3.45.30 \(\int \frac {2 x}{\log ^2(7+6 \log (9+\log (5))+\log ^2(9+\log (5)))} \, dx\)

Optimal. Leaf size=18 \[ \frac {x^2}{\log ^2\left (-2+(3+\log (9+\log (5)))^2\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 30} \begin {gather*} \frac {x^2}{\log ^2\left (7+\log ^2(9+\log (5))+6 \log (9+\log (5))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x)/Log[7 + 6*Log[9 + Log[5]] + Log[9 + Log[5]]^2]^2,x]

[Out]

x^2/Log[7 + 6*Log[9 + Log[5]] + Log[9 + Log[5]]^2]^2

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} &=\frac {2 \int x \, dx}{\log ^2\left (7+6 \log (9+\log (5))+\log ^2(9+\log (5))\right )}\\ &=\frac {x^2}{\log ^2\left (7+6 \log (9+\log (5))+\log ^2(9+\log (5))\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.28 \begin {gather*} \frac {x^2}{\log ^2\left (7+6 \log (9+\log (5))+\log ^2(9+\log (5))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x)/Log[7 + 6*Log[9 + Log[5]] + Log[9 + Log[5]]^2]^2,x]

[Out]

x^2/Log[7 + 6*Log[9 + Log[5]] + Log[9 + Log[5]]^2]^2

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fricas [A]  time = 0.58, size = 23, normalized size = 1.28 \begin {gather*} \frac {x^{2}}{\log \left (\log \left (\log \relax (5) + 9\right )^{2} + 6 \, \log \left (\log \relax (5) + 9\right ) + 7\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/log(log(log(5)+9)^2+6*log(log(5)+9)+7)^2,x, algorithm="fricas")

[Out]

x^2/log(log(log(5) + 9)^2 + 6*log(log(5) + 9) + 7)^2

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giac [A]  time = 0.13, size = 23, normalized size = 1.28 \begin {gather*} \frac {x^{2}}{\log \left (\log \left (\log \relax (5) + 9\right )^{2} + 6 \, \log \left (\log \relax (5) + 9\right ) + 7\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/log(log(log(5)+9)^2+6*log(log(5)+9)+7)^2,x, algorithm="giac")

[Out]

x^2/log(log(log(5) + 9)^2 + 6*log(log(5) + 9) + 7)^2

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maple [A]  time = 0.04, size = 24, normalized size = 1.33

method	result	size
gosper	\(\frac {x^{2}}{\ln \left (\ln \left (\ln \relax (5)+9\right )^{2}+6 \ln \left (\ln \relax (5)+9\right )+7\right )^{2}}\)	\(24\)
default	\(\frac {x^{2}}{\ln \left (\ln \left (\ln \relax (5)+9\right )^{2}+6 \ln \left (\ln \relax (5)+9\right )+7\right )^{2}}\)	\(24\)
norman	\(\frac {x^{2}}{\ln \left (\ln \left (\ln \relax (5)+9\right )^{2}+6 \ln \left (\ln \relax (5)+9\right )+7\right )^{2}}\)	\(24\)
risch	\(\frac {x^{2}}{\ln \left (\ln \left (\ln \relax (5)+9\right )^{2}+6 \ln \left (\ln \relax (5)+9\right )+7\right )^{2}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x/ln(ln(ln(5)+9)^2+6*ln(ln(5)+9)+7)^2,x,method=_RETURNVERBOSE)

[Out]

x^2/ln(ln(ln(5)+9)^2+6*ln(ln(5)+9)+7)^2

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maxima [A]  time = 0.36, size = 23, normalized size = 1.28 \begin {gather*} \frac {x^{2}}{\log \left (\log \left (\log \relax (5) + 9\right )^{2} + 6 \, \log \left (\log \relax (5) + 9\right ) + 7\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/log(log(log(5)+9)^2+6*log(log(5)+9)+7)^2,x, algorithm="maxima")

[Out]

x^2/log(log(log(5) + 9)^2 + 6*log(log(5) + 9) + 7)^2

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mupad [B]  time = 3.25, size = 23, normalized size = 1.28 \begin {gather*} \frac {x^2}{{\ln \left (6\,\ln \left (\ln \relax (5)+9\right )+{\ln \left (\ln \relax (5)+9\right )}^2+7\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x)/log(6*log(log(5) + 9) + log(log(5) + 9)^2 + 7)^2,x)

[Out]

x^2/log(6*log(log(5) + 9) + log(log(5) + 9)^2 + 7)^2

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sympy [A]  time = 0.06, size = 24, normalized size = 1.33 \begin {gather*} \frac {x^{2}}{\log {\left (\log {\left (\log {\relax (5 )} + 9 \right )}^{2} + 7 + 6 \log {\left (\log {\relax (5 )} + 9 \right )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/ln(ln(ln(5)+9)**2+6*ln(ln(5)+9)+7)**2,x)

[Out]

x**2/log(log(log(5) + 9)**2 + 7 + 6*log(log(5) + 9))**2

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