3.95.83 \(\int \frac {-16-8 x^2 \log (x)+(-8 x^2 \log (x)-32 \log (x) \log (\log (x))) \log (\frac {1}{4} (x^2+4 \log (\log (x))))}{(9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))) \log ^3(\frac {1}{4} (x^2+4 \log (\log (x))))} \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{9 x^4 \log ^2\left (\frac {x^2}{4}+\log (\log (x))\right )} \]

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Rubi [F]  time = 2.28, 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 {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16 - 8*x^2*Log[x] + (-8*x^2*Log[x] - 32*Log[x]*Log[Log[x]])*Log[(x^2 + 4*Log[Log[x]])/4])/((9*x^7*Log[x]
+ 36*x^5*Log[x]*Log[Log[x]])*Log[(x^2 + 4*Log[Log[x]])/4]^3),x]

[Out]

(8*Defer[Int][1/(x^3*(-x^2 - 4*Log[Log[x]])*Log[(x^2 + 4*Log[Log[x]])/4]^3), x])/9 + (16*Defer[Int][1/(x^5*Log
[x]*(-x^2 - 4*Log[Log[x]])*Log[(x^2 + 4*Log[Log[x]])/4]^3), x])/9 - (8*Defer[Int][1/(x^5*Log[(x^2 + 4*Log[Log[
x]])/4]^2), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{9 x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {1}{9} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {1}{9} \int \left (\frac {8 \left (-2-x^2 \log (x)\right )}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}-\frac {8}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}\right ) \, dx\\ &=\frac {8}{9} \int \frac {-2-x^2 \log (x)}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx-\frac {8}{9} \int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {8}{9} \int \left (\frac {1}{x^3 \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}+\frac {2}{x^5 \log (x) \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}\right ) \, dx-\frac {8}{9} \int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ &=\frac {8}{9} \int \frac {1}{x^3 \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx-\frac {8}{9} \int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx+\frac {16}{9} \int \frac {1}{x^5 \log (x) \left (-x^2-4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{9 x^4 \log ^2\left (\frac {x^2}{4}+\log (\log (x))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 8*x^2*Log[x] + (-8*x^2*Log[x] - 32*Log[x]*Log[Log[x]])*Log[(x^2 + 4*Log[Log[x]])/4])/((9*x^7*
Log[x] + 36*x^5*Log[x]*Log[Log[x]])*Log[(x^2 + 4*Log[Log[x]])/4]^3),x]

[Out]

2/(9*x^4*Log[x^2/4 + Log[Log[x]]]^2)

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fricas [A]  time = 0.57, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{9 \, x^{4} \log \left (\frac {1}{4} \, x^{2} + \log \left (\log \relax (x)\right )\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*log(x)*log(log(x))-8*x^2*log(x))*log(log(log(x))+1/4*x^2)-8*x^2*log(x)-16)/(36*x^5*log(x)*log(
log(x))+9*x^7*log(x))/log(log(log(x))+1/4*x^2)^3,x, algorithm="fricas")

[Out]

2/9/(x^4*log(1/4*x^2 + log(log(x)))^2)

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giac [B]  time = 0.23, size = 104, normalized size = 4.95 \begin {gather*} \frac {2 \, {\left (x^{2} \log \relax (x) + 2\right )}}{9 \, {\left (4 \, x^{6} \log \relax (2)^{2} \log \relax (x) - 4 \, x^{6} \log \relax (2) \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right ) \log \relax (x) + x^{6} \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right )^{2} \log \relax (x) + 8 \, x^{4} \log \relax (2)^{2} - 8 \, x^{4} \log \relax (2) \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right ) + 2 \, x^{4} \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*log(x)*log(log(x))-8*x^2*log(x))*log(log(log(x))+1/4*x^2)-8*x^2*log(x)-16)/(36*x^5*log(x)*log(
log(x))+9*x^7*log(x))/log(log(log(x))+1/4*x^2)^3,x, algorithm="giac")

[Out]

2/9*(x^2*log(x) + 2)/(4*x^6*log(2)^2*log(x) - 4*x^6*log(2)*log(x^2 + 4*log(log(x)))*log(x) + x^6*log(x^2 + 4*l
og(log(x)))^2*log(x) + 8*x^4*log(2)^2 - 8*x^4*log(2)*log(x^2 + 4*log(log(x))) + 2*x^4*log(x^2 + 4*log(log(x)))
^2)

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maple [A]  time = 0.25, size = 18, normalized size = 0.86

method	result	size
risch	\(\frac {2}{9 x^{4} \ln \left (\ln \left (\ln \relax (x )\right )+\frac {x^{2}}{4}\right )^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*ln(x)*ln(ln(x))-8*x^2*ln(x))*ln(ln(ln(x))+1/4*x^2)-8*x^2*ln(x)-16)/(36*x^5*ln(x)*ln(ln(x))+9*x^7*ln(
x))/ln(ln(ln(x))+1/4*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

2/9/x^4/ln(ln(ln(x))+1/4*x^2)^2

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maxima [B]  time = 0.49, size = 47, normalized size = 2.24 \begin {gather*} \frac {2}{9 \, {\left (4 \, x^{4} \log \relax (2)^{2} - 4 \, x^{4} \log \relax (2) \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right ) + x^{4} \log \left (x^{2} + 4 \, \log \left (\log \relax (x)\right )\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*log(x)*log(log(x))-8*x^2*log(x))*log(log(log(x))+1/4*x^2)-8*x^2*log(x)-16)/(36*x^5*log(x)*log(
log(x))+9*x^7*log(x))/log(log(log(x))+1/4*x^2)^3,x, algorithm="maxima")

[Out]

2/9/(4*x^4*log(2)^2 - 4*x^4*log(2)*log(x^2 + 4*log(log(x))) + x^4*log(x^2 + 4*log(log(x)))^2)

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mupad [B]  time = 10.76, size = 519, normalized size = 24.71 \begin {gather*} \frac {4}{9\,x^4}-\frac {\frac {2\,\ln \relax (x)\,\left (4\,\ln \left (\ln \relax (x)\right )+x^2\right )}{9\,x^4\,\left (x^2\,\ln \relax (x)+2\right )}-\frac {2\,\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )\,\ln \relax (x)\,\left (4\,\ln \left (\ln \relax (x)\right )+x^2\right )\,\left (4\,\ln \left (\ln \relax (x)\right )-2\,x^4\,{\ln \relax (x)}^2-16\,\ln \left (\ln \relax (x)\right )\,\ln \relax (x)+x^2-12\,x^2\,\ln \left (\ln \relax (x)\right )\,{\ln \relax (x)}^2+4\right )}{9\,x^4\,{\left (x^2\,\ln \relax (x)+2\right )}^3}}{\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )}+\frac {\frac {2}{9\,x^4}+\frac {2\,\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )\,\ln \relax (x)\,\left (4\,\ln \left (\ln \relax (x)\right )+x^2\right )}{9\,x^4\,\left (x^2\,\ln \relax (x)+2\right )}}{{\ln \left (\ln \left (\ln \relax (x)\right )+\frac {x^2}{4}\right )}^2}-\frac {\ln \left (\ln \relax (x)\right )\,\left (\ln \relax (x)\,\left (x^2\,\left (\frac {16\,\left (x^2-20\right )}{9\,x^6}+\frac {416}{9\,x^6}\right )-\frac {64}{9\,x^4}\right )-\frac {40\,{\ln \relax (x)}^3}{9}-\frac {32\,\left (x^2-4\right )}{9\,x^6}+\frac {32\,\left (x^2-20\right )}{9\,x^6}-\frac {32\,{\ln \relax (x)}^2}{9\,x^2}+\frac {512}{9\,x^6}\right )}{x^6\,{\ln \relax (x)}^3+6\,x^4\,{\ln \relax (x)}^2+12\,x^2\,\ln \relax (x)+8}-\frac {{\ln \left (\ln \relax (x)\right )}^2\,\left (\ln \relax (x)\,\left (x^2\,\left (\frac {32\,\left (x^2-20\right )}{9\,x^8}+\frac {896}{9\,x^8}\right )-\frac {256}{9\,x^6}\right )-\frac {64\,\left (x^2-4\right )}{9\,x^8}+\frac {64\,\left (x^2-20\right )}{9\,x^8}-\frac {32\,{\ln \relax (x)}^3}{3\,x^2}-\frac {128\,{\ln \relax (x)}^2}{9\,x^4}+\frac {1024}{9\,x^8}\right )}{x^6\,{\ln \relax (x)}^3+6\,x^4\,{\ln \relax (x)}^2+12\,x^2\,\ln \relax (x)+8}+\frac {8\,\left (x^2-4\right )}{3\,x^3\,\left (4\,x-x^3\right )\,\left (x^2\,\ln \relax (x)+2\right )}-\frac {4\,\left (x^5-8\,x^3+16\,x\right )}{9\,x^4\,\left (4\,x-x^3\right )\,\left (x^6\,{\ln \relax (x)}^3+6\,x^4\,{\ln \relax (x)}^2+12\,x^2\,\ln \relax (x)+8\right )}+\frac {2\,\left (x^4-24\,x^2+80\right )}{9\,x^3\,\left (4\,x-x^3\right )\,\left (x^4\,{\ln \relax (x)}^2+4\,x^2\,\ln \relax (x)+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^2*log(x) + log(log(log(x)) + x^2/4)*(8*x^2*log(x) + 32*log(log(x))*log(x)) + 16)/(log(log(log(x)) +
x^2/4)^3*(9*x^7*log(x) + 36*x^5*log(log(x))*log(x))),x)

[Out]

4/(9*x^4) - ((2*log(x)*(4*log(log(x)) + x^2))/(9*x^4*(x^2*log(x) + 2)) - (2*log(log(log(x)) + x^2/4)*log(x)*(4
*log(log(x)) + x^2)*(4*log(log(x)) - 2*x^4*log(x)^2 - 16*log(log(x))*log(x) + x^2 - 12*x^2*log(log(x))*log(x)^
2 + 4))/(9*x^4*(x^2*log(x) + 2)^3))/log(log(log(x)) + x^2/4) + (2/(9*x^4) + (2*log(log(log(x)) + x^2/4)*log(x)
*(4*log(log(x)) + x^2))/(9*x^4*(x^2*log(x) + 2)))/log(log(log(x)) + x^2/4)^2 - (log(log(x))*(log(x)*(x^2*((16*
(x^2 - 20))/(9*x^6) + 416/(9*x^6)) - 64/(9*x^4)) - (40*log(x)^3)/9 - (32*(x^2 - 4))/(9*x^6) + (32*(x^2 - 20))/
(9*x^6) - (32*log(x)^2)/(9*x^2) + 512/(9*x^6)))/(12*x^2*log(x) + 6*x^4*log(x)^2 + x^6*log(x)^3 + 8) - (log(log
(x))^2*(log(x)*(x^2*((32*(x^2 - 20))/(9*x^8) + 896/(9*x^8)) - 256/(9*x^6)) - (64*(x^2 - 4))/(9*x^8) + (64*(x^2
- 20))/(9*x^8) - (32*log(x)^3)/(3*x^2) - (128*log(x)^2)/(9*x^4) + 1024/(9*x^8)))/(12*x^2*log(x) + 6*x^4*log(x
)^2 + x^6*log(x)^3 + 8) + (8*(x^2 - 4))/(3*x^3*(4*x - x^3)*(x^2*log(x) + 2)) - (4*(16*x - 8*x^3 + x^5))/(9*x^4
*(4*x - x^3)*(12*x^2*log(x) + 6*x^4*log(x)^2 + x^6*log(x)^3 + 8)) + (2*(x^4 - 24*x^2 + 80))/(9*x^3*(4*x - x^3)
*(4*x^2*log(x) + x^4*log(x)^2 + 4))

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sympy [A]  time = 0.48, size = 19, normalized size = 0.90 \begin {gather*} \frac {2}{9 x^{4} \log {\left (\frac {x^{2}}{4} + \log {\left (\log {\relax (x )} \right )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*ln(x)*ln(ln(x))-8*x**2*ln(x))*ln(ln(ln(x))+1/4*x**2)-8*x**2*ln(x)-16)/(36*x**5*ln(x)*ln(ln(x))
+9*x**7*ln(x))/ln(ln(ln(x))+1/4*x**2)**3,x)

[Out]

2/(9*x**4*log(x**2/4 + log(log(x)))**2)

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