3.47.66 \(\int \frac {e^{-x-\frac {2 e^{-x} (e^{\frac {1}{x}} x^2-x^3)}{\log (x)}} (-5 e^{\frac {1}{x}} x+5 x^2+(-15 x^2+5 x^3+e^{\frac {1}{x}} (-5+10 x-5 x^2)) \log (x)+e^{x+\frac {2 e^{-x} (e^{\frac {1}{x}} x^2-x^3)}{\log (x)}} \log ^2(x))}{\log ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac {5}{2} e^{-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}}+x \]

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Rubi [F]  time = 35.04, 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-\frac {2 e^{-x} \left (e^{\frac {1}{x}} x^2-x^3\right )}{\log (x)}} \left (-5 e^{\frac {1}{x}} x+5 x^2+\left (-15 x^2+5 x^3+e^{\frac {1}{x}} \left (-5+10 x-5 x^2\right )\right ) \log (x)+e^{x+\frac {2 e^{-x} \left (e^{\frac {1}{x}} x^2-x^3\right )}{\log (x)}} \log ^2(x)\right )}{\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x - (2*(E^x^(-1)*x^2 - x^3))/(E^x*Log[x]))*(-5*E^x^(-1)*x + 5*x^2 + (-15*x^2 + 5*x^3 + E^x^(-1)*(-5 +
10*x - 5*x^2))*Log[x] + E^(x + (2*(E^x^(-1)*x^2 - x^3))/(E^x*Log[x]))*Log[x]^2))/Log[x]^2,x]

[Out]

x - 5*Defer[Int][(E^(x^(-1) - x - (2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]))*x)/Log[x]^2, x] + 5*Defer[Int][(E^(-x -
(2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]))*x^2)/Log[x]^2, x] - 5*Defer[Int][E^(x^(-1) - x - (2*(E^x^(-1) - x)*x^2)/
(E^x*Log[x]))/Log[x], x] + 10*Defer[Int][(E^(x^(-1) - x - (2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]))*x)/Log[x], x] -
15*Defer[Int][(E^(-x - (2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]))*x^2)/Log[x], x] - 5*Defer[Int][(E^(x^(-1) - x - (
2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]))*x^2)/Log[x], x] + 5*Defer[Int][(E^(-x - (2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]
))*x^3)/Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {e^{-x} x \left (2 e^{\frac {1}{x}} x-2 x^2+e^x \log (x)\right )}{\log (x)}\right ) \left (-5 e^{\frac {1}{x}} x+5 x^2+\left (-15 x^2+5 x^3+e^{\frac {1}{x}} \left (-5+10 x-5 x^2\right )\right ) \log (x)+e^{x+\frac {2 e^{-x} \left (e^{\frac {1}{x}} x^2-x^3\right )}{\log (x)}} \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=\int \left (\exp \left (x+\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}-\frac {e^{-x} x \left (2 e^{\frac {1}{x}} x-2 x^2+e^x \log (x)\right )}{\log (x)}\right )-\frac {5 \exp \left (-\frac {e^{-x} x \left (2 e^{\frac {1}{x}} x-2 x^2+e^x \log (x)\right )}{\log (x)}\right ) \left (e^{\frac {1}{x}} x-x^2+e^{\frac {1}{x}} \log (x)-2 e^{\frac {1}{x}} x \log (x)+3 x^2 \log (x)+e^{\frac {1}{x}} x^2 \log (x)-x^3 \log (x)\right )}{\log ^2(x)}\right ) \, dx\\ &=-\left (5 \int \frac {\exp \left (-\frac {e^{-x} x \left (2 e^{\frac {1}{x}} x-2 x^2+e^x \log (x)\right )}{\log (x)}\right ) \left (e^{\frac {1}{x}} x-x^2+e^{\frac {1}{x}} \log (x)-2 e^{\frac {1}{x}} x \log (x)+3 x^2 \log (x)+e^{\frac {1}{x}} x^2 \log (x)-x^3 \log (x)\right )}{\log ^2(x)} \, dx\right )+\int \exp \left (x+\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}-\frac {e^{-x} x \left (2 e^{\frac {1}{x}} x-2 x^2+e^x \log (x)\right )}{\log (x)}\right ) \, dx\\ &=-\left (5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} \left (\left (e^{\frac {1}{x}}-x\right ) x+\left (e^{\frac {1}{x}} (-1+x)^2-(-3+x) x^2\right ) \log (x)\right )}{\log ^2(x)} \, dx\right )+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=x-5 \int \left (-\frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2 (1-3 \log (x)+x \log (x))}{\log ^2(x)}+\frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} \left (x+\log (x)-2 x \log (x)+x^2 \log (x)\right )}{\log ^2(x)}\right ) \, dx\\ &=x+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2 (1-3 \log (x)+x \log (x))}{\log ^2(x)} \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} \left (x+\log (x)-2 x \log (x)+x^2 \log (x)\right )}{\log ^2(x)} \, dx\\ &=x+5 \int \left (\frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log ^2(x)}+\frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} (-3+x) x^2}{\log (x)}\right ) \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} \left (x+(-1+x)^2 \log (x)\right )}{\log ^2(x)} \, dx\\ &=x-5 \int \left (\frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x}{\log ^2(x)}+\frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} (-1+x)^2}{\log (x)}\right ) \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log ^2(x)} \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} (-3+x) x^2}{\log (x)} \, dx\\ &=x+5 \int \left (-\frac {3 e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log (x)}+\frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^3}{\log (x)}\right ) \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x}{\log ^2(x)} \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log ^2(x)} \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} (-1+x)^2}{\log (x)} \, dx\\ &=x-5 \int \left (\frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}}}{\log (x)}-\frac {2 e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x}{\log (x)}+\frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log (x)}\right ) \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x}{\log ^2(x)} \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log ^2(x)} \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^3}{\log (x)} \, dx-15 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log (x)} \, dx\\ &=x-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x}{\log ^2(x)} \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log ^2(x)} \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}}}{\log (x)} \, dx-5 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log (x)} \, dx+5 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^3}{\log (x)} \, dx+10 \int \frac {e^{\frac {1}{x}-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x}{\log (x)} \, dx-15 \int \frac {e^{-x-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}} x^2}{\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 31, normalized size = 1.00 \begin {gather*} -\frac {5}{2} e^{-\frac {2 e^{-x} \left (e^{\frac {1}{x}}-x\right ) x^2}{\log (x)}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x - (2*(E^x^(-1)*x^2 - x^3))/(E^x*Log[x]))*(-5*E^x^(-1)*x + 5*x^2 + (-15*x^2 + 5*x^3 + E^x^(-1)
*(-5 + 10*x - 5*x^2))*Log[x] + E^(x + (2*(E^x^(-1)*x^2 - x^3))/(E^x*Log[x]))*Log[x]^2))/Log[x]^2,x]

[Out]

-5/(2*E^((2*(E^x^(-1) - x)*x^2)/(E^x*Log[x]))) + x

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fricas [B]  time = 1.03, size = 82, normalized size = 2.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, x e^{\left (-x - \frac {2 \, x^{3} e^{\left (-x\right )} - 2 \, x^{2} e^{\left (-x + \frac {1}{x}\right )} - x \log \relax (x)}{\log \relax (x)}\right )} - 5\right )} e^{\left (x + \frac {2 \, x^{3} e^{\left (-x\right )} - 2 \, x^{2} e^{\left (-x + \frac {1}{x}\right )} - x \log \relax (x)}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*log(x)^2*exp((x^2*exp(1/x)-x^3)/exp(x)/log(x))^2+((-5*x^2+10*x-5)*exp(1/x)+5*x^3-15*x^2)*log
(x)-5*x*exp(1/x)+5*x^2)/exp(x)/log(x)^2/exp((x^2*exp(1/x)-x^3)/exp(x)/log(x))^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e^{\left (x - \frac {2 \, {\left (x^{3} - x^{2} e^{\frac {1}{x}}\right )} e^{\left (-x\right )}}{\log \relax (x)}\right )} \log \relax (x)^{2} + 5 \, x^{2} - 5 \, x e^{\frac {1}{x}} + 5 \, {\left (x^{3} - 3 \, x^{2} - {\left (x^{2} - 2 \, x + 1\right )} e^{\frac {1}{x}}\right )} \log \relax (x)\right )} e^{\left (-x + \frac {2 \, {\left (x^{3} - x^{2} e^{\frac {1}{x}}\right )} e^{\left (-x\right )}}{\log \relax (x)}\right )}}{\log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*log(x)^2*exp((x^2*exp(1/x)-x^3)/exp(x)/log(x))^2+((-5*x^2+10*x-5)*exp(1/x)+5*x^3-15*x^2)*log
(x)-5*x*exp(1/x)+5*x^2)/exp(x)/log(x)^2/exp((x^2*exp(1/x)-x^3)/exp(x)/log(x))^2,x, algorithm="giac")

[Out]

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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left ({\mathrm e}^{x} \ln \relax (x )^{2} {\mathrm e}^{\frac {2 \left (x^{2} {\mathrm e}^{\frac {1}{x}}-x^{3}\right ) {\mathrm e}^{-x}}{\ln \relax (x )}}+\left (\left (-5 x^{2}+10 x -5\right ) {\mathrm e}^{\frac {1}{x}}+5 x^{3}-15 x^{2}\right ) \ln \relax (x )-5 x \,{\mathrm e}^{\frac {1}{x}}+5 x^{2}\right ) {\mathrm e}^{-x} {\mathrm e}^{-\frac {2 \left (x^{2} {\mathrm e}^{\frac {1}{x}}-x^{3}\right ) {\mathrm e}^{-x}}{\ln \relax (x )}}}{\ln \relax (x )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*ln(x)^2*exp((x^2*exp(1/x)-x^3)/exp(x)/ln(x))^2+((-5*x^2+10*x-5)*exp(1/x)+5*x^3-15*x^2)*ln(x)-5*x*e
xp(1/x)+5*x^2)/exp(x)/ln(x)^2/exp((x^2*exp(1/x)-x^3)/exp(x)/ln(x))^2,x)

[Out]

int((exp(x)*ln(x)^2*exp((x^2*exp(1/x)-x^3)/exp(x)/ln(x))^2+((-5*x^2+10*x-5)*exp(1/x)+5*x^3-15*x^2)*ln(x)-5*x*e
xp(1/x)+5*x^2)/exp(x)/ln(x)^2/exp((x^2*exp(1/x)-x^3)/exp(x)/ln(x))^2,x)

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maxima [A]  time = 0.57, size = 36, normalized size = 1.16 \begin {gather*} x - \frac {5}{2} \, e^{\left (\frac {2 \, x^{3} e^{\left (-x\right )}}{\log \relax (x)} - \frac {2 \, x^{2} e^{\left (-x + \frac {1}{x}\right )}}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*log(x)^2*exp((x^2*exp(1/x)-x^3)/exp(x)/log(x))^2+((-5*x^2+10*x-5)*exp(1/x)+5*x^3-15*x^2)*log
(x)-5*x*exp(1/x)+5*x^2)/exp(x)/log(x)^2/exp((x^2*exp(1/x)-x^3)/exp(x)/log(x))^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.59, size = 36, normalized size = 1.16 \begin {gather*} x-\frac {5\,{\mathrm {e}}^{\frac {2\,x^3\,{\mathrm {e}}^{-x}}{\ln \relax (x)}}\,{\mathrm {e}}^{-\frac {2\,x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{1/x}}{\ln \relax (x)}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*exp(-(2*exp(-x)*(x^2*exp(1/x) - x^3))/log(x))*(5*x*exp(1/x) + log(x)*(exp(1/x)*(5*x^2 - 10*x + 5
) + 15*x^2 - 5*x^3) - 5*x^2 - exp((2*exp(-x)*(x^2*exp(1/x) - x^3))/log(x))*exp(x)*log(x)^2))/log(x)^2,x)

[Out]

x - (5*exp((2*x^3*exp(-x))/log(x))*exp(-(2*x^2*exp(-x)*exp(1/x))/log(x)))/2

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sympy [A]  time = 1.46, size = 27, normalized size = 0.87 \begin {gather*} x - \frac {5 e^{- \frac {2 \left (- x^{3} + x^{2} e^{\frac {1}{x}}\right ) e^{- x}}{\log {\relax (x )}}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*ln(x)**2*exp((x**2*exp(1/x)-x**3)/exp(x)/ln(x))**2+((-5*x**2+10*x-5)*exp(1/x)+5*x**3-15*x**2
)*ln(x)-5*x*exp(1/x)+5*x**2)/exp(x)/ln(x)**2/exp((x**2*exp(1/x)-x**3)/exp(x)/ln(x))**2,x)

[Out]

x - 5*exp(-2*(-x**3 + x**2*exp(1/x))*exp(-x)/log(x))/2

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