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

Optimal. Leaf size=20 \[ \frac {1}{3 \log \left (\frac {2 (1+x)}{\log \left (\frac {2}{x}\right )}\right )} \]

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Rubi [F]  time = 1.48, 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-x \log \left (\frac {2}{x}\right )}{\left (3 x+3 x^2\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {2+2 x}{\log \left (\frac {2}{x}\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{3 \log \left (\frac {2 (1+x)}{\log \left (\frac {2}{x}\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(3*Log[(2*(1 + x))/Log[2/x]])

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fricas [A]  time = 0.90, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3 \, \log \left (\frac {2 \, {\left (x + 1\right )}}{\log \left (\frac {2}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/log(2*(x + 1)/log(2/x))

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giac [B]  time = 0.21, size = 229, normalized size = 11.45 \begin {gather*} \frac {x \log \relax (2) \log \left (\frac {2}{x}\right ) - x \log \relax (x) \log \left (\frac {2}{x}\right ) + x \log \left (\frac {2}{x}\right ) + \log \left (\frac {2}{x}\right )}{3 \, {\left (x \log \relax (2)^{2} \log \left (\frac {2}{x}\right ) + x \log \relax (2) \log \left (x + 1\right ) \log \left (\frac {2}{x}\right ) - x \log \relax (2) \log \relax (x) \log \left (\frac {2}{x}\right ) - x \log \left (x + 1\right ) \log \relax (x) \log \left (\frac {2}{x}\right ) - x \log \relax (2) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right ) + x \log \relax (x) \log \left (\frac {2}{x}\right ) \log \left (\log \left (\frac {2}{x}\right )\right ) + x \log \relax (2)^{2} + x \log \relax (2) \log \left (x + 1\right ) - x \log \relax (2) \log \relax (x) - x \log \left (x + 1\right ) \log \relax (x) - x \log \relax (2) \log \left (\log \left (\frac {2}{x}\right )\right ) + x \log \relax (x) \log \left (\log \left (\frac {2}{x}\right )\right ) + \log \relax (2)^{2} + \log \relax (2) \log \left (x + 1\right ) - \log \relax (2) \log \relax (x) - \log \left (x + 1\right ) \log \relax (x) - \log \relax (2) \log \left (\log \left (\frac {2}{x}\right )\right ) + \log \relax (x) \log \left (\log \left (\frac {2}{x}\right )\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {-x \ln \left (\frac {2}{x}\right )-x -1}{\left (3 x^{2}+3 x \right ) \ln \left (\frac {2}{x}\right ) \ln \left (\frac {2 x +2}{\ln \left (\frac {2}{x}\right )}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.66, size = 26, normalized size = 1.30 \begin {gather*} -\frac {1}{3 \, {\left (-i \, \pi - \log \relax (2) - \log \left (x + 1\right ) + \log \left (-\log \relax (2) + \log \relax (x)\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(-I*pi - log(2) - log(x + 1) + log(-log(2) + log(x)))

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mupad [B]  time = 1.38, size = 19, normalized size = 0.95 \begin {gather*} \frac {1}{3\,\ln \left (\frac {2\,x+2}{\ln \left (\frac {2}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(3*log((2*x + 2)/log(2/x)))

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sympy [A]  time = 0.31, size = 14, normalized size = 0.70 \begin {gather*} \frac {1}{3 \log {\left (\frac {2 x + 2}{\log {\left (\frac {2}{x} \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(3*log((2*x + 2)/log(2/x)))

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