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

Optimal. Leaf size=15 \[ 1+\log \left (\frac {1}{x+\frac {4+x}{\log (x)}}\right ) \]

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Rubi [F]  time = 0.45, 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 {4+x-x \log (x)-x \log ^2(x)}{\left (4 x+x^2\right ) \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 14, normalized size = 0.93 \begin {gather*} \log (\log (x))-\log (4+x+x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[x]] - Log[4 + x + x*Log[x]]

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fricas [A]  time = 0.83, size = 22, normalized size = 1.47 \begin {gather*} -\log \relax (x) - \log \left (\frac {x \log \relax (x) + x + 4}{x}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 14, normalized size = 0.93 \begin {gather*} -\log \left (x \log \relax (x) + x + 4\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x*log(x) + x + 4) + log(log(x))

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maple [A]  time = 0.03, size = 15, normalized size = 1.00

method	result	size
norman	\(-\ln \left (x \ln \relax (x )+x +4\right )+\ln \left (\ln \relax (x )\right )\)	\(15\)
risch	\(-\ln \relax (x )+\ln \left (\ln \relax (x )\right )-\ln \left (\ln \relax (x )+\frac {4+x}{x}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x*ln(x)+x+4)+ln(ln(x))

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maxima [A]  time = 0.38, size = 22, normalized size = 1.47 \begin {gather*} -\log \relax (x) - \log \left (\frac {x \log \relax (x) + x + 4}{x}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.85, size = 14, normalized size = 0.93 \begin {gather*} \ln \left (\ln \relax (x)\right )-\ln \left (x+x\,\ln \relax (x)+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x)) - log(x + x*log(x) + 4)

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sympy [A]  time = 0.27, size = 20, normalized size = 1.33 \begin {gather*} - \log {\relax (x )} - \log {\left (\log {\relax (x )} + \frac {2 x + 8}{2 x} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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