3.100.41 \(\int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} (6 x+2 x^2+6 x \log (x))+e^{e^x} (-3+3 e^x x) \log (x) \log (\frac {\log (x)}{2})+((6 x^2+2 x^3) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} ((6+2 x) \log (x)+3 \log ^2(x))+e^{e^x} ((12 x+4 x^2) \log (x)+6 x \log ^2(x))) \log (\frac {\log (x)}{2}) \log (\log (\frac {\log (x)}{2}))}{(3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)) \log (\frac {\log (x)}{2})} \, dx\)

Optimal. Leaf size=32 \[ -\frac {x}{e^{e^x}+x}+x \left (1+\frac {x}{3}+\log (x)\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \]

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Rubi [F]  time = 3.27, 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 {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^2 + x^3 + 3*x^2*Log[x] + E^(2*E^x)*(3 + x + 3*Log[x]) + E^E^x*(6*x + 2*x^2 + 6*x*Log[x]) + E^E^x*(-3
+ 3*E^x*x)*Log[x]*Log[Log[x]/2] + ((6*x^2 + 2*x^3)*Log[x] + 3*x^2*Log[x]^2 + E^(2*E^x)*((6 + 2*x)*Log[x] + 3*L
og[x]^2) + E^E^x*((12*x + 4*x^2)*Log[x] + 6*x*Log[x]^2))*Log[Log[x]/2]*Log[Log[Log[x]/2]])/((3*E^(2*E^x)*Log[x
] + 6*E^E^x*x*Log[x] + 3*x^2*Log[x])*Log[Log[x]/2]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.52, size = 34, normalized size = 1.06 \begin {gather*} \frac {1}{3} \left (-\frac {3 x}{e^{e^x}+x}+x (3+x+3 \log (x)) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + x^3 + 3*x^2*Log[x] + E^(2*E^x)*(3 + x + 3*Log[x]) + E^E^x*(6*x + 2*x^2 + 6*x*Log[x]) + E^E^
x*(-3 + 3*E^x*x)*Log[x]*Log[Log[x]/2] + ((6*x^2 + 2*x^3)*Log[x] + 3*x^2*Log[x]^2 + E^(2*E^x)*((6 + 2*x)*Log[x]
+ 3*Log[x]^2) + E^E^x*((12*x + 4*x^2)*Log[x] + 6*x*Log[x]^2))*Log[Log[x]/2]*Log[Log[Log[x]/2]])/((3*E^(2*E^x)
*Log[x] + 6*E^E^x*x*Log[x] + 3*x^2*Log[x])*Log[Log[x]/2]),x]

[Out]

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

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fricas [A]  time = 1.35, size = 52, normalized size = 1.62 \begin {gather*} \frac {{\left (x^{3} + 3 \, x^{2} \log \relax (x) + 3 \, x^{2} + {\left (x^{2} + 3 \, x \log \relax (x) + 3 \, x\right )} e^{\left (e^{x}\right )}\right )} \log \left (\log \left (\frac {1}{2} \, \log \relax (x)\right )\right ) - 3 \, x}{3 \, {\left (x + e^{\left (e^{x}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2+12*x)*log(x))*exp(exp(x))+3*x^2*log
(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*log(x))*log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log
(x))+(3*log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*log(x)+x^3+3*x^2)/(3*log(x)*exp(exp
(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log(x))/log(1/2*log(x)),x, algorithm="fricas")

[Out]

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

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giac [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((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2+12*x)*log(x))*exp(exp(x))+3*x^2*log
(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*log(x))*log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log
(x))+(3*log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*log(x)+x^3+3*x^2)/(3*log(x)*exp(exp
(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log(x))/log(1/2*log(x)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.14, size = 30, normalized size = 0.94

method	result	size
risch	\(\left (x \ln \relax (x )+\frac {x^{2}}{3}+x \right ) \ln \left (\ln \left (\frac {\ln \relax (x )}{2}\right )\right )-\frac {x}{x +{\mathrm e}^{{\mathrm e}^{x}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*ln(x)^2+(2*x+6)*ln(x))*exp(exp(x))^2+(6*x*ln(x)^2+(4*x^2+12*x)*ln(x))*exp(exp(x))+3*x^2*ln(x)^2+(2*x^
3+6*x^2)*ln(x))*ln(1/2*ln(x))*ln(ln(1/2*ln(x)))+(3*exp(x)*x-3)*ln(x)*exp(exp(x))*ln(1/2*ln(x))+(3*ln(x)+3+x)*e
xp(exp(x))^2+(6*x*ln(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*ln(x)+x^3+3*x^2)/(3*ln(x)*exp(exp(x))^2+6*x*ln(x)*exp(exp
(x))+3*x^2*ln(x))/ln(1/2*ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.52, size = 55, normalized size = 1.72 \begin {gather*} \frac {{\left (x^{3} + 3 \, x^{2} \log \relax (x) + 3 \, x^{2} + {\left (x^{2} + 3 \, x \log \relax (x) + 3 \, x\right )} e^{\left (e^{x}\right )}\right )} \log \left (-\log \relax (2) + \log \left (\log \relax (x)\right )\right ) - 3 \, x}{3 \, {\left (x + e^{\left (e^{x}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2+12*x)*log(x))*exp(exp(x))+3*x^2*log
(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*log(x))*log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log
(x))+(3*log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*log(x)+x^3+3*x^2)/(3*log(x)*exp(exp
(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log(x))/log(1/2*log(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.17, size = 40, normalized size = 1.25 \begin {gather*} \ln \left (\ln \left (\frac {\ln \relax (x)}{2}\right )\right )\,\left (\frac {x^3+6\,x^2}{3\,x}-x+x\,\ln \relax (x)\right )-\frac {x}{x+{\mathrm {e}}^{{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*log(x) + exp(2*exp(x))*(x + 3*log(x) + 3) + exp(exp(x))*(6*x + 6*x*log(x) + 2*x^2) + 3*x^2 + x^3 +
log(log(log(x)/2))*log(log(x)/2)*(log(x)*(6*x^2 + 2*x^3) + 3*x^2*log(x)^2 + exp(2*exp(x))*(3*log(x)^2 + log(x)
*(2*x + 6)) + exp(exp(x))*(6*x*log(x)^2 + log(x)*(12*x + 4*x^2))) + exp(exp(x))*log(log(x)/2)*log(x)*(3*x*exp(
x) - 3))/(log(log(x)/2)*(3*x^2*log(x) + 3*exp(2*exp(x))*log(x) + 6*x*exp(exp(x))*log(x))),x)

[Out]

log(log(log(x)/2))*((6*x^2 + x^3)/(3*x) - x + x*log(x)) - x/(x + exp(exp(x)))

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sympy [A]  time = 2.21, size = 27, normalized size = 0.84 \begin {gather*} - \frac {x}{x + e^{e^{x}}} + \left (\frac {x^{2}}{3} + x \log {\relax (x )} + x\right ) \log {\left (\log {\left (\frac {\log {\relax (x )}}{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*ln(x)**2+(2*x+6)*ln(x))*exp(exp(x))**2+(6*x*ln(x)**2+(4*x**2+12*x)*ln(x))*exp(exp(x))+3*x**2*ln
(x)**2+(2*x**3+6*x**2)*ln(x))*ln(1/2*ln(x))*ln(ln(1/2*ln(x)))+(3*exp(x)*x-3)*ln(x)*exp(exp(x))*ln(1/2*ln(x))+(
3*ln(x)+3+x)*exp(exp(x))**2+(6*x*ln(x)+2*x**2+6*x)*exp(exp(x))+3*x**2*ln(x)+x**3+3*x**2)/(3*ln(x)*exp(exp(x))*
*2+6*x*ln(x)*exp(exp(x))+3*x**2*ln(x))/ln(1/2*ln(x)),x)

[Out]

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

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