\(\int \frac {-162-486 x-x^3-3 x^4+(-729 x+9 x^4+(-243 x+3 x^4) \log (\frac {-81+x^3}{x^2})) \log (3+\log (\frac {-81+x^3}{x^2}))}{(-729 x+9 x^4+(-243 x+3 x^4) \log (\frac {-81+x^3}{x^2})) \log ^2(3+\log (\frac {-81+x^3}{x^2}))} \, dx\) [230]

Optimal result

Integrand size = 105, antiderivative size = 19 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {\frac {1}{3}+x}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \] Output:

(x+1/3)/ln(ln(x-81/x^2)+3)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=-\frac {-1-3 x}{3 \log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \] Input:

Integrate[(-162 - 486*x - x^3 - 3*x^4 + (-729*x + 9*x^4 + (-243*x + 3*x^4) 
*Log[(-81 + x^3)/x^2])*Log[3 + Log[(-81 + x^3)/x^2]])/((-729*x + 9*x^4 + ( 
-243*x + 3*x^4)*Log[(-81 + x^3)/x^2])*Log[3 + Log[(-81 + x^3)/x^2]]^2),x]
 

Output:

-1/3*(-1 - 3*x)/Log[3 + Log[-81/x^2 + x]]
 

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.

\(\displaystyle \int \frac {-3 x^4-x^3+\left (9 x^4+\left (3 x^4-243 x\right ) \log \left (\frac {x^3-81}{x^2}\right )-729 x\right ) \log \left (\log \left (\frac {x^3-81}{x^2}\right )+3\right )-486 x-162}{\left (9 x^4+\left (3 x^4-243 x\right ) \log \left (\frac {x^3-81}{x^2}\right )-729 x\right ) \log ^2\left (\log \left (\frac {x^3-81}{x^2}\right )+3\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {3 x^4+x^3-\left (9 x^4+\left (3 x^4-243 x\right ) \log \left (\frac {x^3-81}{x^2}\right )-729 x\right ) \log \left (\log \left (\frac {x^3-81}{x^2}\right )+3\right )+486 x+162}{3 x \left (81-x^3\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (\frac {x^3-81}{x^2}\right )+3\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 x^4+x^3+486 x+3 \left (-3 x^4+243 x+\left (81 x-x^4\right ) \log \left (-\frac {81-x^3}{x^2}\right )\right ) \log \left (\log \left (-\frac {81-x^3}{x^2}\right )+3\right )+162}{x \left (81-x^3\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (-\frac {81-x^3}{x^2}\right )+3\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {1}{3} \int \left (\frac {-3 x^4-x^3-486 x-162}{x \left (x^3-81\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}+\frac {3}{\log \left (\log \left (x-\frac {81}{x^2}\right )+3\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (-3 \int \frac {1}{\left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+\int \frac {1}{\left (-x-3 \sqrt [3]{-3}\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+9 \sqrt [3]{3} \int \frac {1}{\left (3 \sqrt [3]{3}-x\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+\int \frac {1}{\left (3 \sqrt [3]{3}-x\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+\int \frac {1}{\left (3 (-1)^{2/3} \sqrt [3]{3}-x\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+2 \int \frac {1}{x \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+9 \sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{-1} x+3 \sqrt [3]{3}\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+9 \sqrt [3]{3} \int \frac {1}{\left (3 \sqrt [3]{3}-(-1)^{2/3} x\right ) \left (\log \left (x-\frac {81}{x^2}\right )+3\right ) \log ^2\left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx+3 \int \frac {1}{\log \left (\log \left (x-\frac {81}{x^2}\right )+3\right )}dx\right )\)

Input:

Int[(-162 - 486*x - x^3 - 3*x^4 + (-729*x + 9*x^4 + (-243*x + 3*x^4)*Log[( 
-81 + x^3)/x^2])*Log[3 + Log[(-81 + x^3)/x^2]])/((-729*x + 9*x^4 + (-243*x 
 + 3*x^4)*Log[(-81 + x^3)/x^2])*Log[3 + Log[(-81 + x^3)/x^2]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21

method	result	size
parallelrisch	\(\frac {972+2916 x}{2916 \ln \left (\ln \left (\frac {x^{3}-81}{x^{2}}\right )+3\right )}\)	\(23\)

Input:

int((((3*x^4-243*x)*ln((x^3-81)/x^2)+9*x^4-729*x)*ln(ln((x^3-81)/x^2)+3)-3 
*x^4-x^3-486*x-162)/((3*x^4-243*x)*ln((x^3-81)/x^2)+9*x^4-729*x)/ln(ln((x^ 
3-81)/x^2)+3)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2916*(972+2916*x)/ln(ln((x^3-81)/x^2)+3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 \, x + 1}{3 \, \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right )} \] Input:

integrate((((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x)*log(log((x^3-81)/ 
x^2)+3)-3*x^4-x^3-486*x-162)/((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x) 
/log(log((x^3-81)/x^2)+3)^2,x, algorithm="fricas")
 

Output:

1/3*(3*x + 1)/log(log((x^3 - 81)/x^2) + 3)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 x + 1}{3 \log {\left (\log {\left (\frac {x^{3} - 81}{x^{2}} \right )} + 3 \right )}} \] Input:

integrate((((3*x**4-243*x)*ln((x**3-81)/x**2)+9*x**4-729*x)*ln(ln((x**3-81 
)/x**2)+3)-3*x**4-x**3-486*x-162)/((3*x**4-243*x)*ln((x**3-81)/x**2)+9*x** 
4-729*x)/ln(ln((x**3-81)/x**2)+3)**2,x)
 

Output:

(3*x + 1)/(3*log(log((x**3 - 81)/x**2) + 3))
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 \, x + 1}{3 \, \log \left (\log \left (x^{3} - 81\right ) - 2 \, \log \left (x\right ) + 3\right )} \] Input:

integrate((((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x)*log(log((x^3-81)/ 
x^2)+3)-3*x^4-x^3-486*x-162)/((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x) 
/log(log((x^3-81)/x^2)+3)^2,x, algorithm="maxima")
 

Output:

1/3*(3*x + 1)/log(log(x^3 - 81) - 2*log(x) + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).

Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 \, x \log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 9 \, x + \log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3}{3 \, {\left (\log \left (x^{3} - 81\right ) \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right ) - \log \left (x^{2}\right ) \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right ) + 3 \, \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right )\right )}} \] Input:

integrate((((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x)*log(log((x^3-81)/ 
x^2)+3)-3*x^4-x^3-486*x-162)/((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x) 
/log(log((x^3-81)/x^2)+3)^2,x, algorithm="giac")
 

Output:

1/3*(3*x*log((x^3 - 81)/x^2) + 9*x + log((x^3 - 81)/x^2) + 3)/(log(x^3 - 8 
1)*log(log((x^3 - 81)/x^2) + 3) - log(x^2)*log(log((x^3 - 81)/x^2) + 3) + 
3*log(log((x^3 - 81)/x^2) + 3))
 

Mupad [B] (verification not implemented)

Time = 3.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 5.32 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=3\,x-\frac {729\,x}{x^3+162}+\frac {x-\frac {x\,\ln \left (\ln \left (\frac {x^3-81}{x^2}\right )+3\right )\,\left (x^3-81\right )\,\left (\ln \left (\frac {x^3-81}{x^2}\right )+3\right )}{x^3+162}+\frac {1}{3}}{\ln \left (\ln \left (\frac {x^3-81}{x^2}\right )+3\right )}-\frac {\ln \left (\frac {x^3-81}{x^2}\right )\,\left (81\,x-x^4\right )}{x^3+162} \] Input:

int((486*x + log(log((x^3 - 81)/x^2) + 3)*(729*x + log((x^3 - 81)/x^2)*(24 
3*x - 3*x^4) - 9*x^4) + x^3 + 3*x^4 + 162)/(log(log((x^3 - 81)/x^2) + 3)^2 
*(729*x + log((x^3 - 81)/x^2)*(243*x - 3*x^4) - 9*x^4)),x)
 

Output:

3*x - (729*x)/(x^3 + 162) + (x - (x*log(log((x^3 - 81)/x^2) + 3)*(x^3 - 81 
)*(log((x^3 - 81)/x^2) + 3))/(x^3 + 162) + 1/3)/log(log((x^3 - 81)/x^2) + 
3) - (log((x^3 - 81)/x^2)*(81*x - x^4))/(x^3 + 162)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 x +1}{3 \,\mathrm {log}\left (\mathrm {log}\left (\frac {x^{3}-81}{x^{2}}\right )+3\right )} \] Input:

int((((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x)*log(log((x^3-81)/x^2)+3 
)-3*x^4-x^3-486*x-162)/((3*x^4-243*x)*log((x^3-81)/x^2)+9*x^4-729*x)/log(l 
og((x^3-81)/x^2)+3)^2,x)
 

Output:

(3*x + 1)/(3*log(log((x**3 - 81)/x**2) + 3))