\(\int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+(4 x-4 x^3+2 x^4) \log (x)+(-x^2+x^5-x^3 \log (x)) \log (\frac {1-x^3+x \log (x)}{x})}{(4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+(4 x^2+x^3+4 x^4-x^5) \log (x)+(x^3-x^6+x^4 \log (x)) \log (\frac {1-x^3+x \log (x)}{x})) \log ^2(\frac {4+x+4 x^2-x^3+x^2 \log (\frac {1-x^3+x \log (x)}{x})}{x})} \, dx\) [4243]

Optimal result

Integrand size = 199, antiderivative size = 30 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log \left (\frac {4+x}{x}+x \left (4-x+\log \left (\frac {1}{x}-x^2+\log (x)\right )\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {6818} \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log \left (\frac {-x^3+4 x^2+x^2 \log \left (\frac {-x^3+x \log (x)+1}{x}\right )+x+4}{x}\right )} \]

[In]

Int[(4 - 3*x^2 - 3*x^3 + 6*x^5 - 2*x^6 + (4*x - 4*x^3 + 2*x^4)*Log[x] + (-x^2 + x^5 - x^3*Log[x])*Log[(1 - x^3
+ x*Log[x])/x])/((4*x + x^2 + 4*x^3 - 5*x^4 - x^5 - 4*x^6 + x^7 + (4*x^2 + x^3 + 4*x^4 - x^5)*Log[x] + (x^3 -
x^6 + x^4*Log[x])*Log[(1 - x^3 + x*Log[x])/x])*Log[(4 + x + 4*x^2 - x^3 + x^2*Log[(1 - x^3 + x*Log[x])/x])/x]
^2),x]

[Out]

Log[(4 + x + 4*x^2 - x^3 + x^2*Log[(1 - x^3 + x*Log[x])/x])/x]^(-1)

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log \left (1+\frac {4}{x}+4 x-x^2+x \log \left (\frac {1}{x}-x^2+\log (x)\right )\right )} \]

[In]

Integrate[(4 - 3*x^2 - 3*x^3 + 6*x^5 - 2*x^6 + (4*x - 4*x^3 + 2*x^4)*Log[x] + (-x^2 + x^5 - x^3*Log[x])*Log[(1
- x^3 + x*Log[x])/x])/((4*x + x^2 + 4*x^3 - 5*x^4 - x^5 - 4*x^6 + x^7 + (4*x^2 + x^3 + 4*x^4 - x^5)*Log[x] +
(x^3 - x^6 + x^4*Log[x])*Log[(1 - x^3 + x*Log[x])/x])*Log[(4 + x + 4*x^2 - x^3 + x^2*Log[(1 - x^3 + x*Log[x])/
x])/x]^2),x]

[Out]

Log[1 + 4/x + 4*x - x^2 + x*Log[x^(-1) - x^2 + Log[x]]]^(-1)

Maple [C] (warning: unable to verify)

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

Time = 1.51 (sec) , antiderivative size = 3007, normalized size of antiderivative = 100.23

\[\text {output too large to display}\]

[In]

int(((-x^3*ln(x)+x^5-x^2)*ln((x*ln(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*ln(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/((x^4*ln(x)
-x^6+x^3)*ln((x*ln(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*ln(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/ln((x^2*ln((x*
ln(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x)

[Out]

-2*I/(-Pi*csgn(-8-2*x+2*x^3-8*x^2-I*x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2+I*x^2*Pi*csgn(I*(x*ln(x)-x^3+
1))*csgn(I*(x*ln(x)-x^3+1)/x)^2+2*I*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2+I*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*
x^2*ln(-x*ln(x)+x^3-1)+I*x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*x^2*ln(x)-2*I*Pi
*x^2)*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)
-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2
*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*P
i*x^2))^2+Pi*csgn(-8-2*x+2*x^3-8*x^2-I*x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2+I*x^2*Pi*csgn(I*(x*ln(x)-x
^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2+2*I*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2+I*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3
-2*x^2*ln(-x*ln(x)+x^3-1)+I*x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*x^2*ln(x)-2*I
*Pi*x^2)*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln
(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^
3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+
2*Pi*x^2))*csgn(I/x)-Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*
Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*l
n(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)
+2*I*x^2*ln(x)+2*Pi*x^2))^3+Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)
^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn
(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^
3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))^2*csgn(I/x)+Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(
x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)
/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*c
sgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I
*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+
1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))
*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))^2-Pi*csgn(I/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)
*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(
x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-
x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/
x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*l
n(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x
)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))+Pi*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*cs
gn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I
*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x
*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))^3-Pi*csgn(1/x*(-8*I-2*I*x+2*I*x^3-8*I*x^2+x^
2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)^2-2*x^2*Pi
*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^3-1)-x^2*Pi*csgn(I/x)*cs
gn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))^2+Pi+2*I*ln(2)+2*I*ln(x)-2*I*ln(-8*I-
2*I*x+2*I*x^3-8*I*x^2+x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln
(x)-x^3+1)/x)^2-2*x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^2-x^2*Pi*csgn(I*(x*ln(x)-x^3+1)/x)^3-2*I*x^2*ln(-x*ln(x)+x^
3-1)-x^2*Pi*csgn(I/x)*csgn(I*(x*ln(x)-x^3+1))*csgn(I*(x*ln(x)-x^3+1)/x)+2*I*x^2*ln(x)+2*Pi*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log \left (-\frac {x^{3} - x^{2} \log \left (-\frac {x^{3} - x \log \left (x\right ) - 1}{x}\right ) - 4 \, x^{2} - x - 4}{x}\right )} \]

[In]

integrate(((-x^3*log(x)+x^5-x^2)*log((x*log(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*log(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/(
(x^4*log(x)-x^6+x^3)*log((x*log(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*log(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/
log((x^2*log((x*log(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 159.82 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log {\left (\frac {- x^{3} + x^{2} \log {\left (\frac {- x^{3} + x \log {\left (x \right )} + 1}{x} \right )} + 4 x^{2} + x + 4}{x} \right )}} \]

[In]

integrate(((-x**3*ln(x)+x**5-x**2)*ln((x*ln(x)-x**3+1)/x)+(2*x**4-4*x**3+4*x)*ln(x)-2*x**6+6*x**5-3*x**3-3*x**
2+4)/((x**4*ln(x)-x**6+x**3)*ln((x*ln(x)-x**3+1)/x)+(-x**5+4*x**4+x**3+4*x**2)*ln(x)+x**7-4*x**6-x**5-5*x**4+4
*x**3+x**2+4*x)/ln((x**2*ln((x*ln(x)-x**3+1)/x)-x**3+4*x**2+x+4)/x)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log \left (-x^{3} + x^{2} {\left (\log \left (-x^{3} + x \log \left (x\right ) + 1\right ) + 4\right )} - x^{2} \log \left (x\right ) + x + 4\right ) - \log \left (x\right )} \]

[In]

integrate(((-x^3*log(x)+x^5-x^2)*log((x*log(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*log(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/(
(x^4*log(x)-x^6+x^3)*log((x*log(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*log(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/
log((x^2*log((x*log(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.69 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\log \left (-x^{3} + x^{2} \log \left (-x^{3} + x \log \left (x\right ) + 1\right ) - x^{2} \log \left (x\right ) + 4 \, x^{2} + x + 4\right ) - \log \left (x\right )} \]

[In]

integrate(((-x^3*log(x)+x^5-x^2)*log((x*log(x)-x^3+1)/x)+(2*x^4-4*x^3+4*x)*log(x)-2*x^6+6*x^5-3*x^3-3*x^2+4)/(
(x^4*log(x)-x^6+x^3)*log((x*log(x)-x^3+1)/x)+(-x^5+4*x^4+x^3+4*x^2)*log(x)+x^7-4*x^6-x^5-5*x^4+4*x^3+x^2+4*x)/
log((x^2*log((x*log(x)-x^3+1)/x)-x^3+4*x^2+x+4)/x)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.83 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {4-3 x^2-3 x^3+6 x^5-2 x^6+\left (4 x-4 x^3+2 x^4\right ) \log (x)+\left (-x^2+x^5-x^3 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{\left (4 x+x^2+4 x^3-5 x^4-x^5-4 x^6+x^7+\left (4 x^2+x^3+4 x^4-x^5\right ) \log (x)+\left (x^3-x^6+x^4 \log (x)\right ) \log \left (\frac {1-x^3+x \log (x)}{x}\right )\right ) \log ^2\left (\frac {4+x+4 x^2-x^3+x^2 \log \left (\frac {1-x^3+x \log (x)}{x}\right )}{x}\right )} \, dx=\frac {1}{\ln \left (4\,x+x\,\ln \left (\frac {x\,\ln \left (x\right )-x^3+1}{x}\right )+\frac {4}{x}-x^2+1\right )} \]

[In]

int(-(3*x^2 + 3*x^3 - 6*x^5 + 2*x^6 + log((x*log(x) - x^3 + 1)/x)*(x^3*log(x) + x^2 - x^5) - log(x)*(4*x - 4*x
^3 + 2*x^4) - 4)/(log((x + x^2*log((x*log(x) - x^3 + 1)/x) + 4*x^2 - x^3 + 4)/x)^2*(4*x + log(x)*(4*x^2 + x^3
+ 4*x^4 - x^5) + x^2 + 4*x^3 - 5*x^4 - x^5 - 4*x^6 + x^7 + log((x*log(x) - x^3 + 1)/x)*(x^4*log(x) + x^3 - x^6
))),x)

[Out]

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