3.15.21 \(\int \frac {(-16+4 x) \log (3)}{9+(32-16 x+2 x^2) \log (3)} \, dx\) [1421]

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

[Out]

ln(9+2*ln(3)*(x-4)^2)

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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, 1601} \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 1601

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]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; 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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Maple [A]
time = 1.04, size = 20, normalized size = 1.67

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

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

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Maxima [A]
time = 0.26, size = 19, normalized size = 1.58 \begin {gather*} \log \left (2 \, x^{2} \log \left (3\right ) - 16 \, x \log \left (3\right ) + 32 \, \log \left (3\right ) + 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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Fricas [A]
time = 0.36, size = 15, normalized size = 1.25 \begin {gather*} \log \left (2 \, {\left (x^{2} - 8 \, x + 16\right )} \log \left (3\right ) + 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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Sympy [A]
time = 0.07, size = 22, normalized size = 1.83 \begin {gather*} \log {\left (2 x^{2} \log {\left (3 \right )} - 16 x \log {\left (3 \right )} + 9 + 32 \log {\left (3 \right )} \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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Giac [A]
time = 0.39, size = 18, normalized size = 1.50 \begin {gather*} \log \left (2 \, {\left (x^{2} - 8 \, x\right )} \log \left (3\right ) + 32 \, \log \left (3\right ) + 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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Mupad [B]
time = 0.11, size = 19, normalized size = 1.58 \begin {gather*} \ln \left (2\,\ln \left (3\right )\,x^2-16\,\ln \left (3\right )\,x+32\,\ln \left (3\right )+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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