)*log(abs(x + 1)) - 1/36*(d + e + f + g)*log(abs(x - 1)) + 1/144* (d + 2*e + 4*f + 8*g)*log(abs(x - 2)) - 1/12*((3*d - 4*e + 6*f - 10*g)*x + 5*d - 6*e + 8*f - 12*g)/((x + 2)*(x + 1))
Time = 7.98 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.98 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\ln \left (x-2\right )\,\left (\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}\right )-\ln \left (x+1\right )\,\left (\frac {7\,d}{36}-\frac {13\,e}{36}+\frac {19\,f}{36}-\frac {25\,g}{36}\right )-\ln \left (x-1\right )\,\left (\frac {d}{36}+\frac {e}{36}+\frac {f}{36}+\frac {g}{36}\right )+\ln \left (x+2\right )\,\left (\frac {31\,d}{144}-\frac {25\,e}{72}+\frac {19\,f}{36}-\frac {13\,g}{18}\right )-\frac {\frac {5\,d}{12}-\frac {e}{2}+\frac {2\,f}{3}-g+x\,\left (\frac {d}{4}-\frac {e}{3}+\frac {f}{2}-\frac {5\,g}{6}\right )}{x^2+3\,x+2} \]