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

Optimal. Leaf size=14 \[ 5-\frac {x}{3}+\log \left (\log \left (x+x^x\right )\right ) \]

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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 {gather*} \int \frac {3+x^x (3+3 \log (x))+\left (-x-x^x\right ) \log \left (x+x^x\right )}{\left (3 x+3 x^x\right ) \log \left (x+x^x\right )} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {3+x^x (3+3 \log (x))+\left (-x-x^x\right ) \log \left (x+x^x\right )}{3 \left (x+x^x\right ) \log \left (x+x^x\right )} \, dx\\ &=\frac {1}{3} \int \frac {3+x^x (3+3 \log (x))+\left (-x-x^x\right ) \log \left (x+x^x\right )}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx\\ &=\frac {1}{3} \int \left (-\frac {3 (-1+x+x \log (x))}{\left (x+x^x\right ) \log \left (x+x^x\right )}+\frac {3+3 \log (x)-\log \left (x+x^x\right )}{\log \left (x+x^x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {3+3 \log (x)-\log \left (x+x^x\right )}{\log \left (x+x^x\right )} \, dx-\int \frac {-1+x+x \log (x)}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx\\ &=\frac {1}{3} \int \left (-1+\frac {3 (1+\log (x))}{\log \left (x+x^x\right )}\right ) \, dx-\int \left (-\frac {1}{\left (x+x^x\right ) \log \left (x+x^x\right )}+\frac {x}{\left (x+x^x\right ) \log \left (x+x^x\right )}+\frac {x \log (x)}{\left (x+x^x\right ) \log \left (x+x^x\right )}\right ) \, dx\\ &=-\frac {x}{3}+\int \frac {1}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx-\int \frac {x}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx-\int \frac {x \log (x)}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx+\int \frac {1+\log (x)}{\log \left (x+x^x\right )} \, dx\\ &=-\frac {x}{3}+\int \left (\frac {1}{\log \left (x+x^x\right )}+\frac {\log (x)}{\log \left (x+x^x\right )}\right ) \, dx+\int \frac {1}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx-\int \frac {x}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx-\int \frac {x \log (x)}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx\\ &=-\frac {x}{3}+\int \frac {1}{\log \left (x+x^x\right )} \, dx+\int \frac {1}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx-\int \frac {x}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx+\int \frac {\log (x)}{\log \left (x+x^x\right )} \, dx-\int \frac {x \log (x)}{\left (x+x^x\right ) \log \left (x+x^x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 17, normalized size = 1.21 \begin {gather*} \frac {1}{3} \left (-x+3 \log \left (\log \left (x+x^x\right )\right )\right ) \end {gather*}

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 {gather*} -\frac {1}{3} \, x + \log \left (\log \left (x + x^{x}\right )\right ) \end {gather*}

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 {gather*} -\frac {1}{3} \, x + \log \left (\log \left (x + x^{x}\right )\right ) \end {gather*}

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	\(-\frac {x}{3}+\ln \left (\ln \left (x^{x}+x \right )\right )\)	\(12\)

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 {gather*} -\frac {1}{3} \, x + \log \left (\log \left (x + x^{x}\right )\right ) \end {gather*}

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 {gather*} \ln \left (\ln \left (x+x^x\right )\right )-\frac {x}{3} \end {gather*}

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 {gather*} - \frac {x}{3} + \log {\left (\log {\left (x + e^{x \log {\relax (x )}} \right )} \right )} \end {gather*}

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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