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 $$-1+\frac{x\log (x)}{\log (4+x-\log (x))}$$

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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 = {} $$\begin{array}{c}\int \frac{(-1+x)\log (x)+(-4-x+(-3-x)\log (x)+{\log }^2(x))\log (4+x-\log (x))}{(-4-x+\log (x)){\log }^2(4+x-\log (x))}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{(-1+x)\log (x)}{(4+x-\log (x)){\log }^2(4+x-\log (x))}+\frac{1+\log (x)}{\log (4+x-\log (x))})dx\\ & =-\int \frac{(-1+x)\log (x)}{(4+x-\log (x)){\log }^2(4+x-\log (x))}dx+\int \frac{1+\log (x)}{\log (4+x-\log (x))}dx\\ & =-\int (-\frac{\log (x)}{(4+x-\log (x)){\log }^2(4+x-\log (x))}+\frac{x\log (x)}{(4+x-\log (x)){\log }^2(4+x-\log (x))})dx+\int (\frac{1}{\log (4+x-\log (x))}+\frac{\log (x)}{\log (4+x-\log (x))})dx\\ & =\int \frac{\log (x)}{(4+x-\log (x)){\log }^2(4+x-\log (x))}dx-\int \frac{x\log (x)}{(4+x-\log (x)){\log }^2(4+x-\log (x))}dx+\int \frac{1}{\log (4+x-\log (x))}dx+\int \frac{\log (x)}{\log (4+x-\log (x))}dx\end{array}\end{array}$$

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

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]]

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fricas [A]  time = 0.78, size = 14, normalized size = 0.88 $$\begin{array}{c}\frac{x\log (x)}{\log (x-\log (x)+4)}\end{array}$$

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)

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giac [A]  time = 0.21, size = 14, normalized size = 0.88 $$\begin{array}{c}\frac{x\log (x)}{\log (x-\log (x)+4)}\end{array}$$

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)

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

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

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)

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maxima [A]  time = 0.40, size = 14, normalized size = 0.88 $$\begin{array}{c}\frac{x\log (x)}{\log (x-\log (x)+4)}\end{array}$$

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)

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mupad [B]  time = 4.74, size = 82, normalized size = 5.12 $$\begin{array}{c}x+3\ln (x)+\frac{5}{x-1}-{\ln (x)}^2(\frac{1}{x-1}+1)+\frac{x\ln (x)-\frac{x\ln (x-\ln (x)+4)(\ln (x)+1)(x-\ln (x)+4)}{x-1}}{\ln (x-\ln (x)+4)}+\frac{\ln (x)(x^2+3)}{x-1}\end{array}$$

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)

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sympy [A]  time = 0.35, size = 12, normalized size = 0.75 $$\begin{array}{c}\frac{x\log (x)}{\log (x-\log (x)+4)}\end{array}$$

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)

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