\(\int -\frac {2}{5 x \log (2) \log ^2(4 x^2)} \, dx\) [3244]

Optimal result

Integrand size = 19, antiderivative size = 16 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \log (2) \log \left (4 x^2\right )} \]

[Out]

1/5/ln(4*x^2)/ln(2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 2339, 30} \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \log (2) \log \left (4 x^2\right )} \]

[In]

Int[-2/(5*x*Log[2]*Log[4*x^2]^2),x]

[Out]

1/(5*Log[2]*Log[4*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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \int \frac {1}{x \log ^2\left (4 x^2\right )} \, dx}{5 \log (2)} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (4 x^2\right )\right )}{5 \log (2)} \\ & = \frac {1}{5 \log (2) \log \left (4 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \log (2) \log \left (4 x^2\right )} \]

[In]

Integrate[-2/(5*x*Log[2]*Log[4*x^2]^2),x]

[Out]

1/(5*Log[2]*Log[4*x^2])

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {1}{5 \ln \left (4 x^{2}\right ) \ln \left (2\right )}\)	\(15\)
default	\(\frac {1}{5 \ln \left (4 x^{2}\right ) \ln \left (2\right )}\)	\(15\)
norman	\(\frac {1}{5 \ln \left (4 x^{2}\right ) \ln \left (2\right )}\)	\(15\)
risch	\(\frac {1}{5 \ln \left (4 x^{2}\right ) \ln \left (2\right )}\)	\(15\)
parallelrisch	\(\frac {1}{5 \ln \left (4 x^{2}\right ) \ln \left (2\right )}\)	\(15\)

[In]

int(-2/5/x/ln(2)/ln(4*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/5/ln(4*x^2)/ln(2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \, \log \left (2\right ) \log \left (4 \, x^{2}\right )} \]

[In]

integrate(-2/5/x/log(2)/log(4*x^2)^2,x, algorithm="fricas")

[Out]

1/5/(log(2)*log(4*x^2))

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \log {\left (2 \right )} \log {\left (4 x^{2} \right )}} \]

[In]

integrate(-2/5/x/ln(2)/ln(4*x**2)**2,x)

[Out]

1/(5*log(2)*log(4*x**2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \, \log \left (2\right ) \log \left (4 \, x^{2}\right )} \]

[In]

integrate(-2/5/x/log(2)/log(4*x^2)^2,x, algorithm="maxima")

[Out]

1/5/(log(2)*log(4*x^2))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5 \, \log \left (2\right ) \log \left (4 \, x^{2}\right )} \]

[In]

integrate(-2/5/x/log(2)/log(4*x^2)^2,x, algorithm="giac")

[Out]

1/5/(log(2)*log(4*x^2))

Mupad [B] (verification not implemented)

Time = 9.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int -\frac {2}{5 x \log (2) \log ^2\left (4 x^2\right )} \, dx=\frac {1}{5\,\ln \left (2\right )\,\ln \left (4\,x^2\right )} \]

[In]

int(-2/(5*x*log(2)*log(4*x^2)^2),x)

[Out]

1/(5*log(2)*log(4*x^2))