3.30.32 \(\int \frac {-4+x^2-2 x^3 \log (x)+(4+x^2) \log (x) \log (4 \log (x))}{(4 x^2-x^4) \log (x)+(-4 x+x^3) \log (x) \log (4 \log (x))} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {7 \left (4-x^2\right ) \log (4) (-x+\log (4 \log (x)))}{3 x}\right ) \]

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Rubi [A]  time = 1.13, antiderivative size = 23, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6741, 6725, 446, 72, 6684} \begin {gather*} \log \left (4-x^2\right )-\log (x)+\log (x-\log (4 \log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[4 - x^2] + Log[x - Log[4*Log[x]]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 6684

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && 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 {-4+x^2-2 x^3 \log (x)+\left (4+x^2\right ) \log (x) \log (4 \log (x))}{x \left (4-x^2\right ) \log (x) (x-\log (4 \log (x)))} \, dx\\ &=\int \left (\frac {4+x^2}{x \left (-4+x^2\right )}+\frac {-1+x \log (x)}{x \log (x) (x-\log (4 \log (x)))}\right ) \, dx\\ &=\int \frac {4+x^2}{x \left (-4+x^2\right )} \, dx+\int \frac {-1+x \log (x)}{x \log (x) (x-\log (4 \log (x)))} \, dx\\ &=\log (x-\log (4 \log (x)))+\frac {1}{2} \operatorname {Subst}\left (\int \frac {4+x}{(-4+x) x} \, dx,x,x^2\right )\\ &=\log (x-\log (4 \log (x)))+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2}{-4+x}-\frac {1}{x}\right ) \, dx,x,x^2\right )\\ &=-\log (x)+\log \left (4-x^2\right )+\log (x-\log (4 \log (x)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 23, normalized size = 0.88 \begin {gather*} -\log (x)+\log \left (4-x^2\right )+\log (x-\log (4 \log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[4 - x^2] + Log[x - Log[4*Log[x]]]

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fricas [A]  time = 0.65, size = 21, normalized size = 0.81 \begin {gather*} \log \left (x^{2} - 4\right ) - \log \relax (x) + \log \left (-x + \log \left (4 \, \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 4) - log(x) + log(-x + log(4*log(x)))

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giac [A]  time = 0.35, size = 21, normalized size = 0.81 \begin {gather*} \log \left (x^{2} - 4\right ) - \log \relax (x) + \log \left (-x + \log \left (4 \, \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 4) - log(x) + log(-x + log(4*log(x)))

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maple [A]  time = 0.10, size = 22, normalized size = 0.85

method	result	size
risch	\(-\ln \relax (x )+\ln \left (x^{2}-4\right )+\ln \left (\ln \left (4 \ln \relax (x )\right )-x \right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+4)*ln(x)*ln(4*ln(x))-2*x^3*ln(x)+x^2-4)/((x^3-4*x)*ln(x)*ln(4*ln(x))+(-x^4+4*x^2)*ln(x)),x,method=_R
ETURNVERBOSE)

[Out]

-ln(x)+ln(x^2-4)+ln(ln(4*ln(x))-x)

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maxima [A]  time = 1.01, size = 25, normalized size = 0.96 \begin {gather*} \log \left (x + 2\right ) + \log \left (x - 2\right ) - \log \relax (x) + \log \left (-x + 2 \, \log \relax (2) + \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 2) + log(x - 2) - log(x) + log(-x + 2*log(2) + log(log(x)))

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mupad [B]  time = 1.89, size = 21, normalized size = 0.81 \begin {gather*} \ln \left (\ln \left (4\,\ln \relax (x)\right )-x\right )+\ln \left (x^2-4\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(4*log(x)) - x) + log(x^2 - 4) - log(x)

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sympy [A]  time = 0.34, size = 19, normalized size = 0.73 \begin {gather*} - \log {\relax (x )} + \log {\left (- x + \log {\left (4 \log {\relax (x )} \right )} \right )} + \log {\left (x^{2} - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(-x + log(4*log(x))) + log(x**2 - 4)

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