3.6.53 \(\int \frac {16+2 x^2-6 \log (x)}{25-20 x-6 x^2+4 x^3+x^4+(10 x-4 x^2-2 x^3) \log (6)+x^2 \log ^2(6)+(-30+12 x+6 x^2-6 x \log (6)) \log (x)+9 \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {2}{-2+\frac {5}{x}-x+\log (6)-\frac {3 \log (x)}{x}} \]

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Rubi [F]  time = 0.52, 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 {16+2 x^2-6 \log (x)}{25-20 x-6 x^2+4 x^3+x^4+\left (10 x-4 x^2-2 x^3\right ) \log (6)+x^2 \log ^2(6)+\left (-30+12 x+6 x^2-6 x \log (6)\right ) \log (x)+9 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16 + 2*x^2 - 6*Log[x])/(25 - 20*x - 6*x^2 + 4*x^3 + x^4 + (10*x - 4*x^2 - 2*x^3)*Log[6] + x^2*Log[6]^2 +
(-30 + 12*x + 6*x^2 - 6*x*Log[6])*Log[x] + 9*Log[x]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16+2 x^2-6 \log (x)}{25-20 x+4 x^3+x^4+\left (10 x-4 x^2-2 x^3\right ) \log (6)+x^2 \left (-6+\log ^2(6)\right )+\left (-30+12 x+6 x^2-6 x \log (6)\right ) \log (x)+9 \log ^2(x)} \, dx\\ &=\int \frac {2 \left (8+x^2-3 \log (x)\right )}{\left (5-x^2+x (-2+\log (6))-3 \log (x)\right )^2} \, dx\\ &=2 \int \frac {8+x^2-3 \log (x)}{\left (5-x^2+x (-2+\log (6))-3 \log (x)\right )^2} \, dx\\ &=2 \int \left (\frac {3+2 x^2+x (2-\log (6))}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2}+\frac {1}{5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)}\right ) \, dx\\ &=2 \int \frac {3+2 x^2+x (2-\log (6))}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2} \, dx+2 \int \frac {1}{5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)} \, dx\\ &=2 \int \left (\frac {3}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2}+\frac {2 x^2}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2}+\frac {x (2-\log (6))}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2}\right ) \, dx+2 \int \frac {1}{5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)} \, dx\\ &=2 \int \frac {1}{5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)} \, dx+4 \int \frac {x^2}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2} \, dx+6 \int \frac {1}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2} \, dx+(2 (2-\log (6))) \int \frac {x}{\left (5-x^2-2 x \left (1-\frac {\log (6)}{2}\right )-3 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 22, normalized size = 0.96 \begin {gather*} -\frac {2 x}{-5+2 x+x^2-x \log (6)+3 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 + 2*x^2 - 6*Log[x])/(25 - 20*x - 6*x^2 + 4*x^3 + x^4 + (10*x - 4*x^2 - 2*x^3)*Log[6] + x^2*Log[6
]^2 + (-30 + 12*x + 6*x^2 - 6*x*Log[6])*Log[x] + 9*Log[x]^2),x]

[Out]

(-2*x)/(-5 + 2*x + x^2 - x*Log[6] + 3*Log[x])

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fricas [A]  time = 0.58, size = 22, normalized size = 0.96 \begin {gather*} -\frac {2 \, x}{x^{2} - x \log \relax (6) + 2 \, x + 3 \, \log \relax (x) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(x)+2*x^2+16)/(9*log(x)^2+(-6*x*log(6)+6*x^2+12*x-30)*log(x)+x^2*log(6)^2+(-2*x^3-4*x^2+10*x)
*log(6)+x^4+4*x^3-6*x^2-20*x+25),x, algorithm="fricas")

[Out]

-2*x/(x^2 - x*log(6) + 2*x + 3*log(x) - 5)

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giac [A]  time = 0.39, size = 22, normalized size = 0.96 \begin {gather*} -\frac {2 \, x}{x^{2} - x \log \relax (6) + 2 \, x + 3 \, \log \relax (x) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(x)+2*x^2+16)/(9*log(x)^2+(-6*x*log(6)+6*x^2+12*x-30)*log(x)+x^2*log(6)^2+(-2*x^3-4*x^2+10*x)
*log(6)+x^4+4*x^3-6*x^2-20*x+25),x, algorithm="giac")

[Out]

-2*x/(x^2 - x*log(6) + 2*x + 3*log(x) - 5)

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maple [A]  time = 0.31, size = 24, normalized size = 1.04

method	result	size
norman	\(\frac {2 x}{x \ln \relax (6)-x^{2}-2 x -3 \ln \relax (x )+5}\)	\(24\)
risch	\(\frac {2 x}{x \ln \relax (2)+x \ln \relax (3)-x^{2}-2 x -3 \ln \relax (x )+5}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*ln(x)+2*x^2+16)/(9*ln(x)^2+(-6*x*ln(6)+6*x^2+12*x-30)*ln(x)+x^2*ln(6)^2+(-2*x^3-4*x^2+10*x)*ln(6)+x^4+
4*x^3-6*x^2-20*x+25),x,method=_RETURNVERBOSE)

[Out]

2*x/(x*ln(6)-x^2-2*x-3*ln(x)+5)

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maxima [A]  time = 0.98, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \, x}{x^{2} - x {\left (\log \relax (3) + \log \relax (2) - 2\right )} + 3 \, \log \relax (x) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(x)+2*x^2+16)/(9*log(x)^2+(-6*x*log(6)+6*x^2+12*x-30)*log(x)+x^2*log(6)^2+(-2*x^3-4*x^2+10*x)
*log(6)+x^4+4*x^3-6*x^2-20*x+25),x, algorithm="maxima")

[Out]

-2*x/(x^2 - x*(log(3) + log(2) - 2) + 3*log(x) - 5)

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mupad [B]  time = 0.98, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2\,x}{3\,\ln \relax (x)-x\,\left (\ln \relax (6)-2\right )+x^2-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - 6*log(x) + 16)/(x^2*log(6)^2 - 20*x + 9*log(x)^2 - log(6)*(4*x^2 - 10*x + 2*x^3) - 6*x^2 + 4*x^3
+ x^4 + log(x)*(12*x - 6*x*log(6) + 6*x^2 - 30) + 25),x)

[Out]

-(2*x)/(3*log(x) - x*(log(6) - 2) + x^2 - 5)

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sympy [A]  time = 0.16, size = 22, normalized size = 0.96 \begin {gather*} - \frac {2 x}{x^{2} - x \log {\relax (6 )} + 2 x + 3 \log {\relax (x )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*ln(x)+2*x**2+16)/(9*ln(x)**2+(-6*x*ln(6)+6*x**2+12*x-30)*ln(x)+x**2*ln(6)**2+(-2*x**3-4*x**2+10*
x)*ln(6)+x**4+4*x**3-6*x**2-20*x+25),x)

[Out]

-2*x/(x**2 - x*log(6) + 2*x + 3*log(x) - 5)

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