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

Optimal. Leaf size=23 \[ \log \left ((-x+\log (x))^2 \log ^4\left (\log \left (e^x+x^2+\log (x)\right )\right )\right ) \]

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Rubi [F]  time = 9.05, 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 {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

2*Log[x - Log[x]] + 4*Defer[Int][1/(Log[E^x + x^2 + Log[x]]*Log[Log[E^x + x^2 + Log[x]]]), x] + 4*Defer[Int][1
/(x*(E^x + x^2 + Log[x])*Log[E^x + x^2 + Log[x]]*Log[Log[E^x + x^2 + Log[x]]]), x] + 8*Defer[Int][x/((E^x + x^
2 + Log[x])*Log[E^x + x^2 + Log[x]]*Log[Log[E^x + x^2 + Log[x]]]), x] - 4*Defer[Int][x^2/((E^x + x^2 + Log[x])
*Log[E^x + x^2 + Log[x]]*Log[Log[E^x + x^2 + Log[x]]]), x] - 4*Defer[Int][Log[x]/((E^x + x^2 + Log[x])*Log[E^x
+ x^2 + Log[x]]*Log[Log[E^x + x^2 + Log[x]]]), x]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 24, normalized size = 1.04 \begin {gather*} 2 \log (x-\log (x))+4 \log \left (\log \left (\log \left (e^x+x^2+\log (x)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[x - Log[x]] + 4*Log[Log[Log[E^x + x^2 + Log[x]]]]

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fricas [A]  time = 0.91, size = 23, normalized size = 1.00 \begin {gather*} 2 \, \log \left (-x + \log \relax (x)\right ) + 4 \, \log \left (\log \left (\log \left (x^{2} + e^{x} + \log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.51, size = 23, normalized size = 1.00 \begin {gather*} 2 \, \log \left (-x + \log \relax (x)\right ) + 4 \, \log \left (\log \left (\log \left (x^{2} + e^{x} + \log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 24, normalized size = 1.04

method	result	size
risch	\(2 \ln \left (\ln \relax (x )-x \right )+4 \ln \left (\ln \left (\ln \left (\ln \relax (x )+x^{2}+{\mathrm e}^{x}\right )\right )\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(ln(x)-x)+4*ln(ln(ln(ln(x)+x^2+exp(x))))

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maxima [A]  time = 0.52, size = 23, normalized size = 1.00 \begin {gather*} 2 \, \log \left (-x + \log \relax (x)\right ) + 4 \, \log \left (\log \left (\log \left (x^{2} + e^{x} + \log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.71, size = 23, normalized size = 1.00 \begin {gather*} 4\,\ln \left (\ln \left (\ln \left ({\mathrm {e}}^x+\ln \relax (x)+x^2\right )\right )\right )+2\,\ln \left (\ln \relax (x)-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*log(log(log(exp(x) + log(x) + x^2))) + 2*log(log(x) - x)

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sympy [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((((-2*x+2)*ln(x)+(-2*x+2)*exp(x)-2*x**3+2*x**2)*ln(ln(x)+x**2+exp(x))*ln(ln(ln(x)+x**2+exp(x)))+(4*e
xp(x)*x+8*x**2+4)*ln(x)-4*exp(x)*x**2-8*x**3-4*x)/(x*ln(x)**2+(exp(x)*x+x**3-x**2)*ln(x)-exp(x)*x**2-x**4)/ln(
ln(x)+x**2+exp(x))/ln(ln(ln(x)+x**2+exp(x))),x)

[Out]

Timed out

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