Integrand size = 99, antiderivative size = 30 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=1+\left (e^{\frac {x}{3 \left (x^2-x^3\right )}}+3 x\right )^2+\log (2 x) \] Output:
(3*x+exp(x/(-3*x^3+3*x^2)))^2+1+ln(2*x)
Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=\left (e^{\frac {1}{3 x-3 x^2}}+3 x\right )^2+\log (x) \] Input:
Integrate[(3*x - 6*x^2 + 57*x^3 - 108*x^4 + 54*x^5 + (-2 + 4*x)/E^(2/(-3*x + 3*x^2)) + (-6*x + 30*x^2 - 36*x^3 + 18*x^4)/E^(-3*x + 3*x^2)^(-1))/(3*x ^2 - 6*x^3 + 3*x^4),x]
Output:
(E^(3*x - 3*x^2)^(-1) + 3*x)^2 + Log[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.
\(\displaystyle \int \frac {54 x^5-108 x^4+57 x^3-6 x^2+e^{-\frac {2}{3 x^2-3 x}} (4 x-2)+e^{-\frac {1}{3 x^2-3 x}} \left (18 x^4-36 x^3+30 x^2-6 x\right )+3 x}{3 x^4-6 x^3+3 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {54 x^5-108 x^4+57 x^3-6 x^2+e^{-\frac {2}{3 x^2-3 x}} (4 x-2)+e^{-\frac {1}{3 x^2-3 x}} \left (18 x^4-36 x^3+30 x^2-6 x\right )+3 x}{x^2 \left (3 x^2-6 x+3\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 12 \int -\frac {-54 x^5+108 x^4-57 x^3+6 x^2-3 x+2 e^{\frac {2}{3 \left (x-x^2\right )}} (1-2 x)+6 e^{\frac {1}{3 \left (x-x^2\right )}} \left (-3 x^4+6 x^3-5 x^2+x\right )}{36 (1-x)^2 x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} \int \frac {-54 x^5+108 x^4-57 x^3+6 x^2-3 x+2 e^{\frac {2}{3 \left (x-x^2\right )}} (1-2 x)+6 e^{\frac {1}{3 \left (x-x^2\right )}} \left (-3 x^4+6 x^3-5 x^2+x\right )}{(1-x)^2 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (-\frac {54 x^3}{(x-1)^2}+\frac {108 x^2}{(x-1)^2}-\frac {57 x}{(x-1)^2}+\frac {6}{(x-1)^2}+\frac {6 e^{\frac {1}{(3-3 x) x}} \left (-3 x^3+6 x^2-5 x+1\right )}{(1-x)^2 x}-\frac {3}{(x-1)^2 x}+\frac {2 e^{\frac {2}{(3-3 x) x}} (1-2 x)}{(1-x)^2 x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (18 \int e^{\frac {1}{(3-3 x) x}}dx-6 \int \frac {e^{\frac {1}{(3-3 x) x}}}{1-x}dx+6 \int \frac {e^{\frac {1}{(3-3 x) x}}}{(x-1)^2}dx-6 \int \frac {e^{\frac {1}{(3-3 x) x}}}{x}dx+27 x^2+3 e^{\frac {2}{3 (1-x) x}}+3 \log (x)\right )\) |
Input:
Int[(3*x - 6*x^2 + 57*x^3 - 108*x^4 + 54*x^5 + (-2 + 4*x)/E^(2/(-3*x + 3*x ^2)) + (-6*x + 30*x^2 - 36*x^3 + 18*x^4)/E^(-3*x + 3*x^2)^(-1))/(3*x^2 - 6 *x^3 + 3*x^4),x]
Output:
$Aborted
Time = 1.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
method | result | size |
risch | \(9 x^{2}+\ln \left (x \right )+{\mathrm e}^{-\frac {2}{3 x \left (-1+x \right )}}+6 x \,{\mathrm e}^{-\frac {1}{3 x \left (-1+x \right )}}\) | \(34\) |
parallelrisch | \(9 x^{2}+6 x \,{\mathrm e}^{-\frac {1}{3 x \left (-1+x \right )}}+{\mathrm e}^{-\frac {2}{3 x \left (-1+x \right )}}+\ln \left (x \right )-54\) | \(37\) |
norman | \(\frac {x^{2} {\mathrm e}^{-\frac {2}{3 x^{2}-3 x}}-9 x^{3}+9 x^{4}-x \,{\mathrm e}^{-\frac {2}{3 x^{2}-3 x}}-6 x^{2} {\mathrm e}^{-\frac {1}{3 x^{2}-3 x}}+6 x^{3} {\mathrm e}^{-\frac {1}{3 x^{2}-3 x}}}{x \left (-1+x \right )}+\ln \left (x \right )\) | \(101\) |
parts | \(9 x^{2}+\ln \left (x \right )+\frac {-6 x \,{\mathrm e}^{-\frac {1}{3 x^{2}-3 x}}+6 x^{2} {\mathrm e}^{-\frac {1}{3 x^{2}-3 x}}}{-1+x}+\frac {x^{2} {\mathrm e}^{-\frac {2}{3 x^{2}-3 x}}-x \,{\mathrm e}^{-\frac {2}{3 x^{2}-3 x}}}{x \left (-1+x \right )}\) | \(101\) |
Input:
int(((4*x-2)*exp(-1/(3*x^2-3*x))^2+(18*x^4-36*x^3+30*x^2-6*x)*exp(-1/(3*x^ 2-3*x))+54*x^5-108*x^4+57*x^3-6*x^2+3*x)/(3*x^4-6*x^3+3*x^2),x,method=_RET URNVERBOSE)
Output:
9*x^2+ln(x)+exp(-2/3/x/(-1+x))+6*x*exp(-1/3/x/(-1+x))
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=9 \, x^{2} + 6 \, x e^{\left (-\frac {1}{3 \, {\left (x^{2} - x\right )}}\right )} + e^{\left (-\frac {2}{3 \, {\left (x^{2} - x\right )}}\right )} + \log \left (x\right ) \] Input:
integrate(((4*x-2)*exp(-1/(3*x^2-3*x))^2+(18*x^4-36*x^3+30*x^2-6*x)*exp(-1 /(3*x^2-3*x))+54*x^5-108*x^4+57*x^3-6*x^2+3*x)/(3*x^4-6*x^3+3*x^2),x, algo rithm="fricas")
Output:
9*x^2 + 6*x*e^(-1/3/(x^2 - x)) + e^(-2/3/(x^2 - x)) + log(x)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=9 x^{2} + 6 x e^{- \frac {1}{3 x^{2} - 3 x}} + \log {\left (x \right )} + e^{- \frac {2}{3 x^{2} - 3 x}} \] Input:
integrate(((4*x-2)*exp(-1/(3*x**2-3*x))**2+(18*x**4-36*x**3+30*x**2-6*x)*e xp(-1/(3*x**2-3*x))+54*x**5-108*x**4+57*x**3-6*x**2+3*x)/(3*x**4-6*x**3+3* x**2),x)
Output:
9*x**2 + 6*x*exp(-1/(3*x**2 - 3*x)) + log(x) + exp(-2/(3*x**2 - 3*x))
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=9 \, x^{2} + 6 \, x e^{\left (-\frac {1}{3 \, {\left (x - 1\right )}} + \frac {1}{3 \, x}\right )} + e^{\left (-\frac {2}{3 \, {\left (x - 1\right )}} + \frac {2}{3 \, x}\right )} + \log \left (x\right ) \] Input:
integrate(((4*x-2)*exp(-1/(3*x^2-3*x))^2+(18*x^4-36*x^3+30*x^2-6*x)*exp(-1 /(3*x^2-3*x))+54*x^5-108*x^4+57*x^3-6*x^2+3*x)/(3*x^4-6*x^3+3*x^2),x, algo rithm="maxima")
Output:
9*x^2 + 6*x*e^(-1/3/(x - 1) + 1/3/x) + e^(-2/3/(x - 1) + 2/3/x) + log(x)
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=9 \, x^{2} + 6 \, x e^{\left (-\frac {1}{3 \, {\left (x^{2} - x\right )}}\right )} + e^{\left (-\frac {2}{3 \, {\left (x^{2} - x\right )}}\right )} + \log \left (x\right ) \] Input:
integrate(((4*x-2)*exp(-1/(3*x^2-3*x))^2+(18*x^4-36*x^3+30*x^2-6*x)*exp(-1 /(3*x^2-3*x))+54*x^5-108*x^4+57*x^3-6*x^2+3*x)/(3*x^4-6*x^3+3*x^2),x, algo rithm="giac")
Output:
9*x^2 + 6*x*e^(-1/3/(x^2 - x)) + e^(-2/3/(x^2 - x)) + log(x)
Time = 3.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx={\mathrm {e}}^{\frac {2}{3\,x-3\,x^2}}+\ln \left (x\right )+6\,x\,{\mathrm {e}}^{\frac {1}{3\,x-3\,x^2}}+9\,x^2 \] Input:
int((3*x + exp(2/(3*x - 3*x^2))*(4*x - 2) - exp(1/(3*x - 3*x^2))*(6*x - 30 *x^2 + 36*x^3 - 18*x^4) - 6*x^2 + 57*x^3 - 108*x^4 + 54*x^5)/(3*x^2 - 6*x^ 3 + 3*x^4),x)
Output:
exp(2/(3*x - 3*x^2)) + log(x) + 6*x*exp(1/(3*x - 3*x^2)) + 9*x^2
\[ \int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} \left (-6 x+30 x^2-36 x^3+18 x^4\right )}{3 x^2-6 x^3+3 x^4} \, dx=\frac {-54 e^{\frac {2}{3 x^{2}-3 x}} \left (\int \frac {1}{e^{\frac {2}{3 x^{2}-3 x}} x^{4}-2 e^{\frac {2}{3 x^{2}-3 x}} x^{3}+e^{\frac {2}{3 x^{2}-3 x}} x^{2}}d x \right )+108 e^{\frac {2}{3 x^{2}-3 x}} \left (\int \frac {1}{e^{\frac {2}{3 x^{2}-3 x}} x^{3}-2 e^{\frac {2}{3 x^{2}-3 x}} x^{2}+e^{\frac {2}{3 x^{2}-3 x}} x}d x \right )+2401 e^{\frac {2}{3 x^{2}-3 x}} \mathrm {log}\left (x \right )+21609 e^{\frac {2}{3 x^{2}-3 x}} x^{2}+14406 e^{\frac {1}{3 x^{2}-3 x}} x +2320}{2401 e^{\frac {2}{3 x^{2}-3 x}}} \] Input:
int(((4*x-2)*exp(-1/(3*x^2-3*x))^2+(18*x^4-36*x^3+30*x^2-6*x)*exp(-1/(3*x^ 2-3*x))+54*x^5-108*x^4+57*x^3-6*x^2+3*x)/(3*x^4-6*x^3+3*x^2),x)
Output:
( - 54*e**(2/(3*x**2 - 3*x))*int(1/(e**(2/(3*x**2 - 3*x))*x**4 - 2*e**(2/( 3*x**2 - 3*x))*x**3 + e**(2/(3*x**2 - 3*x))*x**2),x) + 108*e**(2/(3*x**2 - 3*x))*int(1/(e**(2/(3*x**2 - 3*x))*x**3 - 2*e**(2/(3*x**2 - 3*x))*x**2 + e**(2/(3*x**2 - 3*x))*x),x) + 2401*e**(2/(3*x**2 - 3*x))*log(x) + 21609*e* *(2/(3*x**2 - 3*x))*x**2 + 14406*e**(1/(3*x**2 - 3*x))*x + 2320)/(2401*e** (2/(3*x**2 - 3*x)))