3.86.66 \(\int \frac {-1+3 x+256 x^2+(2 x^2-9 x^3-1024 x^4) \log (x)+(3 x+512 x^2) \log (x) \log (\log (x))}{(x^3-3 x^4-256 x^5) \log (x)+(-x+3 x^2+256 x^3) \log (x) \log (\log (x))} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\left (-1+3 x+256 x^2\right ) \left (-x^2+\log (\log (x))\right )\right ) \]

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Rubi [A]  time = 1.38, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6741, 6728, 628, 6684} \begin {gather*} \log \left (-256 x^2-3 x+1\right )+\log \left (x^2-\log (\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x + 256*x^2 + (2*x^2 - 9*x^3 - 1024*x^4)*Log[x] + (3*x + 512*x^2)*Log[x]*Log[Log[x]])/((x^3 - 3*x^
4 - 256*x^5)*Log[x] + (-x + 3*x^2 + 256*x^3)*Log[x]*Log[Log[x]]),x]

[Out]

Log[1 - 3*x - 256*x^2] + Log[x^2 - Log[Log[x]]]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+3 x+256 x^2+\left (2 x^2-9 x^3-1024 x^4\right ) \log (x)+\left (3 x+512 x^2\right ) \log (x) \log (\log (x))}{x \left (1-3 x-256 x^2\right ) \log (x) \left (x^2-\log (\log (x))\right )} \, dx\\ &=\int \left (\frac {3+512 x}{-1+3 x+256 x^2}+\frac {-1+2 x^2 \log (x)}{x \log (x) \left (x^2-\log (\log (x))\right )}\right ) \, dx\\ &=\int \frac {3+512 x}{-1+3 x+256 x^2} \, dx+\int \frac {-1+2 x^2 \log (x)}{x \log (x) \left (x^2-\log (\log (x))\right )} \, dx\\ &=\log \left (1-3 x-256 x^2\right )+\log \left (x^2-\log (\log (x))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 22, normalized size = 1.05 \begin {gather*} \log \left (1-3 x-256 x^2\right )+\log \left (x^2-\log (\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 3*x + 256*x^2 + (2*x^2 - 9*x^3 - 1024*x^4)*Log[x] + (3*x + 512*x^2)*Log[x]*Log[Log[x]])/((x^3
- 3*x^4 - 256*x^5)*Log[x] + (-x + 3*x^2 + 256*x^3)*Log[x]*Log[Log[x]]),x]

[Out]

Log[1 - 3*x - 256*x^2] + Log[x^2 - Log[Log[x]]]

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fricas [A]  time = 0.48, size = 22, normalized size = 1.05 \begin {gather*} \log \left (256 \, x^{2} + 3 \, x - 1\right ) + \log \left (-x^{2} + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^2+3*x)*log(x)*log(log(x))+(-1024*x^4-9*x^3+2*x^2)*log(x)+256*x^2+3*x-1)/((256*x^3+3*x^2-x)*l
og(x)*log(log(x))+(-256*x^5-3*x^4+x^3)*log(x)),x, algorithm="fricas")

[Out]

log(256*x^2 + 3*x - 1) + log(-x^2 + log(log(x)))

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giac [A]  time = 0.19, size = 22, normalized size = 1.05 \begin {gather*} \log \left (256 \, x^{2} + 3 \, x - 1\right ) + \log \left (-x^{2} + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^2+3*x)*log(x)*log(log(x))+(-1024*x^4-9*x^3+2*x^2)*log(x)+256*x^2+3*x-1)/((256*x^3+3*x^2-x)*l
og(x)*log(log(x))+(-256*x^5-3*x^4+x^3)*log(x)),x, algorithm="giac")

[Out]

log(256*x^2 + 3*x - 1) + log(-x^2 + log(log(x)))

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maple [A]  time = 0.08, size = 23, normalized size = 1.10

method	result	size
risch	\(\ln \left (256 x^{2}+3 x -1\right )+\ln \left (\ln \left (\ln \relax (x )\right )-x^{2}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((512*x^2+3*x)*ln(x)*ln(ln(x))+(-1024*x^4-9*x^3+2*x^2)*ln(x)+256*x^2+3*x-1)/((256*x^3+3*x^2-x)*ln(x)*ln(ln
(x))+(-256*x^5-3*x^4+x^3)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

ln(256*x^2+3*x-1)+ln(ln(ln(x))-x^2)

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maxima [A]  time = 0.45, size = 22, normalized size = 1.05 \begin {gather*} \log \left (256 \, x^{2} + 3 \, x - 1\right ) + \log \left (-x^{2} + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^2+3*x)*log(x)*log(log(x))+(-1024*x^4-9*x^3+2*x^2)*log(x)+256*x^2+3*x-1)/((256*x^3+3*x^2-x)*l
og(x)*log(log(x))+(-256*x^5-3*x^4+x^3)*log(x)),x, algorithm="maxima")

[Out]

log(256*x^2 + 3*x - 1) + log(-x^2 + log(log(x)))

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mupad [B]  time = 5.34, size = 22, normalized size = 1.05 \begin {gather*} \ln \left (\ln \left (\ln \relax (x)\right )-x^2\right )+\ln \left (256\,x^2+3\,x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - log(x)*(9*x^3 - 2*x^2 + 1024*x^4) + 256*x^2 + log(log(x))*log(x)*(3*x + 512*x^2) - 1)/(log(x)*(3*x
^4 - x^3 + 256*x^5) - log(log(x))*log(x)*(3*x^2 - x + 256*x^3)),x)

[Out]

log(log(log(x)) - x^2) + log(3*x + 256*x^2 - 1)

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sympy [A]  time = 0.34, size = 20, normalized size = 0.95 \begin {gather*} \log {\left (- x^{2} + \log {\left (\log {\relax (x )} \right )} \right )} + \log {\left (256 x^{2} + 3 x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x**2+3*x)*ln(x)*ln(ln(x))+(-1024*x**4-9*x**3+2*x**2)*ln(x)+256*x**2+3*x-1)/((256*x**3+3*x**2-x
)*ln(x)*ln(ln(x))+(-256*x**5-3*x**4+x**3)*ln(x)),x)

[Out]

log(-x**2 + log(log(x))) + log(256*x**2 + 3*x - 1)

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