3.50.44 \(\int \frac {(6 x^2+2 x^3) \log (1-x)+(3+x-4 x^2) \log ^2(1-x)+((6 x+2 x^2) \log (1-x)+(-x+x^2) \log ^2(1-x)) \log (x)}{-x^3+x^4+(-2 x^2+2 x^3) \log (x)+(-x+x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {(3+x) \log ^2(1-x)}{x+\log (x)} \]

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Rubi [F]  time = 3.21, 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 (6 x^2+2 x^3\right ) \log (1-x)+\left (3+x-4 x^2\right ) \log ^2(1-x)+\left (\left (6 x+2 x^2\right ) \log (1-x)+\left (-x+x^2\right ) \log ^2(1-x)\right ) \log (x)}{-x^3+x^4+\left (-2 x^2+2 x^3\right ) \log (x)+\left (-x+x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((6*x^2 + 2*x^3)*Log[1 - x] + (3 + x - 4*x^2)*Log[1 - x]^2 + ((6*x + 2*x^2)*Log[1 - x] + (-x + x^2)*Log[1
- x]^2)*Log[x])/(-x^3 + x^4 + (-2*x^2 + 2*x^3)*Log[x] + (-x + x^2)*Log[x]^2),x]

[Out]

-4*Defer[Int][Log[1 - x]^2/(x + Log[x])^2, x] - 3*Defer[Int][Log[1 - x]^2/(x*(x + Log[x])^2), x] - Defer[Int][
(x*Log[1 - x]^2)/(x + Log[x])^2, x] + 2*Defer[Int][Log[1 - x]/(x + Log[x]), x] + 8*Defer[Int][Log[1 - x]/((-1
+ x)*(x + Log[x])), x] + Defer[Int][Log[1 - x]^2/(x + Log[x]), x]

Rubi steps

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

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Mathematica [A]  time = 0.32, size = 18, normalized size = 1.00 \begin {gather*} \frac {(3+x) \log ^2(1-x)}{x+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((6*x^2 + 2*x^3)*Log[1 - x] + (3 + x - 4*x^2)*Log[1 - x]^2 + ((6*x + 2*x^2)*Log[1 - x] + (-x + x^2)*
Log[1 - x]^2)*Log[x])/(-x^3 + x^4 + (-2*x^2 + 2*x^3)*Log[x] + (-x + x^2)*Log[x]^2),x]

[Out]

((3 + x)*Log[1 - x]^2)/(x + Log[x])

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fricas [A]  time = 0.98, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (x + 3\right )} \log \left (-x + 1\right )^{2}}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-x)*log(-x+1)^2+(2*x^2+6*x)*log(-x+1))*log(x)+(-4*x^2+x+3)*log(-x+1)^2+(2*x^3+6*x^2)*log(-x+1)
)/((x^2-x)*log(x)^2+(2*x^3-2*x^2)*log(x)+x^4-x^3),x, algorithm="fricas")

[Out]

(x + 3)*log(-x + 1)^2/(x + log(x))

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giac [A]  time = 0.17, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (x + 3\right )} \log \left (-x + 1\right )^{2}}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-x)*log(-x+1)^2+(2*x^2+6*x)*log(-x+1))*log(x)+(-4*x^2+x+3)*log(-x+1)^2+(2*x^3+6*x^2)*log(-x+1)
)/((x^2-x)*log(x)^2+(2*x^3-2*x^2)*log(x)+x^4-x^3),x, algorithm="giac")

[Out]

(x + 3)*log(-x + 1)^2/(x + log(x))

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maple [A]  time = 0.05, size = 19, normalized size = 1.06

method	result	size
risch	\(\frac {\ln \left (1-x \right )^{2} \left (3+x \right )}{x +\ln \relax (x )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-x)*ln(1-x)^2+(2*x^2+6*x)*ln(1-x))*ln(x)+(-4*x^2+x+3)*ln(1-x)^2+(2*x^3+6*x^2)*ln(1-x))/((x^2-x)*ln(x
)^2+(2*x^3-2*x^2)*ln(x)+x^4-x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (x + 3\right )} \log \left (-x + 1\right )^{2}}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-x)*log(-x+1)^2+(2*x^2+6*x)*log(-x+1))*log(x)+(-4*x^2+x+3)*log(-x+1)^2+(2*x^3+6*x^2)*log(-x+1)
)/((x^2-x)*log(x)^2+(2*x^3-2*x^2)*log(x)+x^4-x^3),x, algorithm="maxima")

[Out]

(x + 3)*log(-x + 1)^2/(x + log(x))

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mupad [B]  time = 4.20, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\ln \left (1-x\right )}^2\,\left (x+3\right )}{x+\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(1 - x)^2*(x - 4*x^2 + 3) + log(x)*(log(1 - x)*(6*x + 2*x^2) - log(1 - x)^2*(x - x^2)) + log(1 - x)*(
6*x^2 + 2*x^3))/(log(x)*(2*x^2 - 2*x^3) + log(x)^2*(x - x^2) + x^3 - x^4),x)

[Out]

(log(1 - x)^2*(x + 3))/(x + log(x))

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sympy [A]  time = 0.45, size = 14, normalized size = 0.78 \begin {gather*} \frac {\left (x + 3\right ) \log {\left (1 - x \right )}^{2}}{x + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-x)*ln(-x+1)**2+(2*x**2+6*x)*ln(-x+1))*ln(x)+(-4*x**2+x+3)*ln(-x+1)**2+(2*x**3+6*x**2)*ln(-x+
1))/((x**2-x)*ln(x)**2+(2*x**3-2*x**2)*ln(x)+x**4-x**3),x)

[Out]

(x + 3)*log(1 - x)**2/(x + log(x))

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