3.73.37 \(\int \frac {-40+40 x+6 x^2+18 x^3-x^5+(-4 x-20 x^2+x^4) \log (x)+(8-8 x-4 x^3+4 x^2 \log (x)) \log (x-\log (x))}{-4 x^2+4 x \log (x)} \, dx\)

Optimal. Leaf size=26 \[ -2+e^5-x+\left (-5+\frac {x^2}{4}+\log (x-\log (x))\right )^2 \]

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Rubi [F]  time = 1.58, 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 {-40+40 x+6 x^2+18 x^3-x^5+\left (-4 x-20 x^2+x^4\right ) \log (x)+\left (8-8 x-4 x^3+4 x^2 \log (x)\right ) \log (x-\log (x))}{-4 x^2+4 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-40 + 40*x + 6*x^2 + 18*x^3 - x^5 + (-4*x - 20*x^2 + x^4)*Log[x] + (8 - 8*x - 4*x^3 + 4*x^2*Log[x])*Log[x
- Log[x]])/(-4*x^2 + 4*x*Log[x]),x]

[Out]

-x - (5*x^2)/2 + x^4/16 - 10*Defer[Int][(x - Log[x])^(-1), x] + 10*Defer[Int][1/(x*(x - Log[x])), x] - Defer[I
nt][x/(x - Log[x]), x]/2 + Defer[Int][x^2/(x - Log[x]), x]/2 + 2*Defer[Int][Log[x - Log[x]]/(x - Log[x]), x] -
2*Defer[Int][Log[x - Log[x]]/(x*(x - Log[x])), x] + Defer[Int][(x^2*Log[x - Log[x]])/(x - Log[x]), x] - Defer
[Int][(x*Log[x]*Log[x - Log[x]])/(x - Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-40+40 x+6 x^2+18 x^3-x^5+\left (-4 x-20 x^2+x^4\right ) \log (x)+\left (8-8 x-4 x^3+4 x^2 \log (x)\right ) \log (x-\log (x))}{x (-4 x+4 \log (x))} \, dx\\ &=\int \frac {40-40 x-6 x^2-18 x^3+x^5-\left (-4 x-20 x^2+x^4\right ) \log (x)-\left (8-8 x-4 x^3+4 x^2 \log (x)\right ) \log (x-\log (x))}{4 x (x-\log (x))} \, dx\\ &=\frac {1}{4} \int \frac {40-40 x-6 x^2-18 x^3+x^5-\left (-4 x-20 x^2+x^4\right ) \log (x)-\left (8-8 x-4 x^3+4 x^2 \log (x)\right ) \log (x-\log (x))}{x (x-\log (x))} \, dx\\ &=\frac {1}{4} \int \left (\frac {40-40 x-6 x^2-18 x^3+x^5+4 x \log (x)+20 x^2 \log (x)-x^4 \log (x)}{x (x-\log (x))}+\frac {4 \left (-2+2 x+x^3-x^2 \log (x)\right ) \log (x-\log (x))}{x (x-\log (x))}\right ) \, dx\\ &=\frac {1}{4} \int \frac {40-40 x-6 x^2-18 x^3+x^5+4 x \log (x)+20 x^2 \log (x)-x^4 \log (x)}{x (x-\log (x))} \, dx+\int \frac {\left (-2+2 x+x^3-x^2 \log (x)\right ) \log (x-\log (x))}{x (x-\log (x))} \, dx\\ &=\frac {1}{4} \int \left (-4-20 x+x^3+\frac {2 \left (20-20 x-x^2+x^3\right )}{x (x-\log (x))}\right ) \, dx+\int \left (\frac {2 \log (x-\log (x))}{x-\log (x)}-\frac {2 \log (x-\log (x))}{x (x-\log (x))}+\frac {x^2 \log (x-\log (x))}{x-\log (x)}-\frac {x \log (x) \log (x-\log (x))}{x-\log (x)}\right ) \, dx\\ &=-x-\frac {5 x^2}{2}+\frac {x^4}{16}+\frac {1}{2} \int \frac {20-20 x-x^2+x^3}{x (x-\log (x))} \, dx+2 \int \frac {\log (x-\log (x))}{x-\log (x)} \, dx-2 \int \frac {\log (x-\log (x))}{x (x-\log (x))} \, dx+\int \frac {x^2 \log (x-\log (x))}{x-\log (x)} \, dx-\int \frac {x \log (x) \log (x-\log (x))}{x-\log (x)} \, dx\\ &=-x-\frac {5 x^2}{2}+\frac {x^4}{16}+\frac {1}{2} \int \left (-\frac {20}{x-\log (x)}+\frac {20}{x (x-\log (x))}-\frac {x}{x-\log (x)}+\frac {x^2}{x-\log (x)}\right ) \, dx+2 \int \frac {\log (x-\log (x))}{x-\log (x)} \, dx-2 \int \frac {\log (x-\log (x))}{x (x-\log (x))} \, dx+\int \frac {x^2 \log (x-\log (x))}{x-\log (x)} \, dx-\int \frac {x \log (x) \log (x-\log (x))}{x-\log (x)} \, dx\\ &=-x-\frac {5 x^2}{2}+\frac {x^4}{16}-\frac {1}{2} \int \frac {x}{x-\log (x)} \, dx+\frac {1}{2} \int \frac {x^2}{x-\log (x)} \, dx+2 \int \frac {\log (x-\log (x))}{x-\log (x)} \, dx-2 \int \frac {\log (x-\log (x))}{x (x-\log (x))} \, dx-10 \int \frac {1}{x-\log (x)} \, dx+10 \int \frac {1}{x (x-\log (x))} \, dx+\int \frac {x^2 \log (x-\log (x))}{x-\log (x)} \, dx-\int \frac {x \log (x) \log (x-\log (x))}{x-\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 52, normalized size = 2.00 \begin {gather*} \frac {1}{4} \left (-4 x-10 x^2+\frac {x^4}{4}-40 \log (x-\log (x))+2 x^2 \log (x-\log (x))+4 \log ^2(x-\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40 + 40*x + 6*x^2 + 18*x^3 - x^5 + (-4*x - 20*x^2 + x^4)*Log[x] + (8 - 8*x - 4*x^3 + 4*x^2*Log[x])
*Log[x - Log[x]])/(-4*x^2 + 4*x*Log[x]),x]

[Out]

(-4*x - 10*x^2 + x^4/4 - 40*Log[x - Log[x]] + 2*x^2*Log[x - Log[x]] + 4*Log[x - Log[x]]^2)/4

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fricas [A]  time = 0.84, size = 37, normalized size = 1.42 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {5}{2} \, x^{2} + \frac {1}{2} \, {\left (x^{2} - 20\right )} \log \left (x - \log \relax (x)\right ) + \log \left (x - \log \relax (x)\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2*log(x)-4*x^3-8*x+8)*log(x-log(x))+(x^4-20*x^2-4*x)*log(x)-x^5+18*x^3+6*x^2+40*x-40)/(4*x*log
(x)-4*x^2),x, algorithm="fricas")

[Out]

1/16*x^4 - 5/2*x^2 + 1/2*(x^2 - 20)*log(x - log(x)) + log(x - log(x))^2 - x

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giac [A]  time = 0.23, size = 44, normalized size = 1.69 \begin {gather*} \frac {1}{16} \, x^{4} + \frac {1}{2} \, x^{2} \log \left (x - \log \relax (x)\right ) - \frac {5}{2} \, x^{2} + \log \left (x - \log \relax (x)\right )^{2} - x - 10 \, \log \left (x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2*log(x)-4*x^3-8*x+8)*log(x-log(x))+(x^4-20*x^2-4*x)*log(x)-x^5+18*x^3+6*x^2+40*x-40)/(4*x*log
(x)-4*x^2),x, algorithm="giac")

[Out]

1/16*x^4 + 1/2*x^2*log(x - log(x)) - 5/2*x^2 + log(x - log(x))^2 - x - 10*log(x - log(x))

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maple [A]  time = 0.04, size = 45, normalized size = 1.73

method	result	size
risch	\(\ln \left (x -\ln \relax (x )\right )^{2}+\frac {\ln \left (x -\ln \relax (x )\right ) x^{2}}{2}+\frac {x^{4}}{16}-\frac {5 x^{2}}{2}-x -10 \ln \left (\ln \relax (x )-x \right )\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2*ln(x)-4*x^3-8*x+8)*ln(x-ln(x))+(x^4-20*x^2-4*x)*ln(x)-x^5+18*x^3+6*x^2+40*x-40)/(4*x*ln(x)-4*x^2),
x,method=_RETURNVERBOSE)

[Out]

ln(x-ln(x))^2+1/2*ln(x-ln(x))*x^2+1/16*x^4-5/2*x^2-x-10*ln(ln(x)-x)

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maxima [A]  time = 0.40, size = 37, normalized size = 1.42 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {5}{2} \, x^{2} + \frac {1}{2} \, {\left (x^{2} - 20\right )} \log \left (x - \log \relax (x)\right ) + \log \left (x - \log \relax (x)\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2*log(x)-4*x^3-8*x+8)*log(x-log(x))+(x^4-20*x^2-4*x)*log(x)-x^5+18*x^3+6*x^2+40*x-40)/(4*x*log
(x)-4*x^2),x, algorithm="maxima")

[Out]

1/16*x^4 - 5/2*x^2 + 1/2*(x^2 - 20)*log(x - log(x)) + log(x - log(x))^2 - x

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mupad [B]  time = 4.51, size = 44, normalized size = 1.69 \begin {gather*} \frac {x^2\,\ln \left (x-\ln \relax (x)\right )}{2}-10\,\ln \left (\ln \relax (x)-x\right )-x-\frac {5\,x^2}{2}+\frac {x^4}{16}+{\ln \left (x-\ln \relax (x)\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - log(x))*(8*x - 4*x^2*log(x) + 4*x^3 - 8) - 40*x - 6*x^2 - 18*x^3 + x^5 + log(x)*(4*x + 20*x^2 -
x^4) + 40)/(4*x*log(x) - 4*x^2),x)

[Out]

(x^2*log(x - log(x)))/2 - 10*log(log(x) - x) - x - (5*x^2)/2 + x^4/16 + log(x - log(x))^2

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sympy [B]  time = 0.53, size = 41, normalized size = 1.58 \begin {gather*} \frac {x^{4}}{16} + \frac {x^{2} \log {\left (x - \log {\relax (x )} \right )}}{2} - \frac {5 x^{2}}{2} - x - 10 \log {\left (- x + \log {\relax (x )} \right )} + \log {\left (x - \log {\relax (x )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2*ln(x)-4*x**3-8*x+8)*ln(x-ln(x))+(x**4-20*x**2-4*x)*ln(x)-x**5+18*x**3+6*x**2+40*x-40)/(4*x*
ln(x)-4*x**2),x)

[Out]

x**4/16 + x**2*log(x - log(x))/2 - 5*x**2/2 - x - 10*log(-x + log(x)) + log(x - log(x))**2

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