Integrand size = 69, antiderivative size = 30 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=2 x+\left (1+2 x^2\right ) \left (3-\frac {e^{e^8}}{-x+x^4}\right ) \] Output:
2*x+(2*x^2+1)*(3-exp(exp(4)^2)/(x^4-x))
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=\frac {e^{e^8}}{x}+2 x+6 x^2+\frac {e^{e^8} \left (-2 x-x^2\right )}{-1+x^3} \] Input:
Integrate[(2*x^2 + 12*x^3 - 4*x^5 - 24*x^6 + 2*x^8 + 12*x^9 + E^E^8*(-1 + 2*x^2 + 4*x^3 + 4*x^5))/(x^2 - 2*x^5 + x^8),x]
Output:
E^E^8/x + 2*x + 6*x^2 + (E^E^8*(-2*x - x^2))/(-1 + x^3)
Time = 0.60 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {2026, 1380, 2368, 27, 2019, 2010, 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.
\(\displaystyle \int \frac {12 x^9+2 x^8-24 x^6-4 x^5+12 x^3+2 x^2+e^{e^8} \left (4 x^5+4 x^3+2 x^2-1\right )}{x^8-2 x^5+x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {12 x^9+2 x^8-24 x^6-4 x^5+12 x^3+2 x^2+e^{e^8} \left (4 x^5+4 x^3+2 x^2-1\right )}{x^2 \left (x^6-2 x^3+1\right )}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int \frac {12 x^9+2 x^8-24 x^6-4 x^5+12 x^3+2 x^2-e^{e^8} \left (-4 x^5-4 x^3-2 x^2+1\right )}{x^2 \left (1-x^3\right )^2}dx\) |
\(\Big \downarrow \) 2368 |
\(\displaystyle \frac {1}{3} \int -\frac {3 \left (12 x^6+2 x^5-\left (12+e^{e^8}\right ) x^3-2 x^2+e^{e^8}\right )}{x^2 \left (1-x^3\right )}dx+\frac {e^{e^8} x (x+2)}{1-x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^{e^8} x (x+2)}{1-x^3}-\int \frac {12 x^6+2 x^5-\left (12+e^{e^8}\right ) x^3-2 x^2+e^{e^8}}{x^2 \left (1-x^3\right )}dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \frac {e^{e^8} x (x+2)}{1-x^3}-\int \frac {-12 x^3-2 x^2+e^{e^8}}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {e^{e^8} x (x+2)}{1-x^3}-\int \left (-12 x-2+\frac {e^{e^8}}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{e^8} (x+2) x}{1-x^3}+6 x^2+2 x+\frac {e^{e^8}}{x}\) |
Input:
Int[(2*x^2 + 12*x^3 - 4*x^5 - 24*x^6 + 2*x^8 + 12*x^9 + E^E^8*(-1 + 2*x^2 + 4*x^3 + 4*x^5))/(x^2 - 2*x^5 + x^8),x]
Output:
E^E^8/x + 2*x + 6*x^2 + (E^E^8*x*(2 + x))/(1 - x^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
risch | \(6 x^{2}+2 x +\frac {-2 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}-{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(35\) |
default | \(2 x +6 x^{2}-\frac {{\mathrm e}^{{\mathrm e}^{8}}}{-1+x}-\frac {{\mathrm e}^{{\mathrm e}^{8}}}{x^{2}+x +1}+\frac {{\mathrm e}^{{\mathrm e}^{8}}}{x}\) | \(40\) |
norman | \(\frac {6 x^{6}+2 x^{5}-6 x^{3}+\left (-2 \,{\mathrm e}^{{\mathrm e}^{8}}-2\right ) x^{2}-{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(48\) |
gosper | \(-\frac {-6 x^{6}-2 x^{5}+2 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}+6 x^{3}+2 x^{2}+{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(49\) |
parallelrisch | \(-\frac {-6 x^{6}-2 x^{5}+2 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}+6 x^{3}+2 x^{2}+{\mathrm e}^{{\mathrm e}^{8}}}{x \left (x^{3}-1\right )}\) | \(49\) |
Input:
int(((4*x^5+4*x^3+2*x^2-1)*exp(exp(4)^2)+12*x^9+2*x^8-24*x^6-4*x^5+12*x^3+ 2*x^2)/(x^8-2*x^5+x^2),x,method=_RETURNVERBOSE)
Output:
6*x^2+2*x+(-2*exp(exp(8))*x^2-exp(exp(8)))/x/(x^3-1)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=\frac {6 \, x^{6} + 2 \, x^{5} - 6 \, x^{3} - 2 \, x^{2} - {\left (2 \, x^{2} + 1\right )} e^{\left (e^{8}\right )}}{x^{4} - x} \] Input:
integrate(((4*x^5+4*x^3+2*x^2-1)*exp(exp(4)^2)+12*x^9+2*x^8-24*x^6-4*x^5+1 2*x^3+2*x^2)/(x^8-2*x^5+x^2),x, algorithm="fricas")
Output:
(6*x^6 + 2*x^5 - 6*x^3 - 2*x^2 - (2*x^2 + 1)*e^(e^8))/(x^4 - x)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=6 x^{2} + 2 x + \frac {- 2 x^{2} e^{e^{8}} - e^{e^{8}}}{x^{4} - x} \] Input:
integrate(((4*x**5+4*x**3+2*x**2-1)*exp(exp(4)**2)+12*x**9+2*x**8-24*x**6- 4*x**5+12*x**3+2*x**2)/(x**8-2*x**5+x**2),x)
Output:
6*x**2 + 2*x + (-2*x**2*exp(exp(8)) - exp(exp(8)))/(x**4 - x)
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=6 \, x^{2} + 2 \, x - \frac {2 \, x^{2} e^{\left (e^{8}\right )} + e^{\left (e^{8}\right )}}{x^{4} - x} \] Input:
integrate(((4*x^5+4*x^3+2*x^2-1)*exp(exp(4)^2)+12*x^9+2*x^8-24*x^6-4*x^5+1 2*x^3+2*x^2)/(x^8-2*x^5+x^2),x, algorithm="maxima")
Output:
6*x^2 + 2*x - (2*x^2*e^(e^8) + e^(e^8))/(x^4 - x)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=6 \, x^{2} + 2 \, x - \frac {2 \, x^{2} e^{\left (e^{8}\right )} + e^{\left (e^{8}\right )}}{x^{4} - x} \] Input:
integrate(((4*x^5+4*x^3+2*x^2-1)*exp(exp(4)^2)+12*x^9+2*x^8-24*x^6-4*x^5+1 2*x^3+2*x^2)/(x^8-2*x^5+x^2),x, algorithm="giac")
Output:
6*x^2 + 2*x - (2*x^2*e^(e^8) + e^(e^8))/(x^4 - x)
Time = 0.53 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=2\,x+6\,x^2+\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^8}\,x^2+{\mathrm {e}}^{{\mathrm {e}}^8}}{x-x^4} \] Input:
int((exp(exp(8))*(2*x^2 + 4*x^3 + 4*x^5 - 1) + 2*x^2 + 12*x^3 - 4*x^5 - 24 *x^6 + 2*x^8 + 12*x^9)/(x^2 - 2*x^5 + x^8),x)
Output:
2*x + 6*x^2 + (exp(exp(8)) + 2*x^2*exp(exp(8)))/(x - x^4)
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {2 x^2+12 x^3-4 x^5-24 x^6+2 x^8+12 x^9+e^{e^8} \left (-1+2 x^2+4 x^3+4 x^5\right )}{x^2-2 x^5+x^8} \, dx=\frac {-2 e^{e^{8}} x^{2}-e^{e^{8}}+6 x^{6}+2 x^{5}-6 x^{3}-2 x^{2}}{x \left (x^{3}-1\right )} \] Input:
int(((4*x^5+4*x^3+2*x^2-1)*exp(exp(4)^2)+12*x^9+2*x^8-24*x^6-4*x^5+12*x^3+ 2*x^2)/(x^8-2*x^5+x^2),x)
Output:
( - 2*e**(e**8)*x**2 - e**(e**8) + 6*x**6 + 2*x**5 - 6*x**3 - 2*x**2)/(x*( x**3 - 1))