3.83.2 \(\int \frac {-9 x+3 \log ^2(\frac {1}{x})+(9+9 \log (\frac {1}{x})) \log (\log (4))}{9 x^2+6 x \log (\frac {1}{x})+\log ^2(\frac {1}{x})} \, dx\)

Optimal. Leaf size=22 \[ \frac {3 (x-\log (\log (4)))}{1+\frac {3 x}{\log \left (\frac {1}{x}\right )}} \]

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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 {-9 x+3 \log ^2\left (\frac {1}{x}\right )+\left (9+9 \log \left (\frac {1}{x}\right )\right ) \log (\log (4))}{9 x^2+6 x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

3*x + 9*Log[Log[4]]*Defer[Int][(3*x + Log[x^(-1)])^(-2), x] - 9*(1 + 3*Log[Log[4]])*Defer[Int][x/(3*x + Log[x^
(-1)])^2, x] + 27*Defer[Int][x^2/(3*x + Log[x^(-1)])^2, x] + 9*Log[Log[4]]*Defer[Int][(3*x + Log[x^(-1)])^(-1)
, x] - 18*Defer[Int][x/(3*x + Log[x^(-1)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9 x+3 \log ^2\left (\frac {1}{x}\right )+\left (9+9 \log \left (\frac {1}{x}\right )\right ) \log (\log (4))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx\\ &=\int \left (3+\frac {9 (-1+3 x) (x-\log (\log (4)))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}-\frac {9 (2 x-\log (\log (4)))}{3 x+\log \left (\frac {1}{x}\right )}\right ) \, dx\\ &=3 x+9 \int \frac {(-1+3 x) (x-\log (\log (4)))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx-9 \int \frac {2 x-\log (\log (4))}{3 x+\log \left (\frac {1}{x}\right )} \, dx\\ &=3 x-9 \int \left (\frac {2 x}{3 x+\log \left (\frac {1}{x}\right )}-\frac {\log (\log (4))}{3 x+\log \left (\frac {1}{x}\right )}\right ) \, dx+9 \int \left (\frac {3 x^2}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}+\frac {\log (\log (4))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}-\frac {x (1+3 \log (\log (4)))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}\right ) \, dx\\ &=3 x-18 \int \frac {x}{3 x+\log \left (\frac {1}{x}\right )} \, dx+27 \int \frac {x^2}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx+(9 \log (\log (4))) \int \frac {1}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx+(9 \log (\log (4))) \int \frac {1}{3 x+\log \left (\frac {1}{x}\right )} \, dx-(9 (1+3 \log (\log (4)))) \int \frac {x}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 24, normalized size = 1.09 \begin {gather*} 3 x-\frac {9 x (x-\log (\log (4)))}{3 x+\log \left (\frac {1}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x + 3*Log[x^(-1)]^2 + (9 + 9*Log[x^(-1)])*Log[Log[4]])/(9*x^2 + 6*x*Log[x^(-1)] + Log[x^(-1)]^2)
,x]

[Out]

3*x - (9*x*(x - Log[Log[4]]))/(3*x + Log[x^(-1)])

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fricas [A]  time = 0.52, size = 27, normalized size = 1.23 \begin {gather*} \frac {3 \, {\left (3 \, x \log \left (2 \, \log \relax (2)\right ) + x \log \left (\frac {1}{x}\right )\right )}}{3 \, x + \log \left (\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*log(1/x)+9)*log(2*log(2))+3*log(1/x)^2-9*x)/(log(1/x)^2+6*x*log(1/x)+9*x^2),x, algorithm="fricas
")

[Out]

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

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giac [A]  time = 0.14, size = 31, normalized size = 1.41 \begin {gather*} 3 \, x - \frac {9 \, {\left (x^{2} - x \log \relax (2) - x \log \left (\log \relax (2)\right )\right )}}{3 \, x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*log(1/x)+9)*log(2*log(2))+3*log(1/x)^2-9*x)/(log(1/x)^2+6*x*log(1/x)+9*x^2),x, algorithm="giac")

[Out]

3*x - 9*(x^2 - x*log(2) - x*log(log(2)))/(3*x - log(x))

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maple [A]  time = 0.12, size = 27, normalized size = 1.23

method	result	size
risch	\(3 x +\frac {9 \left (-x +\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right ) x}{\ln \left (\frac {1}{x}\right )+3 x}\)	\(27\)
norman	\(\frac {\left (-3 \ln \relax (2)-3 \ln \left (\ln \relax (2)\right )\right ) \ln \left (\frac {1}{x}\right )+3 x \ln \left (\frac {1}{x}\right )}{\ln \left (\frac {1}{x}\right )+3 x}\)	\(35\)
derivativedivides	\(\frac {3 \ln \left (\frac {1}{x}\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \relax (2)}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \left (\ln \relax (2)\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}\)	\(53\)
default	\(\frac {3 \ln \left (\frac {1}{x}\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \relax (2)}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \left (\ln \relax (2)\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*ln(1/x)+9)*ln(2*ln(2))+3*ln(1/x)^2-9*x)/(ln(1/x)^2+6*x*ln(1/x)+9*x^2),x,method=_RETURNVERBOSE)

[Out]

3*x+9*(-x+ln(2)+ln(ln(2)))*x/(ln(1/x)+3*x)

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maxima [A]  time = 0.49, size = 27, normalized size = 1.23 \begin {gather*} \frac {3 \, {\left (3 \, x {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} - x \log \relax (x)\right )}}{3 \, x - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*log(1/x)+9)*log(2*log(2))+3*log(1/x)^2-9*x)/(log(1/x)^2+6*x*log(1/x)+9*x^2),x, algorithm="maxima
")

[Out]

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

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mupad [B]  time = 5.13, size = 28, normalized size = 1.27 \begin {gather*} 3\,x+\frac {x\,\ln \left ({\ln \relax (4)}^9\right )-9\,x^2}{3\,x+\ln \left (\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*log(2))*(9*log(1/x) + 9) - 9*x + 3*log(1/x)^2)/(6*x*log(1/x) + log(1/x)^2 + 9*x^2),x)

[Out]

3*x + (x*log(log(4)^9) - 9*x^2)/(3*x + log(1/x))

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sympy [A]  time = 0.12, size = 31, normalized size = 1.41 \begin {gather*} 3 x + \frac {- 9 x^{2} + 9 x \log {\left (\log {\relax (2 )} \right )} + 9 x \log {\relax (2 )}}{3 x + \log {\left (\frac {1}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*ln(1/x)+9)*ln(2*ln(2))+3*ln(1/x)**2-9*x)/(ln(1/x)**2+6*x*ln(1/x)+9*x**2),x)

[Out]

3*x + (-9*x**2 + 9*x*log(log(2)) + 9*x*log(2))/(3*x + log(1/x))

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