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

Optimal. Leaf size=27 \[ 1-e^{12}-\log ^2\left (x+\frac {\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \]

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Rubi [A]  time = 0.72, antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 2, number of rules used = 6, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2561, 6741, 12, 6742, 6684, 6686} \begin {gather*} -\log ^2\left (\frac {x^2+\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[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 2561

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 6684

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

Rule 6686

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 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.74 \begin {gather*} -\log ^2\left (x+\frac {\log \left (16 x \log ^2(\log (5))\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.87, size = 22, normalized size = 0.81 \begin {gather*} -\log \left (\frac {x^{2} + \log \left (16 \, x \log \left (\log \relax (5)\right )^{2}\right )}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.20, size = 60, normalized size = 2.22 \begin {gather*} -2 \, {\left (\log \left (x^{2} + \log \left (16 \, \log \left (\log \relax (5)\right )^{2}\right ) + \log \relax (x)\right ) - \log \relax (x)\right )} \log \left (x^{2} + \log \left (16 \, x \log \left (\log \relax (5)\right )^{2}\right )\right ) + \log \left (x^{2} + \log \left (16 \, \log \left (\log \relax (5)\right )^{2}\right ) + \log \relax (x)\right )^{2} - \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 \ln \left (16 x \ln \left (\ln \relax (5)\right )^{2}\right )-2 x^{2}-2\right ) \ln \left (\frac {\ln \left (16 x \ln \left (\ln \relax (5)\right )^{2}\right )+x^{2}}{x}\right )}{x \ln \left (16 x \ln \left (\ln \relax (5)\right )^{2}\right )+x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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)

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maxima [B]  time = 0.48, size = 87, normalized size = 3.22 \begin {gather*} \log \left (x^{2} + 4 \, \log \relax (2) + \log \relax (x) + 2 \, \log \left (\log \left (\log \relax (5)\right )\right )\right )^{2} - 2 \, \log \left (x^{2} + 4 \, \log \relax (2) + \log \relax (x) + 2 \, \log \left (\log \left (\log \relax (5)\right )\right )\right ) \log \relax (x) + \log \relax (x)^{2} - 2 \, {\left (\log \left (x^{2} + 4 \, \log \relax (2) + \log \relax (x) + 2 \, \log \left (\log \left (\log \relax (5)\right )\right )\right ) - \log \relax (x)\right )} \log \left (\frac {x^{2} + \log \left (16 \, x \log \left (\log \relax (5)\right )^{2}\right )}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.34, size = 22, normalized size = 0.81 \begin {gather*} -{\ln \left (\frac {\ln \left (16\,x\,{\ln \left (\ln \relax (5)\right )}^2\right )+x^2}{x}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.33, size = 20, normalized size = 0.74 \begin {gather*} - \log {\left (\frac {x^{2} + \log {\left (16 x \log {\left (\log {\relax (5 )} \right )}^{2} \right )}}{x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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