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

Optimal. Leaf size=32 \[ \log \left (x^2-\log \left (1-\frac {-3+\frac {1-x}{x}-x}{2 x}+x\right )\right ) \]

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Rubi [A]  time = 0.28, antiderivative size = 25, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 3, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6741, 12, 6684} \begin {gather*} \log \left (x^2-\log \left (-\frac {1}{2 x^2}+x+\frac {2}{x}+\frac {3}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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 6684

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

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 3.03, size = 25, normalized size = 0.78 \begin {gather*} \log \left (x^2-\log \left (\frac {3}{2}-\frac {1}{2 x^2}+\frac {2}{x}+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.66, size = 28, normalized size = 0.88 \begin {gather*} \log \left (-x^{2} + \log \left (\frac {2 \, x^{3} + 3 \, x^{2} + 4 \, x - 1}{2 \, x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.25, size = 28, normalized size = 0.88 \begin {gather*} \log \left (-x^{2} + \log \left (\frac {2 \, x^{3} + 3 \, x^{2} + 4 \, x - 1}{2 \, x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.06, size = 29, normalized size = 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	\(\ln \left (-x^{2}+\ln \left (\frac {2 x^{3}+3 x^{2}+4 x -1}{2 x^{2}}\right )\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.50, size = 31, normalized size = 0.97 \begin {gather*} \log \left (-x^{2} - \log \relax (2) + \log \left (2 \, x^{3} + 3 \, x^{2} + 4 \, x - 1\right ) - 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 3.94, size = 25, normalized size = 0.78 \begin {gather*} \ln \left (\ln \left (\frac {x^3+\frac {3\,x^2}{2}+2\,x-\frac {1}{2}}{x^2}\right )-x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.40, size = 26, normalized size = 0.81 \begin {gather*} \log {\left (- x^{2} + \log {\left (\frac {x^{3} + \frac {3 x^{2}}{2} + 2 x - \frac {1}{2}}{x^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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