\(\int \frac {-24+12 x+(36 x-36 \log (x^2)) \log (-x+\log (x^2))+(x^4-x^3 \log (x^2)) \log ^2(-x+\log (x^2))}{(-x^5+x^4 \log (x^2)) \log ^2(-x+\log (x^2))} \, dx\) [8761]

Optimal result

Integrand size = 78, antiderivative size = 22 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=1-\log (x)+\frac {12}{x^3 \log \left (-x+\log \left (x^2\right )\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx \]

[In]

Int[(-24 + 12*x + (36*x - 36*Log[x^2])*Log[-x + Log[x^2]] + (x^4 - x^3*Log[x^2])*Log[-x + Log[x^2]]^2)/((-x^5
+ x^4*Log[x^2])*Log[-x + Log[x^2]]^2),x]

[Out]

-Log[x] + 24*Defer[Int][1/(x^4*(x - Log[x^2])*Log[-x + Log[x^2]]^2), x] - 12*Defer[Int][1/(x^3*(x - Log[x^2])*
Log[-x + Log[x^2]]^2), x] - 36*Defer[Int][1/(x^4*Log[-x + Log[x^2]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{x^4 \left (-x+\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx \\ & = \int \frac {-x^3-\frac {12 (-2+x)}{\left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}-\frac {36}{\log \left (-x+\log \left (x^2\right )\right )}}{x^4} \, dx \\ & = \int \left (-\frac {1}{x}-\frac {12 (-2+x)}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}-\frac {36}{x^4 \log \left (-x+\log \left (x^2\right )\right )}\right ) \, dx \\ & = -\log (x)-12 \int \frac {-2+x}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx-36 \int \frac {1}{x^4 \log \left (-x+\log \left (x^2\right )\right )} \, dx \\ & = -\log (x)-12 \int \left (-\frac {2}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}+\frac {1}{x^3 \left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}\right ) \, dx-36 \int \frac {1}{x^4 \log \left (-x+\log \left (x^2\right )\right )} \, dx \\ & = -\log (x)-12 \int \frac {1}{x^3 \left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx+24 \int \frac {1}{x^4 \left (x-\log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx-36 \int \frac {1}{x^4 \log \left (-x+\log \left (x^2\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=-\log (x)+\frac {12}{x^3 \log \left (-x+\log \left (x^2\right )\right )} \]

[In]

Integrate[(-24 + 12*x + (36*x - 36*Log[x^2])*Log[-x + Log[x^2]] + (x^4 - x^3*Log[x^2])*Log[-x + Log[x^2]]^2)/(
(-x^5 + x^4*Log[x^2])*Log[-x + Log[x^2]]^2),x]

[Out]

-Log[x] + 12/(x^3*Log[-x + Log[x^2]])

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68

method	result	size
parallelrisch	\(\frac {24-\ln \left (\ln \left (x^{2}\right )-x \right ) \ln \left (x^{2}\right ) x^{3}}{2 x^{3} \ln \left (\ln \left (x^{2}\right )-x \right )}\)	\(37\)

[In]

int(((-x^3*ln(x^2)+x^4)*ln(ln(x^2)-x)^2+(-36*ln(x^2)+36*x)*ln(ln(x^2)-x)+12*x-24)/(x^4*ln(x^2)-x^5)/ln(ln(x^2)
-x)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*(24-ln(ln(x^2)-x)*ln(x^2)*x^3)/x^3/ln(ln(x^2)-x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=-\frac {x^{3} \log \left (x^{2}\right ) \log \left (-x + \log \left (x^{2}\right )\right ) - 24}{2 \, x^{3} \log \left (-x + \log \left (x^{2}\right )\right )} \]

[In]

integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(log(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^
5)/log(log(x^2)-x)^2,x, algorithm="fricas")

[Out]

-1/2*(x^3*log(x^2)*log(-x + log(x^2)) - 24)/(x^3*log(-x + log(x^2)))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=- \log {\left (x \right )} + \frac {12}{x^{3} \log {\left (- x + \log {\left (x^{2} \right )} \right )}} \]

[In]

integrate(((-x**3*ln(x**2)+x**4)*ln(ln(x**2)-x)**2+(-36*ln(x**2)+36*x)*ln(ln(x**2)-x)+12*x-24)/(x**4*ln(x**2)-
x**5)/ln(ln(x**2)-x)**2,x)

[Out]

-log(x) + 12/(x**3*log(-x + log(x**2)))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {12}{x^{3} \log \left (-x + 2 \, \log \left (x\right )\right )} - \log \left (x\right ) \]

[In]

integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(log(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^
5)/log(log(x^2)-x)^2,x, algorithm="maxima")

[Out]

12/(x^3*log(-x + 2*log(x))) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {12}{x^{3} \log \left (-x + \log \left (x^{2}\right )\right )} - \log \left (x\right ) \]

[In]

integrate(((-x^3*log(x^2)+x^4)*log(log(x^2)-x)^2+(-36*log(x^2)+36*x)*log(log(x^2)-x)+12*x-24)/(x^4*log(x^2)-x^
5)/log(log(x^2)-x)^2,x, algorithm="giac")

[Out]

12/(x^3*log(-x + log(x^2))) - log(x)

Mupad [B] (verification not implemented)

Time = 13.94 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {-24+12 x+\left (36 x-36 \log \left (x^2\right )\right ) \log \left (-x+\log \left (x^2\right )\right )+\left (x^4-x^3 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )}{\left (-x^5+x^4 \log \left (x^2\right )\right ) \log ^2\left (-x+\log \left (x^2\right )\right )} \, dx=\frac {36}{2\,x^2-x^3}-\frac {36\,x}{2\,x^3-x^4}-\ln \left (x\right )+\frac {12}{x^3\,\ln \left (\ln \left (x^2\right )-x\right )} \]

[In]

int((12*x + log(log(x^2) - x)*(36*x - 36*log(x^2)) - log(log(x^2) - x)^2*(x^3*log(x^2) - x^4) - 24)/(log(log(x
^2) - x)^2*(x^4*log(x^2) - x^5)),x)

[Out]

36/(2*x^2 - x^3) - (36*x)/(2*x^3 - x^4) - log(x) + 12/(x^3*log(log(x^2) - x))