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

Optimal. Leaf size=16 \[ -1+\frac {x \log (x)}{\log (4+x-\log (x))} \]

________________________________________________________________________________________

Rubi [F]  time = 0.88, 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)+\left (-4-x+(-3-x) \log (x)+\log ^2(x)\right ) \log (4+x-\log (x))}{(-4-x+\log (x)) \log ^2(4+x-\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 14, normalized size = 0.88 \begin {gather*} \frac {x \log (x)}{\log (4+x-\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[x])/Log[4 + x - Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.78, size = 14, normalized size = 0.88 \begin {gather*} \frac {x \log \relax (x)}{\log \left (x - \log \relax (x) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+(-3-x)*log(x)-x-4)*log(-log(x)+4+x)+(x-1)*log(x))/(log(x)-x-4)/log(-log(x)+4+x)^2,x, algo
rithm="fricas")

[Out]

x*log(x)/log(x - log(x) + 4)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 14, normalized size = 0.88 \begin {gather*} \frac {x \log \relax (x)}{\log \left (x - \log \relax (x) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+(-3-x)*log(x)-x-4)*log(-log(x)+4+x)+(x-1)*log(x))/(log(x)-x-4)/log(-log(x)+4+x)^2,x, algo
rithm="giac")

[Out]

x*log(x)/log(x - log(x) + 4)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 15, normalized size = 0.94

method	result	size
risch	\(\frac {x \ln \relax (x )}{\ln \left (-\ln \relax (x )+4+x \right )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)^2+(-3-x)*ln(x)-x-4)*ln(-ln(x)+4+x)+(x-1)*ln(x))/(ln(x)-x-4)/ln(-ln(x)+4+x)^2,x,method=_RETURNVERBO
SE)

[Out]

x/ln(-ln(x)+4+x)*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 14, normalized size = 0.88 \begin {gather*} \frac {x \log \relax (x)}{\log \left (x - \log \relax (x) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+(-3-x)*log(x)-x-4)*log(-log(x)+4+x)+(x-1)*log(x))/(log(x)-x-4)/log(-log(x)+4+x)^2,x, algo
rithm="maxima")

[Out]

x*log(x)/log(x - log(x) + 4)

________________________________________________________________________________________

mupad [B]  time = 4.74, size = 82, normalized size = 5.12 \begin {gather*} x+3\,\ln \relax (x)+\frac {5}{x-1}-{\ln \relax (x)}^2\,\left (\frac {1}{x-1}+1\right )+\frac {x\,\ln \relax (x)-\frac {x\,\ln \left (x-\ln \relax (x)+4\right )\,\left (\ln \relax (x)+1\right )\,\left (x-\ln \relax (x)+4\right )}{x-1}}{\ln \left (x-\ln \relax (x)+4\right )}+\frac {\ln \relax (x)\,\left (x^2+3\right )}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(x - 1) - log(x - log(x) + 4)*(x + log(x)*(x + 3) - log(x)^2 + 4))/(log(x - log(x) + 4)^2*(x - lo
g(x) + 4)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.35, size = 12, normalized size = 0.75 \begin {gather*} \frac {x \log {\relax (x )}}{\log {\left (x - \log {\relax (x )} + 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x)/log(x - log(x) + 4)

________________________________________________________________________________________