3.30.7 \(\int \frac {(-40 x^2+40 x^3-10 x^4) \log ^2(x)+(2-3 x+x^2) \log (x-x^2)+\log (x) (-2+5 x-2 x^2+(2-2 x+x^2) \log (x-x^2))}{(120 x^2-120 x^3+30 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=37 \[ \frac {1}{3} \left (-x+\frac {\log ((1-x) x)}{10 \left (-x+\frac {x}{-1+x}\right ) \log (x)}\right ) \]

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Rubi [F]  time = 7.92, 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 {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-40*x^2 + 40*x^3 - 10*x^4)*Log[x]^2 + (2 - 3*x + x^2)*Log[x - x^2] + Log[x]*(-2 + 5*x - 2*x^2 + (2 - 2*x
+ x^2)*Log[x - x^2]))/((120*x^2 - 120*x^3 + 30*x^4)*Log[x]^2),x]

[Out]

-1/3*x + Defer[Int][(1 - 2*x)/((-2 + x)*x^2*Log[x]), x]/30 + Defer[Int][Log[(1 - x)*x]/((2 - x)*Log[x]^2), x]/
20 + (7*Defer[Int][Log[(1 - x)*x]/((-2 + x)*Log[x]^2), x])/120 + Defer[Int][Log[(1 - x)*x]/(x^2*Log[x]^2), x]/
60 - Defer[Int][Log[(1 - x)*x]/(x*Log[x]^2), x]/120 + Defer[Int][Log[(1 - x)*x]/((2 - x)^2*Log[x]), x]/60 + De
fer[Int][Log[(1 - x)*x]/((2 - x)*Log[x]), x]/20 + Defer[Int][Log[(1 - x)*x]/((-2 + x)*Log[x]), x]/20 + Defer[I
nt][Log[(1 - x)*x]/(x^2*Log[x]), x]/60

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{x^2 \left (120-120 x+30 x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{30 (-2+x)^2 x^2 \log ^2(x)} \, dx\\ &=\frac {1}{30} \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{(-2+x)^2 x^2 \log ^2(x)} \, dx\\ &=\frac {1}{30} \int \left (\frac {1-2 x+20 x^2 \log (x)-10 x^3 \log (x)}{(-2+x) x^2 \log (x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x)^2 x^2 \log ^2(x)}\right ) \, dx\\ &=\frac {1}{30} \int \frac {1-2 x+20 x^2 \log (x)-10 x^3 \log (x)}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x)^2 x^2 \log ^2(x)} \, dx\\ &=\frac {1}{30} \int \left (-10+\frac {1-2 x}{(-2+x) x^2 \log (x)}\right ) \, dx+\frac {1}{30} \int \left (\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 (2-x)^2 \log ^2(x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 (2-x) \log ^2(x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 x^2 \log ^2(x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 x \log ^2(x)}\right ) \, dx\\ &=-\frac {x}{3}+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx\\ &=-\frac {x}{3}+\frac {1}{120} \int \left (\frac {\log ((1-x) x)}{\log ^2(x)}+\frac {2 \log ((1-x) x)}{x^2 \log ^2(x)}-\frac {3 \log ((1-x) x)}{x \log ^2(x)}+\frac {\log ((1-x) x)}{\log (x)}+\frac {2 \log ((1-x) x)}{x^2 \log (x)}-\frac {2 \log ((1-x) x)}{x \log (x)}\right ) \, dx+\frac {1}{120} \int \left (-\frac {3 \log ((1-x) x)}{\log ^2(x)}+\frac {2 \log ((1-x) x)}{x \log ^2(x)}+\frac {x \log ((1-x) x)}{\log ^2(x)}-\frac {2 \log ((1-x) x)}{\log (x)}+\frac {2 \log ((1-x) x)}{x \log (x)}+\frac {x \log ((1-x) x)}{\log (x)}\right ) \, dx+\frac {1}{120} \int \left (\frac {2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)}-\frac {3 x \log ((1-x) x)}{(2-x)^2 \log ^2(x)}+\frac {x^2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)}+\frac {2 \log ((1-x) x)}{(2-x)^2 \log (x)}-\frac {2 x \log ((1-x) x)}{(2-x)^2 \log (x)}+\frac {x^2 \log ((1-x) x)}{(2-x)^2 \log (x)}\right ) \, dx+\frac {1}{120} \int \left (\frac {2 \log ((1-x) x)}{(2-x) \log ^2(x)}+\frac {3 x \log ((1-x) x)}{(-2+x) \log ^2(x)}+\frac {x^2 \log ((1-x) x)}{(2-x) \log ^2(x)}+\frac {2 \log ((1-x) x)}{(2-x) \log (x)}+\frac {2 x \log ((1-x) x)}{(-2+x) \log (x)}+\frac {x^2 \log ((1-x) x)}{(2-x) \log (x)}\right ) \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx\\ &=-\frac {x}{3}+\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{120} \int \frac {x \log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\log ((1-x) x)}{\log (x)} \, dx+\frac {1}{120} \int \frac {x \log ((1-x) x)}{\log (x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx-\frac {1}{60} \int \frac {x \log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {x \log ((1-x) x)}{(-2+x) \log (x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx-\frac {1}{40} \int \frac {x \log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{40} \int \frac {x \log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 32, normalized size = 0.86 \begin {gather*} \frac {1}{30} \left (-10 x-\frac {(-1+x) \log (-((-1+x) x))}{(-2+x) x \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-40*x^2 + 40*x^3 - 10*x^4)*Log[x]^2 + (2 - 3*x + x^2)*Log[x - x^2] + Log[x]*(-2 + 5*x - 2*x^2 + (2
- 2*x + x^2)*Log[x - x^2]))/((120*x^2 - 120*x^3 + 30*x^4)*Log[x]^2),x]

[Out]

(-10*x - ((-1 + x)*Log[-((-1 + x)*x)])/((-2 + x)*x*Log[x]))/30

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fricas [A]  time = 0.69, size = 41, normalized size = 1.11 \begin {gather*} -\frac {{\left (x - 1\right )} \log \left (-x^{2} + x\right ) + 10 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x)}{30 \, {\left (x^{2} - 2 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2+5*x-2)*log(x)+(x^2-3*x+2)*log(-x^2+
x))/(30*x^4-120*x^3+120*x^2)/log(x)^2,x, algorithm="fricas")

[Out]

-1/30*((x - 1)*log(-x^2 + x) + 10*(x^3 - 2*x^2)*log(x))/((x^2 - 2*x)*log(x))

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giac [A]  time = 0.34, size = 41, normalized size = 1.11 \begin {gather*} -\frac {1}{3} \, x - \frac {{\left (x - 1\right )} \log \left (-x + 1\right )}{30 \, {\left (x^{2} \log \relax (x) - 2 \, x \log \relax (x)\right )}} - \frac {1}{60 \, {\left (x - 2\right )}} - \frac {1}{60 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2+5*x-2)*log(x)+(x^2-3*x+2)*log(-x^2+
x))/(30*x^4-120*x^3+120*x^2)/log(x)^2,x, algorithm="giac")

[Out]

-1/3*x - 1/30*(x - 1)*log(-x + 1)/(x^2*log(x) - 2*x*log(x)) - 1/60/(x - 2) - 1/60/x

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maple [C]  time = 0.17, size = 259, normalized size = 7.00

method	result	size
risch	\(-\frac {\left (x -1\right ) \ln \left (x -1\right )}{30 x \left (x -2\right ) \ln \relax (x )}-\frac {-i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}+i \pi x \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}-i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )-2 i \pi -i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )-i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i x \right )-2 i \pi x \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+2 i x \pi +2 i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+i \pi x \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \left (x -1\right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right )+20 x^{3} \ln \relax (x )-40 x^{2} \ln \relax (x )+2 x \ln \relax (x )-2 \ln \relax (x )}{60 x \left (x -2\right ) \ln \relax (x )}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^4+40*x^3-40*x^2)*ln(x)^2+((x^2-2*x+2)*ln(-x^2+x)-2*x^2+5*x-2)*ln(x)+(x^2-3*x+2)*ln(-x^2+x))/(30*x^
4-120*x^3+120*x^2)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/30/x*(x-1)/(x-2)/ln(x)*ln(x-1)-1/60*(-I*Pi*csgn(I*x*(x-1))^3+I*Pi*x*csgn(I*x*(x-1))^3-I*Pi*csgn(I*(x-1))*cs
gn(I*x*(x-1))^2-2*I*Pi-I*Pi*x*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))-I*Pi*csgn(I*x)*csgn(I*x*(x-1))^2-2*I*Pi*
x*csgn(I*x*(x-1))^2+2*I*Pi*x+2*I*Pi*csgn(I*x*(x-1))^2+I*Pi*x*csgn(I*(x-1))*csgn(I*x*(x-1))^2+I*Pi*x*csgn(I*x)*
csgn(I*x*(x-1))^2+I*Pi*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))+20*x^3*ln(x)-40*x^2*ln(x)+2*x*ln(x)-2*ln(x))/x/
(x-2)/ln(x)

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maxima [A]  time = 0.46, size = 42, normalized size = 1.14 \begin {gather*} -\frac {{\left (10 \, x^{3} - 20 \, x^{2} + x - 1\right )} \log \relax (x) + {\left (x - 1\right )} \log \left (-x + 1\right )}{30 \, {\left (x^{2} - 2 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2+5*x-2)*log(x)+(x^2-3*x+2)*log(-x^2+
x))/(30*x^4-120*x^3+120*x^2)/log(x)^2,x, algorithm="maxima")

[Out]

-1/30*((10*x^3 - 20*x^2 + x - 1)*log(x) + (x - 1)*log(-x + 1))/((x^2 - 2*x)*log(x))

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mupad [B]  time = 1.94, size = 39, normalized size = 1.05 \begin {gather*} \frac {\frac {\ln \left (x-x^2\right )}{30}-\frac {x\,\ln \left (x-x^2\right )}{30}}{x\,\ln \relax (x)\,\left (x-2\right )}-\frac {x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(5*x - 2*x^2 + log(x - x^2)*(x^2 - 2*x + 2) - 2) - log(x)^2*(40*x^2 - 40*x^3 + 10*x^4) + log(x - x
^2)*(x^2 - 3*x + 2))/(log(x)^2*(120*x^2 - 120*x^3 + 30*x^4)),x)

[Out]

(log(x - x^2)/30 - (x*log(x - x^2))/30)/(x*log(x)*(x - 2)) - x/3

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sympy [A]  time = 0.45, size = 27, normalized size = 0.73 \begin {gather*} - \frac {x}{3} + \frac {\left (1 - x\right ) \log {\left (- x^{2} + x \right )}}{30 x^{2} \log {\relax (x )} - 60 x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**4+40*x**3-40*x**2)*ln(x)**2+((x**2-2*x+2)*ln(-x**2+x)-2*x**2+5*x-2)*ln(x)+(x**2-3*x+2)*ln(-
x**2+x))/(30*x**4-120*x**3+120*x**2)/ln(x)**2,x)

[Out]

-x/3 + (1 - x)*log(-x**2 + x)/(30*x**2*log(x) - 60*x*log(x))

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