3.45.66 \(\int \frac {x+2 x^2+(1+10 x+2 x \log (x)) \log (\log (2))}{x^2+(5 x+x \log (x)) \log (\log (2))} \, dx\)

Optimal. Leaf size=19 \[ -10+2 x+\log (x-(-5-\log (x)) \log (\log (2))) \]

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Rubi [A]  time = 0.16, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6742, 6684} \begin {gather*} 2 x+\log (x+\log (\log (2)) \log (x)+5 \log (\log (2))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + 2*x^2 + (1 + 10*x + 2*x*Log[x])*Log[Log[2]])/(x^2 + (5*x + x*Log[x])*Log[Log[2]]),x]

[Out]

2*x + Log[x + 5*Log[Log[2]] + Log[x]*Log[Log[2]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 18, normalized size = 0.95 \begin {gather*} 2 x+\log (x+5 \log (\log (2))+\log (x) \log (\log (2))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + 2*x^2 + (1 + 10*x + 2*x*Log[x])*Log[Log[2]])/(x^2 + (5*x + x*Log[x])*Log[Log[2]]),x]

[Out]

2*x + Log[x + 5*Log[Log[2]] + Log[x]*Log[Log[2]]]

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fricas [A]  time = 0.51, size = 15, normalized size = 0.79 \begin {gather*} 2 \, x + \log \left ({\left (\log \relax (x) + 5\right )} \log \left (\log \relax (2)\right ) + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+10*x+1)*log(log(2))+2*x^2+x)/((x*log(x)+5*x)*log(log(2))+x^2),x, algorithm="fricas")

[Out]

2*x + log((log(x) + 5)*log(log(2)) + x)

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giac [A]  time = 0.15, size = 18, normalized size = 0.95 \begin {gather*} 2 \, x + \log \left (\log \relax (x) \log \left (\log \relax (2)\right ) + x + 5 \, \log \left (\log \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+10*x+1)*log(log(2))+2*x^2+x)/((x*log(x)+5*x)*log(log(2))+x^2),x, algorithm="giac")

[Out]

2*x + log(log(x)*log(log(2)) + x + 5*log(log(2)))

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maple [A]  time = 0.04, size = 19, normalized size = 1.00

method	result	size
norman	\(2 x +\ln \left (\ln \relax (x ) \ln \left (\ln \relax (2)\right )+5 \ln \left (\ln \relax (2)\right )+x \right )\)	\(19\)
risch	\(2 x +\ln \left (\ln \relax (x )+\frac {5 \ln \left (\ln \relax (2)\right )+x}{\ln \left (\ln \relax (2)\right )}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(x)+10*x+1)*ln(ln(2))+2*x^2+x)/((x*ln(x)+5*x)*ln(ln(2))+x^2),x,method=_RETURNVERBOSE)

[Out]

2*x+ln(ln(x)*ln(ln(2))+5*ln(ln(2))+x)

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maxima [A]  time = 0.47, size = 24, normalized size = 1.26 \begin {gather*} 2 \, x + \log \left (\frac {\log \relax (x) \log \left (\log \relax (2)\right ) + x + 5 \, \log \left (\log \relax (2)\right )}{\log \left (\log \relax (2)\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(x)+10*x+1)*log(log(2))+2*x^2+x)/((x*log(x)+5*x)*log(log(2))+x^2),x, algorithm="maxima")

[Out]

2*x + log((log(x)*log(log(2)) + x + 5*log(log(2)))/log(log(2)))

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mupad [B]  time = 3.37, size = 18, normalized size = 0.95 \begin {gather*} 2\,x+\ln \left (x+5\,\ln \left (\ln \relax (2)\right )+\ln \left (\ln \relax (2)\right )\,\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + log(log(2))*(10*x + 2*x*log(x) + 1) + 2*x^2)/(log(log(2))*(5*x + x*log(x)) + x^2),x)

[Out]

2*x + log(x + 5*log(log(2)) + log(log(2))*log(x))

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sympy [A]  time = 0.15, size = 20, normalized size = 1.05 \begin {gather*} 2 x + \log {\left (\frac {x + 5 \log {\left (\log {\relax (2 )} \right )}}{\log {\left (\log {\relax (2 )} \right )}} + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(x)+10*x+1)*ln(ln(2))+2*x**2+x)/((x*ln(x)+5*x)*ln(ln(2))+x**2),x)

[Out]

2*x + log((x + 5*log(log(2)))/log(log(2)) + log(x))

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