3.20.27 \(\int \frac {e^{-3+e^{e^x}} (e^x (-4 x-4 x^2) \log (4)+e^{e^x} (8 e^x x \log (4)-4 e^{2 x} x^2 \log (4)))+e^{-6+2 e^{e^x}} (-8 \log (4)+4 e^x x \log (4)+e^x (-4 x-4 x^2) \log (4) \log (x))}{x+2 e^{-3+e^{e^x}} x \log (x)+e^{-6+2 e^{e^x}} x \log ^2(x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {4 \left (2-e^x x\right ) \log (4)}{e^{3-e^{e^x}}+\log (x)} \]

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Rubi [F]  time = 10.59, 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^{-3+e^{e^x}} \left (e^x \left (-4 x-4 x^2\right ) \log (4)+e^{e^x} \left (8 e^x x \log (4)-4 e^{2 x} x^2 \log (4)\right )\right )+e^{-6+2 e^{e^x}} \left (-8 \log (4)+4 e^x x \log (4)+e^x \left (-4 x-4 x^2\right ) \log (4) \log (x)\right )}{x+2 e^{-3+e^{e^x}} x \log (x)+e^{-6+2 e^{e^x}} x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-3 + E^E^x)*(E^x*(-4*x - 4*x^2)*Log[4] + E^E^x*(8*E^x*x*Log[4] - 4*E^(2*x)*x^2*Log[4])) + E^(-6 + 2*E^
E^x)*(-8*Log[4] + 4*E^x*x*Log[4] + E^x*(-4*x - 4*x^2)*Log[4]*Log[x]))/(x + 2*E^(-3 + E^E^x)*x*Log[x] + E^(-6 +
2*E^E^x)*x*Log[x]^2),x]

[Out]

8*E^6*Log[4]*Defer[Int][E^(-3 + E^E^x + E^x + x)/(E^3 + E^E^E^x*Log[x])^2, x] - 8*E^6*Log[4]*Defer[Int][E^(2*(
-3 + E^E^x))/(x*(E^3 + E^E^E^x*Log[x])^2), x] - 4*E^6*Log[4]*Defer[Int][(E^(-3 + E^E^x + E^x + 2*x)*x)/(E^3 +
E^E^E^x*Log[x])^2, x] - 4*E^6*Log[4]*Defer[Int][E^(-3 + E^E^x + x)/(Log[x]*(E^3 + E^E^E^x*Log[x])^2), x] - 4*E
^6*Log[4]*Defer[Int][E^(-6 + E^E^x + x)/(E^3 + E^E^E^x*Log[x]), x] - 4*E^6*Log[4]*Defer[Int][(E^(-6 + E^E^x +
x)*x)/(E^3 + E^E^E^x*Log[x]), x] + 4*E^6*Log[4]*Defer[Int][E^(-6 + E^E^x + x)/(Log[x]*(E^3 + E^E^E^x*Log[x])),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^6 \left (e^{-3+e^{e^x}} \left (e^x \left (-4 x-4 x^2\right ) \log (4)+e^{e^x} \left (8 e^x x \log (4)-4 e^{2 x} x^2 \log (4)\right )\right )+e^{-6+2 e^{e^x}} \left (-8 \log (4)+4 e^x x \log (4)+e^x \left (-4 x-4 x^2\right ) \log (4) \log (x)\right )\right )}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ &=e^6 \int \frac {e^{-3+e^{e^x}} \left (e^x \left (-4 x-4 x^2\right ) \log (4)+e^{e^x} \left (8 e^x x \log (4)-4 e^{2 x} x^2 \log (4)\right )\right )+e^{-6+2 e^{e^x}} \left (-8 \log (4)+4 e^x x \log (4)+e^x \left (-4 x-4 x^2\right ) \log (4) \log (x)\right )}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ &=e^6 \int \left (-\frac {8 e^{2 \left (-3+e^{e^x}\right )} \log (4)}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2}-\frac {4 e^{-3+e^{e^x}+e^x+2 x} x \log (4)}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2}-\frac {4 e^{-6+e^{e^x}+x} \log (4) \left (e^3-e^{e^{e^x}}-2 e^{3+e^x}+e^3 x+e^{e^{e^x}} \log (x)+e^{e^{e^x}} x \log (x)\right )}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2}\right ) \, dx\\ &=-\left (\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+e^x+2 x} x}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\right )-\left (4 e^6 \log (4)\right ) \int \frac {e^{-6+e^{e^x}+x} \left (e^3-e^{e^{e^x}}-2 e^{3+e^x}+e^3 x+e^{e^{e^x}} \log (x)+e^{e^{e^x}} x \log (x)\right )}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx-\left (8 e^6 \log (4)\right ) \int \frac {e^{2 \left (-3+e^{e^x}\right )}}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ &=-\left (\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+e^x+2 x} x}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\right )-\left (4 e^6 \log (4)\right ) \int \left (-\frac {e^{-3+e^{e^x}+x} \left (-1+2 e^{e^x} \log (x)\right )}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )^2}+\frac {e^{-6+e^{e^x}+x} (-1+\log (x)+x \log (x))}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )}\right ) \, dx-\left (8 e^6 \log (4)\right ) \int \frac {e^{2 \left (-3+e^{e^x}\right )}}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ &=-\left (\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+e^x+2 x} x}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\right )+\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+x} \left (-1+2 e^{e^x} \log (x)\right )}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx-\left (4 e^6 \log (4)\right ) \int \frac {e^{-6+e^{e^x}+x} (-1+\log (x)+x \log (x))}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )} \, dx-\left (8 e^6 \log (4)\right ) \int \frac {e^{2 \left (-3+e^{e^x}\right )}}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ &=-\left (\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+e^x+2 x} x}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\right )+\left (4 e^6 \log (4)\right ) \int \left (\frac {2 e^{-3+e^{e^x}+e^x+x}}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2}-\frac {e^{-3+e^{e^x}+x}}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )^2}\right ) \, dx-\left (4 e^6 \log (4)\right ) \int \left (\frac {e^{-6+e^{e^x}+x}}{e^3+e^{e^{e^x}} \log (x)}+\frac {e^{-6+e^{e^x}+x} x}{e^3+e^{e^{e^x}} \log (x)}-\frac {e^{-6+e^{e^x}+x}}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )}\right ) \, dx-\left (8 e^6 \log (4)\right ) \int \frac {e^{2 \left (-3+e^{e^x}\right )}}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ &=-\left (\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+e^x+2 x} x}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\right )-\left (4 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+x}}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx-\left (4 e^6 \log (4)\right ) \int \frac {e^{-6+e^{e^x}+x}}{e^3+e^{e^{e^x}} \log (x)} \, dx-\left (4 e^6 \log (4)\right ) \int \frac {e^{-6+e^{e^x}+x} x}{e^3+e^{e^{e^x}} \log (x)} \, dx+\left (4 e^6 \log (4)\right ) \int \frac {e^{-6+e^{e^x}+x}}{\log (x) \left (e^3+e^{e^{e^x}} \log (x)\right )} \, dx+\left (8 e^6 \log (4)\right ) \int \frac {e^{-3+e^{e^x}+e^x+x}}{\left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx-\left (8 e^6 \log (4)\right ) \int \frac {e^{2 \left (-3+e^{e^x}\right )}}{x \left (e^3+e^{e^{e^x}} \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 34, normalized size = 1.21 \begin {gather*} -\frac {4 e^{e^{e^x}} \left (-2+e^x x\right ) \log (4)}{e^3+e^{e^{e^x}} \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-3 + E^E^x)*(E^x*(-4*x - 4*x^2)*Log[4] + E^E^x*(8*E^x*x*Log[4] - 4*E^(2*x)*x^2*Log[4])) + E^(-6
+ 2*E^E^x)*(-8*Log[4] + 4*E^x*x*Log[4] + E^x*(-4*x - 4*x^2)*Log[4]*Log[x]))/(x + 2*E^(-3 + E^E^x)*x*Log[x] + E
^(-6 + 2*E^E^x)*x*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.78, size = 32, normalized size = 1.14 \begin {gather*} -\frac {8 \, {\left (x e^{x} \log \relax (2) - 2 \, \log \relax (2)\right )} e^{\left (e^{\left (e^{x}\right )} - 3\right )}}{e^{\left (e^{\left (e^{x}\right )} - 3\right )} \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-4*x^2-4*x)*log(2)*exp(x)*log(x)+8*x*log(2)*exp(x)-16*log(2))*exp(-3+exp(exp(x)))^2+((-8*x^2*lo
g(2)*exp(x)^2+16*x*log(2)*exp(x))*exp(exp(x))+2*(-4*x^2-4*x)*log(2)*exp(x))*exp(-3+exp(exp(x))))/(x*log(x)^2*e
xp(-3+exp(exp(x)))^2+2*x*log(x)*exp(-3+exp(exp(x)))+x),x, algorithm="fricas")

[Out]

-8*(x*e^x*log(2) - 2*log(2))*e^(e^(e^x) - 3)/(e^(e^(e^x) - 3)*log(x) + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left ({\left ({\left (x^{2} + x\right )} e^{x} \log \relax (2) \log \relax (x) - x e^{x} \log \relax (2) + 2 \, \log \relax (2)\right )} e^{\left (2 \, e^{\left (e^{x}\right )} - 6\right )} + {\left ({\left (x^{2} + x\right )} e^{x} \log \relax (2) + {\left (x^{2} e^{\left (2 \, x\right )} \log \relax (2) - 2 \, x e^{x} \log \relax (2)\right )} e^{\left (e^{x}\right )}\right )} e^{\left (e^{\left (e^{x}\right )} - 3\right )}\right )}}{x e^{\left (2 \, e^{\left (e^{x}\right )} - 6\right )} \log \relax (x)^{2} + 2 \, x e^{\left (e^{\left (e^{x}\right )} - 3\right )} \log \relax (x) + x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-4*x^2-4*x)*log(2)*exp(x)*log(x)+8*x*log(2)*exp(x)-16*log(2))*exp(-3+exp(exp(x)))^2+((-8*x^2*lo
g(2)*exp(x)^2+16*x*log(2)*exp(x))*exp(exp(x))+2*(-4*x^2-4*x)*log(2)*exp(x))*exp(-3+exp(exp(x))))/(x*log(x)^2*e
xp(-3+exp(exp(x)))^2+2*x*log(x)*exp(-3+exp(exp(x)))+x),x, algorithm="giac")

[Out]

integrate(-8*(((x^2 + x)*e^x*log(2)*log(x) - x*e^x*log(2) + 2*log(2))*e^(2*e^(e^x) - 6) + ((x^2 + x)*e^x*log(2
) + (x^2*e^(2*x)*log(2) - 2*x*e^x*log(2))*e^(e^x))*e^(e^(e^x) - 3))/(x*e^(2*e^(e^x) - 6)*log(x)^2 + 2*x*e^(e^(
e^x) - 3)*log(x) + x), x)

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maple [A]  time = 0.11, size = 43, normalized size = 1.54

method	result	size
risch	\(-\frac {8 \left ({\mathrm e}^{x} x -2\right ) \ln \relax (2)}{\ln \relax (x )}+\frac {8 \left ({\mathrm e}^{x} x -2\right ) \ln \relax (2)}{\ln \relax (x ) \left (\ln \relax (x ) {\mathrm e}^{-3+{\mathrm e}^{{\mathrm e}^{x}}}+1\right )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*(-4*x^2-4*x)*ln(2)*exp(x)*ln(x)+8*x*ln(2)*exp(x)-16*ln(2))*exp(-3+exp(exp(x)))^2+((-8*x^2*ln(2)*exp(x)
^2+16*x*ln(2)*exp(x))*exp(exp(x))+2*(-4*x^2-4*x)*ln(2)*exp(x))*exp(-3+exp(exp(x))))/(x*ln(x)^2*exp(-3+exp(exp(
x)))^2+2*x*ln(x)*exp(-3+exp(exp(x)))+x),x,method=_RETURNVERBOSE)

[Out]

-8*(exp(x)*x-2)*ln(2)/ln(x)+8*(exp(x)*x-2)*ln(2)/ln(x)/(ln(x)*exp(-3+exp(exp(x)))+1)

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maxima [A]  time = 0.76, size = 29, normalized size = 1.04 \begin {gather*} -\frac {8 \, {\left (x e^{x} \log \relax (2) - 2 \, \log \relax (2)\right )} e^{\left (e^{\left (e^{x}\right )}\right )}}{e^{\left (e^{\left (e^{x}\right )}\right )} \log \relax (x) + e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-4*x^2-4*x)*log(2)*exp(x)*log(x)+8*x*log(2)*exp(x)-16*log(2))*exp(-3+exp(exp(x)))^2+((-8*x^2*lo
g(2)*exp(x)^2+16*x*log(2)*exp(x))*exp(exp(x))+2*(-4*x^2-4*x)*log(2)*exp(x))*exp(-3+exp(exp(x))))/(x*log(x)^2*e
xp(-3+exp(exp(x)))^2+2*x*log(x)*exp(-3+exp(exp(x)))+x),x, algorithm="maxima")

[Out]

-8*(x*e^x*log(2) - 2*log(2))*e^(e^(e^x))/(e^(e^(e^x))*log(x) + e^3)

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mupad [B]  time = 1.56, size = 109, normalized size = 3.89 \begin {gather*} \frac {8\,\ln \relax (2)\,\left (2\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}-3}+x^2\,{\mathrm {e}}^{2\,x+{\mathrm {e}}^x}-x^2\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,{\mathrm {e}}^x-x\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}-3}\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}-3}\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\ln \relax (x)-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}-3}\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,{\mathrm {e}}^x\,\ln \relax (x)\right )}{\left (x\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\ln \relax (x)+1\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}-3}\,\ln \relax (x)+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*exp(exp(x)) - 6)*(16*log(2) - 8*x*exp(x)*log(2) + 2*exp(x)*log(2)*log(x)*(4*x + 4*x^2)) - exp(exp(
exp(x)) - 3)*(exp(exp(x))*(16*x*exp(x)*log(2) - 8*x^2*exp(2*x)*log(2)) - 2*exp(x)*log(2)*(4*x + 4*x^2)))/(x +
x*exp(2*exp(exp(x)) - 6)*log(x)^2 + 2*x*exp(exp(exp(x)) - 3)*log(x)),x)

[Out]

(8*log(2)*(2*exp(exp(exp(x)) - 3) + x^2*exp(2*x + exp(x)) - x^2*exp(x + exp(x))*exp(x) - x*exp(exp(exp(x)) - 3
)*exp(x) + 2*x*exp(exp(exp(x)) - 3)*exp(x + exp(x))*log(x) - x^2*exp(exp(exp(x)) - 3)*exp(x + exp(x))*exp(x)*l
og(x)))/((x*exp(x + exp(x))*log(x) + 1)*(exp(exp(exp(x)) - 3)*log(x) + 1))

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sympy [B]  time = 0.56, size = 53, normalized size = 1.89 \begin {gather*} - \frac {8 x e^{x} \log {\relax (2 )}}{\log {\relax (x )}} + \frac {16 \log {\relax (2 )}}{\log {\relax (x )}} + \frac {8 x e^{x} \log {\relax (2 )} - 16 \log {\relax (2 )}}{e^{e^{e^{x}} - 3} \log {\relax (x )}^{2} + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(-4*x**2-4*x)*ln(2)*exp(x)*ln(x)+8*x*ln(2)*exp(x)-16*ln(2))*exp(-3+exp(exp(x)))**2+((-8*x**2*ln(
2)*exp(x)**2+16*x*ln(2)*exp(x))*exp(exp(x))+2*(-4*x**2-4*x)*ln(2)*exp(x))*exp(-3+exp(exp(x))))/(x*ln(x)**2*exp
(-3+exp(exp(x)))**2+2*x*ln(x)*exp(-3+exp(exp(x)))+x),x)

[Out]

-8*x*exp(x)*log(2)/log(x) + 16*log(2)/log(x) + (8*x*exp(x)*log(2) - 16*log(2))/(exp(exp(exp(x)) - 3)*log(x)**2
+ log(x))

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