Integrand size = 79, antiderivative size = 24 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {1}{-x+e^{3-e} \left (1-\frac {3 x}{-2+x}\right )} \] Output:
1/((1-3*x/(-2+x))/exp(exp(1)-3)-x)
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {e^e (2-x)}{e^e (-2+x) x+2 e^3 (1+x)} \] Input:
Integrate[(-6*E^(-3 + E) + E^(-6 + 2*E)*(4 - 4*x + x^2))/(4 + 8*x + 4*x^2 + E^(-3 + E)*(-8*x - 4*x^2 + 4*x^3) + E^(-6 + 2*E)*(4*x^2 - 4*x^3 + x^4)), x]
Output:
(E^E*(2 - x))/(E^E*(-2 + x)*x + 2*E^3*(1 + x))
Leaf count is larger than twice the leaf count of optimal. \(119\) vs. \(2(24)=48\).
Time = 0.54 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2459, 1380, 2345, 24}
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.
\(\displaystyle \int \frac {e^{2 e-6} \left (x^2-4 x+4\right )-6 e^{e-3}}{4 x^2+e^{e-3} \left (4 x^3-4 x^2-8 x\right )+e^{2 e-6} \left (x^4-4 x^3+4 x^2\right )+8 x+4} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {e^{2 e-6} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )^2-2 e^{e-6} \left (e^3+e^e\right ) \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )+e^{2 e-6}-4 e^{e-3}+1}{e^{2 e-6} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )^4-2 \left (1-4 e^{e-3}+e^{2 e-6}\right ) \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )^2+e^{-2 (3+e)} \left (e^6+e^{2 e}-4 e^{3+e}\right )^2}d\left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle e^{2 e-6} \int \frac {e^{-6+2 e} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )\right )^2-2 e^{-6+e} \left (e^3+e^e\right ) \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )\right )+e^{-6+2 e}-4 e^{-3+e}+1}{\left (-e^{-6+2 e} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )\right )^2+e^{-6+2 e}-4 e^{-3+e}+1\right )^2}d\left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )\right )\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle e^{2 e-6} \left (-\frac {\int 0d\left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )\right )}{2 \left (1-4 e^{e-3}+e^{2 e-6}\right )}-\frac {e^{12-2 e} \left (e^{e-12} \left (e^3+e^e\right )-e^{-2 (6-e)} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )\right )}{-e^{2 e-6} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )^2+e^{2 e-6}-4 e^{e-3}+1}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {e^6 \left (e^{e-12} \left (e^3+e^e\right )-e^{-2 (6-e)} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )\right )}{-e^{2 e-6} \left (x+\frac {1}{4} e^{6-2 e} \left (4 e^{e-3}-4 e^{2 e-6}\right )\right )^2+e^{2 e-6}-4 e^{e-3}+1}\) |
Input:
Int[(-6*E^(-3 + E) + E^(-6 + 2*E)*(4 - 4*x + x^2))/(4 + 8*x + 4*x^2 + E^(- 3 + E)*(-8*x - 4*x^2 + 4*x^3) + E^(-6 + 2*E)*(4*x^2 - 4*x^3 + x^4)),x]
Output:
-((E^6*(E^(-12 + E)*(E^3 + E^E) - ((E^(6 - 2*E)*(4*E^(-3 + E) - 4*E^(-6 + 2*E)))/4 + x)/E^(2*(6 - E))))/(1 - 4*E^(-3 + E) + E^(-6 + 2*E) - E^(-6 + 2 *E)*((E^(6 - 2*E)*(4*E^(-3 + E) - 4*E^(-6 + 2*E)))/4 + x)^2))
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[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 0.57 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
method | result | size |
gosper | \(-\frac {\left (-2+x \right ) {\mathrm e}^{{\mathrm e}-3}}{{\mathrm e}^{{\mathrm e}-3} x^{2}-2 \,{\mathrm e}^{{\mathrm e}-3} x +2 x +2}\) | \(35\) |
risch | \(\frac {-{\mathrm e}^{{\mathrm e}-3} x +2 \,{\mathrm e}^{{\mathrm e}-3}}{{\mathrm e}^{{\mathrm e}-3} x^{2}-2 \,{\mathrm e}^{{\mathrm e}-3} x +2 x +2}\) | \(42\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{2 \,{\mathrm e}-6} x^{2}-4 \,{\mathrm e}^{2 \,{\mathrm e}-6} x +6 \,{\mathrm e}^{{\mathrm e}-3} x}{2 \left ({\mathrm e}^{{\mathrm e}-3} x^{2}-2 \,{\mathrm e}^{{\mathrm e}-3} x +2 x +2\right )}\) | \(58\) |
norman | \(\frac {-{\mathrm e}^{2 \,{\mathrm e}} {\mathrm e}^{-6} x^{2}+\left (2 \,{\mathrm e}^{2 \,{\mathrm e}} {\mathrm e}^{-6}-3 \,{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-3}\right ) x}{{\mathrm e}^{{\mathrm e}-3} x^{2}-2 \,{\mathrm e}^{{\mathrm e}-3} x +2 x +2}\) | \(62\) |
default | \(\frac {{\mathrm e}^{{\mathrm e}-3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4+{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_Z}^{4}-\left (4 \,{\mathrm e}^{2 \,{\mathrm e}-6}-4 \,{\mathrm e}^{{\mathrm e}-3}\right ) \textit {\_Z}^{3}+\left (4 \,{\mathrm e}^{2 \,{\mathrm e}-6}-4 \,{\mathrm e}^{{\mathrm e}-3}+4\right ) \textit {\_Z}^{2}-\left (8 \,{\mathrm e}^{{\mathrm e}-3}-8\right ) \textit {\_Z} \right )}{\sum }\frac {\left ({\mathrm e}^{{\mathrm e}-3} \textit {\_R}^{2}-4 \,{\mathrm e}^{{\mathrm e}-3} \textit {\_R} +4 \,{\mathrm e}^{{\mathrm e}-3}-6\right ) \ln \left (x -\textit {\_R} \right )}{2+{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_R}^{3}-3 \,{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_R}^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_R} +3 \,{\mathrm e}^{{\mathrm e}-3} \textit {\_R}^{2}-2 \,{\mathrm e}^{{\mathrm e}-3} \textit {\_R} -2 \,{\mathrm e}^{{\mathrm e}-3}+2 \textit {\_R}}\right )}{4}\) | \(179\) |
Input:
int(((x^2-4*x+4)*exp(exp(1)-3)^2-6*exp(exp(1)-3))/((x^4-4*x^3+4*x^2)*exp(e xp(1)-3)^2+(4*x^3-4*x^2-8*x)*exp(exp(1)-3)+4*x^2+8*x+4),x,method=_RETURNVE RBOSE)
Output:
-(-2+x)*exp(exp(1)-3)/(exp(exp(1)-3)*x^2-2*exp(exp(1)-3)*x+2*x+2)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {{\left (x - 2\right )} e^{\left (e - 3\right )}}{{\left (x^{2} - 2 \, x\right )} e^{\left (e - 3\right )} + 2 \, x + 2} \] Input:
integrate(((x^2-4*x+4)*exp(exp(1)-3)^2-6*exp(exp(1)-3))/((x^4-4*x^3+4*x^2) *exp(exp(1)-3)^2+(4*x^3-4*x^2-8*x)*exp(exp(1)-3)+4*x^2+8*x+4),x, algorithm ="fricas")
Output:
-(x - 2)*e^(e - 3)/((x^2 - 2*x)*e^(e - 3) + 2*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.57 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {- x e^{e} + 2 e^{e}}{x^{2} e^{e} + x \left (- 2 e^{e} + 2 e^{3}\right ) + 2 e^{3}} \] Input:
integrate(((x**2-4*x+4)*exp(exp(1)-3)**2-6*exp(exp(1)-3))/((x**4-4*x**3+4* x**2)*exp(exp(1)-3)**2+(4*x**3-4*x**2-8*x)*exp(exp(1)-3)+4*x**2+8*x+4),x)
Output:
(-x*exp(E) + 2*exp(E))/(x**2*exp(E) + x*(-2*exp(E) + 2*exp(3)) + 2*exp(3))
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {x e^{e} - 2 \, e^{e}}{x^{2} e^{e} + 2 \, x {\left (e^{3} - e^{e}\right )} + 2 \, e^{3}} \] Input:
integrate(((x^2-4*x+4)*exp(exp(1)-3)^2-6*exp(exp(1)-3))/((x^4-4*x^3+4*x^2) *exp(exp(1)-3)^2+(4*x^3-4*x^2-8*x)*exp(exp(1)-3)+4*x^2+8*x+4),x, algorithm ="maxima")
Output:
-(x*e^e - 2*e^e)/(x^2*e^e + 2*x*(e^3 - e^e) + 2*e^3)
\[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e - 6\right )} - 6 \, e^{\left (e - 3\right )}}{4 \, x^{2} + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (2 \, e - 6\right )} + 4 \, {\left (x^{3} - x^{2} - 2 \, x\right )} e^{\left (e - 3\right )} + 8 \, x + 4} \,d x } \] Input:
integrate(((x^2-4*x+4)*exp(exp(1)-3)^2-6*exp(exp(1)-3))/((x^4-4*x^3+4*x^2) *exp(exp(1)-3)^2+(4*x^3-4*x^2-8*x)*exp(exp(1)-3)+4*x^2+8*x+4),x, algorithm ="giac")
Output:
integrate(((x^2 - 4*x + 4)*e^(2*e - 6) - 6*e^(e - 3))/(4*x^2 + (x^4 - 4*x^ 3 + 4*x^2)*e^(2*e - 6) + 4*(x^3 - x^2 - 2*x)*e^(e - 3) + 8*x + 4), x)
Time = 6.74 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {x-2}{x^2+\left (2\,{\mathrm {e}}^{3-\mathrm {e}}-2\right )\,x+2\,{\mathrm {e}}^{3-\mathrm {e}}} \] Input:
int(-(6*exp(exp(1) - 3) - exp(2*exp(1) - 6)*(x^2 - 4*x + 4))/(8*x - exp(ex p(1) - 3)*(8*x + 4*x^2 - 4*x^3) + exp(2*exp(1) - 6)*(4*x^2 - 4*x^3 + x^4) + 4*x^2 + 4),x)
Output:
-(x - 2)/(2*exp(3 - exp(1)) + x*(2*exp(3 - exp(1)) - 2) + x^2)
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {e^{e} \left (-e^{e} x^{2}+4 e^{e}-6 e^{3}\right )}{2 e^{2 e} x^{2}-4 e^{2 e} x -2 e^{e} e^{3} x^{2}+8 e^{e} e^{3} x +4 e^{e} e^{3}-4 e^{6} x -4 e^{6}} \] Input:
int(((x^2-4*x+4)*exp(exp(1)-3)^2-6*exp(exp(1)-3))/((x^4-4*x^3+4*x^2)*exp(e xp(1)-3)^2+(4*x^3-4*x^2-8*x)*exp(exp(1)-3)+4*x^2+8*x+4),x)
Output:
(e**e*( - e**e*x**2 + 4*e**e - 6*e**3))/(2*(e**(2*e)*x**2 - 2*e**(2*e)*x - e**e*e**3*x**2 + 4*e**e*e**3*x + 2*e**e*e**3 - 2*e**6*x - 2*e**6))