3.15.21 \(\int \frac {(-16+4 x) \log (3)}{9+(32-16 x+2 x^2) \log (3)} \, dx\)

Optimal. Leaf size=12 \[ \log \left (9+2 (-4+x)^2 \log (3)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {12, 1587} \begin {gather*} \log \left (2 \left (x^2-8 x+16\right ) \log (3)+9\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-16 + 4*x)*Log[3])/(9 + (32 - 16*x + 2*x^2)*Log[3]),x]

[Out]

Log[9 + 2*(16 - 8*x + x^2)*Log[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 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (3) \int \frac {-16+4 x}{9+\left (32-16 x+2 x^2\right ) \log (3)} \, dx\\ &=\log \left (9+2 \left (16-8 x+x^2\right ) \log (3)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} \log \left (9+2 (-4+x)^2 \log (3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-16 + 4*x)*Log[3])/(9 + (32 - 16*x + 2*x^2)*Log[3]),x]

[Out]

Log[9 + 2*(-4 + x)^2*Log[3]]

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fricas [A]  time = 0.52, size = 15, normalized size = 1.25 \begin {gather*} \log \left (2 \, {\left (x^{2} - 8 \, x + 16\right )} \log \relax (3) + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-16)*log(3)/((2*x^2-16*x+32)*log(3)+9),x, algorithm="fricas")

[Out]

log(2*(x^2 - 8*x + 16)*log(3) + 9)

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giac [A]  time = 0.49, size = 18, normalized size = 1.50 \begin {gather*} \log \left (2 \, {\left (x^{2} - 8 \, x\right )} \log \relax (3) + 32 \, \log \relax (3) + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-16)*log(3)/((2*x^2-16*x+32)*log(3)+9),x, algorithm="giac")

[Out]

log(2*(x^2 - 8*x)*log(3) + 32*log(3) + 9)

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maple [A]  time = 0.54, size = 17, normalized size = 1.42

method	result	size
derivativedivides	\(\ln \left (\left (2 x^{2}-16 x +32\right ) \ln \relax (3)+9\right )\)	\(17\)
default	\(\ln \left (2 x^{2} \ln \relax (3)-16 x \ln \relax (3)+32 \ln \relax (3)+9\right )\)	\(20\)
norman	\(\ln \left (2 x^{2} \ln \relax (3)-16 x \ln \relax (3)+32 \ln \relax (3)+9\right )\)	\(20\)
risch	\(\ln \left (2 x^{2} \ln \relax (3)-16 x \ln \relax (3)+32 \ln \relax (3)+9\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x-16)*ln(3)/((2*x^2-16*x+32)*ln(3)+9),x,method=_RETURNVERBOSE)

[Out]

ln((2*x^2-16*x+32)*ln(3)+9)

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maxima [A]  time = 0.39, size = 19, normalized size = 1.58 \begin {gather*} \log \left (2 \, x^{2} \log \relax (3) - 16 \, x \log \relax (3) + 32 \, \log \relax (3) + 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-16)*log(3)/((2*x^2-16*x+32)*log(3)+9),x, algorithm="maxima")

[Out]

log(2*x^2*log(3) - 16*x*log(3) + 32*log(3) + 9)

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mupad [B]  time = 0.11, size = 19, normalized size = 1.58 \begin {gather*} \ln \left (2\,\ln \relax (3)\,x^2-16\,\ln \relax (3)\,x+32\,\ln \relax (3)+9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(4*x - 16))/(log(3)*(2*x^2 - 16*x + 32) + 9),x)

[Out]

log(32*log(3) - 16*x*log(3) + 2*x^2*log(3) + 9)

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sympy [A]  time = 0.14, size = 22, normalized size = 1.83 \begin {gather*} \log {\left (2 x^{2} \log {\relax (3 )} - 16 x \log {\relax (3 )} + 9 + 32 \log {\relax (3 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x-16)*ln(3)/((2*x**2-16*x+32)*ln(3)+9),x)

[Out]

log(2*x**2*log(3) - 16*x*log(3) + 9 + 32*log(3))

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