\(\int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx\) [3560]

Optimal result

Integrand size = 53, antiderivative size = 27 \[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=8+2 x-x (4+x)-\left (x+\frac {2 x}{3 \log (\log (x))}\right )^2 \]

[Out]

2*x-(x+2/3*x/ln(ln(x)))^2+8-(4+x)*x

Rubi [F]

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

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=-2 x-2 x^2-\frac {4 x^2}{9 \log ^2(\log (x))}-\frac {4 x^2}{3 \log (\log (x))} \]

[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]])

Maple [A] (verified)

Time = 3.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
risch	\(-2 x^{2}-2 x -\frac {4 x^{2} \left (3 \ln \left (\ln \left (x \right )\right )+1\right )}{9 \ln \left (\ln \left (x \right )\right )^{2}}\)	\(27\)
parallelrisch	\(\frac {-18 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-12 x^{2} \ln \left (\ln \left (x \right )\right )-18 x \ln \left (\ln \left (x \right )\right )^{2}-4 x^{2}}{9 \ln \left (\ln \left (x \right )\right )^{2}}\)	\(40\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=-\frac {2 \, {\left (6 \, x^{2} \log \left (\log \left (x\right )\right ) + 9 \, {\left (x^{2} + x\right )} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{2}\right )}}{9 \, \log \left (\log \left (x\right )\right )^{2}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=- 2 x^{2} - 2 x + \frac {- 12 x^{2} \log {\left (\log {\left (x \right )} \right )} - 4 x^{2}}{9 \log {\left (\log {\left (x \right )} \right )}^{2}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=-2 \, x^{2} - 2 \, x - \frac {4 \, {\left (3 \, x^{2} \log \left (\log \left (x\right )\right ) + x^{2}\right )}}{9 \, \log \left (\log \left (x\right )\right )^{2}} \]

[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

Giac [F]

\[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=\int { -\frac {2 \, {\left (9 \, {\left (2 \, x + 1\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + 12 \, x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 2 \, {\left (2 \, x \log \left (x\right ) - 3 \, x\right )} \log \left (\log \left (x\right )\right ) - 4 \, x\right )}}{9 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 9.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {8 x+(12 x-8 x \log (x)) \log (\log (x))-24 x \log (x) \log ^2(\log (x))+(-18-36 x) \log (x) \log ^3(\log (x))}{9 \log (x) \log ^3(\log (x))} \, dx=-2\,x-2\,x^2-\frac {4\,x^2}{3\,\ln \left (\ln \left (x\right )\right )}-\frac {4\,x^2}{9\,{\ln \left (\ln \left (x\right )\right )}^2} \]

[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)