3.62.76 \(\int \frac {8 x^2 \log (x)+e^{\frac {2 (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x)))}{x}} (96 x+(144-48 x^2) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x)))}{e^{\frac {6 (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x)))}{x}} x^2 \log (x)-3 e^{\frac {4 (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x)))}{x}} x^3 \log (x)+3 e^{\frac {2 (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x)))}{x}} x^4 \log (x)-x^5 \log (x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {4}{\left (-e^{\frac {2 \left (-x+3 \left (3+(-x+\log (\log (x)))^2\right )\right )}{x}}+x\right )^2} \]

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Rubi [F]  time = 13.52, 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 {8 x^2 \log (x)+\exp \left (\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}\right ) \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{\exp \left (\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}\right ) x^2 \log (x)-3 \exp \left (\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}\right ) x^3 \log (x)+3 \exp \left (\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}\right ) x^4 \log (x)-x^5 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x^2*Log[x] + E^((2*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*(96*x + (144 - 48*x^2)*Log[x
] - 96*Log[Log[x]] + 48*Log[x]*Log[Log[x]]^2))/(E^((6*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*
x^2*Log[x] - 3*E^((4*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*x^3*Log[x] + 3*E^((2*(9 - x + 3*x
^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*x^4*Log[x] - x^5*Log[x]),x]

[Out]

-96*E^6*Defer[Int][Log[x]^35/(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x) + E^2*x*Log[x]^12)^3, x] - 8*E^6*Defer[Int
][Log[x]^36/(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x) + E^2*x*Log[x]^12)^3, x] - 144*E^6*Defer[Int][Log[x]^36/(x*
(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x) + E^2*x*Log[x]^12)^3), x] + 48*E^6*Defer[Int][(x*Log[x]^36)/(-E^(18/x +
6*x + (6*Log[Log[x]]^2)/x) + E^2*x*Log[x]^12)^3, x] + 96*E^4*Defer[Int][Log[x]^23/(x*(-E^(18/x + 6*x + (6*Log
[Log[x]]^2)/x) + E^2*x*Log[x]^12)^2), x] - 48*E^4*Defer[Int][Log[x]^24/(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x)
+ E^2*x*Log[x]^12)^2, x] + 144*E^4*Defer[Int][Log[x]^24/(x^2*(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x) + E^2*x*Lo
g[x]^12)^2), x] + 96*E^6*Defer[Int][(Log[x]^35*Log[Log[x]])/(x*(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x) + E^2*x*
Log[x]^12)^3), x] - 96*E^4*Defer[Int][(Log[x]^23*Log[Log[x]])/(x^2*(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x) + E^
2*x*Log[x]^12)^2), x] - 48*E^6*Defer[Int][(Log[x]^36*Log[Log[x]]^2)/(x*(-E^(18/x + 6*x + (6*Log[Log[x]]^2)/x)
+ E^2*x*Log[x]^12)^3), x] + 48*E^4*Defer[Int][(Log[x]^24*Log[Log[x]]^2)/(x^2*(-E^(18/x + 6*x + (6*Log[Log[x]]^
2)/x) + E^2*x*Log[x]^12)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^4 \log ^{23}(x) \left (e^2 x^2 \log ^{13}(x)+12 e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}} (x-\log (\log (x)))-6 e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}} \log (x) \left (-3+x^2-\log ^2(\log (x))\right )\right )}{x^2 \left (e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}}-e^2 x \log ^{12}(x)\right )^3} \, dx\\ &=\left (8 e^4\right ) \int \frac {\log ^{23}(x) \left (e^2 x^2 \log ^{13}(x)+12 e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}} (x-\log (\log (x)))-6 e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}} \log (x) \left (-3+x^2-\log ^2(\log (x))\right )\right )}{x^2 \left (e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}}-e^2 x \log ^{12}(x)\right )^3} \, dx\\ &=\left (8 e^4\right ) \int \left (\frac {e^2 \log ^{35}(x) \left (-12 x-18 \log (x)-x \log (x)+6 x^2 \log (x)+12 \log (\log (x))-6 \log (x) \log ^2(\log (x))\right )}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}-\frac {6 \log ^{23}(x) \left (-2 x-3 \log (x)+x^2 \log (x)+2 \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2}\right ) \, dx\\ &=-\left (\left (48 e^4\right ) \int \frac {\log ^{23}(x) \left (-2 x-3 \log (x)+x^2 \log (x)+2 \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2} \, dx\right )+\left (8 e^6\right ) \int \frac {\log ^{35}(x) \left (-12 x-18 \log (x)-x \log (x)+6 x^2 \log (x)+12 \log (\log (x))-6 \log (x) \log ^2(\log (x))\right )}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx\\ &=-\left (\left (48 e^4\right ) \int \left (-\frac {2 \log ^{23}(x)}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2}+\frac {\log ^{24}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2}-\frac {3 \log ^{24}(x)}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2}+\frac {2 \log ^{23}(x) \log (\log (x))}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2}-\frac {\log ^{24}(x) \log ^2(\log (x))}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2}\right ) \, dx\right )+\left (8 e^6\right ) \int \left (-\frac {12 \log ^{35}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}-\frac {\log ^{36}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}-\frac {18 \log ^{36}(x)}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}+\frac {6 x \log ^{36}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}+\frac {12 \log ^{35}(x) \log (\log (x))}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}-\frac {6 \log ^{36}(x) \log ^2(\log (x))}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3}\right ) \, dx\\ &=-\left (\left (48 e^4\right ) \int \frac {\log ^{24}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2} \, dx\right )+\left (48 e^4\right ) \int \frac {\log ^{24}(x) \log ^2(\log (x))}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2} \, dx+\left (96 e^4\right ) \int \frac {\log ^{23}(x)}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2} \, dx-\left (96 e^4\right ) \int \frac {\log ^{23}(x) \log (\log (x))}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2} \, dx+\left (144 e^4\right ) \int \frac {\log ^{24}(x)}{x^2 \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^2} \, dx-\left (8 e^6\right ) \int \frac {\log ^{36}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx+\left (48 e^6\right ) \int \frac {x \log ^{36}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx-\left (48 e^6\right ) \int \frac {\log ^{36}(x) \log ^2(\log (x))}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx-\left (96 e^6\right ) \int \frac {\log ^{35}(x)}{\left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx+\left (96 e^6\right ) \int \frac {\log ^{35}(x) \log (\log (x))}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx-\left (144 e^6\right ) \int \frac {\log ^{36}(x)}{x \left (-e^{\frac {18}{x}+6 x+\frac {6 \log ^2(\log (x))}{x}}+e^2 x \log ^{12}(x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 39, normalized size = 1.22 \begin {gather*} \frac {4 e^4 \log ^{24}(x)}{\left (e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}}-e^2 x \log ^{12}(x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^2*Log[x] + E^((2*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*(96*x + (144 - 48*x^2)
*Log[x] - 96*Log[Log[x]] + 48*Log[x]*Log[Log[x]]^2))/(E^((6*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2
))/x)*x^2*Log[x] - 3*E^((4*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*x^3*Log[x] + 3*E^((2*(9 - x
+ 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*x^4*Log[x] - x^5*Log[x]),x]

[Out]

(4*E^4*Log[x]^24)/(E^((6*(3 + x^2 + Log[Log[x]]^2))/x) - E^2*x*Log[x]^12)^2

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fricas [B]  time = 0.91, size = 69, normalized size = 2.16 \begin {gather*} \frac {4}{x^{2} - 2 \, x e^{\left (\frac {2 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \relax (x)\right ) + 3 \, \log \left (\log \relax (x)\right )^{2} - x + 9\right )}}{x}\right )} + e^{\left (\frac {4 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \relax (x)\right ) + 3 \, \log \left (\log \relax (x)\right )^{2} - x + 9\right )}}{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96*x)*exp((3*log(log(x))^2-6*x*log(log
(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^2*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(
x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^
2-x+9)/x)^2-x^5*log(x)),x, algorithm="fricas")

[Out]

4/(x^2 - 2*x*e^(2*(3*x^2 - 6*x*log(log(x)) + 3*log(log(x))^2 - x + 9)/x) + e^(4*(3*x^2 - 6*x*log(log(x)) + 3*l
og(log(x))^2 - x + 9)/x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96*x)*exp((3*log(log(x))^2-6*x*log(log
(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^2*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(
x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^
2-x+9)/x)^2-x^5*log(x)),x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.06, size = 36, normalized size = 1.12

method	result	size
risch	\(\frac {4}{\left (-\frac {{\mathrm e}^{\frac {6 \ln \left (\ln \relax (x )\right )^{2}+6 x^{2}-2 x +18}{x}}}{\ln \relax (x )^{12}}+x \right )^{2}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((48*ln(x)*ln(ln(x))^2-96*ln(ln(x))+(-48*x^2+144)*ln(x)+96*x)*exp((3*ln(ln(x))^2-6*x*ln(ln(x))+3*x^2-x+9)/
x)^2+8*x^2*ln(x))/(x^2*ln(x)*exp((3*ln(ln(x))^2-6*x*ln(ln(x))+3*x^2-x+9)/x)^6-3*x^3*ln(x)*exp((3*ln(ln(x))^2-6
*x*ln(ln(x))+3*x^2-x+9)/x)^4+3*x^4*ln(x)*exp((3*ln(ln(x))^2-6*x*ln(ln(x))+3*x^2-x+9)/x)^2-x^5*ln(x)),x,method=
_RETURNVERBOSE)

[Out]

4/(-1/ln(x)^12*exp(2*(3*ln(ln(x))^2+3*x^2-x+9)/x)+x)^2

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maxima [B]  time = 0.61, size = 69, normalized size = 2.16 \begin {gather*} \frac {4 \, e^{4} \log \relax (x)^{24}}{x^{2} e^{4} \log \relax (x)^{24} - 2 \, x e^{\left (6 \, x + \frac {6 \, \log \left (\log \relax (x)\right )^{2}}{x} + \frac {18}{x} + 2\right )} \log \relax (x)^{12} + e^{\left (12 \, x + \frac {12 \, \log \left (\log \relax (x)\right )^{2}}{x} + \frac {36}{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96*x)*exp((3*log(log(x))^2-6*x*log(log
(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^2*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(
x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^
2-x+9)/x)^2-x^5*log(x)),x, algorithm="maxima")

[Out]

4*e^4*log(x)^24/(x^2*e^4*log(x)^24 - 2*x*e^(6*x + 6*log(log(x))^2/x + 18/x + 2)*log(x)^12 + e^(12*x + 12*log(l
og(x))^2/x + 36/x))

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mupad [B]  time = 4.76, size = 66, normalized size = 2.06 \begin {gather*} \frac {4}{x^2+\frac {{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{\frac {12\,{\ln \left (\ln \relax (x)\right )}^2}{x}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{36/x}}{{\ln \relax (x)}^{24}}-\frac {2\,x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{\frac {6\,{\ln \left (\ln \relax (x)\right )}^2}{x}}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{18/x}}{{\ln \relax (x)}^{12}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^2*log(x) + exp((2*(3*log(log(x))^2 - 6*x*log(log(x)) - x + 3*x^2 + 9))/x)*(96*x - 96*log(log(x)) - l
og(x)*(48*x^2 - 144) + 48*log(log(x))^2*log(x)))/(x^5*log(x) - 3*x^4*exp((2*(3*log(log(x))^2 - 6*x*log(log(x))
- x + 3*x^2 + 9))/x)*log(x) + 3*x^3*exp((4*(3*log(log(x))^2 - 6*x*log(log(x)) - x + 3*x^2 + 9))/x)*log(x) - x
^2*exp((6*(3*log(log(x))^2 - 6*x*log(log(x)) - x + 3*x^2 + 9))/x)*log(x)),x)

[Out]

4/(x^2 + (exp(12*x)*exp((12*log(log(x))^2)/x)*exp(-4)*exp(36/x))/log(x)^24 - (2*x*exp(6*x)*exp((6*log(log(x))^
2)/x)*exp(-2)*exp(18/x))/log(x)^12)

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sympy [B]  time = 0.63, size = 68, normalized size = 2.12 \begin {gather*} \frac {4}{x^{2} - 2 x e^{\frac {2 \left (3 x^{2} - 6 x \log {\left (\log {\relax (x )} \right )} - x + 3 \log {\left (\log {\relax (x )} \right )}^{2} + 9\right )}{x}} + e^{\frac {4 \left (3 x^{2} - 6 x \log {\left (\log {\relax (x )} \right )} - x + 3 \log {\left (\log {\relax (x )} \right )}^{2} + 9\right )}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*ln(x)*ln(ln(x))**2-96*ln(ln(x))+(-48*x**2+144)*ln(x)+96*x)*exp((3*ln(ln(x))**2-6*x*ln(ln(x))+3*
x**2-x+9)/x)**2+8*x**2*ln(x))/(x**2*ln(x)*exp((3*ln(ln(x))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)**6-3*x**3*ln(x)*exp
((3*ln(ln(x))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)**4+3*x**4*ln(x)*exp((3*ln(ln(x))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)
**2-x**5*ln(x)),x)

[Out]

4/(x**2 - 2*x*exp(2*(3*x**2 - 6*x*log(log(x)) - x + 3*log(log(x))**2 + 9)/x) + exp(4*(3*x**2 - 6*x*log(log(x))
- x + 3*log(log(x))**2 + 9)/x))

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