3.28.45 \(\int (1+(-x-2 x \log (x)) \log (\frac {\log (2)}{3})) \, dx\) [2745]

Optimal. Leaf size=17 \[ -1+x-x^2 \log (x) \log \left (\frac {\log (2)}{3}\right ) \]

[Out]

x-1-x^2*ln(x)*ln(1/3*ln(2))

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2341} \begin {gather*} x-x^2 \log \left (\frac {\log (2)}{3}\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2*Log[x]*Log[Log[2]/3]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\log \left (\frac {\log (2)}{3}\right ) \int (-x-2 x \log (x)) \, dx\\ &=x-\frac {1}{2} x^2 \log \left (\frac {\log (2)}{3}\right )-\left (2 \log \left (\frac {\log (2)}{3}\right )\right ) \int x \log (x) \, dx\\ &=x-x^2 \log (x) \log \left (\frac {\log (2)}{3}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} x-x^2 \log (x) \log \left (\frac {\log (2)}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2*Log[x]*Log[Log[2]/3]

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Maple [A]
time = 0.09, size = 15, normalized size = 0.88

method	result	size
default	\(x -x^{2} \ln \left (x \right ) \ln \left (\frac {\ln \left (2\right )}{3}\right )\)	\(15\)
norman	\(x +\left (\ln \left (3\right )-\ln \left (\ln \left (2\right )\right )\right ) x^{2} \ln \left (x \right )\)	\(17\)
risch	\(-\left (-\ln \left (3\right )+\ln \left (\ln \left (2\right )\right )\right ) x^{2} \ln \left (x \right )+x\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x)-x)*ln(1/3*ln(2))+1,x,method=_RETURNVERBOSE)

[Out]

x-x^2*ln(x)*ln(1/3*ln(2))

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Maxima [A]
time = 0.26, size = 14, normalized size = 0.82 \begin {gather*} -x^{2} \log \left (x\right ) \log \left (\frac {1}{3} \, \log \left (2\right )\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-x)*log(1/3*log(2))+1,x, algorithm="maxima")

[Out]

-x^2*log(x)*log(1/3*log(2)) + x

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Fricas [A]
time = 0.35, size = 14, normalized size = 0.82 \begin {gather*} -x^{2} \log \left (x\right ) \log \left (\frac {1}{3} \, \log \left (2\right )\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-x)*log(1/3*log(2))+1,x, algorithm="fricas")

[Out]

-x^2*log(x)*log(1/3*log(2)) + x

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Sympy [A]
time = 0.05, size = 19, normalized size = 1.12 \begin {gather*} x + \left (- x^{2} \log {\left (\log {\left (2 \right )} \right )} + x^{2} \log {\left (3 \right )}\right ) \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + (-x**2*log(log(2)) + x**2*log(3))*log(x)

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Giac [A]
time = 0.39, size = 14, normalized size = 0.82 \begin {gather*} -x^{2} \log \left (x\right ) \log \left (\frac {1}{3} \, \log \left (2\right )\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-x)*log(1/3*log(2))+1,x, algorithm="giac")

[Out]

-x^2*log(x)*log(1/3*log(2)) + x

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Mupad [B]
time = 1.77, size = 14, normalized size = 0.82 \begin {gather*} x-x^2\,\ln \left (\frac {\ln \left (2\right )}{3}\right )\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - log(log(2)/3)*(x + 2*x*log(x)),x)

[Out]

x - x^2*log(log(2)/3)*log(x)

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