3.26.83 \(\int \frac {-72+9 \log (5)+(24-3 \log (5)) \log (\log (9))}{-2 x^2+x^2 \log (\log (9))} \, dx\)

Optimal. Leaf size=26 \[ \frac {(8-\log (5)) \left (-3+2 x-\frac {3}{2-\log (\log (9))}\right )}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6, 12, 30} \begin {gather*} -\frac {3 (8-\log (5)) (3-\log (\log (9)))}{x (2-\log (\log (9)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-72 + 9*Log[5] + (24 - 3*Log[5])*Log[Log[9]])/(-2*x^2 + x^2*Log[Log[9]]),x]

[Out]

(-3*(8 - Log[5])*(3 - Log[Log[9]]))/(x*(2 - Log[Log[9]]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-72+9 \log (5)+(24-3 \log (5)) \log (\log (9))}{x^2 (-2+\log (\log (9)))} \, dx\\ &=\frac {(3 (8-\log (5)) (3-\log (\log (9)))) \int \frac {1}{x^2} \, dx}{2-\log (\log (9))}\\ &=-\frac {3 (8-\log (5)) (3-\log (\log (9)))}{x (2-\log (\log (9)))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.81 \begin {gather*} \frac {3 (-8+\log (5)) (-3+\log (\log (9)))}{x (-2+\log (\log (9)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-72 + 9*Log[5] + (24 - 3*Log[5])*Log[Log[9]])/(-2*x^2 + x^2*Log[Log[9]]),x]

[Out]

(3*(-8 + Log[5])*(-3 + Log[Log[9]]))/(x*(-2 + Log[Log[9]]))

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fricas [A]  time = 0.53, size = 31, normalized size = 1.19 \begin {gather*} \frac {3 \, {\left ({\left (\log \relax (5) - 8\right )} \log \left (2 \, \log \relax (3)\right ) - 3 \, \log \relax (5) + 24\right )}}{x \log \left (2 \, \log \relax (3)\right ) - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(5)+24)*log(2*log(3))+9*log(5)-72)/(x^2*log(2*log(3))-2*x^2),x, algorithm="fricas")

[Out]

3*((log(5) - 8)*log(2*log(3)) - 3*log(5) + 24)/(x*log(2*log(3)) - 2*x)

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giac [A]  time = 0.64, size = 30, normalized size = 1.15 \begin {gather*} \frac {3 \, {\left ({\left (\log \relax (5) - 8\right )} \log \left (2 \, \log \relax (3)\right ) - 3 \, \log \relax (5) + 24\right )}}{x {\left (\log \left (2 \, \log \relax (3)\right ) - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(5)+24)*log(2*log(3))+9*log(5)-72)/(x^2*log(2*log(3))-2*x^2),x, algorithm="giac")

[Out]

3*((log(5) - 8)*log(2*log(3)) - 3*log(5) + 24)/(x*(log(2*log(3)) - 2))

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maple [A]  time = 0.40, size = 33, normalized size = 1.27

method	result	size
default	\(-\frac {\left (-3 \ln \relax (5)+24\right ) \ln \left (2 \ln \relax (3)\right )+9 \ln \relax (5)-72}{\left (\ln \left (2 \ln \relax (3)\right )-2\right ) x}\)	\(33\)
gosper	\(\frac {3 \ln \left (2 \ln \relax (3)\right ) \ln \relax (5)-24 \ln \left (2 \ln \relax (3)\right )-9 \ln \relax (5)+72}{\left (\ln \left (2 \ln \relax (3)\right )-2\right ) x}\)	\(36\)
norman	\(\frac {3 \ln \relax (2) \ln \relax (5)+3 \ln \relax (5) \ln \left (\ln \relax (3)\right )-24 \ln \relax (2)-9 \ln \relax (5)-24 \ln \left (\ln \relax (3)\right )+72}{\left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right ) x}\)	\(41\)
risch	\(\frac {3 \ln \relax (2) \ln \relax (5)}{x \left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right )}+\frac {3 \ln \relax (5) \ln \left (\ln \relax (3)\right )}{x \left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right )}-\frac {24 \ln \relax (2)}{x \left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right )}-\frac {9 \ln \relax (5)}{x \left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right )}-\frac {24 \ln \left (\ln \relax (3)\right )}{x \left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right )}+\frac {72}{x \left (\ln \relax (2)+\ln \left (\ln \relax (3)\right )-2\right )}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*ln(5)+24)*ln(2*ln(3))+9*ln(5)-72)/(x^2*ln(2*ln(3))-2*x^2),x,method=_RETURNVERBOSE)

[Out]

-((-3*ln(5)+24)*ln(2*ln(3))+9*ln(5)-72)/(ln(2*ln(3))-2)/x

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maxima [A]  time = 0.44, size = 30, normalized size = 1.15 \begin {gather*} \frac {3 \, {\left ({\left (\log \relax (5) - 8\right )} \log \left (2 \, \log \relax (3)\right ) - 3 \, \log \relax (5) + 24\right )}}{x {\left (\log \left (2 \, \log \relax (3)\right ) - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(5)+24)*log(2*log(3))+9*log(5)-72)/(x^2*log(2*log(3))-2*x^2),x, algorithm="maxima")

[Out]

3*((log(5) - 8)*log(2*log(3)) - 3*log(5) + 24)/(x*(log(2*log(3)) - 2))

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mupad [B]  time = 1.48, size = 28, normalized size = 1.08 \begin {gather*} \frac {\ln \left (125\right )\,\ln \left (\ln \relax (9)\right )-\ln \left (32768000000000\,{\ln \relax (3)}^{24}\right )+72}{x\,\left (\ln \left (\ln \relax (9)\right )-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2*log(3))*(3*log(5) - 24) - 9*log(5) + 72)/(x^2*log(2*log(3)) - 2*x^2),x)

[Out]

(log(125)*log(log(9)) - log(32768000000000*log(3)^24) + 72)/(x*(log(log(9)) - 2))

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sympy [B]  time = 0.09, size = 49, normalized size = 1.88 \begin {gather*} - \frac {-72 - 3 \log {\relax (2 )} \log {\relax (5 )} - 3 \log {\relax (5 )} \log {\left (\log {\relax (3 )} \right )} + 24 \log {\left (\log {\relax (3 )} \right )} + 9 \log {\relax (5 )} + 24 \log {\relax (2 )}}{x \left (-2 + \log {\left (\log {\relax (3 )} \right )} + \log {\relax (2 )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*ln(5)+24)*ln(2*ln(3))+9*ln(5)-72)/(x**2*ln(2*ln(3))-2*x**2),x)

[Out]

-(-72 - 3*log(2)*log(5) - 3*log(5)*log(log(3)) + 24*log(log(3)) + 9*log(5) + 24*log(2))/(x*(-2 + log(log(3)) +
log(2)))

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