3.36.60 \(\int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx\)

Optimal. Leaf size=27 \[ 8+2 x-x (4+x)-\left (x+\frac {2 x}{3 \log (\log (x))}\right )^2 \]

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Rubi [F]  time = 0.33, 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 {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x + (12*x - 8*x*Log[x])*Log[Log[x]] - 24*x*Log[x]*Log[Log[x]]^2 + (-18 - 36*x)*Log[x]*Log[Log[x]]^3)/(9
*Log[x]*Log[Log[x]]^3),x]

[Out]

-1/2*(1 + 2*x)^2 + (8*Defer[Int][x/(Log[x]*Log[Log[x]]^3), x])/9 - (8*Defer[Int][x/Log[Log[x]]^2, x])/9 + (4*D
efer[Int][x/(Log[x]*Log[Log[x]]^2), x])/3 - (8*Defer[Int][x/Log[Log[x]], x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{\log (x) \log ^3(\log (x))} \, dx\\ &=\frac {1}{9} \int \left (-18 (1+2 x)+\frac {8 x}{\log (x) \log ^3(\log (x))}-\frac {4 x (-3+2 \log (x))}{\log (x) \log ^2(\log (x))}-\frac {24 x}{\log (\log (x))}\right ) \, dx\\ &=-\frac {1}{2} (1+2 x)^2-\frac {4}{9} \int \frac {x (-3+2 \log (x))}{\log (x) \log ^2(\log (x))} \, dx+\frac {8}{9} \int \frac {x}{\log (x) \log ^3(\log (x))} \, dx-\frac {8}{3} \int \frac {x}{\log (\log (x))} \, dx\\ &=-\frac {1}{2} (1+2 x)^2-\frac {4}{9} \int \left (\frac {2 x}{\log ^2(\log (x))}-\frac {3 x}{\log (x) \log ^2(\log (x))}\right ) \, dx+\frac {8}{9} \int \frac {x}{\log (x) \log ^3(\log (x))} \, dx-\frac {8}{3} \int \frac {x}{\log (\log (x))} \, dx\\ &=-\frac {1}{2} (1+2 x)^2+\frac {8}{9} \int \frac {x}{\log (x) \log ^3(\log (x))} \, dx-\frac {8}{9} \int \frac {x}{\log ^2(\log (x))} \, dx+\frac {4}{3} \int \frac {x}{\log (x) \log ^2(\log (x))} \, dx-\frac {8}{3} \int \frac {x}{\log (\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 33, normalized size = 1.22 \begin {gather*} -2 x-2 x^2-\frac {4 x^2}{9 \log ^2(\log (x))}-\frac {4 x^2}{3 \log (\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x + (12*x - 8*x*Log[x])*Log[Log[x]] - 24*x*Log[x]*Log[Log[x]]^2 + (-18 - 36*x)*Log[x]*Log[Log[x]]
^3)/(9*Log[x]*Log[Log[x]]^3),x]

[Out]

-2*x - 2*x^2 - (4*x^2)/(9*Log[Log[x]]^2) - (4*x^2)/(3*Log[Log[x]])

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fricas [A]  time = 0.61, size = 33, normalized size = 1.22 \begin {gather*} -\frac {2 \, {\left (6 \, x^{2} \log \left (\log \relax (x)\right ) + 9 \, {\left (x^{2} + x\right )} \log \left (\log \relax (x)\right )^{2} + 2 \, x^{2}\right )}}{9 \, \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-36*x-18)*log(x)*log(log(x))^3-24*x*log(x)*log(log(x))^2+(-8*x*log(x)+12*x)*log(log(x))+8*x)/l
og(x)/log(log(x))^3,x, algorithm="fricas")

[Out]

-2/9*(6*x^2*log(log(x)) + 9*(x^2 + x)*log(log(x))^2 + 2*x^2)/log(log(x))^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (9 \, {\left (2 \, x + 1\right )} \log \relax (x) \log \left (\log \relax (x)\right )^{3} + 12 \, x \log \relax (x) \log \left (\log \relax (x)\right )^{2} + 2 \, {\left (2 \, x \log \relax (x) - 3 \, x\right )} \log \left (\log \relax (x)\right ) - 4 \, x\right )}}{9 \, \log \relax (x) \log \left (\log \relax (x)\right )^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-36*x-18)*log(x)*log(log(x))^3-24*x*log(x)*log(log(x))^2+(-8*x*log(x)+12*x)*log(log(x))+8*x)/l
og(x)/log(log(x))^3,x, algorithm="giac")

[Out]

integrate(-2/9*(9*(2*x + 1)*log(x)*log(log(x))^3 + 12*x*log(x)*log(log(x))^2 + 2*(2*x*log(x) - 3*x)*log(log(x)
) - 4*x)/(log(x)*log(log(x))^3), x)

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maple [A]  time = 0.03, size = 27, normalized size = 1.00

method	result	size
risch	\(-2 x^{2}-2 x -\frac {4 x^{2} \left (3 \ln \left (\ln \relax (x )\right )+1\right )}{9 \ln \left (\ln \relax (x )\right )^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((-36*x-18)*ln(x)*ln(ln(x))^3-24*x*ln(x)*ln(ln(x))^2+(-8*x*ln(x)+12*x)*ln(ln(x))+8*x)/ln(x)/ln(ln(x))^
3,x,method=_RETURNVERBOSE)

[Out]

-2*x^2-2*x-4/9*x^2*(3*ln(ln(x))+1)/ln(ln(x))^2

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maxima [A]  time = 0.75, size = 28, normalized size = 1.04 \begin {gather*} -2 \, x^{2} - 2 \, x - \frac {4 \, {\left (3 \, x^{2} \log \left (\log \relax (x)\right ) + x^{2}\right )}}{9 \, \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-36*x-18)*log(x)*log(log(x))^3-24*x*log(x)*log(log(x))^2+(-8*x*log(x)+12*x)*log(log(x))+8*x)/l
og(x)/log(log(x))^3,x, algorithm="maxima")

[Out]

-2*x^2 - 2*x - 4/9*(3*x^2*log(log(x)) + x^2)/log(log(x))^2

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mupad [B]  time = 2.17, size = 29, normalized size = 1.07 \begin {gather*} -2\,x-2\,x^2-\frac {4\,x^2}{3\,\ln \left (\ln \relax (x)\right )}-\frac {4\,x^2}{9\,{\ln \left (\ln \relax (x)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x)/9 + (log(log(x))*(12*x - 8*x*log(x)))/9 - (log(log(x))^3*log(x)*(36*x + 18))/9 - (8*x*log(log(x))^2
*log(x))/3)/(log(log(x))^3*log(x)),x)

[Out]

- 2*x - 2*x^2 - (4*x^2)/(3*log(log(x))) - (4*x^2)/(9*log(log(x))^2)

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sympy [A]  time = 0.25, size = 32, normalized size = 1.19 \begin {gather*} - 2 x^{2} - 2 x + \frac {- 12 x^{2} \log {\left (\log {\relax (x )} \right )} - 4 x^{2}}{9 \log {\left (\log {\relax (x )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-36*x-18)*ln(x)*ln(ln(x))**3-24*x*ln(x)*ln(ln(x))**2+(-8*x*ln(x)+12*x)*ln(ln(x))+8*x)/ln(x)/ln
(ln(x))**3,x)

[Out]

-2*x**2 - 2*x + (-12*x**2*log(log(x)) - 4*x**2)/(9*log(log(x))**2)

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