Integrand size = 152, antiderivative size = 26 \begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=e^{-2+\frac {3}{x}} \left (e^{\frac {2}{-3+x+\log (4)}}-x\right ) x \end {dmath*}
\begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=\int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx \end {dmath*}
Integrate[(E^((3 - 2*x)/x + 2/(-3 + x + Log[4]))*(-27 + 27*x - 11*x^2 + x^ 3 + (18 - 12*x + 2*x^2)*Log[4] + (-3 + x)*Log[4]^2) + E^((3 - 2*x)/x)*(27* x - 36*x^2 + 15*x^3 - 2*x^4 + (-18*x + 18*x^2 - 4*x^3)*Log[4] + (3*x - 2*x ^2)*Log[4]^2))/(9*x - 6*x^2 + x^3 + (-6*x + 2*x^2)*Log[4] + x*Log[4]^2),x]
Integrate[(E^((3 - 2*x)/x + 2/(-3 + x + Log[4]))*(-27 + 27*x - 11*x^2 + x^ 3 + (18 - 12*x + 2*x^2)*Log[4] + (-3 + x)*Log[4]^2) + E^((3 - 2*x)/x)*(27* x - 36*x^2 + 15*x^3 - 2*x^4 + (-18*x + 18*x^2 - 4*x^3)*Log[4] + (3*x - 2*x ^2)*Log[4]^2))/(9*x - 6*x^2 + x^3 + (-6*x + 2*x^2)*Log[4] + x*Log[4]^2), x ]
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.
| \(\displaystyle \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{x-3+\log (4)}} \left (x^3-11 x^2+\left (2 x^2-12 x+18\right ) \log (4)+27 x+(x-3) \log ^2(4)-27\right )+e^{\frac {3-2 x}{x}} \left (-2 x^4+15 x^3-36 x^2+\left (3 x-2 x^2\right ) \log ^2(4)+\left (-4 x^3+18 x^2-18 x\right ) \log (4)+27 x\right )}{x^3-6 x^2+\left (2 x^2-6 x\right ) \log (4)+9 x+x \log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{x-3+\log (4)}} \left (x^3-11 x^2+\left (2 x^2-12 x+18\right ) \log (4)+27 x+(x-3) \log ^2(4)-27\right )+e^{\frac {3-2 x}{x}} \left (-2 x^4+15 x^3-36 x^2+\left (3 x-2 x^2\right ) \log ^2(4)+\left (-4 x^3+18 x^2-18 x\right ) \log (4)+27 x\right )}{x^3-6 x^2+\left (2 x^2-6 x\right ) \log (4)+x \left (9+\log ^2(4)\right )}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{x-3+\log (4)}} \left (x^3-11 x^2+\left (2 x^2-12 x+18\right ) \log (4)+27 x+(x-3) \log ^2(4)-27\right )+e^{\frac {3-2 x}{x}} \left (-2 x^4+15 x^3-36 x^2+\left (3 x-2 x^2\right ) \log ^2(4)+\left (-4 x^3+18 x^2-18 x\right ) \log (4)+27 x\right )}{x \left (x^2-2 x (3-\log (4))+(\log (4)-3)^2\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{x-3+\log (4)}} \left (x^3-11 x^2+\left (2 x^2-12 x+18\right ) \log (4)+27 x+(x-3) \log ^2(4)-27\right )+e^{\frac {3-2 x}{x}} \left (-2 x^4+15 x^3-36 x^2+\left (3 x-2 x^2\right ) \log ^2(4)+\left (-4 x^3+18 x^2-18 x\right ) \log (4)+27 x\right )}{x (x-3+\log (4))^2}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^{\frac {3}{x}-2} \left (\frac {e^{\frac {2}{x-3+\log (4)}} \left (x^3+x^2 (\log (16)-11)+x \left (27+\log ^2(4)-12 \log (4)\right )-3 (\log (4)-3)^2\right )}{x (x-3+\log (4))^2}-2 x+3\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {3}{x}+\frac {2}{x-3+\log (4)}-2} \left (x^3-x^2 (11-\log (16))+x (3-\log (4)) (9-\log (4))-3 (3-\log (4))^2\right )}{x (-x+3-\log (4))^2}-2 e^{\frac {3}{x}-2} x+3 e^{\frac {3}{x}-2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int e^{\frac {2}{x+\log (4)-3}-2+\frac {3}{x}}dx-3 \int \frac {e^{\frac {2}{x+\log (4)-3}-2+\frac {3}{x}}}{x}dx-(6-\log (16)) \int \frac {e^{\frac {2}{x+\log (4)-3}-2+\frac {3}{x}}}{(x+\log (4)-3)^2}dx-2 \int \frac {e^{\frac {2}{x+\log (4)-3}-2+\frac {3}{x}}}{x+\log (4)-3}dx-e^{\frac {3}{x}-2} x^2\) |
Int[(E^((3 - 2*x)/x + 2/(-3 + x + Log[4]))*(-27 + 27*x - 11*x^2 + x^3 + (1 8 - 12*x + 2*x^2)*Log[4] + (-3 + x)*Log[4]^2) + E^((3 - 2*x)/x)*(27*x - 36 *x^2 + 15*x^3 - 2*x^4 + (-18*x + 18*x^2 - 4*x^3)*Log[4] + (3*x - 2*x^2)*Lo g[4]^2))/(9*x - 6*x^2 + x^3 + (-6*x + 2*x^2)*Log[4] + x*Log[4]^2),x]
3.5.86.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 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[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.98 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65
| method | result | size |
| parallelrisch | \(-{\mathrm e}^{-\frac {-3+2 x}{x}} x^{2}+x \,{\mathrm e}^{-\frac {-3+2 x}{x}} {\mathrm e}^{\frac {2}{2 \ln \left (2\right )+x -3}}\) | \(43\) |
| risch | \(-{\mathrm e}^{-\frac {-3+2 x}{x}} x^{2}+{\mathrm e}^{-\frac {4 x \ln \left (2\right )+2 x^{2}-6 \ln \left (2\right )-11 x +9}{x \left (2 \ln \left (2\right )+x -3\right )}} x\) | \(54\) |
| norman | \(\frac {x^{2} {\mathrm e}^{\frac {3-2 x}{x}} {\mathrm e}^{\frac {2}{2 \ln \left (2\right )+x -3}}+\left (3-2 \ln \left (2\right )\right ) x^{2} {\mathrm e}^{\frac {3-2 x}{x}}+\left (2 \ln \left (2\right )-3\right ) x \,{\mathrm e}^{\frac {3-2 x}{x}} {\mathrm e}^{\frac {2}{2 \ln \left (2\right )+x -3}}-x^{3} {\mathrm e}^{\frac {3-2 x}{x}}}{2 \ln \left (2\right )+x -3}\) | \(103\) |
int(((4*(-3+x)*ln(2)^2+2*(2*x^2-12*x+18)*ln(2)+x^3-11*x^2+27*x-27)*exp((3- 2*x)/x)*exp(2/(2*ln(2)+x-3))+(4*(-2*x^2+3*x)*ln(2)^2+2*(-4*x^3+18*x^2-18*x )*ln(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*ln(2)^2+2*(2*x^2-6* x)*ln(2)+x^3-6*x^2+9*x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12 \begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=-x^{2} e^{\left (-\frac {2 \, x - 3}{x}\right )} + x e^{\left (-\frac {2 \, x^{2} + 2 \, {\left (2 \, x - 3\right )} \log \left (2\right ) - 11 \, x + 9}{x^{2} + 2 \, x \log \left (2\right ) - 3 \, x}\right )} \end {dmath*}
integrate(((4*(-3+x)*log(2)^2+2*(2*x^2-12*x+18)*log(2)+x^3-11*x^2+27*x-27) *exp((3-2*x)/x)*exp(2/(2*log(2)+x-3))+(4*(-2*x^2+3*x)*log(2)^2+2*(-4*x^3+1 8*x^2-18*x)*log(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*log(2)^2 +2*(2*x^2-6*x)*log(2)+x^3-6*x^2+9*x),x, algorithm=\
-x^2*e^(-(2*x - 3)/x) + x*e^(-(2*x^2 + 2*(2*x - 3)*log(2) - 11*x + 9)/(x^2 + 2*x*log(2) - 3*x))
Time = 1.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=- x^{2} e^{\frac {3 - 2 x}{x}} + x e^{\frac {3 - 2 x}{x}} e^{\frac {2}{x - 3 + 2 \log {\left (2 \right )}}} \end {dmath*}
integrate(((4*(-3+x)*ln(2)**2+2*(2*x**2-12*x+18)*ln(2)+x**3-11*x**2+27*x-2 7)*exp((3-2*x)/x)*exp(2/(2*ln(2)+x-3))+(4*(-2*x**2+3*x)*ln(2)**2+2*(-4*x** 3+18*x**2-18*x)*ln(2)-2*x**4+15*x**3-36*x**2+27*x)*exp((3-2*x)/x))/(4*x*ln (2)**2+2*(2*x**2-6*x)*ln(2)+x**3-6*x**2+9*x),x)
Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=-{\left (x^{2} e^{\frac {3}{x}} - x e^{\left (\frac {2}{x + 2 \, \log \left (2\right ) - 3} + \frac {3}{x}\right )}\right )} e^{\left (-2\right )} \end {dmath*}
integrate(((4*(-3+x)*log(2)^2+2*(2*x^2-12*x+18)*log(2)+x^3-11*x^2+27*x-27) *exp((3-2*x)/x)*exp(2/(2*log(2)+x-3))+(4*(-2*x^2+3*x)*log(2)^2+2*(-4*x^3+1 8*x^2-18*x)*log(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*log(2)^2 +2*(2*x^2-6*x)*log(2)+x^3-6*x^2+9*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.51 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12 \begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=-x^{2} e^{\left (-\frac {2 \, x - 3}{x}\right )} + x e^{\left (-\frac {2 \, x^{2} + 4 \, x \log \left (2\right ) - 11 \, x - 6 \, \log \left (2\right ) + 9}{x^{2} + 2 \, x \log \left (2\right ) - 3 \, x}\right )} \end {dmath*}
integrate(((4*(-3+x)*log(2)^2+2*(2*x^2-12*x+18)*log(2)+x^3-11*x^2+27*x-27) *exp((3-2*x)/x)*exp(2/(2*log(2)+x-3))+(4*(-2*x^2+3*x)*log(2)^2+2*(-4*x^3+1 8*x^2-18*x)*log(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*log(2)^2 +2*(2*x^2-6*x)*log(2)+x^3-6*x^2+9*x),x, algorithm=\
-x^2*e^(-(2*x - 3)/x) + x*e^(-(2*x^2 + 4*x*log(2) - 11*x - 6*log(2) + 9)/( x^2 + 2*x*log(2) - 3*x))
Timed out. \begin {dmath*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx=\text {Hanged} \end {dmath*}
int((exp(-(2*x - 3)/x)*(27*x - 2*log(2)*(18*x - 18*x^2 + 4*x^3) + 4*log(2) ^2*(3*x - 2*x^2) - 36*x^2 + 15*x^3 - 2*x^4) + exp(2/(x + 2*log(2) - 3))*ex p(-(2*x - 3)/x)*(27*x + 4*log(2)^2*(x - 3) + 2*log(2)*(2*x^2 - 12*x + 18) - 11*x^2 + x^3 - 27))/(9*x - 2*log(2)*(6*x - 2*x^2) + 4*x*log(2)^2 - 6*x^2 + x^3),x)