3.98.28 \(\int \frac {e^{\frac {-(\frac {e^{-x}}{x})^{e^{-e^{15/x}}}+x^3}{x}} (2 x^4 \log (\frac {e^{-x}}{x})+(\frac {e^{-x}}{x})^{e^{-e^{15/x}}} (x \log (\frac {e^{-x}}{x})+e^{-e^{15/x}} \log (\frac {e^{-x}}{x}) (x+x^2-15 e^{15/x} \log (\frac {e^{-x}}{x}))))}{x^3 \log (\frac {e^{-x}}{x})} \, dx\)

Optimal. Leaf size=32 \[ e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \]

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Rubi [F]  time = 11.02, 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^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-(1/(E^x*x))^E^(-E^(15/x)) + x^3)/x)*(2*x^4*Log[1/(E^x*x)] + (1/(E^x*x))^E^(-E^(15/x))*(x*Log[1/(E^x*
x)] + (Log[1/(E^x*x)]*(x + x^2 - 15*E^(15/x)*Log[1/(E^x*x)]))/E^E^(15/x))))/(x^3*Log[1/(E^x*x)]),x]

[Out]

-15*Log[1/(E^x*x)]*Defer[Int][(E^(-E^(15/x) + 15/x - (1/(E^x*x))^E^(-E^(15/x))/x + x^2)*(1/(E^x*x))^E^(-E^(15/
x)))/x^3, x] + Defer[Int][(E^(-((1/(E^x*x))^E^(-E^(15/x))/x) + x^2)*(1/(E^x*x))^E^(-E^(15/x)))/x^2, x] + Defer
[Int][(E^(-E^(15/x) - (1/(E^x*x))^E^(-E^(15/x))/x + x^2)*(1/(E^x*x))^E^(-E^(15/x)))/x^2, x] + Defer[Int][(E^(-
E^(15/x) - (1/(E^x*x))^E^(-E^(15/x))/x + x^2)*(1/(E^x*x))^E^(-E^(15/x)))/x, x] + 2*Defer[Int][E^(-((1/(E^x*x))
^E^(-E^(15/x))/x) + x^2)*x, x] - 15*Defer[Int][Defer[Int][(E^(-E^(15/x) + 15/x - (1/(E^x*x))^E^(-E^(15/x))/x +
x^2)*(1/(E^x*x))^E^(-E^(15/x)))/x^3, x], x] - 15*Defer[Int][Defer[Int][(E^(-E^(15/x) + 15/x - (1/(E^x*x))^E^(
-E^(15/x))/x + x^2)*(1/(E^x*x))^E^(-E^(15/x)))/x^3, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (2 x^4+e^{-e^{15/x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \left (1+e^{e^{15/x}}+x\right )-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )}{x^3} \, dx\\ &=\int \left (2 e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x+\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x+e^{e^{15/x}} x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )}{x^3}\right ) \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x+e^{e^{15/x}} x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )}{x^3} \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+\int \left (\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (1+e^{e^{15/x}}+x\right )}{x^2}-\frac {15 e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \log \left (\frac {e^{-x}}{x}\right )}{x^3}\right ) \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx-15 \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \log \left (\frac {e^{-x}}{x}\right )}{x^3} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (1+e^{e^{15/x}}+x\right )}{x^2} \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+15 \int \frac {(-1-x) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x} \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \left (\frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2}+\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} (1+x)}{x^2}\right ) \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+15 \int \left (-\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx-\frac {\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x}\right ) \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} (1+x)}{x^2} \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx-15 \int \left (\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx\right ) \, dx-15 \int \frac {\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x} \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \left (\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2}+\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}\right ) \, dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx\\ &=2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx-15 \int \left (\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx\right ) \, dx-15 \int \frac {\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x} \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 32, normalized size = 1.00 \begin {gather*} e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-(1/(E^x*x))^E^(-E^(15/x)) + x^3)/x)*(2*x^4*Log[1/(E^x*x)] + (1/(E^x*x))^E^(-E^(15/x))*(x*Log[1
/(E^x*x)] + (Log[1/(E^x*x)]*(x + x^2 - 15*E^(15/x)*Log[1/(E^x*x)]))/E^E^(15/x))))/(x^3*Log[1/(E^x*x)]),x]

[Out]

E^(-((1/(E^x*x))^E^(-E^(15/x))/x) + x^2)

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fricas [A]  time = 0.67, size = 32, normalized size = 1.00 \begin {gather*} e^{\left (\frac {x^{3} - e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*exp(15/x)*log(1/exp(x)/x)+x^2+x)*exp(log(log(1/exp(x)/x))-exp(15/x))+x*log(1/exp(x)/x))*exp(e
xp(log(log(1/exp(x)/x))-exp(15/x)))+2*x^4*log(1/exp(x)/x))*exp((-exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+x^3)
/x)/x^3/log(1/exp(x)/x),x, algorithm="fricas")

[Out]

e^((x^3 - e^(e^(-e^(15/x) + log(log(e^(-x)/x)))))/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{4} \log \left (\frac {e^{\left (-x\right )}}{x}\right ) + {\left ({\left (x^{2} - 15 \, e^{\frac {15}{x}} \log \left (\frac {e^{\left (-x\right )}}{x}\right ) + x\right )} e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )} + x \log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )} e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}\right )} e^{\left (\frac {x^{3} - e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}}{x}\right )}}{x^{3} \log \left (\frac {e^{\left (-x\right )}}{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*exp(15/x)*log(1/exp(x)/x)+x^2+x)*exp(log(log(1/exp(x)/x))-exp(15/x))+x*log(1/exp(x)/x))*exp(e
xp(log(log(1/exp(x)/x))-exp(15/x)))+2*x^4*log(1/exp(x)/x))*exp((-exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+x^3)
/x)/x^3/log(1/exp(x)/x),x, algorithm="giac")

[Out]

integrate((2*x^4*log(e^(-x)/x) + ((x^2 - 15*e^(15/x)*log(e^(-x)/x) + x)*e^(-e^(15/x) + log(log(e^(-x)/x))) + x
*log(e^(-x)/x))*e^(e^(-e^(15/x) + log(log(e^(-x)/x)))))*e^((x^3 - e^(e^(-e^(15/x) + log(log(e^(-x)/x)))))/x)/(
x^3*log(e^(-x)/x)), x)

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maple [C]  time = 0.38, size = 130, normalized size = 4.06

method	result	size
risch	\({\mathrm e}^{\frac {-{\mathrm e}^{-\frac {\left (i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+2 \ln \relax (x )+2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-{\mathrm e}^{\frac {15}{x}}}}{2}}+x^{3}}{x}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-15*exp(15/x)*ln(1/exp(x)/x)+x^2+x)*exp(ln(ln(1/exp(x)/x))-exp(15/x))+x*ln(1/exp(x)/x))*exp(exp(ln(ln(1
/exp(x)/x))-exp(15/x)))+2*x^4*ln(1/exp(x)/x))*exp((-exp(exp(ln(ln(1/exp(x)/x))-exp(15/x)))+x^3)/x)/x^3/ln(1/ex
p(x)/x),x,method=_RETURNVERBOSE)

[Out]

exp((-exp(-1/2*(I*Pi*csgn(I/x*exp(-x))^3-I*Pi*csgn(I/x*exp(-x))^2*csgn(I/x)-I*Pi*csgn(I/x*exp(-x))^2*csgn(I*ex
p(-x))+I*Pi*csgn(I/x*exp(-x))*csgn(I/x)*csgn(I*exp(-x))+2*ln(x)+2*ln(exp(x)))*exp(-exp(15/x)))+x^3)/x)

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maxima [A]  time = 0.63, size = 37, normalized size = 1.16 \begin {gather*} e^{\left (x^{2} - \frac {e^{\left (-x e^{\left (-e^{\frac {15}{x}}\right )} - e^{\left (-e^{\frac {15}{x}}\right )} \log \relax (x)\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*exp(15/x)*log(1/exp(x)/x)+x^2+x)*exp(log(log(1/exp(x)/x))-exp(15/x))+x*log(1/exp(x)/x))*exp(e
xp(log(log(1/exp(x)/x))-exp(15/x)))+2*x^4*log(1/exp(x)/x))*exp((-exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+x^3)
/x)/x^3/log(1/exp(x)/x),x, algorithm="maxima")

[Out]

e^(x^2 - e^(-x*e^(-e^(15/x)) - e^(-e^(15/x))*log(x))/x)

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mupad [B]  time = 6.63, size = 37, normalized size = 1.16 \begin {gather*} {\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x\,{\mathrm {e}}^{-{\mathrm {e}}^{15/x}}}\,{\left (\frac {1}{x}\right )}^{{\mathrm {e}}^{-{\mathrm {e}}^{15/x}}}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(exp(exp(log(log(exp(-x)/x)) - exp(15/x))) - x^3)/x)*(2*x^4*log(exp(-x)/x) + exp(exp(log(log(exp(-x)
/x)) - exp(15/x)))*(x*log(exp(-x)/x) + exp(log(log(exp(-x)/x)) - exp(15/x))*(x + x^2 - 15*exp(15/x)*log(exp(-x
)/x)))))/(x^3*log(exp(-x)/x)),x)

[Out]

exp(x^2)*exp(-(exp(-x*exp(-exp(15/x)))*(1/x)^exp(-exp(15/x)))/x)

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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((((-15*exp(15/x)*ln(1/exp(x)/x)+x**2+x)*exp(ln(ln(1/exp(x)/x))-exp(15/x))+x*ln(1/exp(x)/x))*exp(exp(
ln(ln(1/exp(x)/x))-exp(15/x)))+2*x**4*ln(1/exp(x)/x))*exp((-exp(exp(ln(ln(1/exp(x)/x))-exp(15/x)))+x**3)/x)/x*
*3/ln(1/exp(x)/x),x)

[Out]

Timed out

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