3.16.41 \(\int \frac {2 \log (\frac {625}{331776 e^2})}{x-2 x \log (x^2)+x \log ^2(x^2)} \, dx\)

Optimal. Leaf size=19 \[ \frac {\log \left (\frac {625}{331776 e^2}\right )}{1-\log \left (x^2\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 32} \begin {gather*} -\frac {2+\log \left (\frac {331776}{625}\right )}{1-\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*Log[625/(331776*E^2)])/(x - 2*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

-((2 + Log[331776/625])/(1 - Log[x^2]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (2 \left (2+\log \left (\frac {331776}{625}\right )\right )\right ) \int \frac {1}{x-2 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx\right )\\ &=-\left (\left (2+\log \left (\frac {331776}{625}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x)^2} \, dx,x,\log \left (x^2\right )\right )\right )\\ &=-\frac {2+\log \left (\frac {331776}{625}\right )}{1-\log \left (x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} \frac {2+\log \left (\frac {331776}{625}\right )}{-1+\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*Log[625/(331776*E^2)])/(x - 2*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

(2 + Log[331776/625])/(-1 + Log[x^2])

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fricas [A]  time = 0.69, size = 14, normalized size = 0.74 \begin {gather*} -\frac {\log \left (\frac {625}{331776}\right ) - 2}{\log \left (x^{2}\right ) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(625/331776/exp(2))/(x*log(x^2)^2-2*x*log(x^2)+x),x, algorithm="fricas")

[Out]

-(log(625/331776) - 2)/(log(x^2) - 1)

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giac [A]  time = 0.12, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\log \left (\frac {625}{331776} \, e^{\left (-2\right )}\right )}{\log \left (x^{2}\right ) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(625/331776/exp(2))/(x*log(x^2)^2-2*x*log(x^2)+x),x, algorithm="giac")

[Out]

-log(625/331776*e^(-2))/(log(x^2) - 1)

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maple [A]  time = 0.04, size = 18, normalized size = 0.95

method	result	size
default	\(-\frac {\ln \left (\frac {625 \,{\mathrm e}^{-2}}{331776}\right )}{\ln \left (x^{2}\right )-1}\)	\(18\)
norman	\(\frac {2-4 \ln \relax (5)+4 \ln \left (24\right )}{\ln \left (x^{2}\right )-1}\)	\(20\)
risch	\(-\frac {4 \ln \relax (5)}{\ln \left (x^{2}\right )-1}+\frac {12 \ln \relax (2)}{\ln \left (x^{2}\right )-1}+\frac {4 \ln \relax (3)}{\ln \left (x^{2}\right )-1}+\frac {2}{\ln \left (x^{2}\right )-1}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*ln(625/331776/exp(2))/(x*ln(x^2)^2-2*x*ln(x^2)+x),x,method=_RETURNVERBOSE)

[Out]

-ln(625/331776/exp(2))/(ln(x^2)-1)

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maxima [A]  time = 0.52, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\log \left (\frac {625}{331776} \, e^{\left (-2\right )}\right )}{2 \, \log \relax (x) - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(625/331776/exp(2))/(x*log(x^2)^2-2*x*log(x^2)+x),x, algorithm="maxima")

[Out]

-log(625/331776*e^(-2))/(2*log(x) - 1)

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mupad [B]  time = 1.08, size = 14, normalized size = 0.74 \begin {gather*} \frac {\ln \left (\frac {331776\,{\mathrm {e}}^2}{625}\right )}{\ln \left (x^2\right )-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log((625*exp(-2))/331776))/(x - 2*x*log(x^2) + x*log(x^2)^2),x)

[Out]

log((331776*exp(2))/625)/(log(x^2) - 1)

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sympy [A]  time = 0.10, size = 17, normalized size = 0.89 \begin {gather*} \frac {- 4 \log {\relax (5 )} + 2 + 4 \log {\left (24 \right )}}{\log {\left (x^{2} \right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*ln(625/331776/exp(2))/(x*ln(x**2)**2-2*x*ln(x**2)+x),x)

[Out]

(-4*log(5) + 2 + 4*log(24))/(log(x**2) - 1)

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