\(\int \frac {-162-486 x-x^3-3 x^4+(-729 x+9 x^4+(-243 x+3 x^4) \log (\frac {-81+x^3}{x^2})) \log (3+\log (\frac {-81+x^3}{x^2}))}{(-729 x+9 x^4+(-243 x+3 x^4) \log (\frac {-81+x^3}{x^2})) \log ^2(3+\log (\frac {-81+x^3}{x^2}))} \, dx\) [6230]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

-Defer[Int][1/((3 + Log[-81/x^2 + x])*Log[3 + Log[-81/x^2 + x]]^2), x] + Defer[Int][1/((-3*(-3)^(1/3) - x)*(3
+ Log[-81/x^2 + x])*Log[3 + Log[-81/x^2 + x]]^2), x]/3 + Defer[Int][1/((3*3^(1/3) - x)*(3 + Log[-81/x^2 + x])*
Log[3 + Log[-81/x^2 + x]]^2), x]/3 + 3*3^(1/3)*Defer[Int][1/((3*3^(1/3) - x)*(3 + Log[-81/x^2 + x])*Log[3 + Lo
g[-81/x^2 + x]]^2), x] + Defer[Int][1/((3*(-1)^(2/3)*3^(1/3) - x)*(3 + Log[-81/x^2 + x])*Log[3 + Log[-81/x^2 +
x]]^2), x]/3 + (2*Defer[Int][1/(x*(3 + Log[-81/x^2 + x])*Log[3 + Log[-81/x^2 + x]]^2), x])/3 + 3*3^(1/3)*Defe
r[Int][1/((3*3^(1/3) + (-1)^(1/3)*x)*(3 + Log[-81/x^2 + x])*Log[3 + Log[-81/x^2 + x]]^2), x] + 3*3^(1/3)*Defer
[Int][1/((3*3^(1/3) - (-1)^(2/3)*x)*(3 + Log[-81/x^2 + x])*Log[3 + Log[-81/x^2 + x]]^2), x] + Defer[Int][Log[3
+ Log[-81/x^2 + x]]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \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 )}{3 x \left (81-x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx \\ & = \frac {1}{3} \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 )}{x \left (81-x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx \\ & = \frac {1}{3} \int \left (\frac {-162-486 x-x^3-3 x^4}{x \left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}+\frac {3}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {-162-486 x-x^3-3 x^4}{x \left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ & = \frac {1}{3} \int \left (-\frac {3}{\left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}+\frac {2}{x \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}-\frac {3 \left (243+x^2\right )}{\left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ & = \frac {2}{3} \int \frac {1}{x \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-\int \frac {1}{\left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-\int \frac {243+x^2}{\left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ & = \frac {2}{3} \int \frac {1}{x \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-\int \left (\frac {243}{\left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}+\frac {x^2}{\left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}\right ) \, dx-\int \frac {1}{\left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ & = \frac {2}{3} \int \frac {1}{x \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-243 \int \frac {1}{\left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-\int \frac {1}{\left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-\int \frac {x^2}{\left (-81+x^3\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ & = \frac {2}{3} \int \frac {1}{x \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-243 \int \left (-\frac {1}{27\ 3^{2/3} \left (3 \sqrt [3]{3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}-\frac {1}{27\ 3^{2/3} \left (3 \sqrt [3]{3}+\sqrt [3]{-1} x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}-\frac {1}{27\ 3^{2/3} \left (3 \sqrt [3]{3}-(-1)^{2/3} x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}\right ) \, dx-\int \left (-\frac {1}{3 \left (-3 \sqrt [3]{-3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}-\frac {1}{3 \left (3 \sqrt [3]{3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}-\frac {1}{3 \left (3 (-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )}\right ) \, dx-\int \frac {1}{\left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ & = \frac {1}{3} \int \frac {1}{\left (-3 \sqrt [3]{-3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\frac {1}{3} \int \frac {1}{\left (3 \sqrt [3]{3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\frac {1}{3} \int \frac {1}{\left (3 (-1)^{2/3} \sqrt [3]{3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\frac {2}{3} \int \frac {1}{x \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\left (3 \sqrt [3]{3}\right ) \int \frac {1}{\left (3 \sqrt [3]{3}-x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\left (3 \sqrt [3]{3}\right ) \int \frac {1}{\left (3 \sqrt [3]{3}+\sqrt [3]{-1} x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\left (3 \sqrt [3]{3}\right ) \int \frac {1}{\left (3 \sqrt [3]{3}-(-1)^{2/3} x\right ) \left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx-\int \frac {1}{\left (3+\log \left (-\frac {81}{x^2}+x\right )\right ) \log ^2\left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx+\int \frac {1}{\log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=-\frac {-1-3 x}{3 \log \left (3+\log \left (-\frac {81}{x^2}+x\right )\right )} \]

[In]

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
]

[Out]

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

Maple [A] (verified)

Time = 6.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21

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

[In]

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)*l
n((x^3-81)/x^2)+9*x^4-729*x)/ln(ln((x^3-81)/x^2)+3)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 \, x + 1}{3 \, \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right )} \]

[In]

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")

[Out]

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

[In]

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)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 \, x + 1}{3 \, \log \left (\log \left (x^{3} - 81\right ) - 2 \, \log \left (x\right ) + 3\right )} \]

[In]

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")

[Out]

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 = 0.65 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=\frac {3 \, x \log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 9 \, x + \log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3}{3 \, {\left (\log \left (x^{3} - 81\right ) \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right ) - \log \left (x^{2}\right ) \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right ) + 3 \, \log \left (\log \left (\frac {x^{3} - 81}{x^{2}}\right ) + 3\right )\right )}} \]

[In]

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")

[Out]

1/3*(3*x*log((x^3 - 81)/x^2) + 9*x + log((x^3 - 81)/x^2) + 3)/(log(x^3 - 81)*log(log((x^3 - 81)/x^2) + 3) - lo
g(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 = 12.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 5.32 \[ \int \frac {-162-486 x-x^3-3 x^4+\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log \left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )}{\left (-729 x+9 x^4+\left (-243 x+3 x^4\right ) \log \left (\frac {-81+x^3}{x^2}\right )\right ) \log ^2\left (3+\log \left (\frac {-81+x^3}{x^2}\right )\right )} \, dx=3\,x-\frac {729\,x}{x^3+162}+\frac {x-\frac {x\,\ln \left (\ln \left (\frac {x^3-81}{x^2}\right )+3\right )\,\left (x^3-81\right )\,\left (\ln \left (\frac {x^3-81}{x^2}\right )+3\right )}{x^3+162}+\frac {1}{3}}{\ln \left (\ln \left (\frac {x^3-81}{x^2}\right )+3\right )}-\frac {\ln \left (\frac {x^3-81}{x^2}\right )\,\left (81\,x-x^4\right )}{x^3+162} \]

[In]

int((486*x + log(log((x^3 - 81)/x^2) + 3)*(729*x + log((x^3 - 81)/x^2)*(243*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)

[Out]

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)