3.7.97 \(\int \frac {-3 x+3 x^2-3 \log (x)+(-3 x^2-3 x \log (x)) \log (\log (x))+(3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+(-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+(-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx\)

Optimal. Leaf size=24 \[ \frac {3 x}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \]

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Rubi [F]  time = 3.30, 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+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x + 3*x^2 - 3*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[Log[x]] + (3*x + (-3 - 3*x)*Log[x] + 3*Log[x]^2 + (-3
*x + 3*x*Log[x])*Log[Log[x]])*Log[-3*x + 3*Log[x] + 3*x*Log[Log[x]]])/(-(x^3*Log[x]^2) + x^2*Log[x]^3 + x^3*Lo
g[x]^2*Log[Log[x]] + (-2*x^2*Log[x]^2 + 2*x*Log[x]^3 + 2*x^2*Log[x]^2*Log[Log[x]])*Log[-3*x + 3*Log[x] + 3*x*L
og[Log[x]]] + (-(x*Log[x]^2) + Log[x]^3 + x*Log[x]^2*Log[Log[x]])*Log[-3*x + 3*Log[x] + 3*x*Log[Log[x]]]^2),x]

[Out]

-3*Defer[Int][x/((-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2), x] - 3*Defer[I
nt][x/(Log[x]^2*(-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2), x] - 3*Defer[In
t][1/(Log[x]*(-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2), x] + 3*Defer[Int][
x/(Log[x]*(-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2), x] + 3*Defer[Int][x^2
/(Log[x]*(-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2), x] - 3*Defer[Int][(x*L
og[Log[x]])/(Log[x]*(-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2), x] - 3*Defe
r[Int][(x^2*Log[Log[x]])/(Log[x]*(-x + Log[x] + x*Log[Log[x]])*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])^2)
, x] - 3*Defer[Int][1/(Log[x]^2*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])), x] + 3*Defer[Int][1/(Log[x]*(x
+ Log[3*(Log[x] + x*(-1 + Log[Log[x]]))])), x]

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 24, normalized size = 1.00 \begin {gather*} \frac {3 x}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/(Log[x]*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))]))

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fricas [A]  time = 1.09, size = 28, normalized size = 1.17 \begin {gather*} \frac {3 \, x}{x \log \relax (x) + \log \left (3 \, x \log \left (\log \relax (x)\right ) - 3 \, x + 3 \, \log \relax (x)\right ) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x*log(x)-3*x)*log(log(x))+3*log(x)^2+(-3*x-3)*log(x)+3*x)*log(3*x*log(log(x))+3*log(x)-3*x)+(-3
*x*log(x)-3*x^2)*log(log(x))-3*log(x)+3*x^2-3*x)/((x*log(x)^2*log(log(x))+log(x)^3-x*log(x)^2)*log(3*x*log(log
(x))+3*log(x)-3*x)^2+(2*x^2*log(x)^2*log(log(x))+2*x*log(x)^3-2*x^2*log(x)^2)*log(3*x*log(log(x))+3*log(x)-3*x
)+x^3*log(x)^2*log(log(x))+x^2*log(x)^3-x^3*log(x)^2),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x*log(x)-3*x)*log(log(x))+3*log(x)^2+(-3*x-3)*log(x)+3*x)*log(3*x*log(log(x))+3*log(x)-3*x)+(-3
*x*log(x)-3*x^2)*log(log(x))-3*log(x)+3*x^2-3*x)/((x*log(x)^2*log(log(x))+log(x)^3-x*log(x)^2)*log(3*x*log(log
(x))+3*log(x)-3*x)^2+(2*x^2*log(x)^2*log(log(x))+2*x*log(x)^3-2*x^2*log(x)^2)*log(3*x*log(log(x))+3*log(x)-3*x
)+x^3*log(x)^2*log(log(x))+x^2*log(x)^3-x^3*log(x)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(`
w`),0%%%}Si

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maple [A]  time = 0.05, size = 27, normalized size = 1.12

method	result	size
risch	\(\frac {3 x}{\ln \relax (x ) \left (\ln \left (3 x \ln \left (\ln \relax (x )\right )+3 \ln \relax (x )-3 x \right )+x \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/ln(x)/(ln(3*x*ln(ln(x))+3*ln(x)-3*x)+x)

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maxima [A]  time = 0.84, size = 28, normalized size = 1.17 \begin {gather*} \frac {3 \, x}{{\left (x + \log \relax (3)\right )} \log \relax (x) + \log \left (x \log \left (\log \relax (x)\right ) - x + \log \relax (x)\right ) \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x*log(x)-3*x)*log(log(x))+3*log(x)^2+(-3*x-3)*log(x)+3*x)*log(3*x*log(log(x))+3*log(x)-3*x)+(-3
*x*log(x)-3*x^2)*log(log(x))-3*log(x)+3*x^2-3*x)/((x*log(x)^2*log(log(x))+log(x)^3-x*log(x)^2)*log(3*x*log(log
(x))+3*log(x)-3*x)^2+(2*x^2*log(x)^2*log(log(x))+2*x*log(x)^3-2*x^2*log(x)^2)*log(3*x*log(log(x))+3*log(x)-3*x
)+x^3*log(x)^2*log(log(x))+x^2*log(x)^3-x^3*log(x)^2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {3\,x+3\,\ln \relax (x)+\ln \left (\ln \relax (x)\right )\,\left (3\,x\,\ln \relax (x)+3\,x^2\right )-\ln \left (3\,\ln \relax (x)-3\,x+3\,x\,\ln \left (\ln \relax (x)\right )\right )\,\left (3\,x+3\,{\ln \relax (x)}^2-\ln \relax (x)\,\left (3\,x+3\right )-\ln \left (\ln \relax (x)\right )\,\left (3\,x-3\,x\,\ln \relax (x)\right )\right )-3\,x^2}{\ln \left (3\,\ln \relax (x)-3\,x+3\,x\,\ln \left (\ln \relax (x)\right )\right )\,\left (2\,x\,{\ln \relax (x)}^3-2\,x^2\,{\ln \relax (x)}^2+2\,x^2\,\ln \left (\ln \relax (x)\right )\,{\ln \relax (x)}^2\right )+x^2\,{\ln \relax (x)}^3-x^3\,{\ln \relax (x)}^2+{\ln \left (3\,\ln \relax (x)-3\,x+3\,x\,\ln \left (\ln \relax (x)\right )\right )}^2\,\left ({\ln \relax (x)}^3-x\,{\ln \relax (x)}^2+x\,\ln \left (\ln \relax (x)\right )\,{\ln \relax (x)}^2\right )+x^3\,\ln \left (\ln \relax (x)\right )\,{\ln \relax (x)}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.76, size = 29, normalized size = 1.21 \begin {gather*} \frac {3 x}{x \log {\relax (x )} + \log {\relax (x )} \log {\left (3 x \log {\left (\log {\relax (x )} \right )} - 3 x + 3 \log {\relax (x )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x*ln(x)-3*x)*ln(ln(x))+3*ln(x)**2+(-3*x-3)*ln(x)+3*x)*ln(3*x*ln(ln(x))+3*ln(x)-3*x)+(-3*x*ln(x)
-3*x**2)*ln(ln(x))-3*ln(x)+3*x**2-3*x)/((x*ln(x)**2*ln(ln(x))+ln(x)**3-x*ln(x)**2)*ln(3*x*ln(ln(x))+3*ln(x)-3*
x)**2+(2*x**2*ln(x)**2*ln(ln(x))+2*x*ln(x)**3-2*x**2*ln(x)**2)*ln(3*x*ln(ln(x))+3*ln(x)-3*x)+x**3*ln(x)**2*ln(
ln(x))+x**2*ln(x)**3-x**3*ln(x)**2),x)

[Out]

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

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