3.10.2 \(\int \frac {(-40-2 x^2+4 x^3+x^5+8 x^6) \log (\frac {2-x^3}{x})+(16 x^2+16 x^5) \log (\log (\frac {2-x^3}{x}))+(-8 x^2+4 x^5) \log (\frac {2-x^3}{x}) \log ^2(\log (\frac {2-x^3}{x}))}{(-8 x^2+4 x^5) \log (\frac {2-x^3}{x})} \, dx\) [902]

3.10.2.1 Optimal result

Integrand size = 126, antiderivative size = 36 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=1-\frac {5-x}{x}+x^2+x \left (\frac {1}{4}+\log ^2\left (\log \left (\frac {2}{x}-x^2\right )\right )\right ) \]

output

x*(1/4+ln(ln(2/x-x^2))^2)-(5-x)/x+x^2+1
 

3.10.2.2 Mathematica [F]

\[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx \]

input

Integrate[((-40 - 2*x^2 + 4*x^3 + x^5 + 8*x^6)*Log[(2 - x^3)/x] + (16*x^2 
+ 16*x^5)*Log[Log[(2 - x^3)/x]] + (-8*x^2 + 4*x^5)*Log[(2 - x^3)/x]*Log[Lo 
g[(2 - x^3)/x]]^2)/((-8*x^2 + 4*x^5)*Log[(2 - x^3)/x]),x]
 

output

Integrate[((-40 - 2*x^2 + 4*x^3 + x^5 + 8*x^6)*Log[(2 - x^3)/x] + (16*x^2 
+ 16*x^5)*Log[Log[(2 - x^3)/x]] + (-8*x^2 + 4*x^5)*Log[(2 - x^3)/x]*Log[Lo 
g[(2 - x^3)/x]]^2)/((-8*x^2 + 4*x^5)*Log[(2 - x^3)/x]), x]
 

3.10.2.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (4 x^5-8 x^2\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (16 x^5+16 x^2\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (8 x^6+x^5+4 x^3-2 x^2-40\right ) \log \left (\frac {2-x^3}{x}\right )}{\left (4 x^5-8 x^2\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (4 x^5-8 x^2\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (16 x^5+16 x^2\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (8 x^6+x^5+4 x^3-2 x^2-40\right ) \log \left (\frac {2-x^3}{x}\right )}{x^2 \left (4 x^3-8\right ) \log \left (\frac {2-x^3}{x}\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (\log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )+\frac {4 \left (x^3+1\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (x^3-2\right ) \log \left (\frac {2-x^3}{x}\right )}+\frac {5}{x^2}+2 x+\frac {1}{4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )dx+4 \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\log \left (\frac {2-x^3}{x}\right )}dx-2 \sqrt [3]{2} \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (\sqrt [3]{2}-x\right ) \log \left (\frac {2-x^3}{x}\right )}dx-2 \sqrt [3]{2} \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (\sqrt [3]{-1} x+\sqrt [3]{2}\right ) \log \left (\frac {2-x^3}{x}\right )}dx-2 \sqrt [3]{2} \int \frac {\log \left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (\sqrt [3]{2}-(-1)^{2/3} x\right ) \log \left (\frac {2-x^3}{x}\right )}dx+x^2+\frac {x}{4}-\frac {5}{x}\)

input

Int[((-40 - 2*x^2 + 4*x^3 + x^5 + 8*x^6)*Log[(2 - x^3)/x] + (16*x^2 + 16*x 
^5)*Log[Log[(2 - x^3)/x]] + (-8*x^2 + 4*x^5)*Log[(2 - x^3)/x]*Log[Log[(2 - 
 x^3)/x]]^2)/((-8*x^2 + 4*x^5)*Log[(2 - x^3)/x]),x]
 

output

$Aborted
 

3.10.2.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

3.10.2.4 Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(-\frac {-16 {\ln \left (\ln \left (-\frac {x^{3}-2}{x}\right )\right )}^{2} x^{2}+80-16 x^{3}-4 x^{2}}{16 x}\)	\(37\)

input

int(((4*x^5-8*x^2)*ln((-x^3+2)/x)*ln(ln((-x^3+2)/x))^2+(16*x^5+16*x^2)*ln( 
ln((-x^3+2)/x))+(8*x^6+x^5+4*x^3-2*x^2-40)*ln((-x^3+2)/x))/(4*x^5-8*x^2)/l 
n((-x^3+2)/x),x,method=_RETURNVERBOSE)
 

output

-1/16*(-16*ln(ln(-(x^3-2)/x))^2*x^2+80-16*x^3-4*x^2)/x
 

3.10.2.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\frac {4 \, x^{2} \log \left (\log \left (-\frac {x^{3} - 2}{x}\right )\right )^{2} + 4 \, x^{3} + x^{2} - 20}{4 \, x} \]

input

integrate(((4*x^5-8*x^2)*log((-x^3+2)/x)*log(log((-x^3+2)/x))^2+(16*x^5+16 
*x^2)*log(log((-x^3+2)/x))+(8*x^6+x^5+4*x^3-2*x^2-40)*log((-x^3+2)/x))/(4* 
x^5-8*x^2)/log((-x^3+2)/x),x, algorithm=\
 

output

1/4*(4*x^2*log(log(-(x^3 - 2)/x))^2 + 4*x^3 + x^2 - 20)/x
 

3.10.2.6 Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=x^{2} + x \log {\left (\log {\left (\frac {2 - x^{3}}{x} \right )} \right )}^{2} + \frac {x}{4} - \frac {5}{x} \]

input

integrate(((4*x**5-8*x**2)*ln((-x**3+2)/x)*ln(ln((-x**3+2)/x))**2+(16*x**5 
+16*x**2)*ln(ln((-x**3+2)/x))+(8*x**6+x**5+4*x**3-2*x**2-40)*ln((-x**3+2)/ 
x))/(4*x**5-8*x**2)/ln((-x**3+2)/x),x)
 

output

x**2 + x*log(log((2 - x**3)/x))**2 + x/4 - 5/x
 

3.10.2.7 Maxima [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=x \log \left (\log \left (-x^{3} + 2\right ) - \log \left (x\right )\right )^{2} + x^{2} + \frac {1}{4} \, x - \frac {5}{x} \]

input

integrate(((4*x^5-8*x^2)*log((-x^3+2)/x)*log(log((-x^3+2)/x))^2+(16*x^5+16 
*x^2)*log(log((-x^3+2)/x))+(8*x^6+x^5+4*x^3-2*x^2-40)*log((-x^3+2)/x))/(4* 
x^5-8*x^2)/log((-x^3+2)/x),x, algorithm=\
 

output

x*log(log(-x^3 + 2) - log(x))^2 + x^2 + 1/4*x - 5/x
 

3.10.2.8 Giac [F]

\[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\int { \frac {4 \, {\left (x^{5} - 2 \, x^{2}\right )} \log \left (-\frac {x^{3} - 2}{x}\right ) \log \left (\log \left (-\frac {x^{3} - 2}{x}\right )\right )^{2} + {\left (8 \, x^{6} + x^{5} + 4 \, x^{3} - 2 \, x^{2} - 40\right )} \log \left (-\frac {x^{3} - 2}{x}\right ) + 16 \, {\left (x^{5} + x^{2}\right )} \log \left (\log \left (-\frac {x^{3} - 2}{x}\right )\right )}{4 \, {\left (x^{5} - 2 \, x^{2}\right )} \log \left (-\frac {x^{3} - 2}{x}\right )} \,d x } \]

input

integrate(((4*x^5-8*x^2)*log((-x^3+2)/x)*log(log((-x^3+2)/x))^2+(16*x^5+16 
*x^2)*log(log((-x^3+2)/x))+(8*x^6+x^5+4*x^3-2*x^2-40)*log((-x^3+2)/x))/(4* 
x^5-8*x^2)/log((-x^3+2)/x),x, algorithm=\
 

output

integrate(1/4*(4*(x^5 - 2*x^2)*log(-(x^3 - 2)/x)*log(log(-(x^3 - 2)/x))^2 
+ (8*x^6 + x^5 + 4*x^3 - 2*x^2 - 40)*log(-(x^3 - 2)/x) + 16*(x^5 + x^2)*lo 
g(log(-(x^3 - 2)/x)))/((x^5 - 2*x^2)*log(-(x^3 - 2)/x)), x)
 

3.10.2.9 Mupad [B] (verification not implemented)

Time = 11.78 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {\left (-40-2 x^2+4 x^3+x^5+8 x^6\right ) \log \left (\frac {2-x^3}{x}\right )+\left (16 x^2+16 x^5\right ) \log \left (\log \left (\frac {2-x^3}{x}\right )\right )+\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right ) \log ^2\left (\log \left (\frac {2-x^3}{x}\right )\right )}{\left (-8 x^2+4 x^5\right ) \log \left (\frac {2-x^3}{x}\right )} \, dx=\frac {x}{4}+x\,{\ln \left (\ln \left (-\frac {x^3-2}{x}\right )\right )}^2-\frac {5}{x}+x^2 \]

input

int(-(log(-(x^3 - 2)/x)*(4*x^3 - 2*x^2 + x^5 + 8*x^6 - 40) + log(log(-(x^3 
 - 2)/x))*(16*x^2 + 16*x^5) - log(log(-(x^3 - 2)/x))^2*log(-(x^3 - 2)/x)*( 
8*x^2 - 4*x^5))/(log(-(x^3 - 2)/x)*(8*x^2 - 4*x^5)),x)
 

output

x/4 + x*log(log(-(x^3 - 2)/x))^2 - 5/x + x^2