3.31.68 \(\int \frac {6+e^{3+x} (-1+x)+(6 x-e^{3+x} x-x^2) \log (\frac {3 x}{-6+e^{3+x}+x}) \log (\log (\frac {3 x}{-6+e^{3+x}+x}))}{(-6 x^2+e^{3+x} x^2+x^3) \log (\frac {3 x}{-6+e^{3+x}+x}) \log (\log (\frac {3 x}{-6+e^{3+x}+x}))+(-6 x+e^{3+x} x+x^2) \log (\frac {3 x}{-6+e^{3+x}+x}) \log (\log (\frac {3 x}{-6+e^{3+x}+x})) \log (\log (\log (\frac {3 x}{-6+e^{3+x}+x})))} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\frac {2}{x+\log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )}\right ) \]

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Rubi [A]  time = 1.27, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, integrand size = 170, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6688, 6684} \begin {gather*} -\log \left (x+\log \left (\log \left (\log \left (-\frac {3 x}{-x-e^{x+3}+6}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + E^(3 + x)*(-1 + x) + (6*x - E^(3 + x)*x - x^2)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^
(3 + x) + x)]])/((-6*x^2 + E^(3 + x)*x^2 + x^3)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x)
+ x)]] + (-6*x + E^(3 + x)*x + x^2)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x) + x)]]*Log[L
og[Log[(3*x)/(-6 + E^(3 + x) + x)]]]),x]

[Out]

-Log[x + Log[Log[Log[(-3*x)/(6 - E^(3 + x) - x)]]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 21, normalized size = 0.91 \begin {gather*} -\log \left (x+\log \left (\log \left (\log \left (\frac {3 x}{-6+e^{3+x}+x}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + E^(3 + x)*(-1 + x) + (6*x - E^(3 + x)*x - x^2)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-
6 + E^(3 + x) + x)]])/((-6*x^2 + E^(3 + x)*x^2 + x^3)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3
+ x) + x)]] + (-6*x + E^(3 + x)*x + x^2)*Log[(3*x)/(-6 + E^(3 + x) + x)]*Log[Log[(3*x)/(-6 + E^(3 + x) + x)]]
*Log[Log[Log[(3*x)/(-6 + E^(3 + x) + x)]]]),x]

[Out]

-Log[x + Log[Log[Log[(3*x)/(-6 + E^(3 + x) + x)]]]]

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fricas [A]  time = 0.84, size = 20, normalized size = 0.87 \begin {gather*} -\log \left (x + \log \left (\log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + log(log(log(3*x/(x + e^(x + 3) - 6)))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{2} + x e^{\left (x + 3\right )} - 6 \, x\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right ) - {\left (x - 1\right )} e^{\left (x + 3\right )} - 6}{{\left (x^{2} + x e^{\left (x + 3\right )} - 6 \, x\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right ) \log \left (\log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )\right ) + {\left (x^{3} + x^{2} e^{\left (x + 3\right )} - 6 \, x^{2}\right )} \log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right ) \log \left (\log \left (\frac {3 \, x}{x + e^{\left (x + 3\right )} - 6}\right )\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(3+x)*x-x^2+6*x)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp(3+x)+x-6)))+(x-1)*exp(3+x)+6)/((exp(
3+x)*x+x^2-6*x)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp(3+x)+x-6)))*log(log(log(3*x/(exp(3+x)+x-6))))+(x^2*ex
p(3+x)+x^3-6*x^2)*log(3*x/(exp(3+x)+x-6))*log(log(3*x/(exp(3+x)+x-6)))),x, algorithm="giac")

[Out]

integrate(-((x^2 + x*e^(x + 3) - 6*x)*log(3*x/(x + e^(x + 3) - 6))*log(log(3*x/(x + e^(x + 3) - 6))) - (x - 1)
*e^(x + 3) - 6)/((x^2 + x*e^(x + 3) - 6*x)*log(3*x/(x + e^(x + 3) - 6))*log(log(3*x/(x + e^(x + 3) - 6)))*log(
log(log(3*x/(x + e^(x + 3) - 6)))) + (x^3 + x^2*e^(x + 3) - 6*x^2)*log(3*x/(x + e^(x + 3) - 6))*log(log(3*x/(x
+ e^(x + 3) - 6)))), x)

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maple [C]  time = 0.36, size = 93, normalized size = 4.04

method	result	size
risch	\(-\ln \left (x +\ln \left (\ln \left (\ln \relax (3)+\ln \relax (x )-\ln \left ({\mathrm e}^{3+x}+x -6\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right ) \left (-\mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (\frac {i x}{{\mathrm e}^{3+x}+x -6}\right )+\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{3+x}+x -6}\right )\right )}{2}\right )\right )\right )\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x+ln(ln(ln(3)+ln(x)-ln(exp(3+x)+x-6)-1/2*I*Pi*csgn(I*x/(exp(3+x)+x-6))*(-csgn(I*x/(exp(3+x)+x-6))+csgn(I*x
))*(-csgn(I*x/(exp(3+x)+x-6))+csgn(I/(exp(3+x)+x-6))))))

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maxima [A]  time = 1.21, size = 22, normalized size = 0.96 \begin {gather*} -\log \left (x + \log \left (\log \left (\log \relax (3) - \log \left (x + e^{\left (x + 3\right )} - 6\right ) + \log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + log(log(log(3) - log(x + e^(x + 3) - 6) + log(x))))

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mupad [B]  time = 4.19, size = 20, normalized size = 0.87 \begin {gather*} -\ln \left (x+\ln \left (\ln \left (\ln \left (\frac {3\,x}{x+{\mathrm {e}}^{x+3}-6}\right )\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 3)*(x - 1) - log(log((3*x)/(x + exp(x + 3) - 6)))*log((3*x)/(x + exp(x + 3) - 6))*(x*exp(x + 3) -
6*x + x^2) + 6)/(log(log((3*x)/(x + exp(x + 3) - 6)))*log((3*x)/(x + exp(x + 3) - 6))*(x^2*exp(x + 3) - 6*x^2
+ x^3) + log(log((3*x)/(x + exp(x + 3) - 6)))*log(log(log((3*x)/(x + exp(x + 3) - 6))))*log((3*x)/(x + exp(x
+ 3) - 6))*(x*exp(x + 3) - 6*x + x^2)),x)

[Out]

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

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

[Out]

Timed out

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