3.26.49 \(\int \frac {-9+x^2}{x^2 \log (4-\log (\frac {5}{3}))} \, dx\)

Optimal. Leaf size=26 \[ \frac {1+\frac {9-\frac {11 x}{12}}{x}+x}{\log \left (4-\log \left (\frac {5}{3}\right )\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 14} \begin {gather*} \frac {x}{\log \left (4-\log \left (\frac {5}{3}\right )\right )}+\frac {9}{x \log \left (4-\log \left (\frac {5}{3}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 + x^2)/(x^2*Log[4 - Log[5/3]]),x]

[Out]

9/(x*Log[4 - Log[5/3]]) + x/Log[4 - Log[5/3]]

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-9+x^2}{x^2} \, dx}{\log \left (4-\log \left (\frac {5}{3}\right )\right )}\\ &=\frac {\int \left (1-\frac {9}{x^2}\right ) \, dx}{\log \left (4-\log \left (\frac {5}{3}\right )\right )}\\ &=\frac {9}{x \log \left (4-\log \left (\frac {5}{3}\right )\right )}+\frac {x}{\log \left (4-\log \left (\frac {5}{3}\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.73 \begin {gather*} \frac {\frac {9}{x}+x}{\log \left (4-\log \left (\frac {5}{3}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 + x^2)/(x^2*Log[4 - Log[5/3]]),x]

[Out]

(9/x + x)/Log[4 - Log[5/3]]

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fricas [A]  time = 0.70, size = 16, normalized size = 0.62 \begin {gather*} \frac {x^{2} + 9}{x \log \left (\log \left (\frac {3}{5}\right ) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-9)/x^2/log(log(3/5)+4),x, algorithm="fricas")

[Out]

(x^2 + 9)/(x*log(log(3/5) + 4))

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giac [A]  time = 0.35, size = 15, normalized size = 0.58 \begin {gather*} \frac {x + \frac {9}{x}}{\log \left (\log \left (\frac {3}{5}\right ) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-9)/x^2/log(log(3/5)+4),x, algorithm="giac")

[Out]

(x + 9/x)/log(log(3/5) + 4)

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maple [A]  time = 0.04, size = 16, normalized size = 0.62

method	result	size
default	\(\frac {x +\frac {9}{x}}{\ln \left (\ln \left (\frac {3}{5}\right )+4\right )}\)	\(16\)
gosper	\(\frac {x^{2}+9}{x \ln \left (\ln \left (\frac {3}{5}\right )+4\right )}\)	\(17\)
norman	\(\frac {\frac {x^{2}}{\ln \left (\ln \left (\frac {3}{5}\right )+4\right )}+\frac {9}{\ln \left (\ln \left (\frac {3}{5}\right )+4\right )}}{x}\)	\(26\)
risch	\(\frac {x}{\ln \left (\ln \relax (3)-\ln \relax (5)+4\right )}+\frac {9}{\ln \left (\ln \relax (3)-\ln \relax (5)+4\right ) x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-9)/x^2/ln(ln(3/5)+4),x,method=_RETURNVERBOSE)

[Out]

1/ln(ln(3/5)+4)*(x+9/x)

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maxima [A]  time = 0.43, size = 15, normalized size = 0.58 \begin {gather*} \frac {x + \frac {9}{x}}{\log \left (\log \left (\frac {3}{5}\right ) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-9)/x^2/log(log(3/5)+4),x, algorithm="maxima")

[Out]

(x + 9/x)/log(log(3/5) + 4)

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mupad [B]  time = 1.59, size = 16, normalized size = 0.62 \begin {gather*} \frac {x^2+9}{x\,\ln \left (\ln \left (\frac {3}{5}\right )+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 9)/(x^2*log(log(3/5) + 4)),x)

[Out]

(x^2 + 9)/(x*log(log(3/5) + 4))

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sympy [A]  time = 0.07, size = 14, normalized size = 0.54 \begin {gather*} \frac {x + \frac {9}{x}}{\log {\left (- \log {\relax (5 )} + \log {\relax (3 )} + 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-9)/x**2/ln(ln(3/5)+4),x)

[Out]

(x + 9/x)/log(-log(5) + log(3) + 4)

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