3.3.16 \(\int -\frac {4}{-4 x+3 x^2} \, dx\)

Optimal. Leaf size=14 \[ 1-\log \left (\frac {-4+3 x}{x}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 615} \begin {gather*} \log (x)-\log (4-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-4/(-4*x + 3*x^2),x]

[Out]

-Log[4 - 3*x] + Log[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 615

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.36 \begin {gather*} 4 \left (-\frac {1}{4} \log (4-3 x)+\frac {\log (x)}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-4/(-4*x + 3*x^2),x]

[Out]

4*(-1/4*Log[4 - 3*x] + Log[x]/4)

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fricas [A]  time = 0.50, size = 11, normalized size = 0.79 \begin {gather*} -\log \left (3 \, x - 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(3*x - 4) + log(x)

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giac [A]  time = 0.26, size = 13, normalized size = 0.93 \begin {gather*} -\log \left ({\left | 3 \, x - 4 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(3*x - 4)) + log(abs(x))

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maple [A]  time = 0.08, size = 12, normalized size = 0.86

method	result	size
default	\(-\ln \left (3 x -4\right )+\ln \relax (x )\)	\(12\)
norman	\(-\ln \left (3 x -4\right )+\ln \relax (x )\)	\(12\)
risch	\(-\ln \left (3 x -4\right )+\ln \relax (x )\)	\(12\)
meijerg	\(-\ln \left (1-\frac {3 x}{4}\right )+\ln \relax (x )+\ln \relax (3)-2 \ln \relax (2)+i \pi \)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(3*x-4)+ln(x)

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maxima [A]  time = 0.44, size = 11, normalized size = 0.79 \begin {gather*} -\log \left (3 \, x - 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(3*x - 4) + log(x)

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mupad [B]  time = 0.35, size = 8, normalized size = 0.57 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {3\,x}{2}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4/(4*x - 3*x^2),x)

[Out]

2*atanh((3*x)/2 - 1)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.57 \begin {gather*} \log {\relax (x )} - \log {\left (x - \frac {4}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x - 4/3)

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