\(\int \frac {(-8 x+4 x^2) \log (-2+x)+(2 x^2-x^3+(2-x) \log (3)) \log (-2+x) \log ^2(x^2)+(-2 x^2 \log (x^2)+(-x^3+x \log (3)) \log ^2(x^2)) \log (\frac {-2 x+(-x^2+\log (3)) \log (x^2)}{x \log (x^2)})}{(4 x^2-2 x^3) \log (x^2)+(2 x^3-x^4+(-2 x+x^2) \log (3)) \log ^2(x^2)} \, dx\) [4337]

Optimal result

Integrand size = 149, antiderivative size = 24 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log (-2+x) \log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

-(Log[-2 + x]*Log[x/2]) + Log[-2 + x]*Log[(x - Sqrt[Log[3]])/(2 - Sqrt[Log[3]])] + Log[-2 + x]*Log[(x + Sqrt[L
og[3]])/(2 + Sqrt[Log[3]])] - PolyLog[2, 1 - x/2] + PolyLog[2, (2 - x)/(2 - Sqrt[Log[3]])] + PolyLog[2, (2 - x
)/(2 + Sqrt[Log[3]])] - 2*Defer[Int][Log[-2 + x]/(x*Log[x^2]), x] - 2*Defer[Int][Log[-2 + x]/(2*x + x^2*Log[x^
2] - Log[3]*Log[x^2]), x] - 2*Log[3]*Defer[Int][Log[-2 + x]/(x*(2*x + x^2*Log[x^2] - Log[3]*Log[x^2])), x] + 2
*Defer[Int][(x*Log[-2 + x])/(2*x + x^2*Log[x^2] - Log[3]*Log[x^2]), x] + (Log[9]*Defer[Int][Log[-2 + x]/((-x +
Sqrt[Log[3]])*(2*x + x^2*Log[x^2] - Log[3]*Log[x^2])), x])/Sqrt[Log[3]] + (Log[9]*Defer[Int][Log[-2 + x]/((x
+ Sqrt[Log[3]])*(2*x + x^2*Log[x^2] - Log[3]*Log[x^2])), x])/Sqrt[Log[3]] + Defer[Int][Log[-x + Log[3]/x - 2/L
og[x^2]]/(-2 + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-\frac {\log (-2+x) \left (-4 (-2+x) x+\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log ^2\left (x^2\right )\right )}{x \log \left (x^2\right ) \left (2 x+\left (x^2-\log (3)\right ) \log \left (x^2\right )\right )}-\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{2-x} \, dx \\ & = \int \left (\frac {\log (-2+x) \left (8 x-4 x^2-2 x^2 \log ^2\left (x^2\right )+x^3 \log ^2\left (x^2\right )+x \log (3) \log ^2\left (x^2\right )-\log (9) \log ^2\left (x^2\right )\right )}{(-2+x) x \log \left (x^2\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}+\frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x}\right ) \, dx \\ & = \int \frac {\log (-2+x) \left (8 x-4 x^2-2 x^2 \log ^2\left (x^2\right )+x^3 \log ^2\left (x^2\right )+x \log (3) \log ^2\left (x^2\right )-\log (9) \log ^2\left (x^2\right )\right )}{(-2+x) x \log \left (x^2\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = \int \left (\frac {\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) x \left (x^2-\log (3)\right )}-\frac {2 \log (-2+x)}{x \log \left (x^2\right )}+\frac {2 \left (x^2-\log (3)\right ) \log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}-\frac {2 \left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) \left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\left (2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx\right )+2 \int \frac {\left (x^2-\log (3)\right ) \log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-2 \int \frac {\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) \left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\int \frac {\left (-2 x^2+x^3+x \log (3)-\log (9)\right ) \log (-2+x)}{(-2+x) x \left (x^2-\log (3)\right )} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\left (2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx\right )-2 \int \frac {\left (x^2+\log (3)\right ) \log (-2+x)}{\left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+2 \int \left (\frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )}-\frac {\log (3) \log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx+\int \frac {\left (x^2+\log (3)\right ) \log (-2+x)}{x \left (x^2-\log (3)\right )} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\left (2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx\right )+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-2 \int \left (\frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )}+\frac {\log (9) \log (-2+x)}{\left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\int \left (-\frac {\log (-2+x)}{x}+\frac {2 x \log (-2+x)}{x^2-\log (3)}\right ) \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = 2 \int \frac {x \log (-2+x)}{x^2-\log (3)} \, dx-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-(2 \log (9)) \int \frac {\log (-2+x)}{\left (x^2-\log (3)\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-\int \frac {\log (-2+x)}{x} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\log (-2+x) \log \left (\frac {x}{2}\right )+2 \int \left (-\frac {\log (-2+x)}{2 \left (-x+\sqrt {\log (3)}\right )}+\frac {\log (-2+x)}{2 \left (x+\sqrt {\log (3)}\right )}\right ) \, dx-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx-(2 \log (9)) \int \left (-\frac {\log (-2+x)}{2 \left (-x+\sqrt {\log (3)}\right ) \sqrt {\log (3)} \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}-\frac {\log (-2+x)}{2 \left (x+\sqrt {\log (3)}\right ) \sqrt {\log (3)} \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\log (-2+x) \log \left (\frac {x}{2}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}-\int \frac {\log (-2+x)}{-x+\sqrt {\log (3)}} \, dx+\int \frac {\log (-2+x)}{x+\sqrt {\log (3)}} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\log (-2+x) \log \left (\frac {x}{2}\right )+\log (-2+x) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (-2+x) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}-\int \frac {\log \left (\frac {-x+\sqrt {\log (3)}}{-2+\sqrt {\log (3)}}\right )}{-2+x} \, dx-\int \frac {\log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )}{-2+x} \, dx+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ & = -\log (-2+x) \log \left (\frac {x}{2}\right )+\log (-2+x) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (-2+x) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx-\text {Subst}\left (\int \frac {\log \left (1-\frac {x}{-2+\sqrt {\log (3)}}\right )}{x} \, dx,x,-2+x\right )-\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2+\sqrt {\log (3)}}\right )}{x} \, dx,x,-2+x\right ) \\ & = -\log (-2+x) \log \left (\frac {x}{2}\right )+\log (-2+x) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (-2+x) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )-\text {Li}_2\left (1-\frac {x}{2}\right )+\text {Li}_2\left (\frac {-2+x}{-2+\sqrt {\log (3)}}\right )+\text {Li}_2\left (\frac {2-x}{2+\sqrt {\log (3)}}\right )-2 \int \frac {\log (-2+x)}{x \log \left (x^2\right )} \, dx-2 \int \frac {\log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx+2 \int \frac {x \log (-2+x)}{2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )} \, dx-(2 \log (3)) \int \frac {\log (-2+x)}{x \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx+\frac {\log (9) \int \frac {\log (-2+x)}{\left (-x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (-2+x)}{\left (x+\sqrt {\log (3)}\right ) \left (2 x+x^2 \log \left (x^2\right )-\log (3) \log \left (x^2\right )\right )} \, dx}{\sqrt {\log (3)}}+\int \frac {\log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right )}{-2+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log (-2+x) \log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right ) \]

[In]

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

[Out]

Log[-2 + x]*Log[-x + Log[3]/x - 2/Log[x^2]]

Maple [C] (warning: unable to verify)

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

Time = 149.49 (sec) , antiderivative size = 2484, normalized size of antiderivative = 103.50

method	result	size
risch	\(\text {Expression too large to display}\)	\(2484\)

[In]

int((((x*ln(3)-x^3)*ln(x^2)^2-2*x^2*ln(x^2))*ln(((ln(3)-x^2)*ln(x^2)-2*x)/x/ln(x^2))+((2-x)*ln(3)-x^3+2*x^2)*l
n(-2+x)*ln(x^2)^2+(4*x^2-8*x)*ln(-2+x))/(((x^2-2*x)*ln(3)-x^4+2*x^3)*ln(x^2)^2+(-2*x^3+4*x^2)*ln(x^2)),x,metho
d=_RETURNVERBOSE)

[Out]

ln(-2+x)*ln(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*
x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*l
n(x))-ln(-2+x)*ln(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)-ln(x)*ln
(-2+x)-1/2*I*Pi*ln(-2+x)*csgn(I/x)*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x
)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*c
sgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I
*x^2*ln(x)))*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-P
i*x^2*csgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^
2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))+1/2*I*P
i*ln(-2+x)*csgn(I/x)*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2
)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*c
sgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))
^2+1/2*I*Pi*ln(-2+x)*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^
2))*csgn(I*Pi*x^2*csgn(I*x^2)^3-I*ln(3)*Pi*csgn(I*x^2)^3-4*x+I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)-I*ln(3)*Pi*csgn(
I*x)^2*csgn(I*x^2)-2*I*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2+2*I*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*ln(3)*ln(x)-4*x^2
*ln(x))*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*c
sgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi
*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))+1/2*I*Pi*ln(-2
+x)*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I/(Pi*cs
gn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi
*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I
*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))^2-1/2*I*Pi*ln(-2+x)*csgn(I*Pi*x^2*c
sgn(I*x^2)^3-I*ln(3)*Pi*csgn(I*x^2)^3-4*x+I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)-I*ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)-
2*I*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2+2*I*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*ln(3)*ln(x)-4*x^2*ln(x))*csgn(I/(Pi*
csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*
Pi*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn
(I*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))^2-1/2*I*Pi*ln(-2+x)*csgn(I/(Pi*cs
gn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi
*csgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I
*x^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))^3+1/2*I*Pi*ln(-2+x)*csgn(I/(Pi*csgn
(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi*c
sgn(I*x^2)^3-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x
^2)^2-2*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+
Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-P
i*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csg
n(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))^2-1/2*I*Pi*ln(-2+x)*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+
Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-Pi*x^2*csgn(I*x^2)^3+ln(3)*Pi*csgn(I*x^2)^3-4*I*x-P
i*x^2*csgn(I*x)^2*csgn(I*x^2)+ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*ln(3)*Pi*csg
n(I*x)*csgn(I*x^2)^2+4*I*ln(3)*ln(x)-4*I*x^2*ln(x)))^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log \left (x - 2\right ) \log \left (-\frac {{\left (x^{2} - \log \left (3\right )\right )} \log \left (x^{2}\right ) + 2 \, x}{x \log \left (x^{2}\right )}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log {\left (\frac {- 2 x + \left (- x^{2} + \log {\left (3 \right )}\right ) \log {\left (x^{2} \right )}}{x \log {\left (x^{2} \right )}} \right )} \log {\left (x - 2 \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=-{\left (\log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x - 2\right ) + \log \left (-{\left (x^{2} - \log \left (3\right )\right )} \log \left (x\right ) - x\right ) \log \left (x - 2\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log \left (-x^{2} \log \left (x^{2}\right ) + \log \left (3\right ) \log \left (x^{2}\right ) - 2 \, x\right ) \log \left (x - 2\right ) - \log \left (x - 2\right ) \log \left (x\right ) - \log \left (x - 2\right ) \log \left (\log \left (x^{2}\right )\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.56 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\ln \left (-\frac {2\,x-\ln \left (x^2\right )\,\left (\ln \left (3\right )-x^2\right )}{x\,\ln \left (x^2\right )}\right )\,\ln \left (x-2\right ) \]

[In]

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

[Out]

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