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

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

________________________________________________________________________________________

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 {gather*} \int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (4 x-x^5\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x^2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {-8+2 x^4-4 x^4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+5 x \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \left (4-x^4+5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\int \left (\frac {1}{x}-\frac {2}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\frac {-10 x+4 \log \left (x^2\right )+3 x^4 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx\\ &=\log (x)-2 \int \frac {1}{x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx+\int \frac {-10 x+4 \log \left (x^2\right )+3 x^4 \log \left (x^2\right )}{x \log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ &=\log (x)+\int \left (\frac {4}{x \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )}+\frac {3 x^3}{-4+x^4-5 x \log \left (\log \left (x^2\right )\right )}-\frac {10}{\log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\log \left (x^2\right )\right )\\ &=\log (x)+3 \int \frac {x^3}{-4+x^4-5 x \log \left (\log \left (x^2\right )\right )} \, dx+4 \int \frac {1}{x \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx-10 \int \frac {1}{\log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\log \left (x^2\right )\right )\right )\\ &=\log (x)-\log \left (\log \left (\log \left (x^2\right )\right )\right )+3 \int \frac {x^3}{-4+x^4-5 x \log \left (\log \left (x^2\right )\right )} \, dx+4 \int \frac {1}{x \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx-10 \int \frac {1}{\log \left (x^2\right ) \left (-4+x^4-5 x \log \left (\log \left (x^2\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

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

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]]]

________________________________________________________________________________________

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

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)))

________________________________________________________________________________________

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

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)))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 34, normalized size = 1.62 \begin {gather*} \log \relax (x) + \log \left (-\frac {x^{4} - 5 \, x \log \relax (2) - 5 \, x \log \left (\log \relax (x)\right ) - 4}{5 \, x}\right ) - \log \left (\log \relax (2) + \log \left (\log \relax (x)\right )\right ) \end {gather*}

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)))

________________________________________________________________________________________

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

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))

________________________________________________________________________________________

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

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)))

________________________________________________________________________________________