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

Optimal. Leaf size=21 \[ \frac {1}{5} (-5+4 x)+\log \left (\frac {x-\log (3)}{x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1593, 1820} \begin {gather*} \frac {4 x}{5}-\log (x)+\log (x-\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/5 - Log[x] + Log[x - Log[3]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} \frac {4 x}{5}-\log (x)+\log (x-\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/5 - Log[x] + Log[x - Log[3]]

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fricas [A]  time = 1.02, size = 15, normalized size = 0.71 \begin {gather*} \frac {4}{5} \, x + \log \left (x - \log \relax (3)\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*x + log(x - log(3)) - log(x)

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giac [A]  time = 0.20, size = 17, normalized size = 0.81 \begin {gather*} \frac {4}{5} \, x + \log \left ({\left | x - \log \relax (3) \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*x + log(abs(x - log(3))) - log(abs(x))

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maple [A]  time = 0.17, size = 16, normalized size = 0.76

method	result	size
default	\(\frac {4 x}{5}-\ln \relax (x )+\ln \left (-\ln \relax (3)+x \right )\)	\(16\)
norman	\(\frac {4 x}{5}-\ln \relax (x )+\ln \left (\ln \relax (3)-x \right )\)	\(16\)
risch	\(\frac {4 x}{5}-\ln \relax (x )+\ln \left (-\ln \relax (3)+x \right )\)	\(16\)
meijerg	\(-\ln \relax (x )+\ln \left (\ln \relax (3)\right )-i \pi +\ln \left (1-\frac {x}{\ln \relax (3)}\right )-\frac {4 \ln \relax (3) \ln \left (1-\frac {x}{\ln \relax (3)}\right )}{5}-\frac {4 \ln \relax (3) \left (-\frac {x}{\ln \relax (3)}-\ln \left (1-\frac {x}{\ln \relax (3)}\right )\right )}{5}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*x-ln(x)+ln(-ln(3)+x)

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maxima [A]  time = 0.44, size = 15, normalized size = 0.71 \begin {gather*} \frac {4}{5} \, x + \log \left (x - \log \relax (3)\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*x + log(x - log(3)) - log(x)

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mupad [B]  time = 0.08, size = 16, normalized size = 0.76 \begin {gather*} \frac {4\,x}{5}-2\,\mathrm {atanh}\left (\frac {2\,x}{\ln \relax (3)}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*x)/5 - 2*atanh((2*x)/log(3) - 1)

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sympy [A]  time = 0.17, size = 14, normalized size = 0.67 \begin {gather*} \frac {4 x}{5} - \log {\relax (x )} + \log {\left (x - \log {\relax (3 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x/5 - log(x) + log(x - log(3))

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