3.41.97 $\int \frac{3+x^x(3+3\log (x))+(-x-x^x)\log (x+x^x)}{(3x+3x^x)\log (x+x^x)}dx$

Optimal. Leaf size=14 $$5-\frac{x}{3}+\log (\log (x+x^x))$$

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Rubi [F]  time = 0.98, 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 = {} $$\begin{array}{c}\int \frac{3+x^x(3+3\log (x))+(-x-x^x)\log (x+x^x)}{(3x+3x^x)\log (x+x^x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{3+x^x(3+3\log (x))+(-x-x^x)\log (x+x^x)}{3(x+x^x)\log (x+x^x)}dx\\ & =\frac{1}{3}\int \frac{3+x^x(3+3\log (x))+(-x-x^x)\log (x+x^x)}{(x+x^x)\log (x+x^x)}dx\\ & =\frac{1}{3}\int (-\frac{3(-1+x+x\log (x))}{(x+x^x)\log (x+x^x)}+\frac{3+3\log (x)-\log (x+x^x)}{\log (x+x^x)})dx\\ & =\frac{1}{3}\int \frac{3+3\log (x)-\log (x+x^x)}{\log (x+x^x)}dx-\int \frac{-1+x+x\log (x)}{(x+x^x)\log (x+x^x)}dx\\ & =\frac{1}{3}\int (-1+\frac{3(1+\log (x))}{\log (x+x^x)})dx-\int (-\frac{1}{(x+x^x)\log (x+x^x)}+\frac{x}{(x+x^x)\log (x+x^x)}+\frac{x\log (x)}{(x+x^x)\log (x+x^x)})dx\\ & =-\frac{x}{3}+\int \frac{1}{(x+x^x)\log (x+x^x)}dx-\int \frac{x}{(x+x^x)\log (x+x^x)}dx-\int \frac{x\log (x)}{(x+x^x)\log (x+x^x)}dx+\int \frac{1+\log (x)}{\log (x+x^x)}dx\\ & =-\frac{x}{3}+\int (\frac{1}{\log (x+x^x)}+\frac{\log (x)}{\log (x+x^x)})dx+\int \frac{1}{(x+x^x)\log (x+x^x)}dx-\int \frac{x}{(x+x^x)\log (x+x^x)}dx-\int \frac{x\log (x)}{(x+x^x)\log (x+x^x)}dx\\ & =-\frac{x}{3}+\int \frac{1}{\log (x+x^x)}dx+\int \frac{1}{(x+x^x)\log (x+x^x)}dx-\int \frac{x}{(x+x^x)\log (x+x^x)}dx+\int \frac{\log (x)}{\log (x+x^x)}dx-\int \frac{x\log (x)}{(x+x^x)\log (x+x^x)}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.18, size = 17, normalized size = 1.21 $$\begin{array}{c}\frac{1}{3}(-x+3\log (\log (x+x^x)))\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

(-x + 3*Log[Log[x + x^x]])/3

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fricas [A]  time = 0.68, size = 11, normalized size = 0.79 $$\begin{array}{c}-\frac{1}{3}x+\log (\log (x+x^x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x + log(log(x + x^x))

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giac [A]  time = 1.74, size = 11, normalized size = 0.79 $$\begin{array}{c}-\frac{1}{3}x+\log (\log (x+x^x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x + log(log(x + x^x))

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maple [A]  time = 0.05, size = 12, normalized size = 0.86

method	result	size
risch	$-\frac{x}{3}+\ln (\ln (x^x+x))$	$12$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x+ln(ln(x^x+x))

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maxima [A]  time = 0.41, size = 11, normalized size = 0.79 $$\begin{array}{c}-\frac{1}{3}x+\log (\log (x+x^x))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x + log(log(x + x^x))

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mupad [B]  time = 3.46, size = 11, normalized size = 0.79 $$\begin{array}{c}\ln (\ln (x+x^x))-\frac{x}{3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(x + x^x)) - x/3

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sympy [A]  time = 0.76, size = 14, normalized size = 1.00 $$\begin{array}{c}-\frac{x}{3}+\log (\log (x+e^{x\log (x)}))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/3 + log(log(x + exp(x*log(x))))

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