\(\int \frac {33-4 x+12 x^3+3 \log (\frac {x^2}{2})}{27 x-2 x^2+3 x^4+3 x \log (\frac {x^2}{2})} \, dx\) [10140]

Optimal result

Integrand size = 48, antiderivative size = 22 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log \left (x \left (9+x \left (-\frac {2}{3}+x^2\right )+\log \left (\frac {x^2}{2}\right )\right )\right ) \]

[Out]

ln(x*(9+ln(1/2*x^2)+x*(x^2-2/3)))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6816} \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log \left (3 x^4-2 x^2+3 x \log \left (\frac {x^2}{2}\right )+27 x\right ) \]

[In]

Int[(33 - 4*x + 12*x^3 + 3*Log[x^2/2])/(27*x - 2*x^2 + 3*x^4 + 3*x*Log[x^2/2]),x]

[Out]

Log[27*x - 2*x^2 + 3*x^4 + 3*x*Log[x^2/2]]

Rule 6816

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

Rubi steps \begin{align*} \text {integral}& = \log \left (27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log (x)+\log \left (27-2 x+3 x^3+3 \log \left (\frac {x^2}{2}\right )\right ) \]

[In]

Integrate[(33 - 4*x + 12*x^3 + 3*Log[x^2/2])/(27*x - 2*x^2 + 3*x^4 + 3*x*Log[x^2/2]),x]

[Out]

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
risch	\(\ln \left (x \right )+\ln \left (x^{3}-\frac {2 x}{3}+\ln \left (\frac {x^{2}}{2}\right )+9\right )\)	\(19\)
derivativedivides	\(\ln \left (3 x \ln \left (\frac {x^{2}}{2}\right )+3 x^{4}-2 x^{2}+27 x \right )\)	\(25\)
default	\(\ln \left (3 x \ln \left (\frac {x^{2}}{2}\right )+3 x^{4}-2 x^{2}+27 x \right )\)	\(25\)
parallelrisch	\(\frac {\ln \left (\frac {x^{2}}{2}\right )}{2}+\ln \left (x^{3}-\frac {2 x}{3}+\ln \left (\frac {x^{2}}{2}\right )+9\right )\)	\(25\)
norman	\(\frac {\ln \left (\frac {x^{2}}{2}\right )}{2}+\ln \left (3 x^{3}+3 \ln \left (\frac {x^{2}}{2}\right )-2 x +27\right )\)	\(29\)

[In]

int((3*ln(1/2*x^2)+12*x^3-4*x+33)/(3*x*ln(1/2*x^2)+3*x^4-2*x^2+27*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(x^3-2/3*x+ln(1/2*x^2)+9)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log \left (3 \, x^{3} - 2 \, x + 3 \, \log \left (\frac {1}{2} \, x^{2}\right ) + 27\right ) + \frac {1}{2} \, \log \left (\frac {1}{2} \, x^{2}\right ) \]

[In]

integrate((3*log(1/2*x^2)+12*x^3-4*x+33)/(3*x*log(1/2*x^2)+3*x^4-2*x^2+27*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log {\left (x \right )} + \log {\left (x^{3} - \frac {2 x}{3} + \log {\left (\frac {x^{2}}{2} \right )} + 9 \right )} \]

[In]

integrate((3*ln(1/2*x**2)+12*x**3-4*x+33)/(3*x*ln(1/2*x**2)+3*x**4-2*x**2+27*x),x)

[Out]

log(x) + log(x**3 - 2*x/3 + log(x**2/2) + 9)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log \left (3 \, x^{4} - 2 \, x^{2} + 3 \, x \log \left (\frac {1}{2} \, x^{2}\right ) + 27 \, x\right ) \]

[In]

integrate((3*log(1/2*x^2)+12*x^3-4*x+33)/(3*x*log(1/2*x^2)+3*x^4-2*x^2+27*x),x, algorithm="maxima")

[Out]

log(3*x^4 - 2*x^2 + 3*x*log(1/2*x^2) + 27*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\log \left (3 \, x^{3} - 2 \, x + 3 \, \log \left (\frac {1}{2} \, x^{2}\right ) + 27\right ) + \log \left (x\right ) \]

[In]

integrate((3*log(1/2*x^2)+12*x^3-4*x+33)/(3*x*log(1/2*x^2)+3*x^4-2*x^2+27*x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {33-4 x+12 x^3+3 \log \left (\frac {x^2}{2}\right )}{27 x-2 x^2+3 x^4+3 x \log \left (\frac {x^2}{2}\right )} \, dx=\ln \left (\ln \left (\frac {x^2}{2}\right )-\frac {2\,x}{3}+x^3+9\right )+\frac {\ln \left (x^2\right )}{2} \]

[In]

int((3*log(x^2/2) - 4*x + 12*x^3 + 33)/(27*x + 3*x*log(x^2/2) - 2*x^2 + 3*x^4),x)

[Out]

log(log(x^2/2) - (2*x)/3 + x^3 + 9) + log(x^2)/2