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

Optimal. Leaf size=23 \[ \log \left (-2+x-x \left (x^2-\log \left (2 x-x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.24, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6741, 6684} \begin {gather*} \log \left (x^3-x-x \log ((2-x) x)+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[2 - x + x^3 - x*Log[(2 - x)*x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

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

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Mathematica [A]  time = 0.34, size = 19, normalized size = 0.83 \begin {gather*} \log \left (2-x+x^3-x \log (-((-2+x) x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[2 - x + x^3 - x*Log[-((-2 + x)*x)]]

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fricas [A]  time = 0.57, size = 30, normalized size = 1.30 \begin {gather*} \log \relax (x) + \log \left (-\frac {x^{3} - x \log \left (-x^{2} + 2 \, x\right ) - x + 2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 21, normalized size = 0.91 \begin {gather*} \log \left (-x^{3} + x \log \left (-x^{2} + 2 \, x\right ) + x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-x^3 + x*log(-x^2 + 2*x) + x - 2)

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maple [A]  time = 0.05, size = 23, normalized size = 1.00

method	result	size
norman	\(\ln \left (x^{3}-\ln \left (-x^{2}+2 x \right ) x -x +2\right )\)	\(23\)
risch	\(\ln \relax (x )+\ln \left (\ln \left (-x^{2}+2 x \right )-\frac {x^{3}-x +2}{x}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 31, normalized size = 1.35 \begin {gather*} \log \relax (x) + \log \left (-\frac {x^{3} - x \log \relax (x) - x \log \left (-x + 2\right ) - x + 2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.64, size = 26, normalized size = 1.13 \begin {gather*} \ln \left (\ln \left (2\,x-x^2\right )-\frac {2}{x}-x^2+1\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.30, size = 20, normalized size = 0.87 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\left (- x^{2} + 2 x \right )} + \frac {- x^{3} + x - 2}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(log(-x**2 + 2*x) + (-x**3 + x - 2)/x)

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