3.6.76 \(\int \frac {e^{-2 e^x} (5+(-5+10 e^x x) \log (x)+e^{2 e^x} (9+2 x-1280 x^4) \log ^2(x))}{\log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ x \left (9+x-256 x^4-\frac {5 e^{-2 e^x}}{\log (x)}\right ) \]

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Rubi [F]  time = 1.26, 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^{-2 e^x} \left (5+\left (-5+10 e^x x\right ) \log (x)+e^{2 e^x} \left (9+2 x-1280 x^4\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 + (-5 + 10*E^x*x)*Log[x] + E^(2*E^x)*(9 + 2*x - 1280*x^4)*Log[x]^2)/(E^(2*E^x)*Log[x]^2),x]

[Out]

9*x + x^2 - 256*x^5 + 5*Defer[Int][1/(E^(2*E^x)*Log[x]^2), x] - 5*Defer[Int][1/(E^(2*E^x)*Log[x]), x] + 10*Def
er[Int][(E^(-2*E^x + x)*x)/Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {10 e^{-2 e^x+x} x}{\log (x)}-\frac {e^{-2 e^x} \left (-5+5 \log (x)-9 e^{2 e^x} \log ^2(x)-2 e^{2 e^x} x \log ^2(x)+1280 e^{2 e^x} x^4 \log ^2(x)\right )}{\log ^2(x)}\right ) \, dx\\ &=10 \int \frac {e^{-2 e^x+x} x}{\log (x)} \, dx-\int \frac {e^{-2 e^x} \left (-5+5 \log (x)-9 e^{2 e^x} \log ^2(x)-2 e^{2 e^x} x \log ^2(x)+1280 e^{2 e^x} x^4 \log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=10 \int \frac {e^{-2 e^x+x} x}{\log (x)} \, dx-\int \left (-9-2 x+1280 x^4-\frac {5 e^{-2 e^x}}{\log ^2(x)}+\frac {5 e^{-2 e^x}}{\log (x)}\right ) \, dx\\ &=9 x+x^2-256 x^5+5 \int \frac {e^{-2 e^x}}{\log ^2(x)} \, dx-5 \int \frac {e^{-2 e^x}}{\log (x)} \, dx+10 \int \frac {e^{-2 e^x+x} x}{\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.09, size = 23, normalized size = 1.00 \begin {gather*} x \left (9+x-256 x^4-\frac {5 e^{-2 e^x}}{\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + (-5 + 10*E^x*x)*Log[x] + E^(2*E^x)*(9 + 2*x - 1280*x^4)*Log[x]^2)/(E^(2*E^x)*Log[x]^2),x]

[Out]

x*(9 + x - 256*x^4 - 5/(E^(2*E^x)*Log[x]))

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fricas [A]  time = 0.69, size = 37, normalized size = 1.61 \begin {gather*} -\frac {{\left ({\left (256 \, x^{5} - x^{2} - 9 \, x\right )} e^{\left (2 \, e^{x}\right )} \log \relax (x) + 5 \, x\right )} e^{\left (-2 \, e^{x}\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1280*x^4+2*x+9)*log(x)^2*exp(exp(x))^2+(10*exp(x)*x-5)*log(x)+5)/log(x)^2/exp(exp(x))^2,x, algori
thm="fricas")

[Out]

-((256*x^5 - x^2 - 9*x)*e^(2*e^x)*log(x) + 5*x)*e^(-2*e^x)/log(x)

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giac [A]  time = 0.61, size = 46, normalized size = 2.00 \begin {gather*} -\frac {{\left (256 \, x^{5} e^{x} \log \relax (x) - x^{2} e^{x} \log \relax (x) - 9 \, x e^{x} \log \relax (x) + 5 \, x e^{\left (x - 2 \, e^{x}\right )}\right )} e^{\left (-x\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1280*x^4+2*x+9)*log(x)^2*exp(exp(x))^2+(10*exp(x)*x-5)*log(x)+5)/log(x)^2/exp(exp(x))^2,x, algori
thm="giac")

[Out]

-(256*x^5*e^x*log(x) - x^2*e^x*log(x) - 9*x*e^x*log(x) + 5*x*e^(x - 2*e^x))*e^(-x)/log(x)

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maple [A]  time = 0.08, size = 25, normalized size = 1.09

method	result	size
risch	\(-256 x^{5}+x^{2}+9 x -\frac {5 x \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{\ln \relax (x )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1280*x^4+2*x+9)*ln(x)^2*exp(exp(x))^2+(10*exp(x)*x-5)*ln(x)+5)/ln(x)^2/exp(exp(x))^2,x,method=_RETURNVE
RBOSE)

[Out]

-256*x^5+x^2+9*x-5/ln(x)*x*exp(-2*exp(x))

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maxima [A]  time = 0.57, size = 24, normalized size = 1.04 \begin {gather*} -256 \, x^{5} + x^{2} + 9 \, x - \frac {5 \, x e^{\left (-2 \, e^{x}\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1280*x^4+2*x+9)*log(x)^2*exp(exp(x))^2+(10*exp(x)*x-5)*log(x)+5)/log(x)^2/exp(exp(x))^2,x, algori
thm="maxima")

[Out]

-256*x^5 + x^2 + 9*x - 5*x*e^(-2*e^x)/log(x)

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mupad [B]  time = 0.54, size = 24, normalized size = 1.04 \begin {gather*} 9\,x+x^2-256\,x^5-\frac {5\,x\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*exp(x))*(log(x)*(10*x*exp(x) - 5) + exp(2*exp(x))*log(x)^2*(2*x - 1280*x^4 + 9) + 5))/log(x)^2,x)

[Out]

9*x + x^2 - 256*x^5 - (5*x*exp(-2*exp(x)))/log(x)

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sympy [A]  time = 0.29, size = 24, normalized size = 1.04 \begin {gather*} - 256 x^{5} + x^{2} + 9 x - \frac {5 x e^{- 2 e^{x}}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1280*x**4+2*x+9)*ln(x)**2*exp(exp(x))**2+(10*exp(x)*x-5)*ln(x)+5)/ln(x)**2/exp(exp(x))**2,x)

[Out]

-256*x**5 + x**2 + 9*x - 5*x*exp(-2*exp(x))/log(x)

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