3.98.24 \(\int \frac {1+(x^2+x^3+2 x^4) \log (x)-2 \log (x) \log (\log (x))}{(-6 x^3+x^4+x^5) \log (x)+x^3 \log ^2(x)+x \log (x) \log (\log (x))} \, dx\)

Optimal. Leaf size=18 \[ \log \left (6 \left (-6+x+x^2+\log (x)+\frac {\log (\log (x))}{x^2}\right )\right ) \]

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Rubi [A]  time = 0.73, antiderivative size = 34, normalized size of antiderivative = 1.89, number of steps used = 4, number of rules used = 3, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {6741, 6742, 6684} \begin {gather*} \log \left (-x^4-x^3+6 x^2-x^2 \log (x)-\log (\log (x))\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (x^2 + x^3 + 2*x^4)*Log[x] - 2*Log[x]*Log[Log[x]])/((-6*x^3 + x^4 + x^5)*Log[x] + x^3*Log[x]^2 + x*Lo
g[x]*Log[Log[x]]),x]

[Out]

-2*Log[x] + Log[6*x^2 - x^3 - x^4 - x^2*Log[x] - Log[Log[x]]]

Rule 6684

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-\left (x^2+x^3+2 x^4\right ) \log (x)+2 \log (x) \log (\log (x))}{x \log (x) \left (6 x^2-x^3-x^4-x^2 \log (x)-\log (\log (x))\right )} \, dx\\ &=\int \left (-\frac {2}{x}+\frac {1-11 x^2 \log (x)+3 x^3 \log (x)+4 x^4 \log (x)+2 x^2 \log ^2(x)}{x \log (x) \left (-6 x^2+x^3+x^4+x^2 \log (x)+\log (\log (x))\right )}\right ) \, dx\\ &=-2 \log (x)+\int \frac {1-11 x^2 \log (x)+3 x^3 \log (x)+4 x^4 \log (x)+2 x^2 \log ^2(x)}{x \log (x) \left (-6 x^2+x^3+x^4+x^2 \log (x)+\log (\log (x))\right )} \, dx\\ &=-2 \log (x)+\log \left (6 x^2-x^3-x^4-x^2 \log (x)-\log (\log (x))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 27, normalized size = 1.50 \begin {gather*} -2 \log (x)+\log \left (-6 x^2+x^3+x^4+x^2 \log (x)+\log (\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + (x^2 + x^3 + 2*x^4)*Log[x] - 2*Log[x]*Log[Log[x]])/((-6*x^3 + x^4 + x^5)*Log[x] + x^3*Log[x]^2
+ x*Log[x]*Log[Log[x]]),x]

[Out]

-2*Log[x] + Log[-6*x^2 + x^3 + x^4 + x^2*Log[x] + Log[Log[x]]]

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fricas [A]  time = 0.64, size = 27, normalized size = 1.50 \begin {gather*} \log \left (x^{4} + x^{3} + x^{2} \log \relax (x) - 6 \, x^{2} + \log \left (\log \relax (x)\right )\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)*log(log(x))+(2*x^4+x^3+x^2)*log(x)+1)/(x*log(x)*log(log(x))+x^3*log(x)^2+(x^5+x^4-6*x^3)*
log(x)),x, algorithm="fricas")

[Out]

log(x^4 + x^3 + x^2*log(x) - 6*x^2 + log(log(x))) - 2*log(x)

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giac [A]  time = 0.26, size = 27, normalized size = 1.50 \begin {gather*} \log \left (x^{4} + x^{3} + x^{2} \log \relax (x) - 6 \, x^{2} + \log \left (\log \relax (x)\right )\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)*log(log(x))+(2*x^4+x^3+x^2)*log(x)+1)/(x*log(x)*log(log(x))+x^3*log(x)^2+(x^5+x^4-6*x^3)*
log(x)),x, algorithm="giac")

[Out]

log(x^4 + x^3 + x^2*log(x) - 6*x^2 + log(log(x))) - 2*log(x)

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maple [A]  time = 0.03, size = 28, normalized size = 1.56

method	result	size
risch	\(-2 \ln \relax (x )+\ln \left (x^{4}+x^{2} \ln \relax (x )+x^{3}-6 x^{2}+\ln \left (\ln \relax (x )\right )\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(x)*ln(ln(x))+(2*x^4+x^3+x^2)*ln(x)+1)/(x*ln(x)*ln(ln(x))+x^3*ln(x)^2+(x^5+x^4-6*x^3)*ln(x)),x,metho
d=_RETURNVERBOSE)

[Out]

-2*ln(x)+ln(x^4+x^2*ln(x)+x^3-6*x^2+ln(ln(x)))

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maxima [A]  time = 0.42, size = 27, normalized size = 1.50 \begin {gather*} \log \left (x^{4} + x^{3} + x^{2} \log \relax (x) - 6 \, x^{2} + \log \left (\log \relax (x)\right )\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x)*log(log(x))+(2*x^4+x^3+x^2)*log(x)+1)/(x*log(x)*log(log(x))+x^3*log(x)^2+(x^5+x^4-6*x^3)*
log(x)),x, algorithm="maxima")

[Out]

log(x^4 + x^3 + x^2*log(x) - 6*x^2 + log(log(x))) - 2*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {\ln \relax (x)\,\left (2\,x^4+x^3+x^2\right )-2\,\ln \left (\ln \relax (x)\right )\,\ln \relax (x)+1}{x^3\,{\ln \relax (x)}^2+\ln \relax (x)\,\left (x^5+x^4-6\,x^3\right )+x\,\ln \left (\ln \relax (x)\right )\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(x^2 + x^3 + 2*x^4) - 2*log(log(x))*log(x) + 1)/(x^3*log(x)^2 + log(x)*(x^4 - 6*x^3 + x^5) + x*log
(log(x))*log(x)),x)

[Out]

int((log(x)*(x^2 + x^3 + 2*x^4) - 2*log(log(x))*log(x) + 1)/(x^3*log(x)^2 + log(x)*(x^4 - 6*x^3 + x^5) + x*log
(log(x))*log(x)), x)

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sympy [A]  time = 0.41, size = 29, normalized size = 1.61 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (x^{4} + x^{3} + x^{2} \log {\relax (x )} - 6 x^{2} + \log {\left (\log {\relax (x )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(x)*ln(ln(x))+(2*x**4+x**3+x**2)*ln(x)+1)/(x*ln(x)*ln(ln(x))+x**3*ln(x)**2+(x**5+x**4-6*x**3)*
ln(x)),x)

[Out]

-2*log(x) + log(x**4 + x**3 + x**2*log(x) - 6*x**2 + log(log(x)))

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