3.45.14 \(\int \frac {3 x-6 x^2+57 x^3-108 x^4+54 x^5+e^{-\frac {2}{-3 x+3 x^2}} (-2+4 x)+e^{-\frac {1}{-3 x+3 x^2}} (-6 x+30 x^2-36 x^3+18 x^4)}{3 x^2-6 x^3+3 x^4} \, dx\)

Optimal. Leaf size=30 \[ 1+\left (e^{\frac {x}{3 \left (x^2-x^3\right )}}+3 x\right )^2+\log (2 x) \]

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Rubi [F]  time = 1.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

E^(2/(3*(1 - x)*x)) + 9*x^2 + Log[x] + 6*Defer[Int][E^(1/((3 - 3*x)*x)), x] + 2*Defer[Int][E^(1/((3 - 3*x)*x))
/(-1 + x)^2, x] + 2*Defer[Int][E^(1/((3 - 3*x)*x))/(-1 + x), x] - 2*Defer[Int][E^(1/((3 - 3*x)*x))/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\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 )}{x^2 \left (3-6 x+3 x^2\right )} \, dx\\ &=\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 (-1+x)^2 x^2} \, dx\\ &=\frac {1}{3} \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 )}{(-1+x)^2 x^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {6}{(-1+x)^2}+\frac {3}{(-1+x)^2 x}+\frac {57 x}{(-1+x)^2}-\frac {108 x^2}{(-1+x)^2}+\frac {54 x^3}{(-1+x)^2}+\frac {2 e^{\frac {2}{(3-3 x) x}} (-1+2 x)}{(1-x)^2 x^2}+\frac {6 e^{\frac {1}{(3-3 x) x}} \left (-1+5 x-6 x^2+3 x^3\right )}{(1-x)^2 x}\right ) \, dx\\ &=-\frac {2}{1-x}+\frac {2}{3} \int \frac {e^{\frac {2}{(3-3 x) x}} (-1+2 x)}{(1-x)^2 x^2} \, dx+2 \int \frac {e^{\frac {1}{(3-3 x) x}} \left (-1+5 x-6 x^2+3 x^3\right )}{(1-x)^2 x} \, dx+18 \int \frac {x^3}{(-1+x)^2} \, dx+19 \int \frac {x}{(-1+x)^2} \, dx-36 \int \frac {x^2}{(-1+x)^2} \, dx+\int \frac {1}{(-1+x)^2 x} \, dx\\ &=e^{\frac {2}{3 (1-x) x}}-\frac {2}{1-x}+2 \int \left (3 e^{\frac {1}{(3-3 x) x}}+\frac {e^{\frac {1}{(3-3 x) x}}}{(-1+x)^2}+\frac {e^{\frac {1}{(3-3 x) x}}}{-1+x}-\frac {e^{\frac {1}{(3-3 x) x}}}{x}\right ) \, dx+18 \int \left (2+\frac {1}{(-1+x)^2}+\frac {3}{-1+x}+x\right ) \, dx+19 \int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx-36 \int \left (1+\frac {1}{(-1+x)^2}+\frac {2}{-1+x}\right ) \, dx+\int \left (\frac {1}{1-x}+\frac {1}{(-1+x)^2}+\frac {1}{x}\right ) \, dx\\ &=e^{\frac {2}{3 (1-x) x}}+9 x^2+\log (x)+2 \int \frac {e^{\frac {1}{(3-3 x) x}}}{(-1+x)^2} \, dx+2 \int \frac {e^{\frac {1}{(3-3 x) x}}}{-1+x} \, dx-2 \int \frac {e^{\frac {1}{(3-3 x) x}}}{x} \, dx+6 \int e^{\frac {1}{(3-3 x) x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 22, normalized size = 0.73 \begin {gather*} \left (e^{\frac {1}{3 x-3 x^2}}+3 x\right )^2+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

(E^(3*x - 3*x^2)^(-1) + 3*x)^2 + Log[x]

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fricas [A]  time = 0.65, size = 35, normalized size = 1.17 \begin {gather*} 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 \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="fricas")

[Out]

9*x^2 + 6*x*e^(-1/3/(x^2 - x)) + e^(-2/3/(x^2 - x)) + log(x)

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giac [A]  time = 0.21, size = 35, normalized size = 1.17 \begin {gather*} 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 \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="giac")

[Out]

9*x^2 + 6*x*e^(-1/3/(x^2 - x)) + e^(-2/3/(x^2 - x)) + log(x)

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maple [A]  time = 0.25, size = 34, normalized size = 1.13

method	result	size
risch	\(9 x^{2}+\ln \relax (x )+{\mathrm e}^{-\frac {2}{3 x \left (x -1\right )}}+6 x \,{\mathrm e}^{-\frac {1}{3 x \left (x -1\right )}}\)	\(34\)
norman	\(\frac {x^{2} {\mathrm e}^{-\frac {2}{3 x^{2}-3 x}}-9 x^{3}+9 x^{4}-x \,{\mathrm e}^{-\frac {2}{3 x^{2}-3 x}}-6 x^{2} {\mathrm e}^{-\frac {1}{3 x^{2}-3 x}}+6 x^{3} {\mathrm e}^{-\frac {1}{3 x^{2}-3 x}}}{x \left (x -1\right )}+\ln \relax (x )\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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=_RETURNVERBOSE)

[Out]

9*x^2+ln(x)+exp(-2/3/x/(x-1))+6*x*exp(-1/3/x/(x-1))

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maxima [A]  time = 0.43, size = 39, normalized size = 1.30 \begin {gather*} 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 \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="maxima")

[Out]

9*x^2 + 6*x*e^(-1/3/(x - 1) + 1/3/x) + e^(-2/3/(x - 1) + 2/3/x) + log(x)

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mupad [B]  time = 3.15, size = 37, normalized size = 1.23 \begin {gather*} {\mathrm {e}}^{\frac {2}{3\,x-3\,x^2}}+\ln \relax (x)+6\,x\,{\mathrm {e}}^{\frac {1}{3\,x-3\,x^2}}+9\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

exp(2/(3*x - 3*x^2)) + log(x) + 6*x*exp(1/(3*x - 3*x^2)) + 9*x^2

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sympy [A]  time = 0.27, size = 36, normalized size = 1.20 \begin {gather*} 9 x^{2} + 6 x e^{- \frac {1}{3 x^{2} - 3 x}} + \log {\relax (x )} + e^{- \frac {2}{3 x^{2} - 3 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

9*x**2 + 6*x*exp(-1/(3*x**2 - 3*x)) + log(x) + exp(-2/(3*x**2 - 3*x))

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