3.5.41 \(\int -\frac {6}{9 x-6 x \log (x^2)+x \log ^2(x^2)} \, dx\)

Optimal. Leaf size=15 \[ 88 \log (4)+\frac {3}{-3+\log \left (x^2\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 32} \begin {gather*} -\frac {3}{3-\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-6/(9*x - 6*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

-3/(3 - Log[x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (6 \int \frac {1}{9 x-6 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx\right )\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{(-3+x)^2} \, dx,x,\log \left (x^2\right )\right )\right )\\ &=-\frac {3}{3-\log \left (x^2\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 10, normalized size = 0.67 \begin {gather*} \frac {3}{-3+\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-6/(9*x - 6*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

3/(-3 + Log[x^2])

________________________________________________________________________________________

fricas [A]  time = 0.98, size = 10, normalized size = 0.67 \begin {gather*} \frac {3}{\log \left (x^{2}\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6/(x*log(x^2)^2-6*x*log(x^2)+9*x),x, algorithm="fricas")

[Out]

3/(log(x^2) - 3)

________________________________________________________________________________________

giac [A]  time = 0.40, size = 10, normalized size = 0.67 \begin {gather*} \frac {3}{\log \left (x^{2}\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6/(x*log(x^2)^2-6*x*log(x^2)+9*x),x, algorithm="giac")

[Out]

3/(log(x^2) - 3)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 11, normalized size = 0.73

method	result	size
default	\(\frac {3}{\ln \left (x^{2}\right )-3}\)	\(11\)
norman	\(\frac {3}{\ln \left (x^{2}\right )-3}\)	\(11\)
risch	\(\frac {3}{\ln \left (x^{2}\right )-3}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-6/(x*ln(x^2)^2-6*x*ln(x^2)+9*x),x,method=_RETURNVERBOSE)

[Out]

3/(ln(x^2)-3)

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 10, normalized size = 0.67 \begin {gather*} \frac {3}{2 \, \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6/(x*log(x^2)^2-6*x*log(x^2)+9*x),x, algorithm="maxima")

[Out]

3/(2*log(x) - 3)

________________________________________________________________________________________

mupad [B]  time = 0.52, size = 10, normalized size = 0.67 \begin {gather*} \frac {3}{\ln \left (x^2\right )-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-6/(9*x - 6*x*log(x^2) + x*log(x^2)^2),x)

[Out]

3/(log(x^2) - 3)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 7, normalized size = 0.47 \begin {gather*} \frac {3}{\log {\left (x^{2} \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-6/(x*ln(x**2)**2-6*x*ln(x**2)+9*x),x)

[Out]

3/(log(x**2) - 3)

________________________________________________________________________________________