3.8.67 \(\int \frac {-4 x+16 x^4+4 x^5-4 x^6+(-4-4 x+x^2+20 x^4+4 x^5-5 x^6) \log (x)}{8-8 x+2 x^2+(16-16 x+4 x^2) \log (x)+(8-8 x+2 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {x (2+x) \left (1-x^4\right )}{2 (-2+x) (1+\log (x))} \]

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x + 16*x^4 + 4*x^5 - 4*x^6 + (-4 - 4*x + x^2 + 20*x^4 + 4*x^5 - 5*x^6)*Log[x])/(8 - 8*x + 2*x^2 + (16
- 16*x + 4*x^2)*Log[x] + (8 - 8*x + 2*x^2)*Log[x]^2),x]

[Out]

Defer[Int][(-2 - x + 2*x^4 + x^5)/((-2 + x)*(1 + Log[x])^2), x]/2 + Defer[Int][(-4 - 4*x + x^2 + 20*x^4 + 4*x^
5 - 5*x^6)/((-2 + x)^2*(1 + Log[x])), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 \left (x-4 x^4-x^5+x^6\right )+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{2 (2-x)^2 (1+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {-4 \left (x-4 x^4-x^5+x^6\right )+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{(2-x)^2 (1+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {-2-x+2 x^4+x^5}{(-2+x) (1+\log (x))^2}+\frac {-4-4 x+x^2+20 x^4+4 x^5-5 x^6}{(-2+x)^2 (1+\log (x))}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-2-x+2 x^4+x^5}{(-2+x) (1+\log (x))^2} \, dx+\frac {1}{2} \int \frac {-4-4 x+x^2+20 x^4+4 x^5-5 x^6}{(-2+x)^2 (1+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 29, normalized size = 1.12 \begin {gather*} -\frac {x \left (-2-x+2 x^4+x^5\right )}{2 (-2+x) (1+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + 16*x^4 + 4*x^5 - 4*x^6 + (-4 - 4*x + x^2 + 20*x^4 + 4*x^5 - 5*x^6)*Log[x])/(8 - 8*x + 2*x^2
+ (16 - 16*x + 4*x^2)*Log[x] + (8 - 8*x + 2*x^2)*Log[x]^2),x]

[Out]

-1/2*(x*(-2 - x + 2*x^4 + x^5))/((-2 + x)*(1 + Log[x]))

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fricas [A]  time = 0.99, size = 30, normalized size = 1.15 \begin {gather*} -\frac {x^{6} + 2 \, x^{5} - x^{2} - 2 \, x}{2 \, {\left ({\left (x - 2\right )} \log \relax (x) + x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/((2*x^2-8*x+8)*log(x)^2+(4*x^2-16*x+
16)*log(x)+2*x^2-8*x+8),x, algorithm="fricas")

[Out]

-1/2*(x^6 + 2*x^5 - x^2 - 2*x)/((x - 2)*log(x) + x - 2)

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giac [A]  time = 0.40, size = 32, normalized size = 1.23 \begin {gather*} -\frac {x^{6} + 2 \, x^{5} - x^{2} - 2 \, x}{2 \, {\left (x \log \relax (x) + x - 2 \, \log \relax (x) - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/((2*x^2-8*x+8)*log(x)^2+(4*x^2-16*x+
16)*log(x)+2*x^2-8*x+8),x, algorithm="giac")

[Out]

-1/2*(x^6 + 2*x^5 - x^2 - 2*x)/(x*log(x) + x - 2*log(x) - 2)

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maple [A]  time = 0.05, size = 28, normalized size = 1.08

method	result	size
risch	\(-\frac {x \left (x^{5}+2 x^{4}-x -2\right )}{2 \left (x -2\right ) \left (\ln \relax (x )+1\right )}\)	\(28\)
norman	\(\frac {x +\frac {1}{2} x^{2}-x^{5}-\frac {1}{2} x^{6}}{\left (\ln \relax (x )+1\right ) \left (x -2\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*ln(x)-4*x^6+4*x^5+16*x^4-4*x)/((2*x^2-8*x+8)*ln(x)^2+(4*x^2-16*x+16)*ln(x
)+2*x^2-8*x+8),x,method=_RETURNVERBOSE)

[Out]

-1/2*x*(x^5+2*x^4-x-2)/(x-2)/(ln(x)+1)

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maxima [A]  time = 0.48, size = 30, normalized size = 1.15 \begin {gather*} -\frac {x^{6} + 2 \, x^{5} - x^{2} - 2 \, x}{2 \, {\left ({\left (x - 2\right )} \log \relax (x) + x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/((2*x^2-8*x+8)*log(x)^2+(4*x^2-16*x+
16)*log(x)+2*x^2-8*x+8),x, algorithm="maxima")

[Out]

-1/2*(x^6 + 2*x^5 - x^2 - 2*x)/((x - 2)*log(x) + x - 2)

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mupad [B]  time = 0.77, size = 106, normalized size = 4.08 \begin {gather*} \frac {60\,x}{x^2-4\,x+4}-\frac {31\,x}{2}-\frac {\frac {2\,x^2\,\left (-x^5+x^4+4\,x^3-1\right )}{{\left (x-2\right )}^2}-\frac {x\,\ln \relax (x)\,\left (5\,x^6-4\,x^5-20\,x^4-x^2+4\,x+4\right )}{2\,{\left (x-2\right )}^2}}{\ln \relax (x)+1}-16\,x^2-12\,x^3-8\,x^4-\frac {5\,x^5}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + log(x)*(4*x - x^2 - 20*x^4 - 4*x^5 + 5*x^6 + 4) - 16*x^4 - 4*x^5 + 4*x^6)/(log(x)^2*(2*x^2 - 8*x +
8) - 8*x + log(x)*(4*x^2 - 16*x + 16) + 2*x^2 + 8),x)

[Out]

(60*x)/(x^2 - 4*x + 4) - (31*x)/2 - ((2*x^2*(4*x^3 + x^4 - x^5 - 1))/(x - 2)^2 - (x*log(x)*(4*x - x^2 - 20*x^4
- 4*x^5 + 5*x^6 + 4))/(2*(x - 2)^2))/(log(x) + 1) - 16*x^2 - 12*x^3 - 8*x^4 - (5*x^5)/2

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sympy [A]  time = 0.15, size = 27, normalized size = 1.04 \begin {gather*} \frac {- x^{6} - 2 x^{5} + x^{2} + 2 x}{2 x + \left (2 x - 4\right ) \log {\relax (x )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**6+4*x**5+20*x**4+x**2-4*x-4)*ln(x)-4*x**6+4*x**5+16*x**4-4*x)/((2*x**2-8*x+8)*ln(x)**2+(4*x*
*2-16*x+16)*ln(x)+2*x**2-8*x+8),x)

[Out]

(-x**6 - 2*x**5 + x**2 + 2*x)/(2*x + (2*x - 4)*log(x) - 4)

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