3.24.13 \(\int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} (2-2 x+e^{x^2} (3 x-3 x^2)+(e^{x^2} (3 x^2+6 x^4)+e^{x^2} (-3 x-6 x^3) \log (x)) \log (x-\log (x)))}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+(4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3) \log (x)} \, dx\)

Optimal. Leaf size=21 \[ e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} \]

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Rubi [F]  time = 28.34, 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 {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(2 - 2*x + E^x^2*(3*x - 3*x^2) + (E
^x^2*(3*x^2 + 6*x^4) + E^x^2*(-3*x - 6*x^3)*Log[x])*Log[x - Log[x]]))/(-4*x^2 - 12*E^x^2*x^3 - 9*E^(2*x^2)*x^4
+ (4*x + 12*E^x^2*x^2 + 9*E^(2*x^2)*x^3)*Log[x]),x]

[Out]

Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(-1 + (2 + 3*E^x^2*x)^(-1)))/(2 + 3*E^x^2*x), x]
- Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(-1 + (2 + 3*E^x^2*x)^(-1)))/(x*(2 + 3*E^x^2*x)
), x] + 2*Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*Log[x - Log[x]])/(
x*(2 + 3*E^x^2*x)^2), x] + 4*Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*x*(x - Log[x])^(2 + 3*E^x^2*x)^(-
1)*Log[x - Log[x]])/(2 + 3*E^x^2*x)^2, x] - Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(-1 +
(2 + 3*E^x^2*x)^(-1))*Log[x - Log[x]])/(2 + 3*E^x^2*x), x] - 2*Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1
)*x^2*(x - Log[x])^(-1 + (2 + 3*E^x^2*x)^(-1))*Log[x - Log[x]])/(2 + 3*E^x^2*x), x] + Defer[Int][(E^(x - Log[x
])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(-1 + (2 + 3*E^x^2*x)^(-1))*Log[x]*Log[x - Log[x]])/(x*(2 + 3*E^x^2*x)),
x] + 2*Defer[Int][(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*x*(x - Log[x])^(-1 + (2 + 3*E^x^2*x)^(-1))*Log[x]*Log[x
- Log[x]])/(2 + 3*E^x^2*x), x]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 21, normalized size = 1.00 \begin {gather*} e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(2 - 2*x + E^x^2*(3*x - 3*x^2
) + (E^x^2*(3*x^2 + 6*x^4) + E^x^2*(-3*x - 6*x^3)*Log[x])*Log[x - Log[x]]))/(-4*x^2 - 12*E^x^2*x^3 - 9*E^(2*x^
2)*x^4 + (4*x + 12*E^x^2*x^2 + 9*E^(2*x^2)*x^3)*Log[x]),x]

[Out]

E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)

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fricas [A]  time = 0.72, size = 19, normalized size = 0.90 \begin {gather*} e^{\left ({\left (x - \log \relax (x)\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^3-3*x)*exp(x^2)*log(x)+(6*x^4+3*x^2)*exp(x^2))*log(x-log(x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*ex
p(log(x-log(x))/(3*exp(x^2)*x+2))*exp(exp(log(x-log(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+
4*x)*log(x)-9*x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )} + 3 \, {\left ({\left (2 \, x^{3} + x\right )} e^{\left (x^{2}\right )} \log \relax (x) - {\left (2 \, x^{4} + x^{2}\right )} e^{\left (x^{2}\right )}\right )} \log \left (x - \log \relax (x)\right ) + 2 \, x - 2\right )} {\left (x - \log \relax (x)\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )} e^{\left ({\left (x - \log \relax (x)\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )}\right )}}{9 \, x^{4} e^{\left (2 \, x^{2}\right )} + 12 \, x^{3} e^{\left (x^{2}\right )} + 4 \, x^{2} - {\left (9 \, x^{3} e^{\left (2 \, x^{2}\right )} + 12 \, x^{2} e^{\left (x^{2}\right )} + 4 \, x\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^3-3*x)*exp(x^2)*log(x)+(6*x^4+3*x^2)*exp(x^2))*log(x-log(x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*ex
p(log(x-log(x))/(3*exp(x^2)*x+2))*exp(exp(log(x-log(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+
4*x)*log(x)-9*x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x, algorithm="giac")

[Out]

integrate((3*(x^2 - x)*e^(x^2) + 3*((2*x^3 + x)*e^(x^2)*log(x) - (2*x^4 + x^2)*e^(x^2))*log(x - log(x)) + 2*x
- 2)*(x - log(x))^(1/(3*x*e^(x^2) + 2))*e^((x - log(x))^(1/(3*x*e^(x^2) + 2)))/(9*x^4*e^(2*x^2) + 12*x^3*e^(x^
2) + 4*x^2 - (9*x^3*e^(2*x^2) + 12*x^2*e^(x^2) + 4*x)*log(x)), x)

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maple [A]  time = 0.08, size = 20, normalized size = 0.95

method	result	size
risch	\({\mathrm e}^{\left (x -\ln \relax (x )\right )^{\frac {1}{3 \,{\mathrm e}^{x^{2}} x +2}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x^3-3*x)*exp(x^2)*ln(x)+(6*x^4+3*x^2)*exp(x^2))*ln(x-ln(x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*exp(ln(x-ln
(x))/(3*exp(x^2)*x+2))*exp(exp(ln(x-ln(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+4*x)*ln(x)-9*
x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.66, size = 19, normalized size = 0.90 \begin {gather*} e^{\left ({\left (x - \log \relax (x)\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^3-3*x)*exp(x^2)*log(x)+(6*x^4+3*x^2)*exp(x^2))*log(x-log(x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*ex
p(log(x-log(x))/(3*exp(x^2)*x+2))*exp(exp(log(x-log(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+
4*x)*log(x)-9*x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.90, size = 19, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{{\left (x-\ln \relax (x)\right )}^{\frac {1}{3\,x\,{\mathrm {e}}^{x^2}+2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x - log(x))/(3*x*exp(x^2) + 2))*exp(exp(log(x - log(x))/(3*x*exp(x^2) + 2)))*(exp(x^2)*(3*x - 3*
x^2) - 2*x + log(x - log(x))*(exp(x^2)*(3*x^2 + 6*x^4) - exp(x^2)*log(x)*(3*x + 6*x^3)) + 2))/(12*x^3*exp(x^2)
- log(x)*(4*x + 12*x^2*exp(x^2) + 9*x^3*exp(2*x^2)) + 9*x^4*exp(2*x^2) + 4*x^2),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x**3-3*x)*exp(x**2)*ln(x)+(6*x**4+3*x**2)*exp(x**2))*ln(x-ln(x))+(-3*x**2+3*x)*exp(x**2)-2*x+2
)*exp(ln(x-ln(x))/(3*exp(x**2)*x+2))*exp(exp(ln(x-ln(x))/(3*exp(x**2)*x+2)))/((9*x**3*exp(x**2)**2+12*x**2*exp
(x**2)+4*x)*ln(x)-9*x**4*exp(x**2)**2-12*x**3*exp(x**2)-4*x**2),x)

[Out]

Timed out

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