3.37.5 \(\int \frac {\log (\frac {e^3}{x}) (-1+\log (x))+\log (x)}{\log ^2(\frac {e^3}{x}) \log ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ \frac {x}{\log \left (\frac {e^3}{x}\right ) \log (x)} \]

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Rubi [F]  time = 0.25, 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 {\log \left (\frac {e^3}{x}\right ) (-1+\log (x))+\log (x)}{\log ^2\left (\frac {e^3}{x}\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Log[E^3/x]*(-1 + Log[x]) + Log[x])/(Log[E^3/x]^2*Log[x]^2),x]

[Out]

-Defer[Int][1/((3 + Log[x^(-1)])*Log[x]^2), x] + 4*Defer[Int][1/((3 + Log[x^(-1)])^2*Log[x]), x] + Defer[Int][
Log[x^(-1)]/((3 + Log[x^(-1)])^2*Log[x]), x]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\left (3+\log \left (\frac {1}{x}\right )\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/((3 + Log[x^(-1)])*Log[x])

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fricas [A]  time = 0.79, size = 14, normalized size = 0.88 \begin {gather*} -\frac {x}{\log \relax (x)^{2} - 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(log(x)^2 - 3*log(x))

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giac [A]  time = 0.16, size = 14, normalized size = 0.88 \begin {gather*} -\frac {x}{\log \relax (x)^{2} - 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(log(x)^2 - 3*log(x))

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maple [A]  time = 0.25, size = 16, normalized size = 1.00

method	result	size
norman	\(\frac {x}{\ln \left (\frac {{\mathrm e}^{3}}{x}\right ) \ln \relax (x )}\)	\(16\)
risch	\(\frac {2 i x}{\left (-2 i \ln \relax (x )+6 i\right ) \ln \relax (x )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)-1)*ln(exp(3)/x)+ln(x))/ln(x)^2/ln(exp(3)/x)^2,x,method=_RETURNVERBOSE)

[Out]

x/ln(exp(3)/x)/ln(x)

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maxima [A]  time = 0.48, size = 14, normalized size = 0.88 \begin {gather*} -\frac {x}{\log \relax (x)^{2} - 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(log(x)^2 - 3*log(x))

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mupad [B]  time = 2.32, size = 14, normalized size = 0.88 \begin {gather*} \frac {x}{\ln \relax (x)\,\left (\ln \left (\frac {1}{x}\right )+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) + log(exp(3)/x)*(log(x) - 1))/(log(exp(3)/x)^2*log(x)^2),x)

[Out]

x/(log(x)*(log(1/x) + 3))

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sympy [A]  time = 0.24, size = 12, normalized size = 0.75 \begin {gather*} - \frac {x}{\log {\relax (x )}^{2} - 3 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(x)-1)*ln(exp(3)/x)+ln(x))/ln(x)**2/ln(exp(3)/x)**2,x)

[Out]

-x/(log(x)**2 - 3*log(x))

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