\(\int \frac {(-4+4 x^2-x^4+\log ^2(\frac {x^3}{3})+(4-2 x^2) \log (\log (x))-\log ^2(\log (x)))^x (-4+2 x^2+(-8 x^2+4 x^4) \log (x)-6 \log (x) \log (\frac {x^3}{3})+(2+4 x^2 \log (x)) \log (\log (x))+((4-4 x^2+x^4) \log (x)-\log (x) \log ^2(\frac {x^3}{3})+(-4+2 x^2) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))) \log (-4+4 x^2-x^4+\log ^2(\frac {x^3}{3})+(4-2 x^2) \log (\log (x))-\log ^2(\log (x))))}{(4-4 x^2+x^4) \log (x)-\log (x) \log ^2(\frac {x^3}{3})+(-4+2 x^2) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx\) [2194]

Optimal result

Integrand size = 231, antiderivative size = 25 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx \]

[In]

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

[Out]

-2*Log[27]*Defer[Int][(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^(-1 + x), x] + 8*Defer[Int][x^2*(Log[x^3/3]^
2 - (-2 + x^2 + Log[Log[x]])^2)^(-1 + x), x] - 4*Defer[Int][x^4*(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^(-
1 + x), x] + 4*Defer[Int][(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^(-1 + x)/Log[x], x] - 2*Defer[Int][(x^2*
(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^(-1 + x))/Log[x], x] + 6*Defer[Int][Log[x^3]*(Log[x^3/3]^2 - (-2 +
x^2 + Log[Log[x]])^2)^(-1 + x), x] - 4*Defer[Int][x^2*Log[Log[x]]*(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)
^(-1 + x), x] - 2*Defer[Int][(Log[Log[x]]*(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^(-1 + x))/Log[x], x] + D
efer[Int][(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^x*Log[Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \left (4-2 x^2-4 x^2 \left (-2+x^2\right ) \log (x)+6 \log (x) \log \left (\frac {x^3}{3}\right )-\left (2+4 x^2 \log (x)\right ) \log (\log (x))-\log (x) \left (-\log ^2\left (\frac {x^3}{3}\right )+\left (-2+x^2+\log (\log (x))\right )^2\right ) \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )\right )}{\log (x)} \, dx \\ & = \int \left (-\frac {2 \left (-2+x^2-4 x^2 \log (x)+2 x^4 \log (x)+\log (27) \log (x)-3 \log (x) \log \left (x^3\right )+\log (\log (x))+2 x^2 \log (x) \log (\log (x))\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}+\left (2-x^2-\log \left (\frac {x^3}{3}\right )-\log (\log (x))\right ) \left (x^2-2 \left (1-\frac {\log (3)}{2}\right )-\log \left (x^3\right )+\log (\log (x))\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )\right ) \, dx \\ & = -\left (2 \int \frac {\left (-2+x^2-4 x^2 \log (x)+2 x^4 \log (x)+\log (27) \log (x)-3 \log (x) \log \left (x^3\right )+\log (\log (x))+2 x^2 \log (x) \log (\log (x))\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )+\int \left (2-x^2-\log \left (\frac {x^3}{3}\right )-\log (\log (x))\right ) \left (x^2-2 \left (1-\frac {\log (3)}{2}\right )-\log \left (x^3\right )+\log (\log (x))\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \left (\frac {\left (-2+x^2-4 x^2 \log (x)+2 x^4 \log (x)+\log (27) \log (x)-3 \log (x) \log \left (x^3\right )\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}+\frac {\left (1+2 x^2 \log (x)\right ) \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}\right ) \, dx\right )+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \frac {\left (-2+x^2-4 x^2 \log (x)+2 x^4 \log (x)+\log (27) \log (x)-3 \log (x) \log \left (x^3\right )\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )-2 \int \frac {\left (1+2 x^2 \log (x)\right ) \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \left (\frac {\left (-2+x^2-4 x^2 \log (x)+2 x^4 \log (x)+\log (27) \log (x)\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}-3 \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}\right ) \, dx\right )-2 \int \left (2 x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}+\frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}\right ) \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \frac {\left (-2+x^2-4 x^2 \log (x)+2 x^4 \log (x)+\log (27) \log (x)\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )-2 \int \frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx-4 \int x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+6 \int \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )-2 \int \left (-4 x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}+2 x^4 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}+\log (27) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}+\frac {\left (-2+x^2\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}\right ) \, dx-4 \int x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+6 \int \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \frac {\left (-2+x^2\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )-2 \int \frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx-4 \int x^4 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx-4 \int x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+6 \int \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+8 \int x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx-(2 \log (27)) \int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )-2 \int \left (-\frac {2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}+\frac {x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)}\right ) \, dx-4 \int x^4 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx-4 \int x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+6 \int \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+8 \int x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx-(2 \log (27)) \int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ & = -\left (2 \int \frac {x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx\right )-2 \int \frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx-4 \int x^4 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+4 \int \frac {\left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x}}{\log (x)} \, dx-4 \int x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+6 \int \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+8 \int x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx-(2 \log (27)) \int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^{-1+x} \, dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52

\[\left (-\ln \left (\ln \left (x \right )\right )^{2}+\left (-2 x^{2}+4\right ) \ln \left (\ln \left (x \right )\right )+{\left (\ln \left (3\right )-3 \ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}\right )}^{2}-x^{4}+4 x^{2}-4\right )^{x}\]

[In]

int(((ln(x)*ln(ln(x))^2+(2*x^2-4)*ln(x)*ln(ln(x))-ln(x)*ln(1/3*x^3)^2+(x^4-4*x^2+4)*ln(x))*ln(-ln(ln(x))^2+(-2
*x^2+4)*ln(ln(x))+ln(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*ln(x)+2)*ln(ln(x))-6*ln(x)*ln(1/3*x^3)+(4*x^4-8*x^2)*ln(x)
+2*x^2-4)*exp(x*ln(-ln(ln(x))^2+(-2*x^2+4)*ln(ln(x))+ln(1/3*x^3)^2-x^4+4*x^2-4))/(ln(x)*ln(ln(x))^2+(2*x^2-4)*
ln(x)*ln(ln(x))-ln(x)*ln(1/3*x^3)^2+(x^4-4*x^2+4)*ln(x)),x)

[Out]

(-ln(ln(x))^2+(-2*x^2+4)*ln(ln(x))+(ln(3)-3*ln(x)+1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2+1/2*I*Pi*csg
n(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x)))^2-x^4+4*x^2-4)^x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx={\left (-x^{4} + 4 \, x^{2} + \log \left (3\right )^{2} - 6 \, \log \left (3\right ) \log \left (x\right ) + 9 \, \log \left (x\right )^{2} - 2 \, {\left (x^{2} - 2\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right )}^{x} \]

[In]

integrate(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x))*log(
-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/
3*x^3)+(4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/3*x^3)^2-x^4+4*x^2-
4))/(log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x, algorith
m="fricas")

[Out]

(-x^4 + 4*x^2 + log(3)^2 - 6*log(3)*log(x) + 9*log(x)^2 - 2*(x^2 - 2)*log(log(x)) - log(log(x))^2 - 4)^x

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=e^{\left (x \log \left (x^{2} - \log \left (3\right ) + 3 \, \log \left (x\right ) + \log \left (\log \left (x\right )\right ) - 2\right ) + x \log \left (-x^{2} - \log \left (3\right ) + 3 \, \log \left (x\right ) - \log \left (\log \left (x\right )\right ) + 2\right )\right )} \]

[In]

integrate(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x))*log(
-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/
3*x^3)+(4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/3*x^3)^2-x^4+4*x^2-
4))/(log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x, algorith
m="maxima")

[Out]

e^(x*log(x^2 - log(3) + 3*log(x) + log(log(x)) - 2) + x*log(-x^2 - log(3) + 3*log(x) - log(log(x)) + 2))

Giac [F]

\[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\int { -\frac {{\left (2 \, x^{2} - {\left (\log \left (\frac {1}{3} \, x^{3}\right )^{2} \log \left (x\right ) - 2 \, {\left (x^{2} - 2\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - {\left (x^{4} - 4 \, x^{2} + 4\right )} \log \left (x\right )\right )} \log \left (-x^{4} + 4 \, x^{2} + \log \left (\frac {1}{3} \, x^{3}\right )^{2} - 2 \, {\left (x^{2} - 2\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right ) + 4 \, {\left (x^{4} - 2 \, x^{2}\right )} \log \left (x\right ) - 6 \, \log \left (\frac {1}{3} \, x^{3}\right ) \log \left (x\right ) + 2 \, {\left (2 \, x^{2} \log \left (x\right ) + 1\right )} \log \left (\log \left (x\right )\right ) - 4\right )} {\left (-x^{4} + 4 \, x^{2} + \log \left (\frac {1}{3} \, x^{3}\right )^{2} - 2 \, {\left (x^{2} - 2\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right )}^{x}}{\log \left (\frac {1}{3} \, x^{3}\right )^{2} \log \left (x\right ) - 2 \, {\left (x^{2} - 2\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - {\left (x^{4} - 4 \, x^{2} + 4\right )} \log \left (x\right )} \,d x } \]

[In]

integrate(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x))*log(
-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/
3*x^3)+(4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/3*x^3)^2-x^4+4*x^2-
4))/(log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x, algorith
m="giac")

[Out]

integrate(-(2*x^2 - (log(1/3*x^3)^2*log(x) - 2*(x^2 - 2)*log(x)*log(log(x)) - log(x)*log(log(x))^2 - (x^4 - 4*
x^2 + 4)*log(x))*log(-x^4 + 4*x^2 + log(1/3*x^3)^2 - 2*(x^2 - 2)*log(log(x)) - log(log(x))^2 - 4) + 4*(x^4 - 2
*x^2)*log(x) - 6*log(1/3*x^3)*log(x) + 2*(2*x^2*log(x) + 1)*log(log(x)) - 4)*(-x^4 + 4*x^2 + log(1/3*x^3)^2 -
2*(x^2 - 2)*log(log(x)) - log(log(x))^2 - 4)^x/(log(1/3*x^3)^2*log(x) - 2*(x^2 - 2)*log(x)*log(log(x)) - log(x
)*log(log(x))^2 - (x^4 - 4*x^2 + 4)*log(x)), x)

Mupad [B] (verification not implemented)

Time = 10.45 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx={\left (-x^4-2\,x^2\,\ln \left (\ln \left (x\right )\right )+4\,x^2+{\ln \left (x^3\right )}^2-2\,\ln \left (3\right )\,\ln \left (x^3\right )-{\ln \left (\ln \left (x\right )\right )}^2+4\,\ln \left (\ln \left (x\right )\right )+{\ln \left (3\right )}^2-4\right )}^x \]

[In]

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

[Out]

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