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

Optimal. Leaf size=18 \[ -x-\log \left (3-x^2+3 \log (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6742, 6684} \begin {gather*} -\log \left (-x^2+3 \log (x)+3\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - Log[3 - x^2 + 3*Log[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 6742

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 18, normalized size = 1.00 \begin {gather*} -x-\log \left (3-x^2+3 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - Log[3 - x^2 + 3*Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 18, normalized size = 1.00 \begin {gather*} -x - \log \left (-x^{2} + 3 \, \log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(-x^2 + 3*log(x) + 3)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 18, normalized size = 1.00 \begin {gather*} -x - \log \left (-x^{2} + 3 \, \log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(-x^2 + 3*log(x) + 3)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 17, normalized size = 0.94

method	result	size
norman	\(-x -\ln \left (x^{2}-3 \ln \relax (x )-3\right )\)	\(17\)
risch	\(-x -\ln \left (-\frac {x^{2}}{3}+\ln \relax (x )+1\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 16, normalized size = 0.89 \begin {gather*} -x - \log \left (-\frac {1}{3} \, x^{2} + \log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(-1/3*x^2 + log(x) + 1)

________________________________________________________________________________________

mupad [B]  time = 3.47, size = 16, normalized size = 0.89 \begin {gather*} -x-\ln \left (x^2-3\,\ln \relax (x)-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- x - log(x^2 - 3*log(x) - 3)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 14, normalized size = 0.78 \begin {gather*} - x - \log {\left (- \frac {x^{2}}{3} + \log {\relax (x )} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(-x**2/3 + log(x) + 1)

________________________________________________________________________________________