3.88.58 \(\int -\frac {4}{x^2-2 x \log (\frac {3}{2})+\log ^2(\frac {3}{2})} \, dx\) [8758]

Optimal. Leaf size=12 \[ \frac {4}{x-\log \left (\frac {3}{2}\right )} \]

[Out]

4/(x-ln(3/2))

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 27, 32} \begin {gather*} \frac {4}{x-\log \left (\frac {3}{2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(x - Log[3/2])

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {4}{x-\log \left (\frac {3}{2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(x - Log[3/2])

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Maple [A]
time = 1.74, size = 11, normalized size = 0.92

method	result	size
gosper	\(-\frac {4}{\ln \left (\frac {3}{2}\right )-x}\)	\(11\)
default	\(\frac {4}{x -\ln \left (\frac {3}{2}\right )}\)	\(11\)
norman	\(-\frac {4}{\ln \left (\frac {3}{2}\right )-x}\)	\(11\)
risch	\(\frac {4}{x -\ln \left (3\right )+\ln \left (2\right )}\)	\(13\)
meijerg	\(\frac {4 \left (-\ln \left (3\right )+\ln \left (2\right )\right ) x}{\ln \left (\frac {3}{2}\right )^{2} \left (\ln \left (3\right )-\ln \left (2\right )\right ) \left (1-\frac {x}{\ln \left (3\right )-\ln \left (2\right )}\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(x-ln(3/2))

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Maxima [A]
time = 0.25, size = 10, normalized size = 0.83 \begin {gather*} \frac {4}{x - \log \left (\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(x - log(3/2))

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Fricas [A]
time = 0.34, size = 10, normalized size = 0.83 \begin {gather*} \frac {4}{x - \log \left (\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(x - log(3/2))

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Sympy [A]
time = 0.07, size = 8, normalized size = 0.67 \begin {gather*} \frac {4}{x - \log {\left (3 \right )} + \log {\left (2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(x - log(3) + log(2))

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Giac [A]
time = 0.41, size = 10, normalized size = 0.83 \begin {gather*} \frac {4}{x - \log \left (\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(x - log(3/2))

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Mupad [B]
time = 0.09, size = 8, normalized size = 0.67 \begin {gather*} \frac {4}{x+\ln \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(x + log(2/3))

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