\(\int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+(-x+4 x^2+3 x^3+2 x^4) \log (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2})} \, dx\) [4858]

Optimal result

Integrand size = 90, antiderivative size = 32 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log \left (x^2-\log \left (1-\frac {-3+\frac {1-x}{x}-x}{2 x}+x\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6873, 12, 6816} \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log \left (x^2-\log \left (-\frac {1}{2 x^2}+x+\frac {2}{x}+\frac {3}{2}\right )\right ) \]

[In]

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

[Out]

Log[x^2 - Log[3/2 - 1/(2*x^2) + 2/x + x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6873

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

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log \left (x^2-\log \left (\frac {3}{2}-\frac {1}{2 x^2}+\frac {2}{x}+x\right )\right ) \]

[In]

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

[Out]

Log[x^2 - Log[3/2 - 1/(2*x^2) + 2/x + x]]

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91

method	result	size
norman	\(\ln \left (x^{2}-\ln \left (\frac {2 x^{3}+3 x^{2}+4 x -1}{2 x^{2}}\right )\right )\)	\(29\)
risch	\(\ln \left (-x^{2}+\ln \left (\frac {2 x^{3}+3 x^{2}+4 x -1}{2 x^{2}}\right )\right )\)	\(29\)
parallelrisch	\(\ln \left (x^{2}-\ln \left (\frac {2 x^{3}+3 x^{2}+4 x -1}{2 x^{2}}\right )\right )\)	\(29\)

[In]

int((-4*x^5-6*x^4-6*x^3+2*x^2-4*x+2)/((2*x^4+3*x^3+4*x^2-x)*ln(1/2*(2*x^3+3*x^2+4*x-1)/x^2)-2*x^6-3*x^5-4*x^4+
x^3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log \left (-x^{2} + \log \left (\frac {2 \, x^{3} + 3 \, x^{2} + 4 \, x - 1}{2 \, x^{2}}\right )\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log {\left (- x^{2} + \log {\left (\frac {x^{3} + \frac {3 x^{2}}{2} + 2 x - \frac {1}{2}}{x^{2}} \right )} \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log \left (-x^{2} - \log \left (2\right ) + \log \left (2 \, x^{3} + 3 \, x^{2} + 4 \, x - 1\right ) - 2 \, \log \left (x\right )\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\log \left (-x^{2} + \log \left (\frac {2 \, x^{3} + 3 \, x^{2} + 4 \, x - 1}{2 \, x^{2}}\right )\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {2-4 x+2 x^2-6 x^3-6 x^4-4 x^5}{x^3-4 x^4-3 x^5-2 x^6+\left (-x+4 x^2+3 x^3+2 x^4\right ) \log \left (\frac {-1+4 x+3 x^2+2 x^3}{2 x^2}\right )} \, dx=\ln \left (\ln \left (\frac {x^3+\frac {3\,x^2}{2}+2\,x-\frac {1}{2}}{x^2}\right )-x^2\right ) \]

[In]

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

[Out]

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