3.18.5 \(\int \frac {4 x^5+x^3 \log (x)+(-32 x^3 \log (\frac {4}{x})+64 x^3 \log (\frac {4}{x}) \log (x)) \log (\log (\frac {4}{x})) \log (\log (\log (\frac {4}{x}))) \log (\log (\log (\log (\frac {4}{x}))))+(2 x^3 \log (\frac {4}{x})-4 x^3 \log (\frac {4}{x}) \log (x)) \log (\log (\frac {4}{x})) \log (\log (\log (\frac {4}{x}))) \log (\log (\log (\log (\frac {4}{x})))) \log (\log (\log (\log (\log (\frac {4}{x})))))}{(64 x^6 \log (\frac {4}{x})+48 x^4 \log (\frac {4}{x}) \log (x)+12 x^2 \log (\frac {4}{x}) \log ^2(x)+\log (\frac {4}{x}) \log ^3(x)) \log (\log (\frac {4}{x})) \log (\log (\log (\frac {4}{x}))) \log (\log (\log (\log (\frac {4}{x}))))} \, dx\)

Optimal. Leaf size=25 \[ \frac {16-\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4+\frac {\log (x)}{x^2}\right )^2} \]

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Rubi [F]  time = 14.27, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4 x^5+x^3 \log (x)+\left (-32 x^3 \log \left (\frac {4}{x}\right )+64 x^3 \log \left (\frac {4}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )+\left (2 x^3 \log \left (\frac {4}{x}\right )-4 x^3 \log \left (\frac {4}{x}\right ) \log (x)\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (64 x^6 \log \left (\frac {4}{x}\right )+48 x^4 \log \left (\frac {4}{x}\right ) \log (x)+12 x^2 \log \left (\frac {4}{x}\right ) \log ^2(x)+\log \left (\frac {4}{x}\right ) \log ^3(x)\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^5 + x^3*Log[x] + (-32*x^3*Log[4/x] + 64*x^3*Log[4/x]*Log[x])*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log[Log
[Log[Log[4/x]]]] + (2*x^3*Log[4/x] - 4*x^3*Log[4/x]*Log[x])*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log[Log[Log[Log[4
/x]]]]*Log[Log[Log[Log[Log[4/x]]]]])/((64*x^6*Log[4/x] + 48*x^4*Log[4/x]*Log[x] + 12*x^2*Log[4/x]*Log[x]^2 + L
og[4/x]*Log[x]^3)*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log[Log[Log[Log[4/x]]]]),x]

[Out]

-32*Defer[Int][x^3/(4*x^2 + Log[x])^3, x] - 256*Defer[Int][x^5/(4*x^2 + Log[x])^3, x] + 64*Defer[Int][x^3/(4*x
^2 + Log[x])^2, x] + 4*Defer[Int][x^5/(Log[4/x]*(4*x^2 + Log[x])^3*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log[Log[Lo
g[Log[4/x]]]]), x] + Defer[Int][(x^3*Log[x])/(Log[4/x]*(4*x^2 + Log[x])^3*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log
[Log[Log[Log[4/x]]]]), x] + 2*Defer[Int][(x^3*Log[Log[Log[Log[Log[4/x]]]]])/(4*x^2 + Log[x])^3, x] - 4*Defer[I
nt][(x^3*Log[x]*Log[Log[Log[Log[Log[4/x]]]]])/(4*x^2 + Log[x])^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (4 x^2+\log (x)+2 \log \left (\frac {4}{x}\right ) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right ) \left (-16+\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )\right )-4 \log \left (\frac {4}{x}\right ) \log (x) \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right ) \left (-16+\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )\right )\right )}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ &=\int \left (-\frac {32 x^3}{\left (4 x^2+\log (x)\right )^3}+\frac {64 x^3 \log (x)}{\left (4 x^2+\log (x)\right )^3}+\frac {4 x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )}+\frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )}-\frac {2 x^3 (-1+2 \log (x)) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {x^3 (-1+2 \log (x)) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3} \, dx\right )+4 \int \frac {x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx-32 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^3} \, dx+64 \int \frac {x^3 \log (x)}{\left (4 x^2+\log (x)\right )^3} \, dx+\int \frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {x^3 \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3}+\frac {2 x^3 \log (x) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3}\right ) \, dx\right )+4 \int \frac {x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx-32 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^3} \, dx+64 \int \left (-\frac {4 x^5}{\left (4 x^2+\log (x)\right )^3}+\frac {x^3}{\left (4 x^2+\log (x)\right )^2}\right ) \, dx+\int \frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ &=2 \int \frac {x^3 \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3} \, dx+4 \int \frac {x^5}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx-4 \int \frac {x^3 \log (x) \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^3} \, dx-32 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^3} \, dx+64 \int \frac {x^3}{\left (4 x^2+\log (x)\right )^2} \, dx-256 \int \frac {x^5}{\left (4 x^2+\log (x)\right )^3} \, dx+\int \frac {x^3 \log (x)}{\log \left (\frac {4}{x}\right ) \left (4 x^2+\log (x)\right )^3 \log \left (\log \left (\frac {4}{x}\right )\right ) \log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right ) \log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 27, normalized size = 1.08 \begin {gather*} -\frac {x^4 \left (-16+\log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right )\right )}{\left (4 x^2+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^5 + x^3*Log[x] + (-32*x^3*Log[4/x] + 64*x^3*Log[4/x]*Log[x])*Log[Log[4/x]]*Log[Log[Log[4/x]]]*L
og[Log[Log[Log[4/x]]]] + (2*x^3*Log[4/x] - 4*x^3*Log[4/x]*Log[x])*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log[Log[Log
[Log[4/x]]]]*Log[Log[Log[Log[Log[4/x]]]]])/((64*x^6*Log[4/x] + 48*x^4*Log[4/x]*Log[x] + 12*x^2*Log[4/x]*Log[x]
^2 + Log[4/x]*Log[x]^3)*Log[Log[4/x]]*Log[Log[Log[4/x]]]*Log[Log[Log[Log[4/x]]]]),x]

[Out]

-((x^4*(-16 + Log[Log[Log[Log[Log[4/x]]]]]))/(4*x^2 + Log[x])^2)

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fricas [B]  time = 0.95, size = 67, normalized size = 2.68 \begin {gather*} -\frac {x^{4} \log \left (\log \left (\log \left (\log \left (\log \left (\frac {4}{x}\right )\right )\right )\right )\right ) - 16 \, x^{4}}{16 \, x^{4} + 16 \, x^{2} \log \relax (2) + 4 \, \log \relax (2)^{2} - 4 \, {\left (2 \, x^{2} + \log \relax (2)\right )} \log \left (\frac {4}{x}\right ) + \log \left (\frac {4}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*log(4/x)*log(x)+2*x^3*log(4/x))*log(log(4/x))*log(log(log(4/x)))*log(log(log(log(4/x))))*lo
g(log(log(log(log(4/x)))))+(64*x^3*log(4/x)*log(x)-32*x^3*log(4/x))*log(log(4/x))*log(log(log(4/x)))*log(log(l
og(log(4/x))))+x^3*log(x)+4*x^5)/(log(4/x)*log(x)^3+12*x^2*log(4/x)*log(x)^2+48*x^4*log(4/x)*log(x)+64*x^6*log
(4/x))/log(log(4/x))/log(log(log(4/x)))/log(log(log(log(4/x)))),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.21, size = 116, normalized size = 4.64 \begin {gather*} -\frac {{\left (8 \, x^{6} + x^{4}\right )} \log \left (\log \left (\log \left (\log \left (2 \, \log \relax (2) - \log \relax (x)\right )\right )\right )\right )}{128 \, x^{6} + 64 \, x^{4} \log \relax (x) + 16 \, x^{4} + 8 \, x^{2} \log \relax (x)^{2} + 8 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} + \frac {16 \, {\left (8 \, x^{6} + x^{4}\right )}}{128 \, x^{6} + 64 \, x^{4} \log \relax (x) + 16 \, x^{4} + 8 \, x^{2} \log \relax (x)^{2} + 8 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*log(4/x)*log(x)+2*x^3*log(4/x))*log(log(4/x))*log(log(log(4/x)))*log(log(log(log(4/x))))*lo
g(log(log(log(log(4/x)))))+(64*x^3*log(4/x)*log(x)-32*x^3*log(4/x))*log(log(4/x))*log(log(log(4/x)))*log(log(l
og(log(4/x))))+x^3*log(x)+4*x^5)/(log(4/x)*log(x)^3+12*x^2*log(4/x)*log(x)^2+48*x^4*log(4/x)*log(x)+64*x^6*log
(4/x))/log(log(4/x))/log(log(log(4/x)))/log(log(log(log(4/x)))),x, algorithm="giac")

[Out]

-(8*x^6 + x^4)*log(log(log(log(2*log(2) - log(x)))))/(128*x^6 + 64*x^4*log(x) + 16*x^4 + 8*x^2*log(x)^2 + 8*x^
2*log(x) + log(x)^2) + 16*(8*x^6 + x^4)/(128*x^6 + 64*x^4*log(x) + 16*x^4 + 8*x^2*log(x)^2 + 8*x^2*log(x) + lo
g(x)^2)

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maple [A]  time = 0.64, size = 45, normalized size = 1.80

method	result	size
risch	\(-\frac {x^{4} \ln \left (\ln \left (\ln \left (\ln \left (2 \ln \relax (2)-\ln \relax (x )\right )\right )\right )\right )}{\left (4 x^{2}+\ln \relax (x )\right )^{2}}+\frac {16 x^{4}}{\left (4 x^{2}+\ln \relax (x )\right )^{2}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3*ln(4/x)*ln(x)+2*x^3*ln(4/x))*ln(ln(4/x))*ln(ln(ln(4/x)))*ln(ln(ln(ln(4/x))))*ln(ln(ln(ln(ln(4/x))
)))+(64*x^3*ln(4/x)*ln(x)-32*x^3*ln(4/x))*ln(ln(4/x))*ln(ln(ln(4/x)))*ln(ln(ln(ln(4/x))))+x^3*ln(x)+4*x^5)/(ln
(4/x)*ln(x)^3+12*x^2*ln(4/x)*ln(x)^2+48*x^4*ln(4/x)*ln(x)+64*x^6*ln(4/x))/ln(ln(4/x))/ln(ln(ln(4/x)))/ln(ln(ln
(ln(4/x)))),x,method=_RETURNVERBOSE)

[Out]

-x^4/(4*x^2+ln(x))^2*ln(ln(ln(ln(2*ln(2)-ln(x)))))+16*x^4/(4*x^2+ln(x))^2

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maxima [A]  time = 1.27, size = 44, normalized size = 1.76 \begin {gather*} -\frac {x^{4} \log \left (\log \left (\log \left (\log \left (2 \, \log \relax (2) - \log \relax (x)\right )\right )\right )\right ) - 16 \, x^{4}}{16 \, x^{4} + 8 \, x^{2} \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*log(4/x)*log(x)+2*x^3*log(4/x))*log(log(4/x))*log(log(log(4/x)))*log(log(log(log(4/x))))*lo
g(log(log(log(log(4/x)))))+(64*x^3*log(4/x)*log(x)-32*x^3*log(4/x))*log(log(4/x))*log(log(log(4/x)))*log(log(l
og(log(4/x))))+x^3*log(x)+4*x^5)/(log(4/x)*log(x)^3+12*x^2*log(4/x)*log(x)^2+48*x^4*log(4/x)*log(x)+64*x^6*log
(4/x))/log(log(4/x))/log(log(log(4/x)))/log(log(log(log(4/x)))),x, algorithm="maxima")

[Out]

-(x^4*log(log(log(log(2*log(2) - log(x))))) - 16*x^4)/(16*x^4 + 8*x^2*log(x) + log(x)^2)

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mupad [B]  time = 2.94, size = 168, normalized size = 6.72 \begin {gather*} \frac {\frac {16\,x^4}{8\,x^2+1}-\frac {32\,x^4\,\ln \relax (x)}{8\,x^2+1}}{16\,x^4+8\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2}-\frac {\frac {x^4}{8}+\frac {3\,x^2}{64}+\frac {1}{512}}{x^6+\frac {3\,x^4}{8}+\frac {3\,x^2}{64}+\frac {1}{512}}+\frac {\frac {32\,x^4}{{\left (8\,x^2+1\right )}^3}-\frac {128\,x^4\,\ln \relax (x)\,\left (4\,x^2+1\right )}{{\left (8\,x^2+1\right )}^3}}{\ln \relax (x)+4\,x^2}-\frac {x^5\,\ln \left (\ln \left (\ln \left (\ln \left (\ln \left (\frac {4}{x}\right )\right )\right )\right )\right )}{16\,x^5+8\,x^3\,\ln \relax (x)+x\,{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*log(x) + 4*x^5 - log(log(4/x))*log(log(log(4/x)))*log(log(log(log(4/x))))*(32*x^3*log(4/x) - 64*x^3*l
og(4/x)*log(x)) + log(log(4/x))*log(log(log(log(log(4/x)))))*log(log(log(4/x)))*log(log(log(log(4/x))))*(2*x^3
*log(4/x) - 4*x^3*log(4/x)*log(x)))/(log(log(4/x))*log(log(log(4/x)))*log(log(log(log(4/x))))*(log(4/x)*log(x)
^3 + 64*x^6*log(4/x) + 12*x^2*log(4/x)*log(x)^2 + 48*x^4*log(4/x)*log(x))),x)

[Out]

((16*x^4)/(8*x^2 + 1) - (32*x^4*log(x))/(8*x^2 + 1))/(8*x^2*log(x) + log(x)^2 + 16*x^4) - ((3*x^2)/64 + x^4/8
+ 1/512)/((3*x^2)/64 + (3*x^4)/8 + x^6 + 1/512) + ((32*x^4)/(8*x^2 + 1)^3 - (128*x^4*log(x)*(4*x^2 + 1))/(8*x^
2 + 1)^3)/(log(x) + 4*x^2) - (x^5*log(log(log(log(log(4/x))))))/(x*log(x)^2 + 8*x^3*log(x) + 16*x^5)

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sympy [B]  time = 23.31, size = 58, normalized size = 2.32 \begin {gather*} - \frac {x^{4} \log {\left (\log {\left (\log {\left (\log {\left (- \log {\relax (x )} + \log {\relax (4 )} \right )} \right )} \right )} \right )}}{16 x^{4} + 8 x^{2} \log {\relax (x )} + \log {\relax (x )}^{2}} + \frac {16 x^{4}}{16 x^{4} + 8 x^{2} \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3*ln(4/x)*ln(x)+2*x**3*ln(4/x))*ln(ln(4/x))*ln(ln(ln(4/x)))*ln(ln(ln(ln(4/x))))*ln(ln(ln(ln(
ln(4/x)))))+(64*x**3*ln(4/x)*ln(x)-32*x**3*ln(4/x))*ln(ln(4/x))*ln(ln(ln(4/x)))*ln(ln(ln(ln(4/x))))+x**3*ln(x)
+4*x**5)/(ln(4/x)*ln(x)**3+12*x**2*ln(4/x)*ln(x)**2+48*x**4*ln(4/x)*ln(x)+64*x**6*ln(4/x))/ln(ln(4/x))/ln(ln(l
n(4/x)))/ln(ln(ln(ln(4/x)))),x)

[Out]

-x**4*log(log(log(log(-log(x) + log(4)))))/(16*x**4 + 8*x**2*log(x) + log(x)**2) + 16*x**4/(16*x**4 + 8*x**2*l
og(x) + log(x)**2)

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