\(\int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx\) [7409]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5 x \left (x+\frac {\log (3)}{x}\right )}+\frac {\log (5)}{4} \]

[Out]

1/4*ln(5)+4/5/(x+ln(3)/x)/x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.50, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 28, 267} \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5 x^2+\log (243)} \]

[In]

Int[(-8*x)/(5*x^4 + 10*x^2*Log[3] + 5*Log[3]^2),x]

[Out]

4/(5*x^2 + Log[243])

Rule 12

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

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\left (8 \int \frac {x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx\right ) \\ & = -\left (40 \int \frac {x}{\left (5 x^2+5 \log (3)\right )^2} \, dx\right ) \\ & = \frac {4}{5 x^2+\log (243)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.50 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5 \left (x^2+\log (3)\right )} \]

[In]

Integrate[(-8*x)/(5*x^4 + 10*x^2*Log[3] + 5*Log[3]^2),x]

[Out]

4/(5*(x^2 + Log[3]))

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.46

method	result	size
gosper	\(\frac {4}{5 \left (\ln \left (3\right )+x^{2}\right )}\)	\(11\)
default	\(\frac {4}{5 \left (\ln \left (3\right )+x^{2}\right )}\)	\(11\)
norman	\(\frac {4}{5 \left (\ln \left (3\right )+x^{2}\right )}\)	\(11\)
risch	\(\frac {4}{5 \left (\ln \left (3\right )+x^{2}\right )}\)	\(11\)
parallelrisch	\(\frac {4}{5 \left (\ln \left (3\right )+x^{2}\right )}\)	\(11\)

[In]

int(-8*x/(5*ln(3)^2+10*x^2*ln(3)+5*x^4),x,method=_RETURNVERBOSE)

[Out]

4/5/(ln(3)+x^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5 \, {\left (x^{2} + \log \left (3\right )\right )}} \]

[In]

integrate(-8*x/(5*log(3)^2+10*x^2*log(3)+5*x^4),x, algorithm="fricas")

[Out]

4/5/(x^2 + log(3))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {8}{10 x^{2} + 10 \log {\left (3 \right )}} \]

[In]

integrate(-8*x/(5*ln(3)**2+10*x**2*ln(3)+5*x**4),x)

[Out]

8/(10*x**2 + 10*log(3))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5 \, {\left (x^{2} + \log \left (3\right )\right )}} \]

[In]

integrate(-8*x/(5*log(3)^2+10*x^2*log(3)+5*x^4),x, algorithm="maxima")

[Out]

4/5/(x^2 + log(3))

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5 \, {\left (x^{2} + \log \left (3\right )\right )}} \]

[In]

integrate(-8*x/(5*log(3)^2+10*x^2*log(3)+5*x^4),x, algorithm="giac")

[Out]

4/5/(x^2 + log(3))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58 \[ \int -\frac {8 x}{5 x^4+10 x^2 \log (3)+5 \log ^2(3)} \, dx=\frac {4}{5\,\left (x^2+\ln \left (3\right )\right )} \]

[In]

int(-(8*x)/(10*x^2*log(3) + 5*log(3)^2 + 5*x^4),x)

[Out]

4/(5*(log(3) + x^2))