3.54.88 \(\int \frac {(-8 x+8 x^2+(3 x-4 x^2+x^3) \log (x)) \log (\frac {1}{3} (8+(-3+x) \log (x)))+(-6+8 x-2 x^2+(2 x-2 x^2) \log (x)) \log (\log (\frac {1}{3} (8+(-3+x) \log (x))))+(-8 x+(3 x-x^2) \log (x)) \log (\frac {1}{3} (8+(-3+x) \log (x))) \log ^2(\log (\frac {1}{3} (8+(-3+x) \log (x))))+(-6+2 x+2 x \log (x)) \log ^3(\log (\frac {1}{3} (8+(-3+x) \log (x))))}{(16 x+(-6 x+2 x^2) \log (x)) \log (\frac {1}{3} (8+(-3+x) \log (x)))} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{4} \left (1-x+\log ^2\left (\log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )\right )\right )^2 \]

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Rubi [F]  time = 3.95, 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 {\left (-8 x+8 x^2+\left (3 x-4 x^2+x^3\right ) \log (x)\right ) \log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )+\left (-6+8 x-2 x^2+\left (2 x-2 x^2\right ) \log (x)\right ) \log \left (\log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )\right )+\left (-8 x+\left (3 x-x^2\right ) \log (x)\right ) \log \left (\frac {1}{3} (8+(-3+x) \log (x))\right ) \log ^2\left (\log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )\right )+(-6+2 x+2 x \log (x)) \log ^3\left (\log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )\right )}{\left (16 x+\left (-6 x+2 x^2\right ) \log (x)\right ) \log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-8*x + 8*x^2 + (3*x - 4*x^2 + x^3)*Log[x])*Log[(8 + (-3 + x)*Log[x])/3] + (-6 + 8*x - 2*x^2 + (2*x - 2*x
^2)*Log[x])*Log[Log[(8 + (-3 + x)*Log[x])/3]] + (-8*x + (3*x - x^2)*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]*Log[L
og[(8 + (-3 + x)*Log[x])/3]]^2 + (-6 + 2*x + 2*x*Log[x])*Log[Log[(8 + (-3 + x)*Log[x])/3]]^3)/((16*x + (-6*x +
2*x^2)*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]),x]

[Out]

-1/2*x + x^2/4 + 4*Defer[Int][Log[Log[(8 + (-3 + x)*Log[x])/3]]/((8 - 3*Log[x] + x*Log[x])*Log[(8 + (-3 + x)*L
og[x])/3]), x] - 3*Defer[Int][Log[Log[(8 + (-3 + x)*Log[x])/3]]/(x*(8 - 3*Log[x] + x*Log[x])*Log[(8 + (-3 + x)
*Log[x])/3]), x] - Defer[Int][(x*Log[Log[(8 + (-3 + x)*Log[x])/3]])/((8 - 3*Log[x] + x*Log[x])*Log[(8 + (-3 +
x)*Log[x])/3]), x] + Defer[Int][(Log[x]*Log[Log[(8 + (-3 + x)*Log[x])/3]])/((8 - 3*Log[x] + x*Log[x])*Log[(8 +
(-3 + x)*Log[x])/3]), x] - Defer[Int][(x*Log[x]*Log[Log[(8 + (-3 + x)*Log[x])/3]])/((8 - 3*Log[x] + x*Log[x])
*Log[(8 + (-3 + x)*Log[x])/3]), x] - Defer[Int][Log[Log[(8 + (-3 + x)*Log[x])/3]]^2, x]/2 + Defer[Int][Log[Log
[(8 + (-3 + x)*Log[x])/3]]^3/((8 - 3*Log[x] + x*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]), x] - 3*Defer[Int][Log[L
og[(8 + (-3 + x)*Log[x])/3]]^3/(x*(8 - 3*Log[x] + x*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]), x] + Defer[Int][(Lo
g[x]*Log[Log[(8 + (-3 + x)*Log[x])/3]]^3)/((8 - 3*Log[x] + x*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]), x]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 47, normalized size = 1.74 \begin {gather*} \frac {1}{4} \left ((-2+x) x-2 (-1+x) \log ^2\left (\log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )\right )+\log ^4\left (\log \left (\frac {1}{3} (8+(-3+x) \log (x))\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-8*x + 8*x^2 + (3*x - 4*x^2 + x^3)*Log[x])*Log[(8 + (-3 + x)*Log[x])/3] + (-6 + 8*x - 2*x^2 + (2*x
- 2*x^2)*Log[x])*Log[Log[(8 + (-3 + x)*Log[x])/3]] + (-8*x + (3*x - x^2)*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]
*Log[Log[(8 + (-3 + x)*Log[x])/3]]^2 + (-6 + 2*x + 2*x*Log[x])*Log[Log[(8 + (-3 + x)*Log[x])/3]]^3)/((16*x + (
-6*x + 2*x^2)*Log[x])*Log[(8 + (-3 + x)*Log[x])/3]),x]

[Out]

((-2 + x)*x - 2*(-1 + x)*Log[Log[(8 + (-3 + x)*Log[x])/3]]^2 + Log[Log[(8 + (-3 + x)*Log[x])/3]]^4)/4

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fricas [A]  time = 0.52, size = 42, normalized size = 1.56 \begin {gather*} \frac {1}{4} \, \log \left (\log \left (\frac {1}{3} \, {\left (x - 3\right )} \log \relax (x) + \frac {8}{3}\right )\right )^{4} - \frac {1}{2} \, {\left (x - 1\right )} \log \left (\log \left (\frac {1}{3} \, {\left (x - 3\right )} \log \relax (x) + \frac {8}{3}\right )\right )^{2} + \frac {1}{4} \, x^{2} - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x-6)*log(log(1/3*log(x)*(x-3)+8/3))^3+((-x^2+3*x)*log(x)-8*x)*log(1/3*log(x)*(x-3)+8/
3)*log(log(1/3*log(x)*(x-3)+8/3))^2+((-2*x^2+2*x)*log(x)-2*x^2+8*x-6)*log(log(1/3*log(x)*(x-3)+8/3))+((x^3-4*x
^2+3*x)*log(x)+8*x^2-8*x)*log(1/3*log(x)*(x-3)+8/3))/((2*x^2-6*x)*log(x)+16*x)/log(1/3*log(x)*(x-3)+8/3),x, al
gorithm="fricas")

[Out]

1/4*log(log(1/3*(x - 3)*log(x) + 8/3))^4 - 1/2*(x - 1)*log(log(1/3*(x - 3)*log(x) + 8/3))^2 + 1/4*x^2 - 1/2*x

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giac [B]  time = 2.83, size = 54, normalized size = 2.00 \begin {gather*} \frac {1}{4} \, \log \left (-\log \relax (3) + \log \left (x \log \relax (x) - 3 \, \log \relax (x) + 8\right )\right )^{4} - \frac {1}{2} \, {\left (x - 1\right )} \log \left (-\log \relax (3) + \log \left (x \log \relax (x) - 3 \, \log \relax (x) + 8\right )\right )^{2} + \frac {1}{4} \, x^{2} - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x-6)*log(log(1/3*log(x)*(x-3)+8/3))^3+((-x^2+3*x)*log(x)-8*x)*log(1/3*log(x)*(x-3)+8/
3)*log(log(1/3*log(x)*(x-3)+8/3))^2+((-2*x^2+2*x)*log(x)-2*x^2+8*x-6)*log(log(1/3*log(x)*(x-3)+8/3))+((x^3-4*x
^2+3*x)*log(x)+8*x^2-8*x)*log(1/3*log(x)*(x-3)+8/3))/((2*x^2-6*x)*log(x)+16*x)/log(1/3*log(x)*(x-3)+8/3),x, al
gorithm="giac")

[Out]

1/4*log(-log(3) + log(x*log(x) - 3*log(x) + 8))^4 - 1/2*(x - 1)*log(-log(3) + log(x*log(x) - 3*log(x) + 8))^2
+ 1/4*x^2 - 1/2*x

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maple [A]  time = 0.07, size = 44, normalized size = 1.63

method	result	size
risch	\(\frac {\ln \left (\ln \left (\frac {\ln \relax (x ) \left (x -3\right )}{3}+\frac {8}{3}\right )\right )^{4}}{4}+\left (-\frac {x}{2}+\frac {1}{2}\right ) \ln \left (\ln \left (\frac {\ln \relax (x ) \left (x -3\right )}{3}+\frac {8}{3}\right )\right )^{2}+\frac {x^{2}}{4}-\frac {x}{2}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)+2*x-6)*ln(ln(1/3*ln(x)*(x-3)+8/3))^3+((-x^2+3*x)*ln(x)-8*x)*ln(1/3*ln(x)*(x-3)+8/3)*ln(ln(1/3*
ln(x)*(x-3)+8/3))^2+((-2*x^2+2*x)*ln(x)-2*x^2+8*x-6)*ln(ln(1/3*ln(x)*(x-3)+8/3))+((x^3-4*x^2+3*x)*ln(x)+8*x^2-
8*x)*ln(1/3*ln(x)*(x-3)+8/3))/((2*x^2-6*x)*ln(x)+16*x)/ln(1/3*ln(x)*(x-3)+8/3),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(ln(1/3*ln(x)*(x-3)+8/3))^4+(-1/2*x+1/2)*ln(ln(1/3*ln(x)*(x-3)+8/3))^2+1/4*x^2-1/2*x

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maxima [B]  time = 0.49, size = 50, normalized size = 1.85 \begin {gather*} \frac {1}{4} \, \log \left (-\log \relax (3) + \log \left ({\left (x - 3\right )} \log \relax (x) + 8\right )\right )^{4} - \frac {1}{2} \, {\left (x - 1\right )} \log \left (-\log \relax (3) + \log \left ({\left (x - 3\right )} \log \relax (x) + 8\right )\right )^{2} + \frac {1}{4} \, x^{2} - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+2*x-6)*log(log(1/3*log(x)*(x-3)+8/3))^3+((-x^2+3*x)*log(x)-8*x)*log(1/3*log(x)*(x-3)+8/
3)*log(log(1/3*log(x)*(x-3)+8/3))^2+((-2*x^2+2*x)*log(x)-2*x^2+8*x-6)*log(log(1/3*log(x)*(x-3)+8/3))+((x^3-4*x
^2+3*x)*log(x)+8*x^2-8*x)*log(1/3*log(x)*(x-3)+8/3))/((2*x^2-6*x)*log(x)+16*x)/log(1/3*log(x)*(x-3)+8/3),x, al
gorithm="maxima")

[Out]

1/4*log(-log(3) + log((x - 3)*log(x) + 8))^4 - 1/2*(x - 1)*log(-log(3) + log((x - 3)*log(x) + 8))^2 + 1/4*x^2
- 1/2*x

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mupad [B]  time = 4.22, size = 60, normalized size = 2.22 \begin {gather*} {\ln \left (\ln \left (\frac {\ln \relax (x)\,\left (x-3\right )}{3}+\frac {8}{3}\right )\right )}^2\,\left (\frac {\frac {3\,x^2}{2}-\frac {x^3}{2}}{x\,\left (x-3\right )}+\frac {1}{2}\right )-\frac {x}{2}+\frac {{\ln \left (\ln \left (\frac {\ln \relax (x)\,\left (x-3\right )}{3}+\frac {8}{3}\right )\right )}^4}{4}+\frac {x^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log((log(x)*(x - 3))/3 + 8/3))^3*(2*x + 2*x*log(x) - 6) + log((log(x)*(x - 3))/3 + 8/3)*(log(x)*(3*x
- 4*x^2 + x^3) - 8*x + 8*x^2) + log(log((log(x)*(x - 3))/3 + 8/3))*(8*x + log(x)*(2*x - 2*x^2) - 2*x^2 - 6) -
log(log((log(x)*(x - 3))/3 + 8/3))^2*log((log(x)*(x - 3))/3 + 8/3)*(8*x - log(x)*(3*x - x^2)))/(log((log(x)*(x
- 3))/3 + 8/3)*(16*x - log(x)*(6*x - 2*x^2))),x)

[Out]

log(log((log(x)*(x - 3))/3 + 8/3))^2*(((3*x^2)/2 - x^3/2)/(x*(x - 3)) + 1/2) - x/2 + log(log((log(x)*(x - 3))/
3 + 8/3))^4/4 + x^2/4

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sympy [A]  time = 3.53, size = 49, normalized size = 1.81 \begin {gather*} \frac {x^{2}}{4} - \frac {x}{2} + \left (\frac {1}{2} - \frac {x}{2}\right ) \log {\left (\log {\left (\frac {\left (x - 3\right ) \log {\relax (x )}}{3} + \frac {8}{3} \right )} \right )}^{2} + \frac {\log {\left (\log {\left (\frac {\left (x - 3\right ) \log {\relax (x )}}{3} + \frac {8}{3} \right )} \right )}^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)+2*x-6)*ln(ln(1/3*ln(x)*(x-3)+8/3))**3+((-x**2+3*x)*ln(x)-8*x)*ln(1/3*ln(x)*(x-3)+8/3)*ln
(ln(1/3*ln(x)*(x-3)+8/3))**2+((-2*x**2+2*x)*ln(x)-2*x**2+8*x-6)*ln(ln(1/3*ln(x)*(x-3)+8/3))+((x**3-4*x**2+3*x)
*ln(x)+8*x**2-8*x)*ln(1/3*ln(x)*(x-3)+8/3))/((2*x**2-6*x)*ln(x)+16*x)/ln(1/3*ln(x)*(x-3)+8/3),x)

[Out]

x**2/4 - x/2 + (1/2 - x/2)*log(log((x - 3)*log(x)/3 + 8/3))**2 + log(log((x - 3)*log(x)/3 + 8/3))**4/4

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