3.34.14 \(\int \frac {-x+x^2+2 x^3+(-1+x+2 x^2) \log (x)+(1-2 x^2+x^3+(-1-x+2 x^2) \log (x)) \log (-x+x^2)}{(-x^2-x^3+x^4+x^5) \log ^2(-x+x^2)} \, dx\)

Optimal. Leaf size=21 \[ -\frac {x+\log (x)}{\left (x+x^2\right ) \log ((-1+x) x)} \]

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Rubi [F]  time = 3.69, 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 {-x+x^2+2 x^3+\left (-1+x+2 x^2\right ) \log (x)+\left (1-2 x^2+x^3+\left (-1-x+2 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )}{\left (-x^2-x^3+x^4+x^5\right ) \log ^2\left (-x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x + x^2 + 2*x^3 + (-1 + x + 2*x^2)*Log[x] + (1 - 2*x^2 + x^3 + (-1 - x + 2*x^2)*Log[x])*Log[-x + x^2])/(
(-x^2 - x^3 + x^4 + x^5)*Log[-x + x^2]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x-x^2-2 x^3-\left (-1+x+2 x^2\right ) \log (x)-\left (1-2 x^2+x^3+\left (-1-x+2 x^2\right ) \log (x)\right ) \log \left (-x+x^2\right )}{(1-x) x^2 (1+x)^2 \log ^2((-1+x) x)} \, dx\\ &=\int \left (\frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)}-\frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)}+\frac {2 x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)}+\frac {(-1+2 x) \log (x)}{x^2 \left (-1+x^2\right ) \log ^2((-1+x) x)}+\frac {-1-x+x^2+\log (x)+2 x \log (x)}{x^2 (1+x)^2 \log ((-1+x) x)}\right ) \, dx\\ &=2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {(-1+2 x) \log (x)}{x^2 \left (-1+x^2\right ) \log ^2((-1+x) x)} \, dx+\int \frac {-1-x+x^2+\log (x)+2 x \log (x)}{x^2 (1+x)^2 \log ((-1+x) x)} \, dx\\ &=2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \left (\frac {\log (x)}{2 (-1+x) \log ^2((-1+x) x)}+\frac {\log (x)}{x^2 \log ^2((-1+x) x)}-\frac {2 \log (x)}{x \log ^2((-1+x) x)}+\frac {3 \log (x)}{2 (1+x) \log ^2((-1+x) x)}\right ) \, dx+\int \left (\frac {-1-x+x^2+\log (x)+2 x \log (x)}{x^2 \log ((-1+x) x)}-\frac {2 \left (-1-x+x^2+\log (x)+2 x \log (x)\right )}{x \log ((-1+x) x)}+\frac {-1-x+x^2+\log (x)+2 x \log (x)}{(1+x)^2 \log ((-1+x) x)}+\frac {2 \left (-1-x+x^2+\log (x)+2 x \log (x)\right )}{(1+x) \log ((-1+x) x)}\right ) \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx\\ &=\frac {1}{2} \int \frac {\log (x)}{(-1+x) \log ^2((-1+x) x)} \, dx+\frac {3}{2} \int \frac {\log (x)}{(1+x) \log ^2((-1+x) x)} \, dx+2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-2 \int \frac {\log (x)}{x \log ^2((-1+x) x)} \, dx-2 \int \frac {-1-x+x^2+\log (x)+2 x \log (x)}{x \log ((-1+x) x)} \, dx+2 \int \frac {-1-x+x^2+\log (x)+2 x \log (x)}{(1+x) \log ((-1+x) x)} \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ^2((-1+x) x)} \, dx+\int \frac {-1-x+x^2+\log (x)+2 x \log (x)}{x^2 \log ((-1+x) x)} \, dx+\int \frac {-1-x+x^2+\log (x)+2 x \log (x)}{(1+x)^2 \log ((-1+x) x)} \, dx\\ &=\frac {1}{2} \int \frac {\log (x)}{(-1+x) \log ^2((-1+x) x)} \, dx+\frac {3}{2} \int \frac {\log (x)}{(1+x) \log ^2((-1+x) x)} \, dx-2 \int \left (-\frac {1}{\log ((-1+x) x)}-\frac {1}{x \log ((-1+x) x)}+\frac {x}{\log ((-1+x) x)}+\frac {2 \log (x)}{\log ((-1+x) x)}+\frac {\log (x)}{x \log ((-1+x) x)}\right ) \, dx+2 \int \left (-\frac {1}{(1+x) \log ((-1+x) x)}-\frac {x}{(1+x) \log ((-1+x) x)}+\frac {x^2}{(1+x) \log ((-1+x) x)}+\frac {\log (x)}{(1+x) \log ((-1+x) x)}+\frac {2 x \log (x)}{(1+x) \log ((-1+x) x)}\right ) \, dx+2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-2 \int \frac {\log (x)}{x \log ^2((-1+x) x)} \, dx+\int \left (\frac {1}{\log ((-1+x) x)}-\frac {1}{x^2 \log ((-1+x) x)}-\frac {1}{x \log ((-1+x) x)}+\frac {\log (x)}{x^2 \log ((-1+x) x)}+\frac {2 \log (x)}{x \log ((-1+x) x)}\right ) \, dx+\int \left (-\frac {1}{(1+x)^2 \log ((-1+x) x)}-\frac {x}{(1+x)^2 \log ((-1+x) x)}+\frac {x^2}{(1+x)^2 \log ((-1+x) x)}+\frac {\log (x)}{(1+x)^2 \log ((-1+x) x)}+\frac {2 x \log (x)}{(1+x)^2 \log ((-1+x) x)}\right ) \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ^2((-1+x) x)} \, dx\\ &=\frac {1}{2} \int \frac {\log (x)}{(-1+x) \log ^2((-1+x) x)} \, dx+\frac {3}{2} \int \frac {\log (x)}{(1+x) \log ^2((-1+x) x)} \, dx+2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-2 \int \frac {\log (x)}{x \log ^2((-1+x) x)} \, dx+2 \int \frac {1}{\log ((-1+x) x)} \, dx+2 \int \frac {1}{x \log ((-1+x) x)} \, dx-2 \int \frac {x}{\log ((-1+x) x)} \, dx-2 \int \frac {1}{(1+x) \log ((-1+x) x)} \, dx-2 \int \frac {x}{(1+x) \log ((-1+x) x)} \, dx+2 \int \frac {x^2}{(1+x) \log ((-1+x) x)} \, dx+2 \int \frac {x \log (x)}{(1+x)^2 \log ((-1+x) x)} \, dx+2 \int \frac {\log (x)}{(1+x) \log ((-1+x) x)} \, dx-4 \int \frac {\log (x)}{\log ((-1+x) x)} \, dx+4 \int \frac {x \log (x)}{(1+x) \log ((-1+x) x)} \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ^2((-1+x) x)} \, dx+\int \frac {1}{\log ((-1+x) x)} \, dx-\int \frac {1}{x^2 \log ((-1+x) x)} \, dx-\int \frac {1}{x \log ((-1+x) x)} \, dx-\int \frac {1}{(1+x)^2 \log ((-1+x) x)} \, dx-\int \frac {x}{(1+x)^2 \log ((-1+x) x)} \, dx+\int \frac {x^2}{(1+x)^2 \log ((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ((-1+x) x)} \, dx+\int \frac {\log (x)}{(1+x)^2 \log ((-1+x) x)} \, dx\\ &=\frac {1}{2} \int \frac {\log (x)}{(-1+x) \log ^2((-1+x) x)} \, dx+\frac {3}{2} \int \frac {\log (x)}{(1+x) \log ^2((-1+x) x)} \, dx+2 \int \left (-\frac {\log (x)}{(1+x)^2 \log ((-1+x) x)}+\frac {\log (x)}{(1+x) \log ((-1+x) x)}\right ) \, dx+2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-2 \int \frac {\log (x)}{x \log ^2((-1+x) x)} \, dx+2 \int \frac {1}{\log ((-1+x) x)} \, dx+2 \int \frac {1}{x \log ((-1+x) x)} \, dx-2 \int \frac {x}{\log ((-1+x) x)} \, dx-2 \int \frac {1}{(1+x) \log ((-1+x) x)} \, dx-2 \int \frac {x}{(1+x) \log ((-1+x) x)} \, dx+2 \int \frac {x^2}{(1+x) \log ((-1+x) x)} \, dx+2 \int \frac {\log (x)}{(1+x) \log ((-1+x) x)} \, dx+4 \int \left (\frac {\log (x)}{\log ((-1+x) x)}-\frac {\log (x)}{(1+x) \log ((-1+x) x)}\right ) \, dx-4 \int \frac {\log (x)}{\log ((-1+x) x)} \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ^2((-1+x) x)} \, dx+\int \frac {1}{\log ((-1+x) x)} \, dx-\int \frac {1}{x^2 \log ((-1+x) x)} \, dx-\int \frac {1}{x \log ((-1+x) x)} \, dx-\int \frac {1}{(1+x)^2 \log ((-1+x) x)} \, dx-\int \frac {x}{(1+x)^2 \log ((-1+x) x)} \, dx+\int \frac {x^2}{(1+x)^2 \log ((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ((-1+x) x)} \, dx+\int \frac {\log (x)}{(1+x)^2 \log ((-1+x) x)} \, dx\\ &=\frac {1}{2} \int \frac {\log (x)}{(-1+x) \log ^2((-1+x) x)} \, dx+\frac {3}{2} \int \frac {\log (x)}{(1+x) \log ^2((-1+x) x)} \, dx+2 \int \frac {x}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-2 \int \frac {\log (x)}{x \log ^2((-1+x) x)} \, dx+2 \int \frac {1}{\log ((-1+x) x)} \, dx+2 \int \frac {1}{x \log ((-1+x) x)} \, dx-2 \int \frac {x}{\log ((-1+x) x)} \, dx-2 \int \frac {1}{(1+x) \log ((-1+x) x)} \, dx-2 \int \frac {x}{(1+x) \log ((-1+x) x)} \, dx+2 \int \frac {x^2}{(1+x) \log ((-1+x) x)} \, dx-2 \int \frac {\log (x)}{(1+x)^2 \log ((-1+x) x)} \, dx+2 \left (2 \int \frac {\log (x)}{(1+x) \log ((-1+x) x)} \, dx\right )-4 \int \frac {\log (x)}{(1+x) \log ((-1+x) x)} \, dx+\int \frac {1}{(-1+x) (1+x)^2 \log ^2((-1+x) x)} \, dx-\int \frac {1}{(-1+x) x (1+x)^2 \log ^2((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ^2((-1+x) x)} \, dx+\int \frac {1}{\log ((-1+x) x)} \, dx-\int \frac {1}{x^2 \log ((-1+x) x)} \, dx-\int \frac {1}{x \log ((-1+x) x)} \, dx-\int \frac {1}{(1+x)^2 \log ((-1+x) x)} \, dx-\int \frac {x}{(1+x)^2 \log ((-1+x) x)} \, dx+\int \frac {x^2}{(1+x)^2 \log ((-1+x) x)} \, dx+\int \frac {\log (x)}{x^2 \log ((-1+x) x)} \, dx+\int \frac {\log (x)}{(1+x)^2 \log ((-1+x) x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 22, normalized size = 1.05 \begin {gather*} -\frac {x+\log (x)}{x (1+x) \log ((-1+x) x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + x^2 + 2*x^3 + (-1 + x + 2*x^2)*Log[x] + (1 - 2*x^2 + x^3 + (-1 - x + 2*x^2)*Log[x])*Log[-x + x
^2])/((-x^2 - x^3 + x^4 + x^5)*Log[-x + x^2]^2),x]

[Out]

-((x + Log[x])/(x*(1 + x)*Log[(-1 + x)*x]))

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fricas [A]  time = 0.56, size = 23, normalized size = 1.10 \begin {gather*} -\frac {x + \log \relax (x)}{{\left (x^{2} + x\right )} \log \left (x^{2} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-x-1)*log(x)+x^3-2*x^2+1)*log(x^2-x)+(2*x^2+x-1)*log(x)+2*x^3+x^2-x)/(x^5+x^4-x^3-x^2)/log(x
^2-x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.39, size = 33, normalized size = 1.57 \begin {gather*} -\frac {x + \log \relax (x)}{x^{2} \log \left (x - 1\right ) + x^{2} \log \relax (x) + x \log \left (x - 1\right ) + x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-x-1)*log(x)+x^3-2*x^2+1)*log(x^2-x)+(2*x^2+x-1)*log(x)+2*x^3+x^2-x)/(x^5+x^4-x^3-x^2)/log(x
^2-x)^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.08, size = 106, normalized size = 5.05

method	result	size
risch	\(-\frac {2 \left (x +\ln \relax (x )\right )}{\left (x +1\right ) x \left (2 \ln \left (x -1\right )+2 \ln \relax (x )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i x \left (x -1\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \left (x -1\right )\right )^{3}\right )}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-x-1)*ln(x)+x^3-2*x^2+1)*ln(x^2-x)+(2*x^2+x-1)*ln(x)+2*x^3+x^2-x)/(x^5+x^4-x^3-x^2)/ln(x^2-x)^2,x,
method=_RETURNVERBOSE)

[Out]

-2*(x+ln(x))/(x+1)/x/(2*ln(x-1)+2*ln(x)-I*Pi*csgn(I*x)*csgn(I*(x-1))*csgn(I*x*(x-1))+I*Pi*csgn(I*x)*csgn(I*x*(
x-1))^2+I*Pi*csgn(I*(x-1))*csgn(I*x*(x-1))^2-I*Pi*csgn(I*x*(x-1))^3)

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maxima [A]  time = 0.69, size = 27, normalized size = 1.29 \begin {gather*} -\frac {x + \log \relax (x)}{{\left (x^{2} + x\right )} \log \left (x - 1\right ) + {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-x-1)*log(x)+x^3-2*x^2+1)*log(x^2-x)+(2*x^2+x-1)*log(x)+2*x^3+x^2-x)/(x^5+x^4-x^3-x^2)/log(x
^2-x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.67, size = 24, normalized size = 1.14 \begin {gather*} -\frac {x+\ln \relax (x)}{x\,\ln \left (x^2-x\right )\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(x + 2*x^2 - 1) - x + x^2 + 2*x^3 - log(x^2 - x)*(log(x)*(x - 2*x^2 + 1) + 2*x^2 - x^3 - 1))/(log
(x^2 - x)^2*(x^2 + x^3 - x^4 - x^5)),x)

[Out]

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

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sympy [A]  time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} \frac {- x - \log {\relax (x )}}{\left (x^{2} + x\right ) \log {\left (x^{2} - x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-x-1)*ln(x)+x**3-2*x**2+1)*ln(x**2-x)+(2*x**2+x-1)*ln(x)+2*x**3+x**2-x)/(x**5+x**4-x**3-x**
2)/ln(x**2-x)**2,x)

[Out]

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

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