3.82.17 \(\int \frac {(9 x^3+24 x^4+16 x^5) \log (x)+(-1-4 x-4 x^2+(-3 x-8 x^2-8 x^3) \log (x)) \log (\log (4))+(6 x^2+20 x^3+16 x^4) \log (x) \log (\log (x))+(x+4 x^2+4 x^3) \log (x) \log ^2(\log (x))}{(9 x^3+24 x^4+16 x^5) \log (x)+(6 x^2+20 x^3+16 x^4) \log (x) \log (\log (x))+(x+4 x^2+4 x^3) \log (x) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=24 \[ x+\frac {\log (\log (4))}{2 x+\frac {x}{1+2 x}+\log (\log (x))} \]

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Rubi [F]  time = 1.55, 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 {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((9*x^3 + 24*x^4 + 16*x^5)*Log[x] + (-1 - 4*x - 4*x^2 + (-3*x - 8*x^2 - 8*x^3)*Log[x])*Log[Log[4]] + (6*x^
2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x]] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2)/((9*x^3 + 24*x^4 + 16*x^5)
*Log[x] + (6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x]] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2),x]

[Out]

x - 3*Log[Log[4]]*Defer[Int][(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])^(-2), x] - 8*Log[Log[4]]*Defer[Int]
[x/(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])^2, x] - 8*Log[Log[4]]*Defer[Int][x^2/(3*x + 4*x^2 + Log[Log[x
]] + 2*x*Log[Log[x]])^2, x] - 4*Log[Log[4]]*Defer[Int][1/(Log[x]*(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])
^2), x] - Log[Log[4]]*Defer[Int][1/(x*Log[x]*(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])^2), x] - 4*Log[Log[
4]]*Defer[Int][x/(Log[x]*(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-(1+2 x)^2 \log (\log (4))+x \log (x) \left (24 x^3+16 x^4+x^2 (9-8 \log (\log (4)))-3 \log (\log (4))-8 x \log (\log (4))+2 x \left (3+10 x+8 x^2\right ) \log (\log (x))+(1+2 x)^2 \log ^2(\log (x))\right )}{x \log (x) (x (3+4 x)+(1+2 x) \log (\log (x)))^2} \, dx\\ &=\int \left (1-\frac {\left (1+4 x+4 x^2+3 x \log (x)+8 x^2 \log (x)+8 x^3 \log (x)\right ) \log (\log (4))}{x \log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}\right ) \, dx\\ &=x-\log (\log (4)) \int \frac {1+4 x+4 x^2+3 x \log (x)+8 x^2 \log (x)+8 x^3 \log (x)}{x \log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx\\ &=x-\log (\log (4)) \int \left (\frac {3}{\left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}+\frac {8 x}{\left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}+\frac {8 x^2}{\left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}+\frac {4}{\log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}+\frac {1}{x \log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}+\frac {4 x}{\log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2}\right ) \, dx\\ &=x-\log (\log (4)) \int \frac {1}{x \log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx-(3 \log (\log (4))) \int \frac {1}{\left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx-(4 \log (\log (4))) \int \frac {1}{\log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx-(4 \log (\log (4))) \int \frac {x}{\log (x) \left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx-(8 \log (\log (4))) \int \frac {x}{\left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx-(8 \log (\log (4))) \int \frac {x^2}{\left (3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 31, normalized size = 1.29 \begin {gather*} x+\frac {(1+2 x) \log (\log (4))}{3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((9*x^3 + 24*x^4 + 16*x^5)*Log[x] + (-1 - 4*x - 4*x^2 + (-3*x - 8*x^2 - 8*x^3)*Log[x])*Log[Log[4]] +
(6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x]] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2)/((9*x^3 + 24*x^4 + 1
6*x^5)*Log[x] + (6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x]] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2),x]

[Out]

x + ((1 + 2*x)*Log[Log[4]])/(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])

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fricas [A]  time = 0.69, size = 54, normalized size = 2.25 \begin {gather*} \frac {4 \, x^{3} + 3 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (2 \, \log \relax (2)\right ) + {\left (2 \, x^{2} + x\right )} \log \left (\log \relax (x)\right )}{4 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (\log \relax (x)\right ) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log(x)*log(log(x))+((-8*x^3-8*x^2-3*x)*l
og(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x^5+24*x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+2
0*x^3+6*x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.28, size = 40, normalized size = 1.67 \begin {gather*} x + \frac {2 \, x \log \relax (2) + 2 \, x \log \left (\log \relax (2)\right ) + \log \relax (2) + \log \left (\log \relax (2)\right )}{4 \, x^{2} + 2 \, x \log \left (\log \relax (x)\right ) + 3 \, x + \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log(x)*log(log(x))+((-8*x^3-8*x^2-3*x)*l
og(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x^5+24*x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+2
0*x^3+6*x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.07, size = 35, normalized size = 1.46

method	result	size
risch	\(x +\frac {\left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right ) \left (2 x +1\right )}{2 x \ln \left (\ln \relax (x )\right )+4 x^{2}+\ln \left (\ln \relax (x )\right )+3 x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+4*x^2+x)*ln(x)*ln(ln(x))^2+(16*x^4+20*x^3+6*x^2)*ln(x)*ln(ln(x))+((-8*x^3-8*x^2-3*x)*ln(x)-4*x^2-4
*x-1)*ln(2*ln(2))+(16*x^5+24*x^4+9*x^3)*ln(x))/((4*x^3+4*x^2+x)*ln(x)*ln(ln(x))^2+(16*x^4+20*x^3+6*x^2)*ln(x)*
ln(ln(x))+(16*x^5+24*x^4+9*x^3)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.50, size = 57, normalized size = 2.38 \begin {gather*} \frac {4 \, x^{3} + 3 \, x^{2} + 2 \, x {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + {\left (2 \, x^{2} + x\right )} \log \left (\log \relax (x)\right ) + \log \relax (2) + \log \left (\log \relax (2)\right )}{4 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (\log \relax (x)\right ) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log(x)*log(log(x))+((-8*x^3-8*x^2-3*x)*l
og(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x^5+24*x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+2
0*x^3+6*x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \relax (x)\,\left (4\,x^3+4\,x^2+x\right )\,{\ln \left (\ln \relax (x)\right )}^2+\ln \relax (x)\,\left (16\,x^4+20\,x^3+6\,x^2\right )\,\ln \left (\ln \relax (x)\right )+\ln \relax (x)\,\left (16\,x^5+24\,x^4+9\,x^3\right )-\ln \left (2\,\ln \relax (2)\right )\,\left (4\,x+4\,x^2+\ln \relax (x)\,\left (8\,x^3+8\,x^2+3\,x\right )+1\right )}{\ln \relax (x)\,\left (4\,x^3+4\,x^2+x\right )\,{\ln \left (\ln \relax (x)\right )}^2+\ln \relax (x)\,\left (16\,x^4+20\,x^3+6\,x^2\right )\,\ln \left (\ln \relax (x)\right )+\ln \relax (x)\,\left (16\,x^5+24\,x^4+9\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(9*x^3 + 24*x^4 + 16*x^5) - log(2*log(2))*(4*x + 4*x^2 + log(x)*(3*x + 8*x^2 + 8*x^3) + 1) + log(l
og(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x^2 + 20*x^3 + 16*x^4))/(log(x)*(9*x^3 + 24*x^4 +
16*x^5) + log(log(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x^2 + 20*x^3 + 16*x^4)),x)

[Out]

int((log(x)*(9*x^3 + 24*x^4 + 16*x^5) - log(2*log(2))*(4*x + 4*x^2 + log(x)*(3*x + 8*x^2 + 8*x^3) + 1) + log(l
og(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x^2 + 20*x^3 + 16*x^4))/(log(x)*(9*x^3 + 24*x^4 +
16*x^5) + log(log(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x^2 + 20*x^3 + 16*x^4)), x)

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sympy [A]  time = 0.45, size = 42, normalized size = 1.75 \begin {gather*} x + \frac {2 x \log {\left (\log {\relax (2 )} \right )} + 2 x \log {\relax (2 )} + \log {\left (\log {\relax (2 )} \right )} + \log {\relax (2 )}}{4 x^{2} + 3 x + \left (2 x + 1\right ) \log {\left (\log {\relax (x )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+4*x**2+x)*ln(x)*ln(ln(x))**2+(16*x**4+20*x**3+6*x**2)*ln(x)*ln(ln(x))+((-8*x**3-8*x**2-3*x)
*ln(x)-4*x**2-4*x-1)*ln(2*ln(2))+(16*x**5+24*x**4+9*x**3)*ln(x))/((4*x**3+4*x**2+x)*ln(x)*ln(ln(x))**2+(16*x**
4+20*x**3+6*x**2)*ln(x)*ln(ln(x))+(16*x**5+24*x**4+9*x**3)*ln(x)),x)

[Out]

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

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