\(\int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+(4 x^2-6 x^3+4 x^4-6 x^5) \log (x)+(1+2 x^2+x^4) \log ^2(x)} \, dx\) [6188]

Optimal result

Integrand size = 73, antiderivative size = 26 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=i \pi +\log (4)+\frac {1}{-2+3 x-\log (x)-\frac {\log (x)}{x^2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx \]

[In]

Int[(x + x^3 - 3*x^4 - 2*x*Log[x])/(4*x^4 - 12*x^5 + 9*x^6 + (4*x^2 - 6*x^3 + 4*x^4 - 6*x^5)*Log[x] + (1 + 2*x
^2 + x^4)*Log[x]^2),x]

[Out]

6*Defer[Int][(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])^(-2), x] + (2 - 3*I)*Defer[Int][1/((I - x)*(-2*x^2 + 3*x^3
- Log[x] - x^2*Log[x])^2), x] + 5*Defer[Int][x/(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])^2, x] - 6*Defer[Int][x^
2/(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])^2, x] + Defer[Int][x^3/(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])^2, x] -
3*Defer[Int][x^4/(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])^2, x] - (2 + 3*I)*Defer[Int][1/((I + x)*(-2*x^2 + 3*x
^3 - Log[x] - x^2*Log[x])^2), x] - Defer[Int][1/((I - x)*(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])), x] + Defer[I
nt][1/((I + x)*(-2*x^2 + 3*x^3 - Log[x] - x^2*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x+x^3-3 x^4-2 x \log (x)}{\left (x^2 (-2+3 x)-\left (1+x^2\right ) \log (x)\right )^2} \, dx \\ & = \int \left (\frac {2 x}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )}+\frac {x \left (1+6 x^2-9 x^3+x^4-3 x^5\right )}{\left (1+x^2\right ) \left (2 x^2-3 x^3+\log (x)+x^2 \log (x)\right )^2}\right ) \, dx \\ & = 2 \int \frac {x}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx+\int \frac {x \left (1+6 x^2-9 x^3+x^4-3 x^5\right )}{\left (1+x^2\right ) \left (2 x^2-3 x^3+\log (x)+x^2 \log (x)\right )^2} \, dx \\ & = 2 \int \left (-\frac {1}{2 (i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )}+\frac {1}{2 (i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {6}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}+\frac {5 x}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}-\frac {6 x^2}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}+\frac {x^3}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}-\frac {3 x^4}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}-\frac {2 (3+2 x)}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}\right ) \, dx \\ & = -\left (2 \int \frac {3+2 x}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx\right )-3 \int \frac {x^4}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+5 \int \frac {x}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+6 \int \frac {1}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \frac {x^2}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+\int \frac {x^3}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-\int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx+\int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx \\ & = -\left (2 \int \left (\frac {3}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}+\frac {2 x}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}\right ) \, dx\right )-3 \int \frac {x^4}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+5 \int \frac {x}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+6 \int \frac {1}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \frac {x^2}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+\int \frac {x^3}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-\int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx+\int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx \\ & = -\left (3 \int \frac {x^4}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx\right )-4 \int \frac {x}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+5 \int \frac {x}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+6 \int \frac {1}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \frac {x^2}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \frac {1}{\left (1+x^2\right ) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+\int \frac {x^3}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-\int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx+\int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx \\ & = -\left (3 \int \frac {x^4}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx\right )-4 \int \left (-\frac {1}{2 (i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}+\frac {1}{2 (i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}\right ) \, dx+5 \int \frac {x}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+6 \int \frac {1}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \frac {x^2}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \left (\frac {i}{2 (i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}+\frac {i}{2 (i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2}\right ) \, dx+\int \frac {x^3}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-\int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx+\int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx \\ & = -\left (3 i \int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx\right )-3 i \int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+2 \int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-2 \int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-3 \int \frac {x^4}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+5 \int \frac {x}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+6 \int \frac {1}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-6 \int \frac {x^2}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx+\int \frac {x^3}{\left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )^2} \, dx-\int \frac {1}{(i-x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx+\int \frac {1}{(i+x) \left (-2 x^2+3 x^3-\log (x)-x^2 \log (x)\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x^2}{2 x^2-3 x^3+\log (x)+x^2 \log (x)} \]

[In]

Integrate[(x + x^3 - 3*x^4 - 2*x*Log[x])/(4*x^4 - 12*x^5 + 9*x^6 + (4*x^2 - 6*x^3 + 4*x^4 - 6*x^5)*Log[x] + (1
+ 2*x^2 + x^4)*Log[x]^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

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

[In]

int((-2*x*ln(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*ln(x)^2+(-6*x^5+4*x^4-6*x^3+4*x^2)*ln(x)+9*x^6-12*x^5+4*x^4),x,met
hod=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{3 \, x^{3} - 2 \, x^{2} - {\left (x^{2} + 1\right )} \log \left (x\right )} \]

[In]

integrate((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4-6*x^3+4*x^2)*log(x)+9*x^6-12*x^5+4*x
^4),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=- \frac {x^{2}}{- 3 x^{3} + 2 x^{2} + \left (x^{2} + 1\right ) \log {\left (x \right )}} \]

[In]

integrate((-2*x*ln(x)-3*x**4+x**3+x)/((x**4+2*x**2+1)*ln(x)**2+(-6*x**5+4*x**4-6*x**3+4*x**2)*ln(x)+9*x**6-12*
x**5+4*x**4),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{3 \, x^{3} - 2 \, x^{2} - {\left (x^{2} + 1\right )} \log \left (x\right )} \]

[In]

integrate((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4-6*x^3+4*x^2)*log(x)+9*x^6-12*x^5+4*x
^4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x^{2}}{3 \, x^{3} - x^{2} \log \left (x\right ) - 2 \, x^{2} - \log \left (x\right )} \]

[In]

integrate((-2*x*log(x)-3*x^4+x^3+x)/((x^4+2*x^2+1)*log(x)^2+(-6*x^5+4*x^4-6*x^3+4*x^2)*log(x)+9*x^6-12*x^5+4*x
^4),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x+x^3-3 x^4-2 x \log (x)}{4 x^4-12 x^5+9 x^6+\left (4 x^2-6 x^3+4 x^4-6 x^5\right ) \log (x)+\left (1+2 x^2+x^4\right ) \log ^2(x)} \, dx=\int \frac {x-2\,x\,\ln \left (x\right )+x^3-3\,x^4}{{\ln \left (x\right )}^2\,\left (x^4+2\,x^2+1\right )+\ln \left (x\right )\,\left (-6\,x^5+4\,x^4-6\,x^3+4\,x^2\right )+4\,x^4-12\,x^5+9\,x^6} \,d x \]

[In]

int((x - 2*x*log(x) + x^3 - 3*x^4)/(log(x)^2*(2*x^2 + x^4 + 1) + log(x)*(4*x^2 - 6*x^3 + 4*x^4 - 6*x^5) + 4*x^
4 - 12*x^5 + 9*x^6),x)

[Out]

int((x - 2*x*log(x) + x^3 - 3*x^4)/(log(x)^2*(2*x^2 + x^4 + 1) + log(x)*(4*x^2 - 6*x^3 + 4*x^4 - 6*x^5) + 4*x^
4 - 12*x^5 + 9*x^6), x)