3.15.60 \(\int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} (-x^3+e^{\frac {5}{x^2}} (-e x^2+x^3+x^2 \log (5))-e^{\frac {5}{x^2}} \log ^2(x))}{x^2}} (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x))}{4 x^3} \, dx\)

Optimal. Leaf size=30 \[ \frac {5}{4} e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \]

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Rubi [F]  time = 5.83, 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 {\exp \left (-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}\right ) \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-5/x^2 + (-x^3 + E^(5/x^2)*(-(E*x^2) + x^3 + x^2*Log[5]) - E^(5/x^2)*Log[x]^2)/(E^(5/x^2)*x^2))*(-10*x
- x^3 + E^(5/x^2)*x^3 - 2*E^(5/x^2)*Log[x] + 2*E^(5/x^2)*Log[x]^2))/(4*x^3),x]

[Out]

(5*Defer[Int][E^(-E + x - x/E^(5/x^2) - Log[x]^2/x^2), x])/4 - (5*Defer[Int][E^(-E - 5/x^2 + x - x/E^(5/x^2) -
Log[x]^2/x^2), x])/4 - (25*Defer[Int][E^(-E - 5/x^2 + x - x/E^(5/x^2) - Log[x]^2/x^2)/x^2, x])/2 - (5*Defer[I
nt][(E^(-E + x - x/E^(5/x^2) - Log[x]^2/x^2)*Log[x])/x^3, x])/2 + (5*Defer[Int][(E^(-E + x - x/E^(5/x^2) - Log
[x]^2/x^2)*Log[x]^2)/x^3, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {\exp \left (-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}\right ) \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \frac {5 e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10 x+\left (-1+e^{\frac {5}{x^2}}\right ) x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {5}{4} \int \frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10 x+\left (-1+e^{\frac {5}{x^2}}\right ) x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {5}{4} \int \left (\frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10-x^2\right )}{x^2}+\frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (x^3-2 \log (x)+2 \log ^2(x)\right )}{x^3}\right ) \, dx\\ &=\frac {5}{4} \int \frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10-x^2\right )}{x^2} \, dx+\frac {5}{4} \int \frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (x^3-2 \log (x)+2 \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {5}{4} \int \left (-e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}-\frac {10 e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}}{x^2}\right ) \, dx+\frac {5}{4} \int \left (e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}-\frac {2 e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log (x)}{x^3}+\frac {2 e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log ^2(x)}{x^3}\right ) \, dx\\ &=\frac {5}{4} \int e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \, dx-\frac {5}{4} \int e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \, dx-\frac {5}{2} \int \frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log (x)}{x^3} \, dx+\frac {5}{2} \int \frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log ^2(x)}{x^3} \, dx-\frac {25}{2} \int \frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.33, size = 30, normalized size = 1.00 \begin {gather*} \frac {5}{4} e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-5/x^2 + (-x^3 + E^(5/x^2)*(-(E*x^2) + x^3 + x^2*Log[5]) - E^(5/x^2)*Log[x]^2)/(E^(5/x^2)*x^2))*
(-10*x - x^3 + E^(5/x^2)*x^3 - 2*E^(5/x^2)*Log[x] + 2*E^(5/x^2)*Log[x]^2))/(4*x^3),x]

[Out]

(5*E^(-E + x - x/E^(5/x^2) - Log[x]^2/x^2))/4

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fricas [B]  time = 0.57, size = 61, normalized size = 2.03 \begin {gather*} \frac {1}{4} \, e^{\left (-\frac {{\left (x^{3} + e^{\left (\frac {5}{x^{2}}\right )} \log \relax (x)^{2} - {\left (x^{3} - x^{2} e + x^{2} \log \relax (5) - 5\right )} e^{\left (\frac {5}{x^{2}}\right )}\right )} e^{\left (-\frac {5}{x^{2}}\right )}}{x^{2}} + \frac {5}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*exp(5/x^2)*log(x)^2-2*exp(5/x^2)*log(x)+x^3*exp(5/x^2)-x^3-10*x)*exp((-exp(5/x^2)*log(x)^2+(x
^2*log(5)-x^2*exp(1)+x^3)*exp(5/x^2)-x^3)/x^2/exp(5/x^2))/x^3/exp(5/x^2),x, algorithm="fricas")

[Out]

1/4*e^(-(x^3 + e^(5/x^2)*log(x)^2 - (x^3 - x^2*e + x^2*log(5) - 5)*e^(5/x^2))*e^(-5/x^2)/x^2 + 5/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} e^{\left (\frac {5}{x^{2}}\right )} - x^{3} + 2 \, e^{\left (\frac {5}{x^{2}}\right )} \log \relax (x)^{2} - 2 \, e^{\left (\frac {5}{x^{2}}\right )} \log \relax (x) - 10 \, x\right )} e^{\left (-\frac {{\left (x^{3} + e^{\left (\frac {5}{x^{2}}\right )} \log \relax (x)^{2} - {\left (x^{3} - x^{2} e + x^{2} \log \relax (5)\right )} e^{\left (\frac {5}{x^{2}}\right )}\right )} e^{\left (-\frac {5}{x^{2}}\right )}}{x^{2}} - \frac {5}{x^{2}}\right )}}{4 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*exp(5/x^2)*log(x)^2-2*exp(5/x^2)*log(x)+x^3*exp(5/x^2)-x^3-10*x)*exp((-exp(5/x^2)*log(x)^2+(x
^2*log(5)-x^2*exp(1)+x^3)*exp(5/x^2)-x^3)/x^2/exp(5/x^2))/x^3/exp(5/x^2),x, algorithm="giac")

[Out]

integrate(1/4*(x^3*e^(5/x^2) - x^3 + 2*e^(5/x^2)*log(x)^2 - 2*e^(5/x^2)*log(x) - 10*x)*e^(-(x^3 + e^(5/x^2)*lo
g(x)^2 - (x^3 - x^2*e + x^2*log(5))*e^(5/x^2))*e^(-5/x^2)/x^2 - 5/x^2)/x^3, x)

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maple [A]  time = 0.06, size = 35, normalized size = 1.17

method	result	size
risch	\(\frac {5 \,{\mathrm e}^{-\frac {{\mathrm e}^{-\frac {5}{x^{2}}} x^{3}+x^{2} {\mathrm e}-x^{3}+\ln \relax (x )^{2}}{x^{2}}}}{4}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(2*exp(5/x^2)*ln(x)^2-2*exp(5/x^2)*ln(x)+x^3*exp(5/x^2)-x^3-10*x)*exp((-exp(5/x^2)*ln(x)^2+(x^2*ln(5)-
x^2*exp(1)+x^3)*exp(5/x^2)-x^3)/x^2/exp(5/x^2))/x^3/exp(5/x^2),x,method=_RETURNVERBOSE)

[Out]

5/4*exp(-(exp(-5/x^2)*x^3+x^2*exp(1)-x^3+ln(x)^2)/x^2)

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maxima [A]  time = 1.13, size = 27, normalized size = 0.90 \begin {gather*} \frac {5}{4} \, e^{\left (-x e^{\left (-\frac {5}{x^{2}}\right )} + x - \frac {\log \relax (x)^{2}}{x^{2}} - e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*exp(5/x^2)*log(x)^2-2*exp(5/x^2)*log(x)+x^3*exp(5/x^2)-x^3-10*x)*exp((-exp(5/x^2)*log(x)^2+(x
^2*log(5)-x^2*exp(1)+x^3)*exp(5/x^2)-x^3)/x^2/exp(5/x^2))/x^3/exp(5/x^2),x, algorithm="maxima")

[Out]

5/4*e^(-x*e^(-5/x^2) + x - log(x)^2/x^2 - e)

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mupad [B]  time = 1.13, size = 29, normalized size = 0.97 \begin {gather*} \frac {5\,{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-\frac {5}{x^2}}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-\frac {{\ln \relax (x)}^2}{x^2}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(exp(-5/x^2)*(exp(5/x^2)*log(x)^2 - exp(5/x^2)*(x^2*log(5) - x^2*exp(1) + x^3) + x^3))/x^2)*exp(-5/
x^2)*(10*x - 2*exp(5/x^2)*log(x)^2 - x^3*exp(5/x^2) + x^3 + 2*exp(5/x^2)*log(x)))/(4*x^3),x)

[Out]

(5*exp(-exp(1))*exp(-x*exp(-5/x^2))*exp(x)*exp(-log(x)^2/x^2))/4

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sympy [A]  time = 0.57, size = 51, normalized size = 1.70 \begin {gather*} \frac {e^{\frac {\left (- x^{3} + \left (x^{3} - e x^{2} + x^{2} \log {\relax (5 )}\right ) e^{\frac {5}{x^{2}}} - e^{\frac {5}{x^{2}}} \log {\relax (x )}^{2}\right ) e^{- \frac {5}{x^{2}}}}{x^{2}}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(2*exp(5/x**2)*ln(x)**2-2*exp(5/x**2)*ln(x)+x**3*exp(5/x**2)-x**3-10*x)*exp((-exp(5/x**2)*ln(x)*
*2+(x**2*ln(5)-x**2*exp(1)+x**3)*exp(5/x**2)-x**3)/x**2/exp(5/x**2))/x**3/exp(5/x**2),x)

[Out]

exp((-x**3 + (x**3 - E*x**2 + x**2*log(5))*exp(5/x**2) - exp(5/x**2)*log(x)**2)*exp(-5/x**2)/x**2)/4

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