3.30.38 \(\int \frac {-4-6 x-3 x^3-6 x^4-3 x^5+(-8-12 x^3-12 x^4) \log (\log (5))-12 x^3 \log ^2(\log (5))}{3 x^3+6 x^4+3 x^5+(12 x^3+12 x^4) \log (\log (5))+12 x^3 \log ^2(\log (5))} \, dx\)

Optimal. Leaf size=24 \[ -x+\frac {1}{3 x^2 \left (\frac {1+x}{2}+\log (\log (5))\right )} \]

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Rubi [B]  time = 0.11, antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps used = 4, number of rules used = 2, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 2074} \begin {gather*} \frac {2}{3 x^2 (1+2 \log (\log (5)))}-x+\frac {2}{3 (1+2 \log (\log (5)))^2 (x+1+2 \log (\log (5)))}-\frac {2}{3 x (1+2 \log (\log (5)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 6*x - 3*x^3 - 6*x^4 - 3*x^5 + (-8 - 12*x^3 - 12*x^4)*Log[Log[5]] - 12*x^3*Log[Log[5]]^2)/(3*x^3 + 6*
x^4 + 3*x^5 + (12*x^3 + 12*x^4)*Log[Log[5]] + 12*x^3*Log[Log[5]]^2),x]

[Out]

-x - 2/(3*x*(1 + 2*Log[Log[5]])^2) + 2/(3*x^2*(1 + 2*Log[Log[5]])) + 2/(3*(1 + 2*Log[Log[5]])^2*(1 + x + 2*Log
[Log[5]]))

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 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-6 x-6 x^4-3 x^5+\left (-8-12 x^3-12 x^4\right ) \log (\log (5))+x^3 \left (-3-12 \log ^2(\log (5))\right )}{3 x^3+6 x^4+3 x^5+\left (12 x^3+12 x^4\right ) \log (\log (5))+12 x^3 \log ^2(\log (5))} \, dx\\ &=\int \frac {-4-6 x-6 x^4-3 x^5+\left (-8-12 x^3-12 x^4\right ) \log (\log (5))+x^3 \left (-3-12 \log ^2(\log (5))\right )}{6 x^4+3 x^5+\left (12 x^3+12 x^4\right ) \log (\log (5))+x^3 \left (3+12 \log ^2(\log (5))\right )} \, dx\\ &=\int \left (-1+\frac {2}{3 x^2 (1+2 \log (\log (5)))^2}-\frac {4}{3 x^3 (1+2 \log (\log (5)))}-\frac {2}{3 (1+2 \log (\log (5)))^2 (1+x+2 \log (\log (5)))^2}\right ) \, dx\\ &=-x-\frac {2}{3 x (1+2 \log (\log (5)))^2}+\frac {2}{3 x^2 (1+2 \log (\log (5)))}+\frac {2}{3 (1+2 \log (\log (5)))^2 (1+x+2 \log (\log (5)))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 35, normalized size = 1.46 \begin {gather*} -\frac {-2+3 x^4+x^3 (3+6 \log (\log (5)))}{3 x^2 (1+x+2 \log (\log (5)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 6*x - 3*x^3 - 6*x^4 - 3*x^5 + (-8 - 12*x^3 - 12*x^4)*Log[Log[5]] - 12*x^3*Log[Log[5]]^2)/(3*x^
3 + 6*x^4 + 3*x^5 + (12*x^3 + 12*x^4)*Log[Log[5]] + 12*x^3*Log[Log[5]]^2),x]

[Out]

-1/3*(-2 + 3*x^4 + x^3*(3 + 6*Log[Log[5]]))/(x^2*(1 + x + 2*Log[Log[5]]))

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fricas [B]  time = 0.49, size = 39, normalized size = 1.62 \begin {gather*} -\frac {3 \, x^{4} + 6 \, x^{3} \log \left (\log \relax (5)\right ) + 3 \, x^{3} - 2}{3 \, {\left (x^{3} + 2 \, x^{2} \log \left (\log \relax (5)\right ) + x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^3*log(log(5))^2+(-12*x^4-12*x^3-8)*log(log(5))-3*x^5-6*x^4-3*x^3-6*x-4)/(12*x^3*log(log(5))^2
+(12*x^4+12*x^3)*log(log(5))+3*x^5+6*x^4+3*x^3),x, algorithm="fricas")

[Out]

-1/3*(3*x^4 + 6*x^3*log(log(5)) + 3*x^3 - 2)/(x^3 + 2*x^2*log(log(5)) + x^2)

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giac [B]  time = 0.23, size = 61, normalized size = 2.54 \begin {gather*} -x + \frac {2}{3 \, {\left (4 \, \log \left (\log \relax (5)\right )^{2} + 4 \, \log \left (\log \relax (5)\right ) + 1\right )} {\left (x + 2 \, \log \left (\log \relax (5)\right ) + 1\right )}} - \frac {2 \, {\left (x - 2 \, \log \left (\log \relax (5)\right ) - 1\right )}}{3 \, {\left (4 \, \log \left (\log \relax (5)\right )^{2} + 4 \, \log \left (\log \relax (5)\right ) + 1\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^3*log(log(5))^2+(-12*x^4-12*x^3-8)*log(log(5))-3*x^5-6*x^4-3*x^3-6*x-4)/(12*x^3*log(log(5))^2
+(12*x^4+12*x^3)*log(log(5))+3*x^5+6*x^4+3*x^3),x, algorithm="giac")

[Out]

-x + 2/3/((4*log(log(5))^2 + 4*log(log(5)) + 1)*(x + 2*log(log(5)) + 1)) - 2/3*(x - 2*log(log(5)) - 1)/((4*log
(log(5))^2 + 4*log(log(5)) + 1)*x^2)

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maple [A]  time = 0.08, size = 20, normalized size = 0.83

method	result	size
risch	\(-x +\frac {2}{3 x^{2} \left (1+2 \ln \left (\ln \relax (5)\right )+x \right )}\)	\(20\)
gosper	\(\frac {12 x^{2} \ln \left (\ln \relax (5)\right )^{2}-3 x^{4}+12 \ln \left (\ln \relax (5)\right ) x^{2}+3 x^{2}+2}{3 x^{2} \left (1+2 \ln \left (\ln \relax (5)\right )+x \right )}\)	\(46\)
default	\(-x -\frac {2}{3 \left (1+2 \ln \left (\ln \relax (5)\right )\right )^{2} x}+\frac {2}{3 \left (1+2 \ln \left (\ln \relax (5)\right )\right ) x^{2}}+\frac {2}{3 \left (1+2 \ln \left (\ln \relax (5)\right )\right )^{2} \left (1+2 \ln \left (\ln \relax (5)\right )+x \right )}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*x^3*ln(ln(5))^2+(-12*x^4-12*x^3-8)*ln(ln(5))-3*x^5-6*x^4-3*x^3-6*x-4)/(12*x^3*ln(ln(5))^2+(12*x^4+12*
x^3)*ln(ln(5))+3*x^5+6*x^4+3*x^3),x,method=_RETURNVERBOSE)

[Out]

-x+2/3/x^2/(1+2*ln(ln(5))+x)

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maxima [A]  time = 0.72, size = 23, normalized size = 0.96 \begin {gather*} -x + \frac {2}{3 \, {\left (x^{3} + x^{2} {\left (2 \, \log \left (\log \relax (5)\right ) + 1\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^3*log(log(5))^2+(-12*x^4-12*x^3-8)*log(log(5))-3*x^5-6*x^4-3*x^3-6*x-4)/(12*x^3*log(log(5))^2
+(12*x^4+12*x^3)*log(log(5))+3*x^5+6*x^4+3*x^3),x, algorithm="maxima")

[Out]

-x + 2/3/(x^3 + x^2*(2*log(log(5)) + 1))

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mupad [B]  time = 0.13, size = 25, normalized size = 1.04 \begin {gather*} \frac {2}{3\,x^3+\left (6\,\ln \left (\ln \relax (5)\right )+3\right )\,x^2}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + log(log(5))*(12*x^3 + 12*x^4 + 8) + 12*x^3*log(log(5))^2 + 3*x^3 + 6*x^4 + 3*x^5 + 4)/(log(log(5))
*(12*x^3 + 12*x^4) + 12*x^3*log(log(5))^2 + 3*x^3 + 6*x^4 + 3*x^5),x)

[Out]

2/(x^2*(6*log(log(5)) + 3) + 3*x^3) - x

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sympy [A]  time = 0.42, size = 19, normalized size = 0.79 \begin {gather*} - x + \frac {2}{3 x^{3} + x^{2} \left (6 \log {\left (\log {\relax (5 )} \right )} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x**3*ln(ln(5))**2+(-12*x**4-12*x**3-8)*ln(ln(5))-3*x**5-6*x**4-3*x**3-6*x-4)/(12*x**3*ln(ln(5))
**2+(12*x**4+12*x**3)*ln(ln(5))+3*x**5+6*x**4+3*x**3),x)

[Out]

-x + 2/(3*x**3 + x**2*(6*log(log(5)) + 3))

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