Integrand size = 41, antiderivative size = 95 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {d-2 e+4 f-8 g}{12 (2+x)}-\frac {1}{18} (d+e+f+g) \log (1-x)+\frac {1}{48} (d+2 e+4 f+8 g) \log (2-x)+\frac {1}{6} (d-e+f-g) \log (1+x)-\frac {1}{144} (19 d-26 e+28 f-8 g) \log (2+x) \] Output:
(d-2*e+4*f-8*g)/(24+12*x)-1/18*(d+e+f+g)*ln(1-x)+1/48*(d+2*e+4*f+8*g)*ln(2 -x)+1/6*(d-e+f-g)*ln(1+x)-1/144*(19*d-26*e+28*f-8*g)*ln(2+x)
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \left (\frac {12 (d-2 e+4 f-8 g)}{2+x}+24 (d-e+f-g) \log (-1-x)-8 (d+e+f+g) \log (1-x)+3 (d+2 e+4 f+8 g) \log (2-x)+(-19 d+26 e-28 f+8 g) \log (2+x)\right ) \] Input:
Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x ^4)^2,x]
Output:
((12*(d - 2*e + 4*f - 8*g))/(2 + x) + 24*(d - e + f - g)*Log[-1 - x] - 8*( d + e + f + g)*Log[1 - x] + 3*(d + 2*e + 4*f + 8*g)*Log[2 - x] + (-19*d + 26*e - 28*f + 8*g)*Log[2 + x])/144
Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, 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.
| \(\displaystyle \int \frac {\left (x^3-2 x^2-x+2\right ) \left (d+e x+f x^2+g x^3\right )}{\left (x^4-5 x^2+4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3}{(x+2)^2 \left (x^3-2 x^2-x+2\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {-d-e-f-g}{18 (x-1)}+\frac {d+2 e+4 f+8 g}{48 (x-2)}+\frac {d-e+f-g}{6 (x+1)}+\frac {-19 d+26 e-28 f+8 g}{144 (x+2)}+\frac {-d+2 e-4 f+8 g}{12 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d-2 e+4 f-8 g}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f+g)+\frac {1}{48} \log (2-x) (d+2 e+4 f+8 g)+\frac {1}{6} \log (x+1) (d-e+f-g)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f-8 g)\) |
Input:
Int[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4)^2, x]
Output:
(d - 2*e + 4*f - 8*g)/(12*(2 + x)) - ((d + e + f + g)*Log[1 - x])/18 + ((d + 2*e + 4*f + 8*g)*Log[2 - x])/48 + ((d - e + f - g)*Log[1 + x])/6 - ((19 *d - 26*e + 28*f - 8*g)*Log[2 + x])/144
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 = 94, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}\right ) \ln \left (x -2\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}\right ) \ln \left (1+x \right )+\left (-\frac {d}{18}-\frac {e}{18}-\frac {f}{18}-\frac {g}{18}\right ) \ln \left (x -1\right )+\left (\frac {g}{18}-\frac {7 f}{36}+\frac {13 e}{72}-\frac {19 d}{144}\right ) \ln \left (x +2\right )-\frac {-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}+\frac {2 g}{3}}{x +2}\) | \(94\) |
| norman | \(\frac {\left (-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}+\frac {2 g}{3}\right ) x +\left (\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2 g}{3}\right ) x^{3}+\left (\frac {4 g}{3}-\frac {2 f}{3}+\frac {e}{3}-\frac {d}{6}\right ) x^{2}-\frac {4 g}{3}+\frac {2 f}{3}-\frac {e}{3}+\frac {d}{6}}{x^{4}-5 x^{2}+4}+\left (-\frac {d}{18}-\frac {e}{18}-\frac {f}{18}-\frac {g}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}\right ) \ln \left (1+x \right )+\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}\right ) \ln \left (x -2\right )+\left (\frac {g}{18}-\frac {7 f}{36}+\frac {13 e}{72}-\frac {19 d}{144}\right ) \ln \left (x +2\right )\) | \(149\) |
| risch | \(\frac {d}{24+12 x}-\frac {e}{6 \left (x +2\right )}+\frac {f}{3 x +6}-\frac {2 g}{3 \left (x +2\right )}+\frac {\ln \left (1+x \right ) d}{6}-\frac {\ln \left (1+x \right ) e}{6}+\frac {\ln \left (1+x \right ) f}{6}-\frac {\ln \left (1+x \right ) g}{6}+\frac {\ln \left (2-x \right ) d}{48}+\frac {\ln \left (2-x \right ) e}{24}+\frac {\ln \left (2-x \right ) f}{12}+\frac {\ln \left (2-x \right ) g}{6}+\frac {\ln \left (-x -2\right ) g}{18}-\frac {7 \ln \left (-x -2\right ) f}{36}+\frac {13 \ln \left (-x -2\right ) e}{72}-\frac {19 \ln \left (-x -2\right ) d}{144}-\frac {\ln \left (x -1\right ) d}{18}-\frac {\ln \left (x -1\right ) e}{18}-\frac {\ln \left (x -1\right ) f}{18}-\frac {\ln \left (x -1\right ) g}{18}\) | \(162\) |
| parallelrisch | \(\frac {48 f -96 g +12 d -24 e +12 \ln \left (x -2\right ) e +52 \ln \left (x +2\right ) e +24 \ln \left (1+x \right ) x d -24 \ln \left (1+x \right ) x e -19 \ln \left (x +2\right ) x d +26 \ln \left (x +2\right ) x e -28 \ln \left (x +2\right ) x f +3 \ln \left (x -2\right ) x d +24 \ln \left (x -2\right ) f +48 \ln \left (1+x \right ) d -56 \ln \left (x +2\right ) f -16 \ln \left (x -1\right ) d -8 \ln \left (x -1\right ) x e +24 \ln \left (1+x \right ) x f -38 \ln \left (x +2\right ) d +12 \ln \left (x -2\right ) x f +6 \ln \left (x -2\right ) d -16 \ln \left (x -1\right ) e +48 \ln \left (1+x \right ) f -48 \ln \left (1+x \right ) e -16 \ln \left (x -1\right ) f -48 \ln \left (1+x \right ) g +16 \ln \left (x +2\right ) g +6 \ln \left (x -2\right ) x e -8 \ln \left (x -1\right ) x d -8 \ln \left (x -1\right ) x f +48 \ln \left (x -2\right ) g -16 \ln \left (x -1\right ) g +24 \ln \left (x -2\right ) x g -24 \ln \left (1+x \right ) x g -8 \ln \left (x -1\right ) x g +8 \ln \left (x +2\right ) x g}{144 x +288}\) | \(261\) |
Input:
int((x^3-2*x^2-x+2)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x,method=_RETURNVE RBOSE)
Output:
(1/48*d+1/24*e+1/12*f+1/6*g)*ln(x-2)+(1/6*d-1/6*e+1/6*f-1/6*g)*ln(1+x)+(-1 /18*d-1/18*e-1/18*f-1/18*g)*ln(x-1)+(1/18*g-7/36*f+13/72*e-19/144*d)*ln(x+ 2)-(-1/12*d+1/6*e-1/3*f+2/3*g)/(x+2)
Time = 0.55 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.48 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {{\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e + f - g\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e + f + g\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e - 48 \, f + 96 \, g}{144 \, {\left (x + 2\right )}} \] Input:
integrate((x^3-2*x^2-x+2)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm ="fricas")
Output:
-1/144*(((19*d - 26*e + 28*f - 8*g)*x + 38*d - 52*e + 56*f - 16*g)*log(x + 2) - 24*((d - e + f - g)*x + 2*d - 2*e + 2*f - 2*g)*log(x + 1) + 8*((d + e + f + g)*x + 2*d + 2*e + 2*f + 2*g)*log(x - 1) - 3*((d + 2*e + 4*f + 8*g )*x + 2*d + 4*e + 8*f + 16*g)*log(x - 2) - 12*d + 24*e - 48*f + 96*g)/(x + 2)
Timed out. \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((x**3-2*x**2-x+2)*(g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e + 4 \, f - 8 \, g}{12 \, {\left (x + 2\right )}} \] Input:
integrate((x^3-2*x^2-x+2)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm ="maxima")
Output:
-1/144*(19*d - 26*e + 28*f - 8*g)*log(x + 2) + 1/6*(d - e + f - g)*log(x + 1) - 1/18*(d + e + f + g)*log(x - 1) + 1/48*(d + 2*e + 4*f + 8*g)*log(x - 2) + 1/12*(d - 2*e + 4*f - 8*g)/(x + 2)
Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e + f - g\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e + f + g\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e + 4 \, f - 8 \, g}{12 \, {\left (x + 2\right )}} \] Input:
integrate((x^3-2*x^2-x+2)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm ="giac")
Output:
-1/144*(19*d - 26*e + 28*f - 8*g)*log(abs(x + 2)) + 1/6*(d - e + f - g)*lo g(abs(x + 1)) - 1/18*(d + e + f + g)*log(abs(x - 1)) + 1/48*(d + 2*e + 4*f + 8*g)*log(abs(x - 2)) + 1/12*(d - 2*e + 4*f - 8*g)/(x + 2)
Time = 17.95 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2\,g}{3}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}+\frac {f}{18}+\frac {g}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}+\frac {7\,f}{36}-\frac {g}{18}\right ) \] Input:
int(-((d + e*x + f*x^2 + g*x^3)*(x + 2*x^2 - x^3 - 2))/(x^4 - 5*x^2 + 4)^2 ,x)
Output:
(d/12 - e/6 + f/3 - (2*g)/3)/(x + 2) + log(x + 1)*(d/6 - e/6 + f/6 - g/6) - log(x - 1)*(d/18 + e/18 + f/18 + g/18) + log(x - 2)*(d/48 + e/24 + f/12 + g/6) - log(x + 2)*((19*d)/144 - (13*e)/72 + (7*f)/36 - g/18)
Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.79 \[ \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {12 e x +24 \,\mathrm {log}\left (x +1\right ) f x -28 \,\mathrm {log}\left (x +2\right ) f x +12 \,\mathrm {log}\left (x -2\right ) f x -8 \,\mathrm {log}\left (x -1\right ) f x +48 \,\mathrm {log}\left (x -2\right ) g -16 \,\mathrm {log}\left (x -1\right ) g +16 \,\mathrm {log}\left (x +2\right ) g -48 \,\mathrm {log}\left (x +1\right ) g +3 \,\mathrm {log}\left (x -2\right ) d x +6 \,\mathrm {log}\left (x -2\right ) e x -8 \,\mathrm {log}\left (x -1\right ) d x -8 \,\mathrm {log}\left (x -1\right ) e x -19 \,\mathrm {log}\left (x +2\right ) d x +26 \,\mathrm {log}\left (x +2\right ) e x +24 \,\mathrm {log}\left (x +1\right ) d x -24 \,\mathrm {log}\left (x +1\right ) e x +6 \,\mathrm {log}\left (x -2\right ) d -38 \,\mathrm {log}\left (x +2\right ) d +12 \,\mathrm {log}\left (x -2\right ) e +24 \,\mathrm {log}\left (x -2\right ) f -16 \,\mathrm {log}\left (x -1\right ) d -16 \,\mathrm {log}\left (x -1\right ) e -16 \,\mathrm {log}\left (x -1\right ) f +52 \,\mathrm {log}\left (x +2\right ) e -56 \,\mathrm {log}\left (x +2\right ) f +48 \,\mathrm {log}\left (x +1\right ) d -48 \,\mathrm {log}\left (x +1\right ) e +48 \,\mathrm {log}\left (x +1\right ) f +24 \,\mathrm {log}\left (x -2\right ) g x -8 \,\mathrm {log}\left (x -1\right ) g x +8 \,\mathrm {log}\left (x +2\right ) g x -24 \,\mathrm {log}\left (x +1\right ) g x -24 f x +48 g x -6 d x}{144 x +288} \] Input:
int((x^3-2*x^2-x+2)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x)
Output:
(3*log(x - 2)*d*x + 6*log(x - 2)*d + 6*log(x - 2)*e*x + 12*log(x - 2)*e + 12*log(x - 2)*f*x + 24*log(x - 2)*f + 24*log(x - 2)*g*x + 48*log(x - 2)*g - 8*log(x - 1)*d*x - 16*log(x - 1)*d - 8*log(x - 1)*e*x - 16*log(x - 1)*e - 8*log(x - 1)*f*x - 16*log(x - 1)*f - 8*log(x - 1)*g*x - 16*log(x - 1)*g - 19*log(x + 2)*d*x - 38*log(x + 2)*d + 26*log(x + 2)*e*x + 52*log(x + 2)* e - 28*log(x + 2)*f*x - 56*log(x + 2)*f + 8*log(x + 2)*g*x + 16*log(x + 2) *g + 24*log(x + 1)*d*x + 48*log(x + 1)*d - 24*log(x + 1)*e*x - 48*log(x + 1)*e + 24*log(x + 1)*f*x + 48*log(x + 1)*f - 24*log(x + 1)*g*x - 48*log(x + 1)*g - 6*d*x + 12*e*x - 24*f*x + 48*g*x)/(144*(x + 2))