3.10.53 \(\int \frac {(1+x) \log (x)}{-1+x \log (x)} \, dx\)

Optimal. Leaf size=16 \[ x+\log \left (-1-x^2+x (x+\log (x))\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.26, 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 {(1+x) \log (x)}{-1+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 10, normalized size = 0.62 \begin {gather*} x+\log (1-x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*Log[x])/(-1 + x*Log[x]),x]

[Out]

x + Log[1 - x*Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 15, normalized size = 0.94 \begin {gather*} x + \log \relax (x) + \log \left (\frac {x \log \relax (x) - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*log(x)/(x*log(x)-1),x, algorithm="fricas")

[Out]

x + log(x) + log((x*log(x) - 1)/x)

________________________________________________________________________________________

giac [A]  time = 0.30, size = 9, normalized size = 0.56 \begin {gather*} x + \log \left (x \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*log(x)/(x*log(x)-1),x, algorithm="giac")

[Out]

x + log(x*log(x) - 1)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 10, normalized size = 0.62

method	result	size
norman	\(x +\ln \left (x \ln \relax (x )-1\right )\)	\(10\)
risch	\(x +\ln \relax (x )+\ln \left (\ln \relax (x )-\frac {1}{x}\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*ln(x)/(x*ln(x)-1),x,method=_RETURNVERBOSE)

[Out]

x+ln(x*ln(x)-1)

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 15, normalized size = 0.94 \begin {gather*} x + \log \relax (x) + \log \left (\frac {x \log \relax (x) - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*log(x)/(x*log(x)-1),x, algorithm="maxima")

[Out]

x + log(x) + log((x*log(x) - 1)/x)

________________________________________________________________________________________

mupad [B]  time = 0.74, size = 9, normalized size = 0.56 \begin {gather*} x+\ln \left (x\,\ln \relax (x)-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(x + 1))/(x*log(x) - 1),x)

[Out]

x + log(x*log(x) - 1)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 12, normalized size = 0.75 \begin {gather*} x + \log {\relax (x )} + \log {\left (\log {\relax (x )} - \frac {1}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+1)*ln(x)/(x*ln(x)-1),x)

[Out]

x + log(x) + log(log(x) - 1/x)

________________________________________________________________________________________