\(\int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx\) [5460]

Optimal result

Integrand size = 35, antiderivative size = 18 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-x-\log \left (3-x^2+3 \log (x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6874, 6816} \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-\log \left (-x^2+3 \log (x)+3\right )-x \]

[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 6816

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

Rule 6874

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

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-x-\log \left (3-x^2+3 \log (x)\right ) \]

[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]]

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
norman	\(-x -\ln \left (x^{2}-3 \ln \left (x \right )-3\right )\)	\(17\)
risch	\(-x -\ln \left (-\frac {x^{2}}{3}+\ln \left (x \right )+1\right )\)	\(17\)
parallelrisch	\(-x -\ln \left (x^{2}-3 \ln \left (x \right )-3\right )\)	\(17\)
default	\(-x -\ln \left (3+3 \ln \left (x \right )-x^{2}\right )\)	\(19\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-x - \log \left (-x^{2} + 3 \, \log \left (x\right ) + 3\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=- x - \log {\left (- \frac {x^{2}}{3} + \log {\left (x \right )} + 1 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-x - \log \left (-\frac {1}{3} \, x^{2} + \log \left (x\right ) + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-x - \log \left (-x^{2} + 3 \, \log \left (x\right ) + 3\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 11.74 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3-3 x+2 x^2+x^3-3 x \log (x)}{3 x-x^3+3 x \log (x)} \, dx=-x-\ln \left (x^2-3\,\ln \left (x\right )-3\right ) \]

[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)