3.81.84 \(\int \frac {2+2 x \log (x)-11 x \log ^2(x)}{-2 x \log (x)+(11 x+e^x x) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ 22-x+\log \left (2+\frac {1}{4} \left (3+e^x-\frac {2}{\log (x)}\right )\right ) \]

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Rubi [F]  time = 0.84, 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 {2+2 x \log (x)-11 x \log ^2(x)}{-2 x \log (x)+\left (11 x+e^x x\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 2*x*Log[x] - 11*x*Log[x]^2)/(-2*x*Log[x] + (11*x + E^x*x)*Log[x]^2),x]

[Out]

2*Defer[Int][(-2 + 11*Log[x] + E^x*Log[x])^(-1), x] + 2*Defer[Int][1/(x*Log[x]*(-2 + 11*Log[x] + E^x*Log[x])),
x] - 11*Defer[Int][Log[x]/(-2 + 11*Log[x] + E^x*Log[x]), x]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 23, normalized size = 1.00 \begin {gather*} -x-\log (\log (x))+\log \left (2-11 \log (x)-e^x \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*x*Log[x] - 11*x*Log[x]^2)/(-2*x*Log[x] + (11*x + E^x*x)*Log[x]^2),x]

[Out]

-x - Log[Log[x]] + Log[2 - 11*Log[x] - E^x*Log[x]]

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fricas [A]  time = 1.01, size = 31, normalized size = 1.35 \begin {gather*} -x + \log \left (\frac {{\left (e^{x} + 11\right )} \log \relax (x) - 2}{e^{x} + 11}\right ) + \log \left (e^{x} + 11\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11*x*log(x)^2+2*x*log(x)+2)/((exp(x)*x+11*x)*log(x)^2-2*x*log(x)),x, algorithm="fricas")

[Out]

-x + log(((e^x + 11)*log(x) - 2)/(e^x + 11)) + log(e^x + 11) - log(log(x))

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giac [A]  time = 0.18, size = 21, normalized size = 0.91 \begin {gather*} -x + \log \left (e^{x} \log \relax (x) + 11 \, \log \relax (x) - 2\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11*x*log(x)^2+2*x*log(x)+2)/((exp(x)*x+11*x)*log(x)^2-2*x*log(x)),x, algorithm="giac")

[Out]

-x + log(e^x*log(x) + 11*log(x) - 2) - log(log(x))

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maple [A]  time = 0.04, size = 22, normalized size = 0.96

method	result	size
norman	\(-x -\ln \left (\ln \relax (x )\right )+\ln \left ({\mathrm e}^{x} \ln \relax (x )+11 \ln \relax (x )-2\right )\)	\(22\)
risch	\(-x +\ln \left ({\mathrm e}^{x}+11\right )-\ln \left (\ln \relax (x )\right )+\ln \left (\ln \relax (x )-\frac {2}{{\mathrm e}^{x}+11}\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-11*x*ln(x)^2+2*x*ln(x)+2)/((exp(x)*x+11*x)*ln(x)^2-2*x*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-x-ln(ln(x))+ln(exp(x)*ln(x)+11*ln(x)-2)

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maxima [A]  time = 0.42, size = 21, normalized size = 0.91 \begin {gather*} -x + \log \left (\frac {e^{x} \log \relax (x) + 11 \, \log \relax (x) - 2}{\log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11*x*log(x)^2+2*x*log(x)+2)/((exp(x)*x+11*x)*log(x)^2-2*x*log(x)),x, algorithm="maxima")

[Out]

-x + log((e^x*log(x) + 11*log(x) - 2)/log(x))

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mupad [B]  time = 5.60, size = 21, normalized size = 0.91 \begin {gather*} \ln \left (11\,\ln \relax (x)+{\mathrm {e}}^x\,\ln \relax (x)-2\right )-x-\ln \left (\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*log(x) - 11*x*log(x)^2 + 2)/(log(x)^2*(11*x + x*exp(x)) - 2*x*log(x)),x)

[Out]

log(11*log(x) + exp(x)*log(x) - 2) - x - log(log(x))

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sympy [A]  time = 0.45, size = 15, normalized size = 0.65 \begin {gather*} - x + \log {\left (\frac {11 \log {\relax (x )} - 2}{\log {\relax (x )}} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-11*x*ln(x)**2+2*x*ln(x)+2)/((exp(x)*x+11*x)*ln(x)**2-2*x*ln(x)),x)

[Out]

-x + log((11*log(x) - 2)/log(x) + exp(x))

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