3.78.23 \(\int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} (x-3 x^2+2 x^3-x^4)+(x^2-2 x^3+x^4) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} (-4 x+12 x^3-8 x^4)+e^{\frac {2}{-x+x^2}} (x^2-2 x^3+x^4)+(4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} (-2 x^2+4 x^3-2 x^4)) \log (x)+(x^2-2 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {x}{4-e^{\frac {1}{-x+x^2}}+\frac {2}{x}+\log (x)} \]

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Rubi [F]  time = 11.72, 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 {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x - 5*x^2 - 2*x^3 + 3*x^4 + E^(-x + x^2)^(-1)*(x - 3*x^2 + 2*x^3 - x^4) + (x^2 - 2*x^3 + x^4)*Log[x])/(
4 + 8*x - 12*x^2 - 16*x^3 + 16*x^4 + E^(-x + x^2)^(-1)*(-4*x + 12*x^3 - 8*x^4) + E^(2/(-x + x^2))*(x^2 - 2*x^3
+ x^4) + (4*x - 12*x^3 + 8*x^4 + E^(-x + x^2)^(-1)*(-2*x^2 + 4*x^3 - 2*x^4))*Log[x] + (x^2 - 2*x^3 + x^4)*Log
[x]^2),x]

[Out]

-8*Defer[Int][(-2 - 4*x + E^(-x + x^2)^(-1)*x - x*Log[x])^(-2), x] - 6*Defer[Int][1/((-1 + x)^2*(-2 - 4*x + E^
(-x + x^2)^(-1)*x - x*Log[x])^2), x] - 16*Defer[Int][1/((-1 + x)*(-2 - 4*x + E^(-x + x^2)^(-1)*x - x*Log[x])^2
), x] + 2*Defer[Int][x/(-2 - 4*x + E^(-x + x^2)^(-1)*x - x*Log[x])^2, x] - Defer[Int][x^2/(-2 - 4*x + E^(-x +
x^2)^(-1)*x - x*Log[x])^2, x] - Defer[Int][1/((-1 + x)^2*(-2 - 4*x + E^(-x + x^2)^(-1)*x - x*Log[x])), x] - 2*
Defer[Int][1/((-1 + x)*(-2 - 4*x + E^(-x + x^2)^(-1)*x - x*Log[x])), x] - Defer[Int][x/(-2 - 4*x + E^(-x + x^2
)^(-1)*x - x*Log[x]), x] - 2*Defer[Int][Log[x]/(2 + 4*x - E^(-x + x^2)^(-1)*x + x*Log[x])^2, x] - Defer[Int][L
og[x]/((-1 + x)^2*(2 + 4*x - E^(-x + x^2)^(-1)*x + x*Log[x])^2), x] - 3*Defer[Int][Log[x]/((-1 + x)*(2 + 4*x -
E^(-x + x^2)^(-1)*x + x*Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left ((-1+x)^2 (4+3 x)-e^{\frac {1}{-x+x^2}} \left (-1+3 x-2 x^2+x^3\right )+(-1+x)^2 x \log (x)\right )}{(1-x)^2 \left (2-\left (-4+e^{\frac {1}{-x+x^2}}\right ) x+x \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {-1+3 x-2 x^2+x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )}-\frac {-2-2 x+13 x^2-4 x^3+x^4-x \log (x)+2 x^2 \log (x)}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx\\ &=-\int \frac {-1+3 x-2 x^2+x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {-2-2 x+13 x^2-4 x^3+x^4-x \log (x)+2 x^2 \log (x)}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx\\ &=-\int \left (\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )}+\frac {2}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )}+\frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)}\right ) \, dx-\int \left (-\frac {2}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}-\frac {2 x}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {13 x^2}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}-\frac {4 x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {x^4}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}-\frac {x \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {2 x^2 \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}\right ) \, dx\\ &=2 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+2 \int \frac {x}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-2 \int \frac {x^2 \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx+4 \int \frac {x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-13 \int \frac {x^2}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {x^4}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)} \, dx+\int \frac {x \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx\\ &=2 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx+2 \int \left (\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx-2 \int \left (\frac {\log (x)}{\left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {2 \log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}\right ) \, dx+4 \int \left (\frac {2}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {3}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx-13 \int \left (\frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {2}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)} \, dx-\int \left (\frac {3}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {4}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {2 x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {x^2}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx+\int \left (\frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {\log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}\right ) \, dx\\ &=2 \left (2 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx\right )+2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-2 \int \frac {\log (x)}{\left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx-2 \int \frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx-3 \int \frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+4 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-4 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+4 \int \frac {x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-4 \int \frac {\log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx+8 \int \frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+12 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-13 \int \frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-13 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-26 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {x^2}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)} \, dx+\int \frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx+\int \frac {\log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.10, size = 28, normalized size = 1.08 \begin {gather*} \frac {x^2}{2-\left (-4+e^{\frac {1}{(-1+x) x}}\right ) x+x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x - 5*x^2 - 2*x^3 + 3*x^4 + E^(-x + x^2)^(-1)*(x - 3*x^2 + 2*x^3 - x^4) + (x^2 - 2*x^3 + x^4)*Log
[x])/(4 + 8*x - 12*x^2 - 16*x^3 + 16*x^4 + E^(-x + x^2)^(-1)*(-4*x + 12*x^3 - 8*x^4) + E^(2/(-x + x^2))*(x^2 -
2*x^3 + x^4) + (4*x - 12*x^3 + 8*x^4 + E^(-x + x^2)^(-1)*(-2*x^2 + 4*x^3 - 2*x^4))*Log[x] + (x^2 - 2*x^3 + x^
4)*Log[x]^2),x]

[Out]

x^2/(2 - (-4 + E^(1/((-1 + x)*x)))*x + x*Log[x])

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fricas [A]  time = 0.74, size = 29, normalized size = 1.12 \begin {gather*} -\frac {x^{2}}{x e^{\left (\frac {1}{x^{2} - x}\right )} - x \log \relax (x) - 4 \, x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-2*x^3+x^2)*log(x)+(-x^4+2*x^3-3*x^2+x)*exp(1/(x^2-x))+3*x^4-2*x^3-5*x^2+4*x)/((x^4-2*x^3+x^2)*
log(x)^2+((-2*x^4+4*x^3-2*x^2)*exp(1/(x^2-x))+8*x^4-12*x^3+4*x)*log(x)+(x^4-2*x^3+x^2)*exp(1/(x^2-x))^2+(-8*x^
4+12*x^3-4*x)*exp(1/(x^2-x))+16*x^4-16*x^3-12*x^2+8*x+4),x, algorithm="fricas")

[Out]

-x^2/(x*e^(1/(x^2 - x)) - x*log(x) - 4*x - 2)

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giac [A]  time = 0.29, size = 29, normalized size = 1.12 \begin {gather*} -\frac {x^{2}}{x e^{\left (\frac {1}{x^{2} - x}\right )} - x \log \relax (x) - 4 \, x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-2*x^3+x^2)*log(x)+(-x^4+2*x^3-3*x^2+x)*exp(1/(x^2-x))+3*x^4-2*x^3-5*x^2+4*x)/((x^4-2*x^3+x^2)*
log(x)^2+((-2*x^4+4*x^3-2*x^2)*exp(1/(x^2-x))+8*x^4-12*x^3+4*x)*log(x)+(x^4-2*x^3+x^2)*exp(1/(x^2-x))^2+(-8*x^
4+12*x^3-4*x)*exp(1/(x^2-x))+16*x^4-16*x^3-12*x^2+8*x+4),x, algorithm="giac")

[Out]

-x^2/(x*e^(1/(x^2 - x)) - x*log(x) - 4*x - 2)

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maple [A]  time = 0.04, size = 30, normalized size = 1.15

method	result	size
risch	\(-\frac {x^{2}}{{\mathrm e}^{\frac {1}{x \left (x -1\right )}} x -x \ln \relax (x )-4 x -2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4-2*x^3+x^2)*ln(x)+(-x^4+2*x^3-3*x^2+x)*exp(1/(x^2-x))+3*x^4-2*x^3-5*x^2+4*x)/((x^4-2*x^3+x^2)*ln(x)^2
+((-2*x^4+4*x^3-2*x^2)*exp(1/(x^2-x))+8*x^4-12*x^3+4*x)*ln(x)+(x^4-2*x^3+x^2)*exp(1/(x^2-x))^2+(-8*x^4+12*x^3-
4*x)*exp(1/(x^2-x))+16*x^4-16*x^3-12*x^2+8*x+4),x,method=_RETURNVERBOSE)

[Out]

-x^2/(exp(1/x/(x-1))*x-x*ln(x)-4*x-2)

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maxima [A]  time = 0.43, size = 35, normalized size = 1.35 \begin {gather*} -\frac {x^{2} e^{\frac {1}{x}}}{x e^{\left (\frac {1}{x - 1}\right )} - {\left (x \log \relax (x) + 4 \, x + 2\right )} e^{\frac {1}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-2*x^3+x^2)*log(x)+(-x^4+2*x^3-3*x^2+x)*exp(1/(x^2-x))+3*x^4-2*x^3-5*x^2+4*x)/((x^4-2*x^3+x^2)*
log(x)^2+((-2*x^4+4*x^3-2*x^2)*exp(1/(x^2-x))+8*x^4-12*x^3+4*x)*log(x)+(x^4-2*x^3+x^2)*exp(1/(x^2-x))^2+(-8*x^
4+12*x^3-4*x)*exp(1/(x^2-x))+16*x^4-16*x^3-12*x^2+8*x+4),x, algorithm="maxima")

[Out]

-x^2*e^(1/x)/(x*e^(1/(x - 1)) - (x*log(x) + 4*x + 2)*e^(1/x))

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mupad [B]  time = 5.53, size = 153, normalized size = 5.88 \begin {gather*} -\frac {x\,{\left (x^3-2\,x^2+x\right )}^2\,\left (-x^4+4\,x^3-13\,x^2+2\,x+2\right )+x\,\ln \relax (x)\,\left (x-2\,x^2\right )\,{\left (x^3-2\,x^2+x\right )}^2}{{\left (x-1\right )}^2\,\left (4\,x-x\,{\mathrm {e}}^{-\frac {1}{x-x^2}}+x\,\ln \relax (x)+2\right )\,\left (4\,x^3\,\ln \relax (x)-x^2\,\ln \relax (x)-2\,x-5\,x^4\,\ln \relax (x)+2\,x^5\,\ln \relax (x)+2\,x^2+15\,x^3-32\,x^4+22\,x^5-6\,x^6+x^7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + exp(-1/(x - x^2))*(x - 3*x^2 + 2*x^3 - x^4) + log(x)*(x^2 - 2*x^3 + x^4) - 5*x^2 - 2*x^3 + 3*x^4)/(
8*x - exp(-1/(x - x^2))*(4*x - 12*x^3 + 8*x^4) + log(x)*(4*x - exp(-1/(x - x^2))*(2*x^2 - 4*x^3 + 2*x^4) - 12*
x^3 + 8*x^4) + exp(-2/(x - x^2))*(x^2 - 2*x^3 + x^4) - 12*x^2 - 16*x^3 + 16*x^4 + log(x)^2*(x^2 - 2*x^3 + x^4)
+ 4),x)

[Out]

-(x*(x - 2*x^2 + x^3)^2*(2*x - 13*x^2 + 4*x^3 - x^4 + 2) + x*log(x)*(x - 2*x^2)*(x - 2*x^2 + x^3)^2)/((x - 1)^
2*(4*x - x*exp(-1/(x - x^2)) + x*log(x) + 2)*(4*x^3*log(x) - x^2*log(x) - 2*x - 5*x^4*log(x) + 2*x^5*log(x) +
2*x^2 + 15*x^3 - 32*x^4 + 22*x^5 - 6*x^6 + x^7))

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sympy [A]  time = 0.50, size = 24, normalized size = 0.92 \begin {gather*} - \frac {x^{2}}{x e^{\frac {1}{x^{2} - x}} - x \log {\relax (x )} - 4 x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4-2*x**3+x**2)*ln(x)+(-x**4+2*x**3-3*x**2+x)*exp(1/(x**2-x))+3*x**4-2*x**3-5*x**2+4*x)/((x**4-2
*x**3+x**2)*ln(x)**2+((-2*x**4+4*x**3-2*x**2)*exp(1/(x**2-x))+8*x**4-12*x**3+4*x)*ln(x)+(x**4-2*x**3+x**2)*exp
(1/(x**2-x))**2+(-8*x**4+12*x**3-4*x)*exp(1/(x**2-x))+16*x**4-16*x**3-12*x**2+8*x+4),x)

[Out]

-x**2/(x*exp(1/(x**2 - x)) - x*log(x) - 4*x - 2)

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