3.35.17 \(\int \frac {32 x^2+(12-16 x^2) \log (3-4 x^2)}{-9 x^2+12 x^4+(-3 x+4 x^3) \log (3-4 x^2)} \, dx\)

Optimal. Leaf size=17 \[ 4 \log \left (3+\frac {\log \left (3-4 x^2\right )}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.36, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6741, 6712, 31} \begin {gather*} 4 \log \left (\frac {\log \left (3-4 x^2\right )}{x}+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32*x^2 + (12 - 16*x^2)*Log[3 - 4*x^2])/(-9*x^2 + 12*x^4 + (-3*x + 4*x^3)*Log[3 - 4*x^2]),x]

[Out]

4*Log[3 + Log[3 - 4*x^2]/x]

Rule 31

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

Rule 6712

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ
[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-32 x^2-\left (12-16 x^2\right ) \log \left (3-4 x^2\right )}{x \left (3-4 x^2\right ) \left (3 x+\log \left (3-4 x^2\right )\right )} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {1}{3+x} \, dx,x,\frac {\log \left (3-4 x^2\right )}{x}\right )\\ &=4 \log \left (3+\frac {\log \left (3-4 x^2\right )}{x}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.28, size = 20, normalized size = 1.18 \begin {gather*} 4 \left (-\log (x)+\log \left (3 x+\log \left (3-4 x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32*x^2 + (12 - 16*x^2)*Log[3 - 4*x^2])/(-9*x^2 + 12*x^4 + (-3*x + 4*x^3)*Log[3 - 4*x^2]),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 20, normalized size = 1.18 \begin {gather*} 4 \, \log \left (3 \, x + \log \left (-4 \, x^{2} + 3\right )\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2+12)*log(-4*x^2+3)+32*x^2)/((4*x^3-3*x)*log(-4*x^2+3)+12*x^4-9*x^2),x, algorithm="fricas")

[Out]

4*log(3*x + log(-4*x^2 + 3)) - 4*log(x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 20, normalized size = 1.18 \begin {gather*} 4 \, \log \left (3 \, x + \log \left (-4 \, x^{2} + 3\right )\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2+12)*log(-4*x^2+3)+32*x^2)/((4*x^3-3*x)*log(-4*x^2+3)+12*x^4-9*x^2),x, algorithm="giac")

[Out]

4*log(3*x + log(-4*x^2 + 3)) - 4*log(x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 21, normalized size = 1.24

method	result	size
norman	\(-4 \ln \relax (x )+4 \ln \left (\ln \left (-4 x^{2}+3\right )+3 x \right )\)	\(21\)
risch	\(-4 \ln \relax (x )+4 \ln \left (\ln \left (-4 x^{2}+3\right )+3 x \right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^2+12)*ln(-4*x^2+3)+32*x^2)/((4*x^3-3*x)*ln(-4*x^2+3)+12*x^4-9*x^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 20, normalized size = 1.18 \begin {gather*} 4 \, \log \left (3 \, x + \log \left (-4 \, x^{2} + 3\right )\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2+12)*log(-4*x^2+3)+32*x^2)/((4*x^3-3*x)*log(-4*x^2+3)+12*x^4-9*x^2),x, algorithm="maxima")

[Out]

4*log(3*x + log(-4*x^2 + 3)) - 4*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 20, normalized size = 1.18 \begin {gather*} 4\,\ln \left (3\,x+\ln \left (3-4\,x^2\right )\right )-4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x^2 - log(3 - 4*x^2)*(16*x^2 - 12))/(log(3 - 4*x^2)*(3*x - 4*x^3) + 9*x^2 - 12*x^4),x)

[Out]

4*log(3*x + log(3 - 4*x^2)) - 4*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 19, normalized size = 1.12 \begin {gather*} - 4 \log {\relax (x )} + 4 \log {\left (3 x + \log {\left (3 - 4 x^{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**2+12)*ln(-4*x**2+3)+32*x**2)/((4*x**3-3*x)*ln(-4*x**2+3)+12*x**4-9*x**2),x)

[Out]

-4*log(x) + 4*log(3*x + log(3 - 4*x**2))

________________________________________________________________________________________