3.17.46 $\int \frac{4\log (x)+(-2x+6x^3+2x\log (x))\log (x^2)+(2-4\log (x))\log (x^2)\log (\log (x^2))}{(3x^4-x^2\log (x))\log (x^2)+x\log (x)\log (x^2)\log (\log (x^2))}dx$

Optimal. Leaf size=24 $$5+\log \left({(-3x+\frac{\log (x)(x-\log (\log \left(x^2\right)))}{x^2})}^2\right)$$

________________________________________________________________________________________

Rubi [F]  time = 2.79, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{4\log (x)+(-2x+6x^3+2x\log (x))\log \left(x^2\right)+(2-4\log (x))\log \left(x^2\right)\log (\log \left(x^2\right))}{(3x^4-x^2\log (x))\log \left(x^2\right)+x\log (x)\log \left(x^2\right)\log (\log \left(x^2\right))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(4*Log[x] + (-2*x + 6*x^3 + 2*x*Log[x])*Log[x^2] + (2 - 4*Log[x])*Log[x^2]*Log[Log[x^2]])/((3*x^4 - x^2*Lo
g[x])*Log[x^2] + x*Log[x]*Log[x^2]*Log[Log[x^2]]),x]

[Out]

-4*Log[x] + 2*Log[Log[x]] + 6*Defer[Int][x^2/(Log[x]*(-3*x^3 + x*Log[x] - Log[x]*Log[Log[x^2]])), x] + 2*Defer
[Int][Log[x]/(-3*x^3 + x*Log[x] - Log[x]*Log[Log[x^2]]), x] + 18*Defer[Int][x^2/(3*x^3 - x*Log[x] + Log[x]*Log
[Log[x^2]]), x] + 4*Defer[Int][Log[x]/(x*Log[x^2]*(3*x^3 - x*Log[x] + Log[x]*Log[Log[x^2]])), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{4\log (x)+(-2x+6x^3+2x\log (x))\log \left(x^2\right)+(2-4\log (x))\log \left(x^2\right)\log (\log \left(x^2\right))}{x\log \left(x^2\right)(3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right)))}dx\\ & =\int (-\frac{2(-1+2\log (x))}{x\log (x)}+\frac{2(2{\log }^2(x)-3x^3\log \left(x^2\right)+9x^3\log (x)\log \left(x^2\right)-x{\log }^2(x)\log \left(x^2\right))}{x\log (x)\log \left(x^2\right)(3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right)))})dx\\ & =-(2\int \frac{-1+2\log (x)}{x\log (x)}dx)+2\int \frac{2{\log }^2(x)-3x^3\log \left(x^2\right)+9x^3\log (x)\log \left(x^2\right)-x{\log }^2(x)\log \left(x^2\right)}{x\log (x)\log \left(x^2\right)(3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right)))}dx\\ & =2\int (\frac{3x^2}{\log (x)(-3x^3+x\log (x)-\log (x)\log (\log \left(x^2\right)))}+\frac{\log (x)}{-3x^3+x\log (x)-\log (x)\log (\log \left(x^2\right))}+\frac{9x^2}{3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right))}+\frac{2\log (x)}{x\log \left(x^2\right)(3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right)))})dx-2\mathrm{Subst}(\int \frac{-1+2x}{x}dx,x,\log (x))\\ & =2\int \frac{\log (x)}{-3x^3+x\log (x)-\log (x)\log (\log \left(x^2\right))}dx-2\mathrm{Subst}(\int (2-\frac{1}{x})dx,x,\log (x))+4\int \frac{\log (x)}{x\log \left(x^2\right)(3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right)))}dx+6\int \frac{x^2}{\log (x)(-3x^3+x\log (x)-\log (x)\log (\log \left(x^2\right)))}dx+18\int \frac{x^2}{3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right))}dx\\ & =-4\log (x)+2\log (\log (x))+2\int \frac{\log (x)}{-3x^3+x\log (x)-\log (x)\log (\log \left(x^2\right))}dx+4\int \frac{\log (x)}{x\log \left(x^2\right)(3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right)))}dx+6\int \frac{x^2}{\log (x)(-3x^3+x\log (x)-\log (x)\log (\log \left(x^2\right)))}dx+18\int \frac{x^2}{3x^3-x\log (x)+\log (x)\log (\log \left(x^2\right))}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [B]  time = 0.39, size = 63, normalized size = 2.62 $$\begin{array}{c}2(-2\log (x)+\log (-6x^3+2x(\log (x)-\frac{\log \left(x^2\right)}{2})+x\log \left(x^2\right)-2(\log (x)-\frac{\log \left(x^2\right)}{2})\log (\log \left(x^2\right))-\log \left(x^2\right)\log (\log \left(x^2\right))))\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-2*Log[x] + Log[-6*x^3 + 2*x*(Log[x] - Log[x^2]/2) + x*Log[x^2] - 2*(Log[x] - Log[x^2]/2)*Log[Log[x^2]] - L
og[x^2]*Log[Log[x^2]]])

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 39, normalized size = 1.62 $$\begin{array}{c}-4\log (x)+2\log (2\log (x))+2\log \left(\frac{3x^3-x\log (x)+\log (x)\log (2\log (x))}{\log (x)}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x)+2)*log(x^2)*log(log(x^2))+(2*x*log(x)+6*x^3-2*x)*log(x^2)+4*log(x))/(x*log(x)*log(x^2)*l
og(log(x^2))+(-x^2*log(x)+3*x^4)*log(x^2)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{2((2\log (x)-1)\log \left(x^2\right)\log (\log \left(x^2\right))-(3x^3+x\log (x)-x)\log \left(x^2\right)-2\log (x))}{x\log \left(x^2\right)\log (x)\log (\log \left(x^2\right))+(3x^4-x^2\log (x))\log \left(x^2\right)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x)+2)*log(x^2)*log(log(x^2))+(2*x*log(x)+6*x^3-2*x)*log(x^2)+4*log(x))/(x*log(x)*log(x^2)*l
og(log(x^2))+(-x^2*log(x)+3*x^4)*log(x^2)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.10, size = 65, normalized size = 2.71

method	result	size
risch	$-4\ln (x)+2\ln (\ln (x))+2\ln (\ln (2\ln (x)-\frac{i\pi \mathrm{csgn}\left(i{x}^2\right){(-\mathrm{csgn}\left(i{x}^2\right)+\mathrm{csgn}\left(ix\right))}^2}{2})+\frac{x(3x^2-\ln (x))}{\ln (x)})$	$65$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*ln(x)+2)*ln(x^2)*ln(ln(x^2))+(2*x*ln(x)+6*x^3-2*x)*ln(x^2)+4*ln(x))/(x*ln(x)*ln(x^2)*ln(ln(x^2))+(-x^
2*ln(x)+3*x^4)*ln(x^2)),x,method=_RETURNVERBOSE)

[Out]

-4*ln(x)+2*ln(ln(x))+2*ln(ln(2*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2)+x*(3*x^2-ln(x))/ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 40, normalized size = 1.67 $$\begin{array}{c}-4\log (x)+2\log \left(\frac{3x^3-(x-\log (2))\log (x)+\log (x)\log (\log (x))}{\log (x)}\right)+2\log (\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(x)+2)*log(x^2)*log(log(x^2))+(2*x*log(x)+6*x^3-2*x)*log(x^2)+4*log(x))/(x*log(x)*log(x^2)*l
og(log(x^2))+(-x^2*log(x)+3*x^4)*log(x^2)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 $$\begin{array}{c}\int -\frac{4\ln (x)+\ln \left(x^2\right)(2x\ln (x)-2x+6x^3)-\ln \left(x^2\right)\ln (\ln \left(x^2\right))(4\ln (x)-2)}{\ln \left(x^2\right)(x^2\ln (x)-3x^4)-x\ln \left(x^2\right)\ln (\ln \left(x^2\right))\ln (x)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(x) + log(x^2)*(2*x*log(x) - 2*x + 6*x^3) - log(x^2)*log(log(x^2))*(4*log(x) - 2))/(log(x^2)*(x^2*l
og(x) - 3*x^4) - x*log(x^2)*log(log(x^2))*log(x)),x)

[Out]

int(-(4*log(x) + log(x^2)*(2*x*log(x) - 2*x + 6*x^3) - log(x^2)*log(log(x^2))*(4*log(x) - 2))/(log(x^2)*(x^2*l
og(x) - 3*x^4) - x*log(x^2)*log(log(x^2))*log(x)), x)

________________________________________________________________________________________

sympy [A]  time = 0.44, size = 34, normalized size = 1.42 $$\begin{array}{c}-4\log (x)+2\log (\frac{3x^3-x\log (x)}{\log (x)}+\log (2\log (x)))+2\log (\log (x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________