\(\int \frac {e^{\frac {1}{2} (-1-(-10 x^2+2 x^3) \log (\log (\log (x))))} (5 x-x^2+(10 x-3 x^2) \log (x) \log (\log (x)) \log (\log (\log (x))))}{20 \log (x) \log (\log (x))} \, dx\) [3773]

Optimal result

Integrand size = 66, antiderivative size = 22 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {1}{20} e^{-\frac {1}{2}-(-5+x) x^2 \log (\log (\log (x)))} \]

[Out]

1/20/exp(1/2+ln(ln(ln(x)))*x^2*(-5+x))

Rubi [F]

\[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\int \frac {\exp \left (\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )\right ) \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx \]

[In]

Int[(E^((-1 - (-10*x^2 + 2*x^3)*Log[Log[Log[x]]])/2)*(5*x - x^2 + (10*x - 3*x^2)*Log[x]*Log[Log[x]]*Log[Log[Lo
g[x]]]))/(20*Log[x]*Log[Log[x]]),x]

[Out]

Defer[Int][(x*Log[Log[x]]^(-1 + 5*x^2 - x^3))/Log[x], x]/(4*Sqrt[E]) - Defer[Int][(x^2*Log[Log[x]]^(-1 + 5*x^2
- x^3))/Log[x], x]/(20*Sqrt[E]) + Defer[Int][x*Log[Log[x]]^((5 - x)*x^2)*Log[Log[Log[x]]], x]/(2*Sqrt[E]) - (
3*Defer[Int][x^2*Log[Log[x]]^((5 - x)*x^2)*Log[Log[Log[x]]], x])/(20*Sqrt[E])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{20} \int \frac {\exp \left (\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )\right ) \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{\log (x) \log (\log (x))} \, dx \\ & = \frac {1}{20} \int \frac {x \log ^{-1+5 x^2-x^3}(\log (x)) (5-x-(-10+3 x) \log (x) \log (\log (x)) \log (\log (\log (x))))}{\sqrt {e} \log (x)} \, dx \\ & = \frac {\int \frac {x \log ^{-1+5 x^2-x^3}(\log (x)) (5-x-(-10+3 x) \log (x) \log (\log (x)) \log (\log (\log (x))))}{\log (x)} \, dx}{20 \sqrt {e}} \\ & = \frac {\int \left (-\frac {(-5+x) x \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)}-x (-10+3 x) \log ^{5 x^2-x^3}(\log (x)) \log (\log (\log (x)))\right ) \, dx}{20 \sqrt {e}} \\ & = -\frac {\int \frac {(-5+x) x \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)} \, dx}{20 \sqrt {e}}-\frac {\int x (-10+3 x) \log ^{5 x^2-x^3}(\log (x)) \log (\log (\log (x))) \, dx}{20 \sqrt {e}} \\ & = -\frac {\int \left (-\frac {5 x \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)}+\frac {x^2 \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)}\right ) \, dx}{20 \sqrt {e}}-\frac {\int x (-10+3 x) \log ^{(5-x) x^2}(\log (x)) \log (\log (\log (x))) \, dx}{20 \sqrt {e}} \\ & = -\frac {\int \frac {x^2 \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)} \, dx}{20 \sqrt {e}}-\frac {\int \left (-10 x \log ^{(5-x) x^2}(\log (x)) \log (\log (\log (x)))+3 x^2 \log ^{(5-x) x^2}(\log (x)) \log (\log (\log (x)))\right ) \, dx}{20 \sqrt {e}}+\frac {\int \frac {x \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)} \, dx}{4 \sqrt {e}} \\ & = -\frac {\int \frac {x^2 \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)} \, dx}{20 \sqrt {e}}-\frac {3 \int x^2 \log ^{(5-x) x^2}(\log (x)) \log (\log (\log (x))) \, dx}{20 \sqrt {e}}+\frac {\int \frac {x \log ^{-1+5 x^2-x^3}(\log (x))}{\log (x)} \, dx}{4 \sqrt {e}}+\frac {\int x \log ^{(5-x) x^2}(\log (x)) \log (\log (\log (x))) \, dx}{2 \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {\log ^{-\left ((-5+x) x^2\right )}(\log (x))}{20 \sqrt {e}} \]

[In]

Integrate[(E^((-1 - (-10*x^2 + 2*x^3)*Log[Log[Log[x]]])/2)*(5*x - x^2 + (10*x - 3*x^2)*Log[x]*Log[Log[x]]*Log[
Log[Log[x]]]))/(20*Log[x]*Log[Log[x]]),x]

[Out]

1/(20*Sqrt[E]*Log[Log[x]]^((-5 + x)*x^2))

Maple [A] (verified)

Time = 4.69 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

method	result	size
risch	\(\frac {\ln \left (\ln \left (x \right )\right )^{-x^{2} \left (-5+x \right )} {\mathrm e}^{-\frac {1}{2}}}{20}\)	\(18\)
parallelrisch	\(\frac {{\mathrm e}^{\left (-x^{3}+5 x^{2}\right ) \ln \left (\ln \left (\ln \left (x \right )\right )\right )-\frac {1}{2}}}{20}\)	\(25\)

[In]

int(1/20*((-3*x^2+10*x)*ln(x)*ln(ln(x))*ln(ln(ln(x)))-x^2+5*x)/ln(x)/ln(ln(x))/exp(1/2*(2*x^3-10*x^2)*ln(ln(ln
(x)))+1/2),x,method=_RETURNVERBOSE)

[Out]

1/20/(ln(ln(x))^(x^2*(-5+x)))*exp(-1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {1}{20} \, e^{\left (-{\left (x^{3} - 5 \, x^{2}\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) - \frac {1}{2}\right )} \]

[In]

integrate(1/20*((-3*x^2+10*x)*log(x)*log(log(x))*log(log(log(x)))-x^2+5*x)/log(x)/log(log(x))/exp(1/2*(2*x^3-1
0*x^2)*log(log(log(x)))+1/2),x, algorithm="fricas")

[Out]

1/20*e^(-(x^3 - 5*x^2)*log(log(log(x))) - 1/2)

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {e^{- \left (x^{3} - 5 x^{2}\right ) \log {\left (\log {\left (\log {\left (x \right )} \right )} \right )} - \frac {1}{2}}}{20} \]

[In]

integrate(1/20*((-3*x**2+10*x)*ln(x)*ln(ln(x))*ln(ln(ln(x)))-x**2+5*x)/ln(x)/ln(ln(x))/exp(1/2*(2*x**3-10*x**2
)*ln(ln(ln(x)))+1/2),x)

[Out]

exp(-(x**3 - 5*x**2)*log(log(log(x))) - 1/2)/20

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {1}{20} \, e^{\left (-x^{3} \log \left (\log \left (\log \left (x\right )\right )\right ) + 5 \, x^{2} \log \left (\log \left (\log \left (x\right )\right )\right ) - \frac {1}{2}\right )} \]

[In]

integrate(1/20*((-3*x^2+10*x)*log(x)*log(log(x))*log(log(log(x)))-x^2+5*x)/log(x)/log(log(x))/exp(1/2*(2*x^3-1
0*x^2)*log(log(log(x)))+1/2),x, algorithm="maxima")

[Out]

1/20*e^(-x^3*log(log(log(x))) + 5*x^2*log(log(log(x))) - 1/2)

Giac [A] (verification not implemented)

none

Time = 1.71 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {1}{20} \, e^{\left (-x^{3} \log \left (\log \left (\log \left (x\right )\right )\right ) + 5 \, x^{2} \log \left (\log \left (\log \left (x\right )\right )\right ) - \frac {1}{2}\right )} \]

[In]

integrate(1/20*((-3*x^2+10*x)*log(x)*log(log(x))*log(log(log(x)))-x^2+5*x)/log(x)/log(log(x))/exp(1/2*(2*x^3-1
0*x^2)*log(log(log(x)))+1/2),x, algorithm="giac")

[Out]

1/20*e^(-x^3*log(log(log(x))) + 5*x^2*log(log(log(x))) - 1/2)

Mupad [B] (verification not implemented)

Time = 9.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {1}{2} \left (-1-\left (-10 x^2+2 x^3\right ) \log (\log (\log (x)))\right )} \left (5 x-x^2+\left (10 x-3 x^2\right ) \log (x) \log (\log (x)) \log (\log (\log (x)))\right )}{20 \log (x) \log (\log (x))} \, dx=\frac {{\ln \left (\ln \left (x\right )\right )}^{5\,x^2-x^3}\,{\mathrm {e}}^{-\frac {1}{2}}}{20} \]

[In]

int((exp((log(log(log(x)))*(10*x^2 - 2*x^3))/2 - 1/2)*(x/4 - x^2/20 + (log(log(x))*log(log(log(x)))*log(x)*(10
*x - 3*x^2))/20))/(log(log(x))*log(x)),x)

[Out]

(log(log(x))^(5*x^2 - x^3)*exp(-1/2))/20