3.25.8 \(\int -\frac {2}{2 x+x \log (x^2)} \, dx\)

Optimal. Leaf size=21 \[ 3-\log \left (\frac {2+\log \left (x^2\right )}{16-e^3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.43, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 31} \begin {gather*} -\log \left (\log \left (x^2\right )+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2/(2*x + x*Log[x^2]),x]

[Out]

-Log[2 + 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 31

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 9, normalized size = 0.43 \begin {gather*} -\log \left (2+\log \left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2/(2*x + x*Log[x^2]),x]

[Out]

-Log[2 + Log[x^2]]

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fricas [A]  time = 0.78, size = 9, normalized size = 0.43 \begin {gather*} -\log \left (\log \left (x^{2}\right ) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(x^2) + 2)

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giac [A]  time = 0.23, size = 12, normalized size = 0.57 \begin {gather*} -\frac {1}{2} \, \log \left (4 \, {\left (\log \left ({\left | x \right |}\right ) + 1\right )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(4*(log(abs(x)) + 1)^2)

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maple [A]  time = 0.02, size = 10, normalized size = 0.48

method	result	size
default	\(-\ln \left (2+\ln \left (x^{2}\right )\right )\)	\(10\)
norman	\(-\ln \left (2+\ln \left (x^{2}\right )\right )\)	\(10\)
risch	\(-\ln \left (2+\ln \left (x^{2}\right )\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(2+ln(x^2))

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maxima [A]  time = 0.49, size = 7, normalized size = 0.33 \begin {gather*} -\log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(x) + 1)

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mupad [B]  time = 1.42, size = 9, normalized size = 0.43 \begin {gather*} -\ln \left (\ln \left (x^2\right )+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/(2*x + x*log(x^2)),x)

[Out]

-log(log(x^2) + 2)

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sympy [A]  time = 0.10, size = 8, normalized size = 0.38 \begin {gather*} - \log {\left (\log {\left (x^{2} \right )} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(x**2) + 2)

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