3.57.63 \(\int \frac {-3 x^2+3 x^3 \log (x)+(-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{(-x^3 \log (x)+x^2 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))} \, dx\)

Optimal. Leaf size=23 \[ 3 \left (-\frac {4+x}{x}+\frac {x}{\log (-x+\log (\log (x)))}\right ) \]

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Rubi [F]  time = 1.36, 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+3 x^3 \log (x)+\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))+(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{\left (-x^3 \log (x)+x^2 \log (x) \log (\log (x))\right ) \log ^2(-x+\log (\log (x)))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x^2 + 3*x^3*Log[x] + (-3*x^3*Log[x] + 3*x^2*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x]]] + (-12*x*Log[x]
+ 12*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x]]]^2)/((-(x^3*Log[x]) + x^2*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x
]]]^2),x]

[Out]

-12/x - 3*Defer[Int][x/((x - Log[Log[x]])*Log[-x + Log[Log[x]]]^2), x] + 3*Defer[Int][1/(Log[x]*(x - Log[Log[x
]])*Log[-x + Log[Log[x]]]^2), x] + 3*Defer[Int][Log[-x + Log[Log[x]]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x^2-3 x^3 \log (x)-\left (-3 x^3 \log (x)+3 x^2 \log (x) \log (\log (x))\right ) \log (-x+\log (\log (x)))-(-12 x \log (x)+12 \log (x) \log (\log (x))) \log ^2(-x+\log (\log (x)))}{x^2 \log (x) (x-\log (\log (x))) \log ^2(-x+\log (\log (x)))} \, dx\\ &=\int \left (\frac {12}{x^2}-\frac {3 (-1+x \log (x))}{\log (x) (x-\log (\log (x))) \log ^2(-x+\log (\log (x)))}+\frac {3}{\log (-x+\log (\log (x)))}\right ) \, dx\\ &=-\frac {12}{x}-3 \int \frac {-1+x \log (x)}{\log (x) (x-\log (\log (x))) \log ^2(-x+\log (\log (x)))} \, dx+3 \int \frac {1}{\log (-x+\log (\log (x)))} \, dx\\ &=-\frac {12}{x}-3 \int \left (\frac {x}{(x-\log (\log (x))) \log ^2(-x+\log (\log (x)))}-\frac {1}{\log (x) (x-\log (\log (x))) \log ^2(-x+\log (\log (x)))}\right ) \, dx+3 \int \frac {1}{\log (-x+\log (\log (x)))} \, dx\\ &=-\frac {12}{x}-3 \int \frac {x}{(x-\log (\log (x))) \log ^2(-x+\log (\log (x)))} \, dx+3 \int \frac {1}{\log (x) (x-\log (\log (x))) \log ^2(-x+\log (\log (x)))} \, dx+3 \int \frac {1}{\log (-x+\log (\log (x)))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 20, normalized size = 0.87 \begin {gather*} 3 \left (-\frac {4}{x}+\frac {x}{\log (-x+\log (\log (x)))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^2 + 3*x^3*Log[x] + (-3*x^3*Log[x] + 3*x^2*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x]]] + (-12*x*L
og[x] + 12*Log[x]*Log[Log[x]])*Log[-x + Log[Log[x]]]^2)/((-(x^3*Log[x]) + x^2*Log[x]*Log[Log[x]])*Log[-x + Log
[Log[x]]]^2),x]

[Out]

3*(-4/x + x/Log[-x + Log[Log[x]]])

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fricas [A]  time = 0.55, size = 29, normalized size = 1.26 \begin {gather*} \frac {3 \, {\left (x^{2} - 4 \, \log \left (-x + \log \left (\log \relax (x)\right )\right )\right )}}{x \log \left (-x + \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2*log(x)*log(log(x))-3*x^3*log(x))*lo
g(log(log(x))-x)+3*x^3*log(x)-3*x^2)/(x^2*log(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x, algorithm="fr
icas")

[Out]

3*(x^2 - 4*log(-x + log(log(x))))/(x*log(-x + log(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(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2*log(x)*log(log(x))-3*x^3*log(x))*lo
g(log(log(x))-x)+3*x^3*log(x)-3*x^2)/(x^2*log(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x, algorithm="gi
ac")

[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%%%}Done

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maple [A]  time = 0.03, size = 20, normalized size = 0.87

method	result	size
risch	\(-\frac {12}{x}+\frac {3 x}{\ln \left (\ln \left (\ln \relax (x )\right )-x \right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*ln(x)*ln(ln(x))-12*x*ln(x))*ln(ln(ln(x))-x)^2+(3*x^2*ln(x)*ln(ln(x))-3*x^3*ln(x))*ln(ln(ln(x))-x)+3*x
^3*ln(x)-3*x^2)/(x^2*ln(x)*ln(ln(x))-x^3*ln(x))/ln(ln(ln(x))-x)^2,x,method=_RETURNVERBOSE)

[Out]

-12/x+3*x/ln(ln(ln(x))-x)

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maxima [A]  time = 0.39, size = 29, normalized size = 1.26 \begin {gather*} \frac {3 \, {\left (x^{2} - 4 \, \log \left (-x + \log \left (\log \relax (x)\right )\right )\right )}}{x \log \left (-x + \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*log(x)*log(log(x))-12*x*log(x))*log(log(log(x))-x)^2+(3*x^2*log(x)*log(log(x))-3*x^3*log(x))*lo
g(log(log(x))-x)+3*x^3*log(x)-3*x^2)/(x^2*log(x)*log(log(x))-x^3*log(x))/log(log(log(x))-x)^2,x, algorithm="ma
xima")

[Out]

3*(x^2 - 4*log(-x + log(log(x))))/(x*log(-x + log(log(x))))

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mupad [B]  time = 3.84, size = 90, normalized size = 3.91 \begin {gather*} 3\,x-3\,\ln \left (\ln \relax (x)\right )-\frac {3\,\ln \left (\ln \relax (x)\right )}{x\,\ln \relax (x)-1}+\frac {3\,x-\frac {3\,x\,\ln \relax (x)\,\ln \left (\ln \left (\ln \relax (x)\right )-x\right )\,\left (x-\ln \left (\ln \relax (x)\right )\right )}{x\,\ln \relax (x)-1}}{\ln \left (\ln \left (\ln \relax (x)\right )-x\right )}-\frac {12}{x}+\frac {3\,\left (x^2+x\right )}{\left (x\,\ln \relax (x)-1\right )\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x - 3*log(log(x)) - (3*log(log(x)))/(x*log(x) - 1) + (3*x - (3*x*log(x)*log(log(log(x)) - x)*(x - log(log(x)
)))/(x*log(x) - 1))/log(log(log(x)) - x) - 12/x + (3*(x + x^2))/((x*log(x) - 1)*(x + 1))

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sympy [A]  time = 0.45, size = 14, normalized size = 0.61 \begin {gather*} \frac {3 x}{\log {\left (- x + \log {\left (\log {\relax (x )} \right )} \right )}} - \frac {12}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*ln(x)*ln(ln(x))-12*x*ln(x))*ln(ln(ln(x))-x)**2+(3*x**2*ln(x)*ln(ln(x))-3*x**3*ln(x))*ln(ln(ln(x
))-x)+3*x**3*ln(x)-3*x**2)/(x**2*ln(x)*ln(ln(x))-x**3*ln(x))/ln(ln(ln(x))-x)**2,x)

[Out]

3*x/log(-x + log(log(x))) - 12/x

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