3.73.13 $\int \frac{-8+2x^4-4x^4\log (x^2)\log (\log (x^2))+5x\log (x^2){\log }^2(\log (x^2))}{(4x-x^5)\log (x^2)\log (\log (x^2))+5x^2\log (x^2){\log }^2(\log (x^2))}dx$

Optimal. Leaf size=21 $$\log (-x+\frac{-4+x^4}{5\log (\log \left(x^2\right))})$$

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Rubi [F]  time = 2.04, 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 = {} $$\begin{array}{c}\int \frac{-8+2x^4-4x^4\log \left(x^2\right)\log (\log \left(x^2\right))+5x\log \left(x^2\right){\log }^2(\log \left(x^2\right))}{(4x-x^5)\log \left(x^2\right)\log (\log \left(x^2\right))+5x^2\log \left(x^2\right){\log }^2(\log \left(x^2\right))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-8 + 2*x^4 - 4*x^4*Log[x^2]*Log[Log[x^2]] + 5*x*Log[x^2]*Log[Log[x^2]]^2)/((4*x - x^5)*Log[x^2]*Log[Log[x
^2]] + 5*x^2*Log[x^2]*Log[Log[x^2]]^2),x]

[Out]

Log[x] - Log[Log[Log[x^2]]] + 4*Defer[Int][1/(x*(-4 + x^4 - 5*x*Log[Log[x^2]])), x] + 3*Defer[Int][x^3/(-4 + x
^4 - 5*x*Log[Log[x^2]]), x] - 10*Defer[Int][1/(Log[x^2]*(-4 + x^4 - 5*x*Log[Log[x^2]])), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-8+2x^4-4x^4\log \left(x^2\right)\log (\log \left(x^2\right))+5x\log \left(x^2\right){\log }^2(\log \left(x^2\right))}{x\log \left(x^2\right)\log (\log \left(x^2\right))(4-x^4+5x\log (\log \left(x^2\right)))}dx\\ & =\int (\frac{1}{x}-\frac{2}{x\log \left(x^2\right)\log (\log \left(x^2\right))}+\frac{-10x+4\log \left(x^2\right)+3x^4\log \left(x^2\right)}{x\log \left(x^2\right)(-4+x^4-5x\log (\log \left(x^2\right)))})dx\\ & =\log (x)-2\int \frac{1}{x\log \left(x^2\right)\log (\log \left(x^2\right))}dx+\int \frac{-10x+4\log \left(x^2\right)+3x^4\log \left(x^2\right)}{x\log \left(x^2\right)(-4+x^4-5x\log (\log \left(x^2\right)))}dx\\ & =\log (x)+\int (\frac{4}{x(-4+x^4-5x\log (\log \left(x^2\right)))}+\frac{3x^3}{-4+x^4-5x\log (\log \left(x^2\right))}-\frac{10}{\log \left(x^2\right)(-4+x^4-5x\log (\log \left(x^2\right)))})dx-\mathrm{Subst}(\int \frac{1}{x\log (x)}dx,x,\log \left(x^2\right))\\ & =\log (x)+3\int \frac{x^3}{-4+x^4-5x\log (\log \left(x^2\right))}dx+4\int \frac{1}{x(-4+x^4-5x\log (\log \left(x^2\right)))}dx-10\int \frac{1}{\log \left(x^2\right)(-4+x^4-5x\log (\log \left(x^2\right)))}dx-\mathrm{Subst}(\int \frac{1}{x}dx,x,\log (\log \left(x^2\right)))\\ & =\log (x)-\log (\log (\log \left(x^2\right)))+3\int \frac{x^3}{-4+x^4-5x\log (\log \left(x^2\right))}dx+4\int \frac{1}{x(-4+x^4-5x\log (\log \left(x^2\right)))}dx-10\int \frac{1}{\log \left(x^2\right)(-4+x^4-5x\log (\log \left(x^2\right)))}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.16, size = 25, normalized size = 1.19 $$\begin{array}{c}-\log (\log (\log \left(x^2\right)))+\log (4-x^4+5x\log (\log \left(x^2\right)))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 2*x^4 - 4*x^4*Log[x^2]*Log[Log[x^2]] + 5*x*Log[x^2]*Log[Log[x^2]]^2)/((4*x - x^5)*Log[x^2]*Log
[Log[x^2]] + 5*x^2*Log[x^2]*Log[Log[x^2]]^2),x]

[Out]

-Log[Log[Log[x^2]]] + Log[4 - x^4 + 5*x*Log[Log[x^2]]]

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fricas [A]  time = 0.65, size = 34, normalized size = 1.62 $$\begin{array}{c}\frac{1}{2}\log \left(x^2\right)+\log (-\frac{x^4-5x\log (\log \left(x^2\right))-4}{x})-\log (\log (\log \left(x^2\right)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*log(x^2)*log(log(x^2))^2-4*x^4*log(x^2)*log(log(x^2))+2*x^4-8)/(5*x^2*log(x^2)*log(log(x^2))^2+
(-x^5+4*x)*log(x^2)*log(log(x^2))),x, algorithm="fricas")

[Out]

1/2*log(x^2) + log(-(x^4 - 5*x*log(log(x^2)) - 4)/x) - log(log(log(x^2)))

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giac [A]  time = 0.30, size = 25, normalized size = 1.19 $$\begin{array}{c}\log (-x^4+5x\log (\log \left(x^2\right))+4)-\log (\log (\log \left(x^2\right)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*log(x^2)*log(log(x^2))^2-4*x^4*log(x^2)*log(log(x^2))+2*x^4-8)/(5*x^2*log(x^2)*log(log(x^2))^2+
(-x^5+4*x)*log(x^2)*log(log(x^2))),x, algorithm="giac")

[Out]

log(-x^4 + 5*x*log(log(x^2)) + 4) - log(log(log(x^2)))

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 $$\int \frac{5x\ln \left(x^2\right)\ln {(\ln \left(x^2\right))}^2-4x^4\ln \left(x^2\right)\ln (\ln \left(x^2\right))+2x^4-8}{5x^2\ln \left(x^2\right)\ln {(\ln \left(x^2\right))}^2+(-x^5+4x)\ln \left(x^2\right)\ln (\ln \left(x^2\right))}dx$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x*ln(x^2)*ln(ln(x^2))^2-4*x^4*ln(x^2)*ln(ln(x^2))+2*x^4-8)/(5*x^2*ln(x^2)*ln(ln(x^2))^2+(-x^5+4*x)*ln(x
^2)*ln(ln(x^2))),x)

[Out]

int((5*x*ln(x^2)*ln(ln(x^2))^2-4*x^4*ln(x^2)*ln(ln(x^2))+2*x^4-8)/(5*x^2*ln(x^2)*ln(ln(x^2))^2+(-x^5+4*x)*ln(x
^2)*ln(ln(x^2))),x)

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maxima [A]  time = 0.49, size = 34, normalized size = 1.62 $$\begin{array}{c}\log (x)+\log (-\frac{x^4-5x\log (2)-5x\log (\log (x))-4}{5x})-\log (\log (2)+\log (\log (x)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*log(x^2)*log(log(x^2))^2-4*x^4*log(x^2)*log(log(x^2))+2*x^4-8)/(5*x^2*log(x^2)*log(log(x^2))^2+
(-x^5+4*x)*log(x^2)*log(log(x^2))),x, algorithm="maxima")

[Out]

log(x) + log(-1/5*(x^4 - 5*x*log(2) - 5*x*log(log(x)) - 4)/x) - log(log(2) + log(log(x)))

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mupad [B]  time = 4.73, size = 78, normalized size = 3.71 $$\begin{array}{c}\ln \left(\frac{20x\ln (\ln \left(x^2\right))-4x^4+16}{\ln \left(x^2\right)}\right)-\ln \left(\frac{\ln (\ln \left(x^2\right))(4\ln \left(x^2\right)-10x+3x^4\ln \left(x^2\right))}{\ln \left(x^2\right)}\right)+\ln (4\ln \left(x^2\right)-10x+3x^4\ln \left(x^2\right))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4 + 5*x*log(x^2)*log(log(x^2))^2 - 4*x^4*log(x^2)*log(log(x^2)) - 8)/(5*x^2*log(x^2)*log(log(x^2))^2
+ log(x^2)*log(log(x^2))*(4*x - x^5)),x)

[Out]

log((20*x*log(log(x^2)) - 4*x^4 + 16)/log(x^2)) - log((log(log(x^2))*(4*log(x^2) - 10*x + 3*x^4*log(x^2)))/log
(x^2)) + log(4*log(x^2) - 10*x + 3*x^4*log(x^2))

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sympy [A]  time = 0.75, size = 29, normalized size = 1.38 $$\begin{array}{c}\log (x)+\log (\log (\log \left(x^2\right))+\frac{8-2x^4}{10x})-\log (\log (\log \left(x^2\right)))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*ln(x**2)*ln(ln(x**2))**2-4*x**4*ln(x**2)*ln(ln(x**2))+2*x**4-8)/(5*x**2*ln(x**2)*ln(ln(x**2))**
2+(-x**5+4*x)*ln(x**2)*ln(ln(x**2))),x)

[Out]

log(x) + log(log(log(x**2)) + (8 - 2*x**4)/(10*x)) - log(log(log(x**2)))

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