\(\int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log (x^2)}{-81 x+72 x^2-9 x^3-5 x^4+x^5+(-3 x+x^2) \log (x^2)} \, dx\) [10050]

Optimal result

Integrand size = 62, antiderivative size = 16 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=\log \left (4+x+\frac {-9+\log \left (x^2\right )}{(-3+x)^2}\right ) \]

[Out]

ln(x+4+(ln(x^2)-9)/(-3+x)^2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6873, 6874, 6816} \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=\log \left (x^3-2 x^2+\log \left (x^2\right )-15 x+27\right )-2 \log (3-x) \]

[In]

Int[(-6 - 7*x + 27*x^2 - 9*x^3 + x^4 - 2*x*Log[x^2])/(-81*x + 72*x^2 - 9*x^3 - 5*x^4 + x^5 + (-3*x + x^2)*Log[
x^2]),x]

[Out]

-2*Log[3 - x] + Log[27 - 15*x - 2*x^2 + x^3 + Log[x^2]]

Rule 6816

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {6+7 x-27 x^2+9 x^3-x^4+2 x \log \left (x^2\right )}{(3-x) x \left (27-15 x-2 x^2+x^3+\log \left (x^2\right )\right )} \, dx \\ & = \int \left (-\frac {2}{-3+x}+\frac {2-15 x-4 x^2+3 x^3}{x \left (27-15 x-2 x^2+x^3+\log \left (x^2\right )\right )}\right ) \, dx \\ & = -2 \log (3-x)+\int \frac {2-15 x-4 x^2+3 x^3}{x \left (27-15 x-2 x^2+x^3+\log \left (x^2\right )\right )} \, dx \\ & = -2 \log (3-x)+\log \left (27-15 x-2 x^2+x^3+\log \left (x^2\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=-2 \log (3-x)+\log \left (27-15 x-2 x^2+x^3+\log \left (x^2\right )\right ) \]

[In]

Integrate[(-6 - 7*x + 27*x^2 - 9*x^3 + x^4 - 2*x*Log[x^2])/(-81*x + 72*x^2 - 9*x^3 - 5*x^4 + x^5 + (-3*x + x^2
)*Log[x^2]),x]

[Out]

-2*Log[3 - x] + Log[27 - 15*x - 2*x^2 + x^3 + Log[x^2]]

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62

method	result	size
norman	\(-2 \ln \left (-3+x \right )+\ln \left (x^{3}-2 x^{2}-15 x +\ln \left (x^{2}\right )+27\right )\)	\(26\)
risch	\(-2 \ln \left (-3+x \right )+\ln \left (x^{3}-2 x^{2}-15 x +\ln \left (x^{2}\right )+27\right )\)	\(26\)
parallelrisch	\(-2 \ln \left (-3+x \right )+\ln \left (x^{3}-2 x^{2}-15 x +\ln \left (x^{2}\right )+27\right )\)	\(26\)

[In]

int((-2*x*ln(x^2)+x^4-9*x^3+27*x^2-7*x-6)/((x^2-3*x)*ln(x^2)+x^5-5*x^4-9*x^3+72*x^2-81*x),x,method=_RETURNVERB
OSE)

[Out]

-2*ln(-3+x)+ln(x^3-2*x^2-15*x+ln(x^2)+27)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=\log \left (x^{3} - 2 \, x^{2} - 15 \, x + \log \left (x^{2}\right ) + 27\right ) - 2 \, \log \left (x - 3\right ) \]

[In]

integrate((-2*x*log(x^2)+x^4-9*x^3+27*x^2-7*x-6)/((x^2-3*x)*log(x^2)+x^5-5*x^4-9*x^3+72*x^2-81*x),x, algorithm
="fricas")

[Out]

log(x^3 - 2*x^2 - 15*x + log(x^2) + 27) - 2*log(x - 3)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=- 2 \log {\left (x - 3 \right )} + \log {\left (x^{3} - 2 x^{2} - 15 x + \log {\left (x^{2} \right )} + 27 \right )} \]

[In]

integrate((-2*x*ln(x**2)+x**4-9*x**3+27*x**2-7*x-6)/((x**2-3*x)*ln(x**2)+x**5-5*x**4-9*x**3+72*x**2-81*x),x)

[Out]

-2*log(x - 3) + log(x**3 - 2*x**2 - 15*x + log(x**2) + 27)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=\log \left (\frac {1}{2} \, x^{3} - x^{2} - \frac {15}{2} \, x + \log \left (x\right ) + \frac {27}{2}\right ) - 2 \, \log \left (x - 3\right ) \]

[In]

integrate((-2*x*log(x^2)+x^4-9*x^3+27*x^2-7*x-6)/((x^2-3*x)*log(x^2)+x^5-5*x^4-9*x^3+72*x^2-81*x),x, algorithm
="maxima")

[Out]

log(1/2*x^3 - x^2 - 15/2*x + log(x) + 27/2) - 2*log(x - 3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=\log \left (x^{3} - 2 \, x^{2} - 15 \, x + \log \left (x^{2}\right ) + 27\right ) - 2 \, \log \left (x - 3\right ) \]

[In]

integrate((-2*x*log(x^2)+x^4-9*x^3+27*x^2-7*x-6)/((x^2-3*x)*log(x^2)+x^5-5*x^4-9*x^3+72*x^2-81*x),x, algorithm
="giac")

[Out]

log(x^3 - 2*x^2 - 15*x + log(x^2) + 27) - 2*log(x - 3)

Mupad [B] (verification not implemented)

Time = 15.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-6-7 x+27 x^2-9 x^3+x^4-2 x \log \left (x^2\right )}{-81 x+72 x^2-9 x^3-5 x^4+x^5+\left (-3 x+x^2\right ) \log \left (x^2\right )} \, dx=\ln \left (\ln \left (x^2\right )-15\,x-2\,x^2+x^3+27\right )-2\,\ln \left (x-3\right ) \]

[In]

int((7*x + 2*x*log(x^2) - 27*x^2 + 9*x^3 - x^4 + 6)/(81*x + log(x^2)*(3*x - x^2) - 72*x^2 + 9*x^3 + 5*x^4 - x^
5),x)

[Out]

log(log(x^2) - 15*x - 2*x^2 + x^3 + 27) - 2*log(x - 3)