\(\int \frac {(-2-2 x^2+2 \log (16 x \log ^2(\log (5)))) \log (\frac {x^2+\log (16 x \log ^2(\log (5)))}{x})}{x^3+x \log (16 x \log ^2(\log (5)))} \, dx\) [5055]

Optimal result

Integrand size = 54, antiderivative size = 27 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=1-e^{12}-\log ^2\left (x+\frac {\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \]

[Out]

1-ln(x+ln(16*x*ln(ln(5))^2)/x)^2-exp(6)^2

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 2, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2641, 6873, 12, 6874, 6816, 6818} \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=-\log ^2\left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \]

[In]

Int[((-2 - 2*x^2 + 2*Log[16*x*Log[Log[5]]^2])*Log[(x^2 + Log[16*x*Log[Log[5]]^2])/x])/(x^3 + x*Log[16*x*Log[Lo
g[5]]^2]),x]

[Out]

-Log[(x^2 + Log[16*x*Log[Log[5]]^2])/x]^2

Rule 12

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

Rule 2641

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6816

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

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x \left (x^2+\log \left (16 x \log ^2(\log (5))\right )\right )} \, dx \\ & = -\log ^2\left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=-\log ^2\left (x+\frac {\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \]

[In]

Integrate[((-2 - 2*x^2 + 2*Log[16*x*Log[Log[5]]^2])*Log[(x^2 + Log[16*x*Log[Log[5]]^2])/x])/(x^3 + x*Log[16*x*
Log[Log[5]]^2]),x]

[Out]

-Log[x + Log[16*x*Log[Log[5]]^2]/x]^2

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
default	\(-\ln \left (\frac {4 \ln \left (2\right )+2 \ln \left (\ln \left (\ln \left (5\right )\right )\right )+\ln \left (x \right )+x^{2}}{x}\right )^{2}\)	\(26\)

[In]

int((2*ln(16*x*ln(ln(5))^2)-2*x^2-2)*ln((ln(16*x*ln(ln(5))^2)+x^2)/x)/(x*ln(16*x*ln(ln(5))^2)+x^3),x,method=_R
ETURNVERBOSE)

[Out]

-ln((4*ln(2)+2*ln(ln(ln(5)))+ln(x)+x^2)/x)^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=-\log \left (\frac {x^{2} + \log \left (16 \, x \log \left (\log \left (5\right )\right )^{2}\right )}{x}\right )^{2} \]

[In]

integrate((2*log(16*x*log(log(5))^2)-2*x^2-2)*log((log(16*x*log(log(5))^2)+x^2)/x)/(x*log(16*x*log(log(5))^2)+
x^3),x, algorithm="fricas")

[Out]

-log((x^2 + log(16*x*log(log(5))^2))/x)^2

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=- \log {\left (\frac {x^{2} + \log {\left (16 x \log {\left (\log {\left (5 \right )} \right )}^{2} \right )}}{x} \right )}^{2} \]

[In]

integrate((2*ln(16*x*ln(ln(5))**2)-2*x**2-2)*ln((ln(16*x*ln(ln(5))**2)+x**2)/x)/(x*ln(16*x*ln(ln(5))**2)+x**3)
,x)

[Out]

-log((x**2 + log(16*x*log(log(5))**2))/x)**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (26) = 52\).

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=\log \left (x^{2} + 4 \, \log \left (2\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (\log \left (5\right )\right )\right )\right )^{2} - 2 \, \log \left (x^{2} + 4 \, \log \left (2\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (\log \left (5\right )\right )\right )\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (\log \left (x^{2} + 4 \, \log \left (2\right ) + \log \left (x\right ) + 2 \, \log \left (\log \left (\log \left (5\right )\right )\right )\right ) - \log \left (x\right )\right )} \log \left (\frac {x^{2} + \log \left (16 \, x \log \left (\log \left (5\right )\right )^{2}\right )}{x}\right ) \]

[In]

integrate((2*log(16*x*log(log(5))^2)-2*x^2-2)*log((log(16*x*log(log(5))^2)+x^2)/x)/(x*log(16*x*log(log(5))^2)+
x^3),x, algorithm="maxima")

[Out]

log(x^2 + 4*log(2) + log(x) + 2*log(log(log(5))))^2 - 2*log(x^2 + 4*log(2) + log(x) + 2*log(log(log(5))))*log(
x) + log(x)^2 - 2*(log(x^2 + 4*log(2) + log(x) + 2*log(log(log(5)))) - log(x))*log((x^2 + log(16*x*log(log(5))
^2))/x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=-2 \, {\left (\log \left (x^{2} + \log \left (16 \, \log \left (\log \left (5\right )\right )^{2}\right ) + \log \left (x\right )\right ) - \log \left (x\right )\right )} \log \left (x^{2} + \log \left (16 \, x \log \left (\log \left (5\right )\right )^{2}\right )\right ) + \log \left (x^{2} + \log \left (16 \, \log \left (\log \left (5\right )\right )^{2}\right ) + \log \left (x\right )\right )^{2} - \log \left (x\right )^{2} \]

[In]

integrate((2*log(16*x*log(log(5))^2)-2*x^2-2)*log((log(16*x*log(log(5))^2)+x^2)/x)/(x*log(16*x*log(log(5))^2)+
x^3),x, algorithm="giac")

[Out]

-2*(log(x^2 + log(16*log(log(5))^2) + log(x)) - log(x))*log(x^2 + log(16*x*log(log(5))^2)) + log(x^2 + log(16*
log(log(5))^2) + log(x))^2 - log(x)^2

Mupad [B] (verification not implemented)

Time = 13.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-2-2 x^2+2 \log \left (16 x \log ^2(\log (5))\right )\right ) \log \left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right )}{x^3+x \log \left (16 x \log ^2(\log (5))\right )} \, dx=-{\ln \left (\frac {\ln \left (16\,x\,{\ln \left (\ln \left (5\right )\right )}^2\right )+x^2}{x}\right )}^2 \]

[In]

int(-(log((log(16*x*log(log(5))^2) + x^2)/x)*(2*x^2 - 2*log(16*x*log(log(5))^2) + 2))/(x*log(16*x*log(log(5))^
2) + x^3),x)

[Out]

-log((log(16*x*log(log(5))^2) + x^2)/x)^2