3.21.76 \(\int \frac {e^{\frac {25+10 x^3 \log (x)+(-5 x^4+x^6) \log ^2(x)}{x^4 \log ^2(x)}} (-50+(-100-10 x^3) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x))}{(-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+(-5 x^4+x^6) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log (5-e^{\frac {25+10 x^3 \log (x)+(-5 x^4+x^6) \log ^2(x)}{x^4 \log ^2(x)}}) (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+(-5 x^4+x^6) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x))) \log (20+\log (5-e^{\frac {25+10 x^3 \log (x)+(-5 x^4+x^6) \log ^2(x)}{x^4 \log ^2(x)}}))} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\log \left (20+\log \left (5-e^{-5+\left (x+\frac {5}{x^2 \log (x)}\right )^2}\right )\right )\right ) \]

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Rubi [F]  time = 26.22, 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 {\exp \left (\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}\right ) \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 \exp \left (\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}\right ) x^5 \log ^3(x)+\log \left (5-\exp \left (\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}\right )\right ) \left (-5 x^5 \log ^3(x)+\exp \left (\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}\right ) x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-\exp \left (\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*(-50 + (-100 - 10*x^3)*Log[x] - 10*x^3*
Log[x]^2 + 2*x^6*Log[x]^3))/((-100*x^5*Log[x]^3 + 20*E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Lo
g[x]^2))*x^5*Log[x]^3 + Log[5 - E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))]*(-5*x^5*Log
[x]^3 + E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*x^5*Log[x]^3))*Log[20 + Log[5 - E^((
25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))]]),x]

[Out]

2*Defer[Int][(E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2))*x)/((-5*E^5 + E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2)))*(20 +
Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))])*Log[20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^
2))]]), x] - 50*Defer[Int][E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2))/((-5*E^5 + E^((5 + x^3*Log[x])^2/(x^4*Log[x]^
2)))*x^5*(20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))])*Log[x]^3*Log[20 + Log[5 - E^(-5 + (5 + x^3
*Log[x])^2/(x^4*Log[x]^2))]]), x] - 100*Defer[Int][E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2))/((-5*E^5 + E^((5 + x^
3*Log[x])^2/(x^4*Log[x]^2)))*x^5*(20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))])*Log[x]^2*Log[20 +
Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))]]), x] - 10*Defer[Int][E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2)
)/((-5*E^5 + E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2)))*x^2*(20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2
))])*Log[x]^2*Log[20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))]]), x] - 10*Defer[Int][E^((5 + x^3*L
og[x])^2/(x^4*Log[x]^2))/((-5*E^5 + E^((5 + x^3*Log[x])^2/(x^4*Log[x]^2)))*x^2*(20 + Log[5 - E^(-5 + (5 + x^3*
Log[x])^2/(x^4*Log[x]^2))])*Log[x]*Log[20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}} \left (25+5 \left (10+x^3\right ) \log (x)+5 x^3 \log ^2(x)-x^6 \log ^3(x)\right )}{\left (5 e^5-e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^3(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx\\ &=2 \int \frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}} \left (25+5 \left (10+x^3\right ) \log (x)+5 x^3 \log ^2(x)-x^6 \log ^3(x)\right )}{\left (5 e^5-e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^3(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx\\ &=2 \int \left (\frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}} x}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )}-\frac {25 e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^3(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )}-\frac {50 e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^2(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )}-\frac {5 e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^2 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^2(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )}-\frac {5 e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^2 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log (x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )}\right ) \, dx\\ &=2 \int \frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}} x}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx-10 \int \frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^2 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^2(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx-10 \int \frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^2 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log (x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx-50 \int \frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^3(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx-100 \int \frac {e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right ) \log ^2(x) \log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 31, normalized size = 1.19 \begin {gather*} \log \left (\log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*(-50 + (-100 - 10*x^3)*Log[x] - 1
0*x^3*Log[x]^2 + 2*x^6*Log[x]^3))/((-100*x^5*Log[x]^3 + 20*E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(
x^4*Log[x]^2))*x^5*Log[x]^3 + Log[5 - E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))]*(-5*x
^5*Log[x]^3 + E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*x^5*Log[x]^3))*Log[20 + Log[5
- E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))]]),x]

[Out]

Log[Log[20 + Log[5 - E^(-5 + (5 + x^3*Log[x])^2/(x^4*Log[x]^2))]]]

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fricas [A]  time = 0.85, size = 41, normalized size = 1.58 \begin {gather*} \log \left (\log \left (\log \left (-e^{\left (\frac {10 \, x^{3} \log \relax (x) + {\left (x^{6} - 5 \, x^{4}\right )} \log \relax (x)^{2} + 25}{x^{4} \log \relax (x)^{2}}\right )} + 5\right ) + 20\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6*log(x)^3-10*x^3*log(x)^2+(-10*x^3-100)*log(x)-50)*exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)
/x^4/log(x)^2)/((x^5*log(x)^3*exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-5*x^5*log(x)^3)*log(-e
xp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20*x^5*log(x)^3*exp(((x^6-5*x^4)*log(x)^2+10*x^3*l
og(x)+25)/x^4/log(x)^2)-100*x^5*log(x)^3)/log(log(-exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5
)+20),x, algorithm="fricas")

[Out]

log(log(log(-e^((10*x^3*log(x) + (x^6 - 5*x^4)*log(x)^2 + 25)/(x^4*log(x)^2)) + 5) + 20))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6*log(x)^3-10*x^3*log(x)^2+(-10*x^3-100)*log(x)-50)*exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)
/x^4/log(x)^2)/((x^5*log(x)^3*exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-5*x^5*log(x)^3)*log(-e
xp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20*x^5*log(x)^3*exp(((x^6-5*x^4)*log(x)^2+10*x^3*l
og(x)+25)/x^4/log(x)^2)-100*x^5*log(x)^3)/log(log(-exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5
)+20),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.06, size = 45, normalized size = 1.73

method	result	size
risch	\(\ln \left (\ln \left (\ln \left (-{\mathrm e}^{\frac {x^{6} \ln \relax (x )^{2}-5 x^{4} \ln \relax (x )^{2}+10 x^{3} \ln \relax (x )+25}{x^{4} \ln \relax (x )^{2}}}+5\right )+20\right )\right )\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6*ln(x)^3-10*x^3*ln(x)^2+(-10*x^3-100)*ln(x)-50)*exp(((x^6-5*x^4)*ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^
2)/((x^5*ln(x)^3*exp(((x^6-5*x^4)*ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)-5*x^5*ln(x)^3)*ln(-exp(((x^6-5*x^4)*ln
(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)+5)+20*x^5*ln(x)^3*exp(((x^6-5*x^4)*ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)-1
00*x^5*ln(x)^3)/ln(ln(-exp(((x^6-5*x^4)*ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)+5)+20),x,method=_RETURNVERBOSE)

[Out]

ln(ln(ln(-exp((x^6*ln(x)^2-5*x^4*ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)+5)+20))

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maxima [A]  time = 0.72, size = 35, normalized size = 1.35 \begin {gather*} \log \left (\log \left (\log \left (5 \, e^{5} - e^{\left (x^{2} + \frac {10}{x \log \relax (x)} + \frac {25}{x^{4} \log \relax (x)^{2}}\right )}\right ) + 15\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6*log(x)^3-10*x^3*log(x)^2+(-10*x^3-100)*log(x)-50)*exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)
/x^4/log(x)^2)/((x^5*log(x)^3*exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-5*x^5*log(x)^3)*log(-e
xp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20*x^5*log(x)^3*exp(((x^6-5*x^4)*log(x)^2+10*x^3*l
og(x)+25)/x^4/log(x)^2)-100*x^5*log(x)^3)/log(log(-exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5
)+20),x, algorithm="maxima")

[Out]

log(log(log(5*e^5 - e^(x^2 + 10/(x*log(x)) + 25/(x^4*log(x)^2))) + 15))

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mupad [B]  time = 3.59, size = 35, normalized size = 1.35 \begin {gather*} \ln \left (\ln \left (\ln \left (5-{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{\frac {10}{x\,\ln \relax (x)}}\,{\mathrm {e}}^{\frac {25}{x^4\,{\ln \relax (x)}^2}}\right )+20\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))*(10*x^3*log(x)^2 - 2*x^6*log(x)^3 + log
(x)*(10*x^3 + 100) + 50))/(log(log(5 - exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))) + 20
)*(log(5 - exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2)))*(5*x^5*log(x)^3 - x^5*exp((10*x^
3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))*log(x)^3) + 100*x^5*log(x)^3 - 20*x^5*exp((10*x^3*log(
x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))*log(x)^3)),x)

[Out]

log(log(log(5 - exp(x^2)*exp(-5)*exp(10/(x*log(x)))*exp(25/(x^4*log(x)^2))) + 20))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**6*ln(x)**3-10*x**3*ln(x)**2+(-10*x**3-100)*ln(x)-50)*exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)
+25)/x**4/ln(x)**2)/((x**5*ln(x)**3*exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)-5*x**5*ln(x)*
*3)*ln(-exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)+5)+20*x**5*ln(x)**3*exp(((x**6-5*x**4)*ln
(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)-100*x**5*ln(x)**3)/ln(ln(-exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25
)/x**4/ln(x)**2)+5)+20),x)

[Out]

Timed out

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