3.82.80 \(\int -\frac {4}{(2 x+\log ^2(2)) \log ^2(2 x+\log ^2(2))} \, dx\) [8180]

Optimal. Leaf size=19 \[ -\frac {1}{3}+\log (4)+\frac {2}{\log \left (2 x+\log ^2(2)\right )} \]

[Out]

2*ln(2)-1/3+2/ln(ln(2)^2+2*x)

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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 2437, 2339, 30} \begin {gather*} \frac {2}{\log \left (2 x+\log ^2(2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-4/((2*x + Log[2]^2)*Log[2*x + Log[2]^2]^2),x]

[Out]

2/Log[2*x + Log[2]^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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{\log \left (2 x+\log ^2(2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-4/((2*x + Log[2]^2)*Log[2*x + Log[2]^2]^2),x]

[Out]

2/Log[2*x + Log[2]^2]

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Maple [A]
time = 0.82, size = 14, normalized size = 0.74

method	result	size
derivativedivides	\(\frac {2}{\ln \left (\ln \left (2\right )^{2}+2 x \right )}\)	\(14\)
default	\(\frac {2}{\ln \left (\ln \left (2\right )^{2}+2 x \right )}\)	\(14\)
norman	\(\frac {2}{\ln \left (\ln \left (2\right )^{2}+2 x \right )}\)	\(14\)
risch	\(\frac {2}{\ln \left (\ln \left (2\right )^{2}+2 x \right )}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/ln(ln(2)^2+2*x)

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{\log \left (\log \left (2\right )^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/log(log(2)^2 + 2*x)

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Fricas [A]
time = 0.34, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{\log \left (\log \left (2\right )^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/log(log(2)^2 + 2*x)

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.53 \begin {gather*} \frac {2}{\log {\left (2 x + \log {\left (2 \right )}^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4/(ln(2)**2+2*x)/ln(ln(2)**2+2*x)**2,x)

[Out]

2/log(2*x + log(2)**2)

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Giac [A]
time = 0.42, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{\log \left (\log \left (2\right )^{2} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/log(log(2)^2 + 2*x)

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Mupad [B]
time = 0.33, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{\ln \left (2\,x+{\ln \left (2\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4/(log(2*x + log(2)^2)^2*(2*x + log(2)^2)),x)

[Out]

2/log(2*x + log(2)^2)

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