\(\int \frac {-8+(-x-2 x^2) \log (\frac {1}{2} (4 x^2-x^2 \log (3))) \log ^2(\log (\frac {1}{2} (4 x^2-x^2 \log (3))))}{16 x \log (\frac {1}{2} (4 x^2-x^2 \log (3)))+(-8 x-8 x^2-8 x^3) \log (\frac {1}{2} (4 x^2-x^2 \log (3))) \log (\log (\frac {1}{2} (4 x^2-x^2 \log (3))))+(x+2 x^2+3 x^3+2 x^4+x^5) \log (\frac {1}{2} (4 x^2-x^2 \log (3))) \log ^2(\log (\frac {1}{2} (4 x^2-x^2 \log (3))))} \, dx\) [9165]

Optimal result

Integrand size = 188, antiderivative size = 28 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {1}{1+x (1+x)-\frac {4}{\log \left (\log \left (\frac {1}{2} x^2 (4-\log (3))\right )\right )}} \]

[Out]

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

Rubi [F]

\[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx \]

[In]

Int[(-8 + (-x - 2*x^2)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]]^2)/(16*x*Log[(4*x^2 - x^2*
Log[3])/2] + (-8*x - 8*x^2 - 8*x^3)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]] + (x + 2*x^2
+ 3*x^3 + 2*x^4 + x^5)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]]^2),x]

[Out]

(1 + x + x^2)^(-1) - 16*Defer[Int][1/((1 + x + x^2)^2*(4 - (1 + x + x^2)*Log[Log[-1/2*(x^2*(-4 + Log[3]))]])^2
), x] - 32*Defer[Int][x/((1 + x + x^2)^2*(4 - (1 + x + x^2)*Log[Log[-1/2*(x^2*(-4 + Log[3]))]])^2), x] - 8*Def
er[Int][1/(x*Log[-1/2*(x^2*(-4 + Log[3]))]*(4 - (1 + x + x^2)*Log[Log[-1/2*(x^2*(-4 + Log[3]))]])^2), x] - 8*D
efer[Int][1/((1 + x + x^2)^2*(-4 + (1 + x + x^2)*Log[Log[-1/2*(x^2*(-4 + Log[3]))]])), x] - 16*Defer[Int][x/((
1 + x + x^2)^2*(-4 + (1 + x + x^2)*Log[Log[-1/2*(x^2*(-4 + Log[3]))]])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-8-x (1+2 x) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \log ^2\left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (4-\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx \\ & = \int \left (\frac {-1-2 x}{\left (1+x+x^2\right )^2}-\frac {8 \left (1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{x \left (1+x+x^2\right )^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2}-\frac {8 (1+2 x)}{\left (1+x+x^2\right )^2 \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )}\right ) \, dx \\ & = -\left (8 \int \frac {1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )}{x \left (1+x+x^2\right )^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx\right )-8 \int \frac {1+2 x}{\left (1+x+x^2\right )^2 \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )} \, dx+\int \frac {-1-2 x}{\left (1+x+x^2\right )^2} \, dx \\ & = \frac {1}{1+x+x^2}-8 \int \frac {\left (1+x+x^2\right )^2+2 x (1+2 x) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )}{x \left (1+x+x^2\right )^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (4-\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx-8 \int \frac {-1-2 x}{\left (1+x+x^2\right )^2 \left (4-\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )} \, dx \\ & = \frac {1}{1+x+x^2}-8 \int \left (\frac {1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )}{x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2}-\frac {(1+x) \left (1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{\left (1+x+x^2\right )^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2}-\frac {(1+x) \left (1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{\left (1+x+x^2\right ) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2}\right ) \, dx-8 \int \left (\frac {1}{\left (1+x+x^2\right )^2 \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )}+\frac {2 x}{\left (1+x+x^2\right )^2 \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )}\right ) \, dx \\ & = \frac {1}{1+x+x^2}-8 \int \frac {1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )}{x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx+8 \int \frac {(1+x) \left (1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{\left (1+x+x^2\right )^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx+8 \int \frac {(1+x) \left (1+2 x+3 x^2+2 x^3+x^4+2 x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )+4 x^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{\left (1+x+x^2\right ) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx-8 \int \frac {1}{\left (1+x+x^2\right )^2 \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )} \, dx-16 \int \frac {x}{\left (1+x+x^2\right )^2 \left (-4+\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )+x^2 \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )} \, dx \\ & = \frac {1}{1+x+x^2}-8 \int \frac {\left (1+x+x^2\right )^2+2 x (1+2 x) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )}{x \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (4-\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx+8 \int \frac {(1+x) \left (\left (1+x+x^2\right )^2+2 x (1+2 x) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{\left (1+x+x^2\right )^2 \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (4-\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx+8 \int \frac {(1+x) \left (\left (1+x+x^2\right )^2+2 x (1+2 x) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{\left (1+x+x^2\right ) \log \left (-\frac {1}{2} x^2 (-4+\log (3))\right ) \left (4-\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )^2} \, dx-8 \int \frac {1}{\left (1+x+x^2\right )^2 \left (-4+\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )} \, dx-16 \int \frac {x}{\left (1+x+x^2\right )^2 \left (-4+\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{-4+\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )} \]

[In]

Integrate[(-8 + (-x - 2*x^2)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]]^2)/(16*x*Log[(4*x^2
- x^2*Log[3])/2] + (-8*x - 8*x^2 - 8*x^3)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]] + (x +
2*x^2 + 3*x^3 + 2*x^4 + x^5)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]]^2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(26)=52\).

Time = 3.63 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(\frac {\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right )}{\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right ) x^{2}+\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right ) x +\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right )-4}\)	\(56\)

[In]

int(((-2*x^2-x)*ln(-1/2*x^2*ln(3)+2*x^2)*ln(ln(-1/2*x^2*ln(3)+2*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+x)*ln(-1/2*
x^2*ln(3)+2*x^2)*ln(ln(-1/2*x^2*ln(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*ln(-1/2*x^2*ln(3)+2*x^2)*ln(ln(-1/2*x^2*ln(
3)+2*x^2))+16*x*ln(-1/2*x^2*ln(3)+2*x^2)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\log \left (\log \left (-\frac {1}{2} \, x^{2} \log \left (3\right ) + 2 \, x^{2}\right )\right )}{{\left (x^{2} + x + 1\right )} \log \left (\log \left (-\frac {1}{2} \, x^{2} \log \left (3\right ) + 2 \, x^{2}\right )\right ) - 4} \]

[In]

integrate(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+
x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*
log(log(-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).

Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {4}{- 4 x^{2} - 4 x + \left (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1\right ) \log {\left (\log {\left (- \frac {x^{2} \log {\left (3 \right )}}{2} + 2 x^{2} \right )} \right )} - 4} + \frac {1}{x^{2} + x + 1} \]

[In]

integrate(((-2*x**2-x)*ln(-1/2*x**2*ln(3)+2*x**2)*ln(ln(-1/2*x**2*ln(3)+2*x**2))**2-8)/((x**5+2*x**4+3*x**3+2*
x**2+x)*ln(-1/2*x**2*ln(3)+2*x**2)*ln(ln(-1/2*x**2*ln(3)+2*x**2))**2+(-8*x**3-8*x**2-8*x)*ln(-1/2*x**2*ln(3)+2
*x**2)*ln(ln(-1/2*x**2*ln(3)+2*x**2))+16*x*ln(-1/2*x**2*ln(3)+2*x**2)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\log \left (-\log \left (2\right ) + 2 \, \log \left (x\right ) + \log \left (-\log \left (3\right ) + 4\right )\right )}{{\left (x^{2} + x + 1\right )} \log \left (-\log \left (2\right ) + 2 \, \log \left (x\right ) + \log \left (-\log \left (3\right ) + 4\right )\right ) - 4} \]

[In]

integrate(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+
x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*
log(log(-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (24) = 48\).

Time = 0.95 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {4}{x^{4} \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) + 2 \, x^{3} \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) + 3 \, x^{2} \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) - 4 \, x^{2} + 2 \, x \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) - 4 \, x + \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) - 4} + \frac {1}{x^{2} + x + 1} \]

[In]

integrate(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+
x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*
log(log(-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x, algorithm="giac")

[Out]

4/(x^4*log(-log(2) + log(-x^2*log(3) + 4*x^2)) + 2*x^3*log(-log(2) + log(-x^2*log(3) + 4*x^2)) + 3*x^2*log(-lo
g(2) + log(-x^2*log(3) + 4*x^2)) - 4*x^2 + 2*x*log(-log(2) + log(-x^2*log(3) + 4*x^2)) - 4*x + log(-log(2) + l
og(-x^2*log(3) + 4*x^2)) - 4) + 1/(x^2 + x + 1)

Mupad [F(-1)]

Timed out. \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\int -\frac {\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\,\left (2\,x^2+x\right )\,{\ln \left (\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\right )}^2+8}{\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\,\left (x^5+2\,x^4+3\,x^3+2\,x^2+x\right )\,{\ln \left (\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\right )}^2-\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\,\left (8\,x^3+8\,x^2+8\,x\right )\,\ln \left (\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\right )+16\,x\,\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )} \,d x \]

[In]

int(-(log(log(2*x^2 - (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 2*x^2) + 8)/(16*x*log(2*x^2 - (x^2*l
og(3))/2) - log(log(2*x^2 - (x^2*log(3))/2))*log(2*x^2 - (x^2*log(3))/2)*(8*x + 8*x^2 + 8*x^3) + log(log(2*x^2
- (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 2*x^2 + 3*x^3 + 2*x^4 + x^5)),x)

[Out]

int(-(log(log(2*x^2 - (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 2*x^2) + 8)/(16*x*log(2*x^2 - (x^2*l
og(3))/2) - log(log(2*x^2 - (x^2*log(3))/2))*log(2*x^2 - (x^2*log(3))/2)*(8*x + 8*x^2 + 8*x^3) + log(log(2*x^2
- (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 2*x^2 + 3*x^3 + 2*x^4 + x^5)), x)