3.24.50 \(\int \frac {-8-36 x+8 x \log (\log (2))}{80 x^2+360 x^3+405 x^4+(-80 x^3-180 x^4) \log (\log (2))+20 x^4 \log ^2(\log (2))} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{5 x^2 \left (\frac {9}{2}+\frac {2}{x}-\log (\log (2))\right )} \]

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Rubi [A]  time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.65, number of steps used = 5, number of rules used = 4, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6, 1680, 12, 261} \begin {gather*} -\frac {2 (9-2 \log (\log (2)))}{5 \left (x^2 \left (-(9-2 \log (\log (2)))^2\right )-4 x (9-2 \log (\log (2)))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 - 36*x + 8*x*Log[Log[2]])/(80*x^2 + 360*x^3 + 405*x^4 + (-80*x^3 - 180*x^4)*Log[Log[2]] + 20*x^4*Log[L
og[2]]^2),x]

[Out]

(-2*(9 - 2*Log[Log[2]]))/(5*(-4*x*(9 - 2*Log[Log[2]]) - x^2*(9 - 2*Log[Log[2]])^2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 261

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]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8+x (-36+8 \log (\log (2)))}{80 x^2+360 x^3+405 x^4+\left (-80 x^3-180 x^4\right ) \log (\log (2))+20 x^4 \log ^2(\log (2))} \, dx\\ &=\int \frac {-8+x (-36+8 \log (\log (2)))}{80 x^2+360 x^3+\left (-80 x^3-180 x^4\right ) \log (\log (2))+x^4 \left (405+20 \log ^2(\log (2))\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {4 x (-9+2 \log (\log (2)))^3}{5 \left (4-x^2 (9-2 \log (\log (2)))^2\right )^2} \, dx,x,x+\frac {360-80 \log (\log (2))}{4 \left (405-180 \log (\log (2))+20 \log ^2(\log (2))\right )}\right )\\ &=-\left (\frac {1}{5} \left (4 (9-2 \log (\log (2)))^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (4-x^2 (9-2 \log (\log (2)))^2\right )^2} \, dx,x,x+\frac {360-80 \log (\log (2))}{4 \left (405-180 \log (\log (2))+20 \log ^2(\log (2))\right )}\right )\right )\\ &=\frac {2}{5 \left (4 x+x^2 (9-2 \log (\log (2)))\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \frac {4}{5 \left (8 x+18 x^2-4 x^2 \log (\log (2))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 36*x + 8*x*Log[Log[2]])/(80*x^2 + 360*x^3 + 405*x^4 + (-80*x^3 - 180*x^4)*Log[Log[2]] + 20*x^4
*Log[Log[2]]^2),x]

[Out]

4/(5*(8*x + 18*x^2 - 4*x^2*Log[Log[2]]))

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fricas [A]  time = 0.55, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2}{5 \, {\left (2 \, x^{2} \log \left (\log \relax (2)\right ) - 9 \, x^{2} - 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(log(2))-36*x-8)/(20*x^4*log(log(2))^2+(-180*x^4-80*x^3)*log(log(2))+405*x^4+360*x^3+80*x^2)
,x, algorithm="fricas")

[Out]

-2/5/(2*x^2*log(log(2)) - 9*x^2 - 4*x)

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giac [A]  time = 0.22, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2}{5 \, {\left (2 \, x^{2} \log \left (\log \relax (2)\right ) - 9 \, x^{2} - 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(log(2))-36*x-8)/(20*x^4*log(log(2))^2+(-180*x^4-80*x^3)*log(log(2))+405*x^4+360*x^3+80*x^2)
,x, algorithm="giac")

[Out]

-2/5/(2*x^2*log(log(2)) - 9*x^2 - 4*x)

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maple [A]  time = 0.10, size = 19, normalized size = 0.83

method	result	size
gosper	\(-\frac {2}{5 x \left (2 x \ln \left (\ln \relax (2)\right )-9 x -4\right )}\)	\(19\)
norman	\(-\frac {2}{5 x \left (2 x \ln \left (\ln \relax (2)\right )-9 x -4\right )}\)	\(19\)
risch	\(-\frac {2}{5 x \left (2 x \ln \left (\ln \relax (2)\right )-9 x -4\right )}\)	\(19\)
default	\(-\frac {\left (2 \ln \left (\ln \relax (2)\right )-9\right )^{2}}{10 \left (-2 \ln \left (\ln \relax (2)\right )+9\right ) \left (-2 x \ln \left (\ln \relax (2)\right )+9 x +4\right )}+\frac {1}{10 x}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x*ln(ln(2))-36*x-8)/(20*x^4*ln(ln(2))^2+(-180*x^4-80*x^3)*ln(ln(2))+405*x^4+360*x^3+80*x^2),x,method=_R
ETURNVERBOSE)

[Out]

-2/5/x/(2*x*ln(ln(2))-9*x-4)

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maxima [A]  time = 0.44, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} {\left (2 \, \log \left (\log \relax (2)\right ) - 9\right )} - 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(log(2))-36*x-8)/(20*x^4*log(log(2))^2+(-180*x^4-80*x^3)*log(log(2))+405*x^4+360*x^3+80*x^2)
,x, algorithm="maxima")

[Out]

-2/5/(x^2*(2*log(log(2)) - 9) - 4*x)

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mupad [B]  time = 0.14, size = 20, normalized size = 0.87 \begin {gather*} \frac {2}{20\,x-x^2\,\left (10\,\ln \left (\ln \relax (2)\right )-45\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(36*x - 8*x*log(log(2)) + 8)/(20*x^4*log(log(2))^2 - log(log(2))*(80*x^3 + 180*x^4) + 80*x^2 + 360*x^3 +
405*x^4),x)

[Out]

2/(20*x - x^2*(10*log(log(2)) - 45))

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sympy [A]  time = 0.49, size = 17, normalized size = 0.74 \begin {gather*} - \frac {2}{x^{2} \left (-45 + 10 \log {\left (\log {\relax (2 )} \right )}\right ) - 20 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*ln(ln(2))-36*x-8)/(20*x**4*ln(ln(2))**2+(-180*x**4-80*x**3)*ln(ln(2))+405*x**4+360*x**3+80*x**2
),x)

[Out]

-2/(x**2*(-45 + 10*log(log(2))) - 20*x)

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