3.29.94 \(\int \frac {4-20 x-32 x^2-9 x^3+(-8 x-8 x^2-2 x^3) \log (x)+(4+4 x+x^2) \log (\log (2))}{4 x-6 x^2-12 x^3-4 x^4+(-4 x^2-4 x^3-x^4) \log (x)+(4 x+4 x^2+x^3) \log (\log (2))} \, dx\)

Optimal. Leaf size=24 \[ \log \left (x-x^2 \left (4-\frac {3}{2+x}+\log (x)\right )+x \log (\log (2))\right ) \]

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Rubi [F]  time = 1.90, 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 {4-20 x-32 x^2-9 x^3+\left (-8 x-8 x^2-2 x^3\right ) \log (x)+\left (4+4 x+x^2\right ) \log (\log (2))}{4 x-6 x^2-12 x^3-4 x^4+\left (-4 x^2-4 x^3-x^4\right ) \log (x)+\left (4 x+4 x^2+x^3\right ) \log (\log (2))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 - 20*x - 32*x^2 - 9*x^3 + (-8*x - 8*x^2 - 2*x^3)*Log[x] + (4 + 4*x + x^2)*Log[Log[2]])/(4*x - 6*x^2 - 1
2*x^3 - 4*x^4 + (-4*x^2 - 4*x^3 - x^4)*Log[x] + (4*x + 4*x^2 + x^3)*Log[Log[2]]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+20 x+32 x^2+9 x^3-\left (-8 x-8 x^2-2 x^3\right ) \log (x)-\left (4+4 x+x^2\right ) \log (\log (2))}{x (2+x) \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )} \, dx\\ &=\int \left (\frac {2}{x}+\frac {x^3+4 (1+\log (\log (2)))+4 x (2+\log (\log (2)))+x^2 (8+\log (\log (2)))}{x (2+x) \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )}\right ) \, dx\\ &=2 \log (x)+\int \frac {x^3+4 (1+\log (\log (2)))+4 x (2+\log (\log (2)))+x^2 (8+\log (\log (2)))}{x (2+x) \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )} \, dx\\ &=2 \log (x)+\int \left (\frac {6}{(-2-x) \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )}+\frac {x}{4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))}+\frac {6 \left (1+\frac {1}{6} \log (\log (2))\right )}{4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))}+\frac {2 (1+\log (\log (2)))}{x \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )}\right ) \, dx\\ &=2 \log (x)+6 \int \frac {1}{(-2-x) \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )} \, dx+(2 (1+\log (\log (2)))) \int \frac {1}{x \left (4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))\right )} \, dx+(6+\log (\log (2))) \int \frac {1}{4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))} \, dx+\int \frac {x}{4 x^2+2 x \log (x)+x^2 \log (x)+4 x \left (1-\frac {1}{4} \log (\log (2))\right )-2 (1+\log (\log (2)))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 46, normalized size = 1.92 \begin {gather*} 2 \log (x)-\log (x (2+x))+\log \left (2-4 x-4 x^2-2 x \log (x)-x^2 \log (x)+2 \log (\log (2))+x \log (\log (2))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 20*x - 32*x^2 - 9*x^3 + (-8*x - 8*x^2 - 2*x^3)*Log[x] + (4 + 4*x + x^2)*Log[Log[2]])/(4*x - 6*x
^2 - 12*x^3 - 4*x^4 + (-4*x^2 - 4*x^3 - x^4)*Log[x] + (4*x + 4*x^2 + x^3)*Log[Log[2]]),x]

[Out]

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

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fricas [A]  time = 0.77, size = 44, normalized size = 1.83 \begin {gather*} 2 \, \log \relax (x) + \log \left (\frac {4 \, x^{2} + {\left (x^{2} + 2 \, x\right )} \log \relax (x) - {\left (x + 2\right )} \log \left (\log \relax (2)\right ) + 4 \, x - 2}{x^{2} + 2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+4*x+4)*log(log(2))+(-2*x^3-8*x^2-8*x)*log(x)-9*x^3-32*x^2-20*x+4)/((x^3+4*x^2+4*x)*log(log(2))
+(-x^4-4*x^3-4*x^2)*log(x)-4*x^4-12*x^3-6*x^2+4*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.23, size = 42, normalized size = 1.75 \begin {gather*} \log \left (-x^{2} \log \relax (x) - 4 \, x^{2} - 2 \, x \log \relax (x) + x \log \left (\log \relax (2)\right ) - 4 \, x + 2 \, \log \left (\log \relax (2)\right ) + 2\right ) - \log \left (x + 2\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+4*x+4)*log(log(2))+(-2*x^3-8*x^2-8*x)*log(x)-9*x^3-32*x^2-20*x+4)/((x^3+4*x^2+4*x)*log(log(2))
+(-x^4-4*x^3-4*x^2)*log(x)-4*x^4-12*x^3-6*x^2+4*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.13, size = 40, normalized size = 1.67

method	result	size
risch	\(2 \ln \relax (x )+\ln \left (\ln \relax (x )-\frac {x \ln \left (\ln \relax (2)\right )-4 x^{2}+2 \ln \left (\ln \relax (2)\right )-4 x +2}{x \left (2+x \right )}\right )\)	\(40\)
norman	\(\ln \relax (x )-\ln \left (2+x \right )+\ln \left (-x^{2} \ln \relax (x )+x \ln \left (\ln \relax (2)\right )-4 x^{2}-2 x \ln \relax (x )+2 \ln \left (\ln \relax (2)\right )-4 x +2\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+4*x+4)*ln(ln(2))+(-2*x^3-8*x^2-8*x)*ln(x)-9*x^3-32*x^2-20*x+4)/((x^3+4*x^2+4*x)*ln(ln(2))+(-x^4-4*x^
3-4*x^2)*ln(x)-4*x^4-12*x^3-6*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.74, size = 46, normalized size = 1.92 \begin {gather*} 2 \, \log \relax (x) + \log \left (\frac {4 \, x^{2} - x {\left (\log \left (\log \relax (2)\right ) - 4\right )} + {\left (x^{2} + 2 \, x\right )} \log \relax (x) - 2 \, \log \left (\log \relax (2)\right ) - 2}{x^{2} + 2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+4*x+4)*log(log(2))+(-2*x^3-8*x^2-8*x)*log(x)-9*x^3-32*x^2-20*x+4)/((x^3+4*x^2+4*x)*log(log(2))
+(-x^4-4*x^3-4*x^2)*log(x)-4*x^4-12*x^3-6*x^2+4*x),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {20\,x-\ln \left (\ln \relax (2)\right )\,\left (x^2+4\,x+4\right )+32\,x^2+9\,x^3+\ln \relax (x)\,\left (2\,x^3+8\,x^2+8\,x\right )-4}{6\,x^2-\ln \left (\ln \relax (2)\right )\,\left (x^3+4\,x^2+4\,x\right )-4\,x+12\,x^3+4\,x^4+\ln \relax (x)\,\left (x^4+4\,x^3+4\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x - log(log(2))*(4*x + x^2 + 4) + 32*x^2 + 9*x^3 + log(x)*(8*x + 8*x^2 + 2*x^3) - 4)/(6*x^2 - log(log(
2))*(4*x + 4*x^2 + x^3) - 4*x + 12*x^3 + 4*x^4 + log(x)*(4*x^2 + 4*x^3 + x^4)),x)

[Out]

int((20*x - log(log(2))*(4*x + x^2 + 4) + 32*x^2 + 9*x^3 + log(x)*(8*x + 8*x^2 + 2*x^3) - 4)/(6*x^2 - log(log(
2))*(4*x + 4*x^2 + x^3) - 4*x + 12*x^3 + 4*x^4 + log(x)*(4*x^2 + 4*x^3 + x^4)), x)

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sympy [A]  time = 0.46, size = 39, normalized size = 1.62 \begin {gather*} 2 \log {\relax (x )} + \log {\left (\log {\relax (x )} + \frac {4 x^{2} - x \log {\left (\log {\relax (2 )} \right )} + 4 x - 2 - 2 \log {\left (\log {\relax (2 )} \right )}}{x^{2} + 2 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+4*x+4)*ln(ln(2))+(-2*x**3-8*x**2-8*x)*ln(x)-9*x**3-32*x**2-20*x+4)/((x**3+4*x**2+4*x)*ln(ln(2
))+(-x**4-4*x**3-4*x**2)*ln(x)-4*x**4-12*x**3-6*x**2+4*x),x)

[Out]

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

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