3.39.42 \(\int \frac {2 e^{\frac {e^4-x^2}{x}} x-x^2+(-3 x^2+e^{\frac {e^4-x^2}{x}} (-2 e^4+4 x-2 x^2)) \log (x)}{(2 e^{\frac {e^4-x^2}{x}} x^2-x^3) \log (x)} \, dx\)

Optimal. Leaf size=31 \[ \log \left (\frac {3}{2} \left (-e^{\frac {e^4}{x}-x}+\frac {x}{2}\right ) x^2 \log (x)\right ) \]

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Rubi [F]  time = 1.94, 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 {2 e^{\frac {e^4-x^2}{x}} x-x^2+\left (-3 x^2+e^{\frac {e^4-x^2}{x}} \left (-2 e^4+4 x-2 x^2\right )\right ) \log (x)}{\left (2 e^{\frac {e^4-x^2}{x}} x^2-x^3\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.45, size = 30, normalized size = 0.97 \begin {gather*} -x+2 \log (x)+\log \left (2 e^{\frac {e^4}{x}}-e^x x\right )+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 29, normalized size = 0.94 \begin {gather*} 2 \, \log \relax (x) + \log \left (-x + 2 \, e^{\left (-\frac {x^{2} - e^{4}}{x}\right )}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(4)-2*x^2+4*x)*exp((exp(4)-x^2)/x)-3*x^2)*log(x)+2*x*exp((exp(4)-x^2)/x)-x^2)/(2*x^2*exp((e
xp(4)-x^2)/x)-x^3)/log(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 27, normalized size = 0.87 \begin {gather*} \log \left (x - 2 \, e^{\left (-\frac {x^{2} - e^{4}}{x}\right )}\right ) + 2 \, \log \relax (x) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(4)-2*x^2+4*x)*exp((exp(4)-x^2)/x)-3*x^2)*log(x)+2*x*exp((exp(4)-x^2)/x)-x^2)/(2*x^2*exp((e
xp(4)-x^2)/x)-x^3)/log(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 27, normalized size = 0.87

method	result	size
norman	\(2 \ln \relax (x )+\ln \left (\ln \relax (x )\right )+\ln \left (x -2 \,{\mathrm e}^{\frac {{\mathrm e}^{4}-x^{2}}{x}}\right )\)	\(27\)
risch	\(\ln \left (-\frac {x}{2}+{\mathrm e}^{\frac {{\mathrm e}^{4}-x^{2}}{x}}\right )+2 \ln \relax (x )+\ln \left (\ln \relax (x )\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 25, normalized size = 0.81 \begin {gather*} -x + \log \left (-\frac {1}{2} \, x e^{x} + e^{\left (\frac {e^{4}}{x}\right )}\right ) + 2 \, \log \relax (x) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(4)-2*x^2+4*x)*exp((exp(4)-x^2)/x)-3*x^2)*log(x)+2*x*exp((exp(4)-x^2)/x)-x^2)/(2*x^2*exp((e
xp(4)-x^2)/x)-x^3)/log(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.60, size = 24, normalized size = 0.77 \begin {gather*} \ln \left (\ln \relax (x)\right )+\ln \left (x-2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x}-x}\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.37, size = 24, normalized size = 0.77 \begin {gather*} 2 \log {\relax (x )} + \log {\left (- \frac {x}{2} + e^{\frac {- x^{2} + e^{4}}{x}} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(4)-2*x**2+4*x)*exp((exp(4)-x**2)/x)-3*x**2)*ln(x)+2*x*exp((exp(4)-x**2)/x)-x**2)/(2*x**2*e
xp((exp(4)-x**2)/x)-x**3)/ln(x),x)

[Out]

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

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