3.45.44 \(\int \frac {2 \log (2)+(4+8 x+4 x^2-2 \log (2)) \log (x)+(8+8 x-2 \log (2)) \log (x) \log (\log (x))+4 \log (x) \log ^2(\log (x))}{(1+2 x+x^2) \log (x)+(2+2 x) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=19 \[ 61-x \left (-4+\frac {2 \log (2)}{1+x+\log (\log (x))}\right ) \]

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Rubi [F]  time = 0.40, 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 {2 \log (2)+\left (4+8 x+4 x^2-2 \log (2)\right ) \log (x)+(8+8 x-2 \log (2)) \log (x) \log (\log (x))+4 \log (x) \log ^2(\log (x))}{\left (1+2 x+x^2\right ) \log (x)+(2+2 x) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 19, normalized size = 1.00 \begin {gather*} 2 \left (2 x-\frac {x \log (2)}{1+x+\log (\log (x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 30, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} - x \log \relax (2) + 2 \, x \log \left (\log \relax (x)\right ) + 2 \, x\right )}}{x + \log \left (\log \relax (x)\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)*log(log(x))^2+(-2*log(2)+8*x+8)*log(x)*log(log(x))+(-2*log(2)+4*x^2+8*x+4)*log(x)+2*log(2)
)/(log(x)*log(log(x))^2+(2*x+2)*log(x)*log(log(x))+(x^2+2*x+1)*log(x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 17, normalized size = 0.89 \begin {gather*} 4 \, x - \frac {2 \, x \log \relax (2)}{x + \log \left (\log \relax (x)\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)*log(log(x))^2+(-2*log(2)+8*x+8)*log(x)*log(log(x))+(-2*log(2)+4*x^2+8*x+4)*log(x)+2*log(2)
)/(log(x)*log(log(x))^2+(2*x+2)*log(x)*log(log(x))+(x^2+2*x+1)*log(x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 18, normalized size = 0.95

method	result	size
risch	\(4 x -\frac {2 x \ln \relax (2)}{1+\ln \left (\ln \relax (x )\right )+x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(x)*ln(ln(x))^2+(-2*ln(2)+8*x+8)*ln(x)*ln(ln(x))+(-2*ln(2)+4*x^2+8*x+4)*ln(x)+2*ln(2))/(ln(x)*ln(ln(x
))^2+(2*x+2)*ln(x)*ln(ln(x))+(x^2+2*x+1)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 29, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} - x {\left (\log \relax (2) - 2\right )} + 2 \, x \log \left (\log \relax (x)\right )\right )}}{x + \log \left (\log \relax (x)\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)*log(log(x))^2+(-2*log(2)+8*x+8)*log(x)*log(log(x))+(-2*log(2)+4*x^2+8*x+4)*log(x)+2*log(2)
)/(log(x)*log(log(x))^2+(2*x+2)*log(x)*log(log(x))+(x^2+2*x+1)*log(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.43, size = 32, normalized size = 1.68 \begin {gather*} \frac {4\,x+\ln \relax (4)+\ln \left (\ln \relax (x)\right )\,\ln \relax (4)+4\,x\,\ln \left (\ln \relax (x)\right )+4\,x^2}{x+\ln \left (\ln \relax (x)\right )+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2) + log(x)*(8*x - 2*log(2) + 4*x^2 + 4) + 4*log(log(x))^2*log(x) + log(log(x))*log(x)*(8*x - 2*log
(2) + 8))/(log(x)*(2*x + x^2 + 1) + log(log(x))^2*log(x) + log(log(x))*log(x)*(2*x + 2)),x)

[Out]

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

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sympy [A]  time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} 4 x - \frac {2 x \log {\relax (2 )}}{x + \log {\left (\log {\relax (x )} \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(x)*ln(ln(x))**2+(-2*ln(2)+8*x+8)*ln(x)*ln(ln(x))+(-2*ln(2)+4*x**2+8*x+4)*ln(x)+2*ln(2))/(ln(x)
*ln(ln(x))**2+(2*x+2)*ln(x)*ln(ln(x))+(x**2+2*x+1)*ln(x)),x)

[Out]

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

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