3.99.55 \(\int \frac {-4-x^2-\log (2 x)}{(5 x-x^3+x \log (2 x)) \log (\frac {-5+x^2-\log (2 x)}{x})} \, dx\)

Optimal. Leaf size=15 \[ \log \left (\log \left (x-\frac {5+\log (2 x)}{x}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 18, normalized size of antiderivative = 1.20, number of steps used = 1, number of rules used = 1, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6684} \begin {gather*} \log \left (\log \left (-\frac {-x^2+\log (2 x)+5}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[-((5 - x^2 + Log[2*x])/x)]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (-\frac {5-x^2+\log (2 x)}{x}\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.27, size = 17, normalized size = 1.13 \begin {gather*} \log \left (\log \left (\frac {-5+x^2-\log (2 x)}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[(-5 + x^2 - Log[2*x])/x]]

________________________________________________________________________________________

fricas [A]  time = 0.98, size = 17, normalized size = 1.13 \begin {gather*} \log \left (\log \left (\frac {x^{2} - \log \left (2 \, x\right ) - 5}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log((x^2 - log(2*x) - 5)/x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + \log \left (2 \, x\right ) + 4}{{\left (x^{3} - x \log \left (2 \, x\right ) - 5 \, x\right )} \log \left (\frac {x^{2} - \log \left (2 \, x\right ) - 5}{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2 + log(2*x) + 4)/((x^3 - x*log(2*x) - 5*x)*log((x^2 - log(2*x) - 5)/x)), x)

________________________________________________________________________________________

maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {-\ln \left (2 x \right )-x^{2}-4}{\left (x \ln \left (2 x \right )-x^{3}+5 x \right ) \ln \left (\frac {-\ln \left (2 x \right )+x^{2}-5}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(2*x)-x^2-4)/(x*ln(2*x)-x^3+5*x)/ln((-ln(2*x)+x^2-5)/x),x)

[Out]

int((-ln(2*x)-x^2-4)/(x*ln(2*x)-x^3+5*x)/ln((-ln(2*x)+x^2-5)/x),x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 20, normalized size = 1.33 \begin {gather*} \log \left (\log \left (x^{2} - \log \relax (2) - \log \relax (x) - 5\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x^2 - log(2) - log(x) - 5) - log(x))

________________________________________________________________________________________

mupad [B]  time = 6.13, size = 18, normalized size = 1.20 \begin {gather*} \ln \left (\ln \left (-\frac {\ln \left (2\,x\right )-x^2+5}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(-(log(2*x) - x^2 + 5)/x))

________________________________________________________________________________________

sympy [A]  time = 0.45, size = 14, normalized size = 0.93 \begin {gather*} \log {\left (\log {\left (\frac {x^{2} - \log {\left (2 x \right )} - 5}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log((x**2 - log(2*x) - 5)/x))

________________________________________________________________________________________