3.2.43 \(\int \frac {-2 x^2+(-4 x-2 x^2+2 x^3) \log (2-x)+((-8 x^2+4 x^3) \log (2-x)+(-8 x+4 x^2) \log (2-x) \log (\frac {x}{\log (2-x)})) \log (x+\log (\frac {x}{\log (2-x)}))}{(-2 x+x^2) \log (2-x)+(-2+x) \log (2-x) \log (\frac {x}{\log (2-x)})} \, dx\)

Optimal. Leaf size=19 \[ 2 x^2 \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \]

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Rubi [F]  time = 2.65, 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 {-2 x^2+\left (-4 x-2 x^2+2 x^3\right ) \log (2-x)+\left (\left (-8 x^2+4 x^3\right ) \log (2-x)+\left (-8 x+4 x^2\right ) \log (2-x) \log \left (\frac {x}{\log (2-x)}\right )\right ) \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}{\left (-2 x+x^2\right ) \log (2-x)+(-2+x) \log (2-x) \log \left (\frac {x}{\log (2-x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^2 + (-4*x - 2*x^2 + 2*x^3)*Log[2 - x] + ((-8*x^2 + 4*x^3)*Log[2 - x] + (-8*x + 4*x^2)*Log[2 - x]*Log
[x/Log[2 - x]])*Log[x + Log[x/Log[2 - x]]])/((-2*x + x^2)*Log[2 - x] + (-2 + x)*Log[2 - x]*Log[x/Log[2 - x]]),
x]

[Out]

2*Defer[Int][x/(x + Log[x/Log[2 - x]]), x] + 2*Defer[Int][x^2/(x + Log[x/Log[2 - x]]), x] - 4*Defer[Int][1/(Lo
g[2 - x]*(x + Log[x/Log[2 - x]])), x] - 8*Defer[Int][1/((-2 + x)*Log[2 - x]*(x + Log[x/Log[2 - x]])), x] - 2*D
efer[Int][x/(Log[2 - x]*(x + Log[x/Log[2 - x]])), x] + 4*Defer[Int][x*Log[x + Log[x/Log[2 - x]]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (x-(-2+x) \log (2-x) \left (1+x+2 \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )\right )\right )}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=2 \int \frac {x \left (x-(-2+x) \log (2-x) \left (1+x+2 \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )\right )\right )}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=2 \int \left (\frac {x \left (-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)\right )}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+2 x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )\right ) \, dx\\ &=2 \int \frac {x \left (-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)\right )}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \frac {x \left (x-\left (-2-x+x^2\right ) \log (2-x)\right )}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \left (\frac {-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {2 \left (-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)\right )}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \frac {-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int \frac {-x-2 \log (2-x)-x \log (2-x)+x^2 \log (2-x)}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \frac {-x+\left (-2-x+x^2\right ) \log (2-x)}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int \frac {x-\left (-2-x+x^2\right ) \log (2-x)}{(2-x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=2 \int \left (-\frac {2}{x+\log \left (\frac {x}{\log (2-x)}\right )}-\frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )}+\frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )}-\frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int \left (-\frac {2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}-\frac {x}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {x^2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}-\frac {x}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx\\ &=-\left (2 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+2 \int \frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \int \frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-4 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-4 \int \frac {x}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int \frac {x^2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-4 \int \frac {x}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx-8 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+2 \int \frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \int \frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-4 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-4 \int \left (\frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )}+\frac {2}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int \left (\frac {2}{x+\log \left (\frac {x}{\log (2-x)}\right )}+\frac {4}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )}\right ) \, dx-4 \int \left (\frac {1}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}+\frac {2}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )}\right ) \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx-8 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+2 \int \frac {x^2}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \int \frac {x}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx-2 \left (4 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx\right )+4 \int \frac {x}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-4 \int \frac {1}{\log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+4 \int x \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \, dx+8 \int \frac {1}{x+\log \left (\frac {x}{\log (2-x)}\right )} \, dx-2 \left (8 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\right )-8 \int \frac {1}{(-2+x) \log (2-x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx+16 \int \frac {1}{(-2+x) \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 19, normalized size = 1.00 \begin {gather*} 2 x^2 \log \left (x+\log \left (\frac {x}{\log (2-x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^2 + (-4*x - 2*x^2 + 2*x^3)*Log[2 - x] + ((-8*x^2 + 4*x^3)*Log[2 - x] + (-8*x + 4*x^2)*Log[2 -
x]*Log[x/Log[2 - x]])*Log[x + Log[x/Log[2 - x]]])/((-2*x + x^2)*Log[2 - x] + (-2 + x)*Log[2 - x]*Log[x/Log[2 -
x]]),x]

[Out]

2*x^2*Log[x + Log[x/Log[2 - x]]]

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fricas [A]  time = 0.89, size = 19, normalized size = 1.00 \begin {gather*} 2 \, x^{2} \log \left (x + \log \left (\frac {x}{\log \left (-x + 2\right )}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2-8*x)*log(2-x)*log(x/log(2-x))+(4*x^3-8*x^2)*log(2-x))*log(log(x/log(2-x))+x)+(2*x^3-2*x^2-4
*x)*log(2-x)-2*x^2)/((x-2)*log(2-x)*log(x/log(2-x))+(x^2-2*x)*log(2-x)),x, algorithm="fricas")

[Out]

2*x^2*log(x + log(x/log(-x + 2)))

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giac [A]  time = 1.45, size = 19, normalized size = 1.00 \begin {gather*} 2 \, x^{2} \log \left (x + \log \relax (x) - \log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2-8*x)*log(2-x)*log(x/log(2-x))+(4*x^3-8*x^2)*log(2-x))*log(log(x/log(2-x))+x)+(2*x^3-2*x^2-4
*x)*log(2-x)-2*x^2)/((x-2)*log(2-x)*log(x/log(2-x))+(x^2-2*x)*log(2-x)),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.15, size = 86, normalized size = 4.53

method	result	size
risch	\(2 x^{2} \ln \left (\ln \relax (x )-\ln \left (\ln \left (2-x \right )\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x}{\ln \left (2-x \right )}\right ) \left (-\mathrm {csgn}\left (\frac {i x}{\ln \left (2-x \right )}\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (\frac {i x}{\ln \left (2-x \right )}\right )+\mathrm {csgn}\left (\frac {i}{\ln \left (2-x \right )}\right )\right )}{2}+x \right )\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^2-8*x)*ln(2-x)*ln(x/ln(2-x))+(4*x^3-8*x^2)*ln(2-x))*ln(ln(x/ln(2-x))+x)+(2*x^3-2*x^2-4*x)*ln(2-x)-2
*x^2)/((x-2)*ln(2-x)*ln(x/ln(2-x))+(x^2-2*x)*ln(2-x)),x,method=_RETURNVERBOSE)

[Out]

2*x^2*ln(ln(x)-ln(ln(2-x))-1/2*I*Pi*csgn(I*x/ln(2-x))*(-csgn(I*x/ln(2-x))+csgn(I*x))*(-csgn(I*x/ln(2-x))+csgn(
I/ln(2-x)))+x)

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maxima [A]  time = 0.77, size = 19, normalized size = 1.00 \begin {gather*} 2 \, x^{2} \log \left (x + \log \relax (x) - \log \left (\log \left (-x + 2\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2-8*x)*log(2-x)*log(x/log(2-x))+(4*x^3-8*x^2)*log(2-x))*log(log(x/log(2-x))+x)+(2*x^3-2*x^2-4
*x)*log(2-x)-2*x^2)/((x-2)*log(2-x)*log(x/log(2-x))+(x^2-2*x)*log(2-x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.14, size = 19, normalized size = 1.00 \begin {gather*} 2\,x^2\,\ln \left (x+\ln \left (\frac {x}{\ln \left (2-x\right )}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2 - x)*(4*x + 2*x^2 - 2*x^3) + log(x + log(x/log(2 - x)))*(log(2 - x)*(8*x^2 - 4*x^3) + log(x/log(2 -
x))*log(2 - x)*(8*x - 4*x^2)) + 2*x^2)/(log(2 - x)*(2*x - x^2) - log(x/log(2 - x))*log(2 - x)*(x - 2)),x)

[Out]

2*x^2*log(x + log(x/log(2 - x)))

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sympy [A]  time = 3.63, size = 15, normalized size = 0.79 \begin {gather*} 2 x^{2} \log {\left (x + \log {\left (\frac {x}{\log {\left (2 - x \right )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**2-8*x)*ln(2-x)*ln(x/ln(2-x))+(4*x**3-8*x**2)*ln(2-x))*ln(ln(x/ln(2-x))+x)+(2*x**3-2*x**2-4*x
)*ln(2-x)-2*x**2)/((x-2)*ln(2-x)*ln(x/ln(2-x))+(x**2-2*x)*ln(2-x)),x)

[Out]

2*x**2*log(x + log(x/log(2 - x)))

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