3.30.93 \(\int -\frac {2 \log ^2(\log (5)) \log (4-x+5 \log (\log (4)) \log (\log (25)))}{4-x+5 \log (\log (4)) \log (\log (25))} \, dx\)

Optimal. Leaf size=22 \[ \log ^2(\log (5)) \log ^2(4-x+5 \log (\log (4)) \log (\log (25))) \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 2390, 2301} \begin {gather*} \log ^2(\log (5)) \log ^2(-x+4+5 \log (\log (4)) \log (\log (25))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*Log[Log[5]]^2*Log[4 - x + 5*Log[Log[4]]*Log[Log[25]]])/(4 - x + 5*Log[Log[4]]*Log[Log[25]]),x]

[Out]

Log[Log[5]]^2*Log[4 - x + 5*Log[Log[4]]*Log[Log[25]]]^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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (2 \log ^2(\log (5))\right ) \int \frac {\log (4-x+5 \log (\log (4)) \log (\log (25)))}{4-x+5 \log (\log (4)) \log (\log (25))} \, dx\right )\\ &=\left (2 \log ^2(\log (5))\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,4-x+5 \log (\log (4)) \log (\log (25))\right )\\ &=\log ^2(\log (5)) \log ^2(4-x+5 \log (\log (4)) \log (\log (25)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \log ^2(\log (5)) \log ^2(4-x+5 \log (\log (4)) \log (\log (25))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*Log[Log[5]]^2*Log[4 - x + 5*Log[Log[4]]*Log[Log[25]]])/(4 - x + 5*Log[Log[4]]*Log[Log[25]]),x]

[Out]

Log[Log[5]]^2*Log[4 - x + 5*Log[Log[4]]*Log[Log[25]]]^2

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fricas [A]  time = 0.52, size = 33, normalized size = 1.50 \begin {gather*} \log \left (5 \, \log \relax (2) \log \left (2 \, \log \relax (2)\right ) + 5 \, \log \left (2 \, \log \relax (2)\right ) \log \left (\log \relax (5)\right ) - x + 4\right )^{2} \log \left (\log \relax (5)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*log(log(5))^2*log(5*log(2*log(2))*log(2*log(5))-x+4)/(5*log(2*log(2))*log(2*log(5))-x+4),x, algor
ithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 42, normalized size = 1.91 \begin {gather*} \log \left (5 \, \log \relax (2)^{2} + 5 \, \log \relax (2) \log \left (\log \relax (5)\right ) + 5 \, \log \relax (2) \log \left (\log \relax (2)\right ) + 5 \, \log \left (\log \relax (5)\right ) \log \left (\log \relax (2)\right ) - x + 4\right )^{2} \log \left (\log \relax (5)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*log(log(5))^2*log(5*log(2*log(2))*log(2*log(5))-x+4)/(5*log(2*log(2))*log(2*log(5))-x+4),x, algor
ithm="giac")

[Out]

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

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maple [A]  time = 0.51, size = 27, normalized size = 1.23

method	result	size
derivativedivides	\(\ln \left (5 \ln \left (2 \ln \relax (2)\right ) \ln \left (2 \ln \relax (5)\right )-x +4\right )^{2} \ln \left (\ln \relax (5)\right )^{2}\)	\(27\)
default	\(\ln \left (5 \ln \left (2 \ln \relax (2)\right ) \ln \left (2 \ln \relax (5)\right )-x +4\right )^{2} \ln \left (\ln \relax (5)\right )^{2}\)	\(27\)
norman	\(\ln \left (5 \ln \left (2 \ln \relax (2)\right ) \ln \left (2 \ln \relax (5)\right )-x +4\right )^{2} \ln \left (\ln \relax (5)\right )^{2}\)	\(27\)
risch	\(\ln \left (\ln \relax (5)\right )^{2} \ln \left (5 \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right ) \left (\ln \relax (2)+\ln \left (\ln \relax (5)\right )\right )-x +4\right )^{2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 26, normalized size = 1.18 \begin {gather*} \log \left (5 \, \log \left (2 \, \log \relax (5)\right ) \log \left (2 \, \log \relax (2)\right ) - x + 4\right )^{2} \log \left (\log \relax (5)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*log(log(5))^2*log(5*log(2*log(2))*log(2*log(5))-x+4)/(5*log(2*log(2))*log(2*log(5))-x+4),x, algor
ithm="maxima")

[Out]

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

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mupad [B]  time = 1.93, size = 23, normalized size = 1.05 \begin {gather*} {\ln \left (\ln \left ({\ln \relax (4)}^5\right )\,\ln \left (\ln \left (25\right )\right )-x+4\right )}^2\,{\ln \left (\ln \relax (5)\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(log(4)^5)*log(log(25)) - x + 4)^2*log(log(5))^2

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sympy [A]  time = 0.16, size = 27, normalized size = 1.23 \begin {gather*} \log {\left (- x + 5 \log {\left (2 \log {\relax (2 )} \right )} \log {\left (2 \log {\relax (5 )} \right )} + 4 \right )}^{2} \log {\left (\log {\relax (5 )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*ln(ln(5))**2*ln(5*ln(2*ln(2))*ln(2*ln(5))-x+4)/(5*ln(2*ln(2))*ln(2*ln(5))-x+4),x)

[Out]

log(-x + 5*log(2*log(2))*log(2*log(5)) + 4)**2*log(log(5))**2

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