\(\int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log (x^3+x^2 \log (x))+(-x-\log (x)) \log (x^3+x^2 \log (x)) \log (\frac {1}{4 x \log (x^3+x^2 \log (x))})}{(x^3+x^2 \log (x)) \log (x^3+x^2 \log (x))} \, dx\) [7958]

Optimal result

Integrand size = 96, antiderivative size = 23 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=\frac {\log \left (\frac {1}{4 x \log \left (x^2 (x+\log (x))\right )}\right )}{x} \]

[Out]

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

Rubi [F]

\[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=\int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx \]

[In]

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

[Out]

Log[1/(4*x*Log[x^3 + x^2*Log[x]])]/x + Defer[Int][1/(x^2*(-x - Log[x])*Log[x^3 + x^2*Log[x]]), x] + 3*Defer[In
t][1/(x*(-x - Log[x])*Log[x^3 + x^2*Log[x]]), x] + 2*Defer[Int][Log[x]/(x^2*(-x - Log[x])*Log[x^3 + x^2*Log[x]
]), x] + Defer[Int][1/(x^2*(x + Log[x])*Log[x^3 + x^2*Log[x]]), x] + 3*Defer[Int][1/(x*(x + Log[x])*Log[x^3 +
x^2*Log[x]]), x] + 2*Defer[Int][Log[x]/(x^2*(x + Log[x])*Log[x^3 + x^2*Log[x]]), x]

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=\frac {\log \left (\frac {1}{4 x \log \left (x^2 (x+\log (x))\right )}\right )}{x} \]

[In]

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

[Out]

Log[1/(4*x*Log[x^2*(x + Log[x])])]/x

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {1}{4 x \ln \left (x^{2} \left (x +\ln \left (x \right )\right )\right )}\right )}{x}\)	\(22\)
risch	\(\text {Expression too large to display}\)	\(2102\)

[In]

int(((-x-ln(x))*ln(x^2*ln(x)+x^3)*ln(1/4/x/ln(x^2*ln(x)+x^3))+(-x-ln(x))*ln(x^2*ln(x)+x^3)-2*ln(x)-3*x-1)/(x^2
*ln(x)+x^3)/ln(x^2*ln(x)+x^3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=\frac {\log \left (\frac {1}{4 \, x \log \left (x^{3} + x^{2} \log \left (x\right )\right )}\right )}{x} \]

[In]

integrate(((-x-log(x))*log(x^2*log(x)+x^3)*log(1/4/x/log(x^2*log(x)+x^3))+(-x-log(x))*log(x^2*log(x)+x^3)-2*lo
g(x)-3*x-1)/(x^2*log(x)+x^3)/log(x^2*log(x)+x^3),x, algorithm="fricas")

[Out]

log(1/4/(x*log(x^3 + x^2*log(x))))/x

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=\frac {\log {\left (\frac {1}{4 x \log {\left (x^{3} + x^{2} \log {\left (x \right )} \right )}} \right )}}{x} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=-\frac {2 \, \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (x + \log \left (x\right )\right ) + 2 \, \log \left (x\right )\right )}{x} \]

[In]

integrate(((-x-log(x))*log(x^2*log(x)+x^3)*log(1/4/x/log(x^2*log(x)+x^3))+(-x-log(x))*log(x^2*log(x)+x^3)-2*lo
g(x)-3*x-1)/(x^2*log(x)+x^3)/log(x^2*log(x)+x^3),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=-\frac {\log \left (x\right )}{x} - \frac {\log \left (4 \, \log \left (x + \log \left (x\right )\right ) + 8 \, \log \left (x\right )\right )}{x} \]

[In]

integrate(((-x-log(x))*log(x^2*log(x)+x^3)*log(1/4/x/log(x^2*log(x)+x^3))+(-x-log(x))*log(x^2*log(x)+x^3)-2*lo
g(x)-3*x-1)/(x^2*log(x)+x^3)/log(x^2*log(x)+x^3),x, algorithm="giac")

[Out]

-log(x)/x - log(4*log(x + log(x)) + 8*log(x))/x

Mupad [B] (verification not implemented)

Time = 14.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\right )} \, dx=\frac {\ln \left (\frac {1}{4\,x\,\ln \left (x^2\,\ln \left (x\right )+x^3\right )}\right )}{x} \]

[In]

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

[Out]

log(1/(4*x*log(x^2*log(x) + x^3)))/x