3.43.12 \(\int \frac {e^{-\frac {2 e^2 (4-4 x+x^2)}{100-120 x+36 x^2+(20-12 x) \log (x)+\log ^2(x)}} (-2000 x+3600 x^2-2160 x^3+432 x^4+e^2 (-16 x+32 x^2-12 x^3)+(-600 x+720 x^2-216 x^3+e^2 (-8 x^2+4 x^3)) \log (x)+(-60 x+36 x^2) \log ^2(x)-2 x \log ^3(x))}{1000-1800 x+1080 x^2-216 x^3+(300-360 x+108 x^2) \log (x)+(30-18 x) \log ^2(x)+\log ^3(x)} \, dx\)

Optimal. Leaf size=30 \[ 5-e^{-\frac {2 e^2}{\left (5+\frac {x-\log (x)}{-2+x}\right )^2}} x^2 \]

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Rubi [F]  time = 8.76, 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 {2 e^2 \left (4-4 x+x^2\right )}{100-120 x+36 x^2+(20-12 x) \log (x)+\log ^2(x)}\right ) \left (-2000 x+3600 x^2-2160 x^3+432 x^4+e^2 \left (-16 x+32 x^2-12 x^3\right )+\left (-600 x+720 x^2-216 x^3+e^2 \left (-8 x^2+4 x^3\right )\right ) \log (x)+\left (-60 x+36 x^2\right ) \log ^2(x)-2 x \log ^3(x)\right )}{1000-1800 x+1080 x^2-216 x^3+\left (300-360 x+108 x^2\right ) \log (x)+(30-18 x) \log ^2(x)+\log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2000*x + 3600*x^2 - 2160*x^3 + 432*x^4 + E^2*(-16*x + 32*x^2 - 12*x^3) + (-600*x + 720*x^2 - 216*x^3 + E
^2*(-8*x^2 + 4*x^3))*Log[x] + (-60*x + 36*x^2)*Log[x]^2 - 2*x*Log[x]^3)/(E^((2*E^2*(4 - 4*x + x^2))/(100 - 120
*x + 36*x^2 + (20 - 12*x)*Log[x] + Log[x]^2))*(1000 - 1800*x + 1080*x^2 - 216*x^3 + (300 - 360*x + 108*x^2)*Lo
g[x] + (30 - 18*x)*Log[x]^2 + Log[x]^3)),x]

[Out]

-2*Defer[Int][x/E^((2*E^2*(-2 + x)^2)/(10 - 6*x + Log[x])^2), x] + 16*Defer[Int][(E^(2 - (2*E^2*(-2 + x)^2)/(1
0 - 6*x + Log[x])^2)*x)/(-10 + 6*x - Log[x])^3, x] - 112*Defer[Int][(E^(2 - (2*E^2*(-2 + x)^2)/(10 - 6*x + Log
[x])^2)*x^2)/(-10 + 6*x - Log[x])^3, x] + 100*Defer[Int][(E^(2 - (2*E^2*(-2 + x)^2)/(10 - 6*x + Log[x])^2)*x^3
)/(-10 + 6*x - Log[x])^3, x] - 24*Defer[Int][(E^(2 - (2*E^2*(-2 + x)^2)/(10 - 6*x + Log[x])^2)*x^4)/(-10 + 6*x
- Log[x])^3, x] - 8*Defer[Int][(E^(2 - (2*E^2*(-2 + x)^2)/(10 - 6*x + Log[x])^2)*x^2)/(-10 + 6*x - Log[x])^2,
x] + 4*Defer[Int][(E^(2 - (2*E^2*(-2 + x)^2)/(10 - 6*x + Log[x])^2)*x^3)/(-10 + 6*x - Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x \left (8 (-5+3 x)^3-2 e^2 \left (4-8 x+3 x^2\right )+2 \left (-150-2 \left (-90+e^2\right ) x+\left (-54+e^2\right ) x^2\right ) \log (x)+6 (-5+3 x) \log ^2(x)-\log ^3(x)\right )}{(10-6 x+\log (x))^3} \, dx\\ &=2 \int \frac {e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x \left (8 (-5+3 x)^3-2 e^2 \left (4-8 x+3 x^2\right )+2 \left (-150-2 \left (-90+e^2\right ) x+\left (-54+e^2\right ) x^2\right ) \log (x)+6 (-5+3 x) \log ^2(x)-\log ^3(x)\right )}{(10-6 x+\log (x))^3} \, dx\\ &=2 \int \left (-e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x-\frac {2 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} (-2+x)^2 x (-1+6 x)}{(-10+6 x-\log (x))^3}+\frac {2 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} (-2+x) x^2}{(-10+6 x-\log (x))^2}\right ) \, dx\\ &=-\left (2 \int e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x \, dx\right )-4 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} (-2+x)^2 x (-1+6 x)}{(-10+6 x-\log (x))^3} \, dx+4 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} (-2+x) x^2}{(-10+6 x-\log (x))^2} \, dx\\ &=-\left (2 \int e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x \, dx\right )-4 \int \left (-\frac {4 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x}{(-10+6 x-\log (x))^3}+\frac {28 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^2}{(-10+6 x-\log (x))^3}-\frac {25 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^3}{(-10+6 x-\log (x))^3}+\frac {6 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^4}{(-10+6 x-\log (x))^3}\right ) \, dx+4 \int \left (-\frac {2 e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^2}{(-10+6 x-\log (x))^2}+\frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^3}{(-10+6 x-\log (x))^2}\right ) \, dx\\ &=-\left (2 \int e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x \, dx\right )+4 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^3}{(-10+6 x-\log (x))^2} \, dx-8 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^2}{(-10+6 x-\log (x))^2} \, dx+16 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x}{(-10+6 x-\log (x))^3} \, dx-24 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^4}{(-10+6 x-\log (x))^3} \, dx+100 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^3}{(-10+6 x-\log (x))^3} \, dx-112 \int \frac {e^{2-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^2}{(-10+6 x-\log (x))^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 26, normalized size = 0.87 \begin {gather*} -e^{-\frac {2 e^2 (-2+x)^2}{(10-6 x+\log (x))^2}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2000*x + 3600*x^2 - 2160*x^3 + 432*x^4 + E^2*(-16*x + 32*x^2 - 12*x^3) + (-600*x + 720*x^2 - 216*x
^3 + E^2*(-8*x^2 + 4*x^3))*Log[x] + (-60*x + 36*x^2)*Log[x]^2 - 2*x*Log[x]^3)/(E^((2*E^2*(4 - 4*x + x^2))/(100
- 120*x + 36*x^2 + (20 - 12*x)*Log[x] + Log[x]^2))*(1000 - 1800*x + 1080*x^2 - 216*x^3 + (300 - 360*x + 108*x
^2)*Log[x] + (30 - 18*x)*Log[x]^2 + Log[x]^3)),x]

[Out]

-(x^2/E^((2*E^2*(-2 + x)^2)/(10 - 6*x + Log[x])^2))

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fricas [A]  time = 0.93, size = 43, normalized size = 1.43 \begin {gather*} -x^{2} e^{\left (-\frac {2 \, {\left (x^{2} - 4 \, x + 4\right )} e^{2}}{36 \, x^{2} - 4 \, {\left (3 \, x - 5\right )} \log \relax (x) + \log \relax (x)^{2} - 120 \, x + 100}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)^3+(36*x^2-60*x)*log(x)^2+((4*x^3-8*x^2)*exp(1)^2-216*x^3+720*x^2-600*x)*log(x)+(-12*x^3
+32*x^2-16*x)*exp(1)^2+432*x^4-2160*x^3+3600*x^2-2000*x)/(log(x)^3+(-18*x+30)*log(x)^2+(108*x^2-360*x+300)*log
(x)-216*x^3+1080*x^2-1800*x+1000)/exp((x^2-4*x+4)*exp(1)^2/(log(x)^2+(-12*x+20)*log(x)+36*x^2-120*x+100))^2,x,
algorithm="fricas")

[Out]

-x^2*e^(-2*(x^2 - 4*x + 4)*e^2/(36*x^2 - 4*(3*x - 5)*log(x) + log(x)^2 - 120*x + 100))

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giac [B]  time = 1.69, size = 71, normalized size = 2.37 \begin {gather*} -x^{2} e^{\left (\frac {2 \, {\left (11 \, x^{2} e^{2} - 12 \, x e^{2} \log \relax (x) + e^{2} \log \relax (x)^{2} - 20 \, x e^{2} + 20 \, e^{2} \log \relax (x)\right )}}{25 \, {\left (36 \, x^{2} - 12 \, x \log \relax (x) + \log \relax (x)^{2} - 120 \, x + 20 \, \log \relax (x) + 100\right )}} - \frac {2}{25} \, e^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)^3+(36*x^2-60*x)*log(x)^2+((4*x^3-8*x^2)*exp(1)^2-216*x^3+720*x^2-600*x)*log(x)+(-12*x^3
+32*x^2-16*x)*exp(1)^2+432*x^4-2160*x^3+3600*x^2-2000*x)/(log(x)^3+(-18*x+30)*log(x)^2+(108*x^2-360*x+300)*log
(x)-216*x^3+1080*x^2-1800*x+1000)/exp((x^2-4*x+4)*exp(1)^2/(log(x)^2+(-12*x+20)*log(x)+36*x^2-120*x+100))^2,x,
algorithm="giac")

[Out]

-x^2*e^(2/25*(11*x^2*e^2 - 12*x*e^2*log(x) + e^2*log(x)^2 - 20*x*e^2 + 20*e^2*log(x))/(36*x^2 - 12*x*log(x) +
log(x)^2 - 120*x + 20*log(x) + 100) - 2/25*e^2)

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maple [A]  time = 0.05, size = 25, normalized size = 0.83

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x)^3+(36*x^2-60*x)*ln(x)^2+((4*x^3-8*x^2)*exp(1)^2-216*x^3+720*x^2-600*x)*ln(x)+(-12*x^3+32*x^2-1
6*x)*exp(1)^2+432*x^4-2160*x^3+3600*x^2-2000*x)/(ln(x)^3+(-18*x+30)*ln(x)^2+(108*x^2-360*x+300)*ln(x)-216*x^3+
1080*x^2-1800*x+1000)/exp((x^2-4*x+4)*exp(1)^2/(ln(x)^2+(-12*x+20)*ln(x)+36*x^2-120*x+100))^2,x,method=_RETURN
VERBOSE)

[Out]

-x^2*exp(-2*(x-2)^2*exp(2)/(10+ln(x)-6*x)^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, \int \frac {{\left (216 \, x^{4} - x \log \relax (x)^{3} - 1080 \, x^{3} + 6 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \relax (x)^{2} + 1800 \, x^{2} - 2 \, {\left (3 \, x^{3} - 8 \, x^{2} + 4 \, x\right )} e^{2} - 2 \, {\left (54 \, x^{3} - 180 \, x^{2} - {\left (x^{3} - 2 \, x^{2}\right )} e^{2} + 150 \, x\right )} \log \relax (x) - 1000 \, x\right )} e^{\left (-\frac {2 \, {\left (x^{2} - 4 \, x + 4\right )} e^{2}}{36 \, x^{2} - 4 \, {\left (3 \, x - 5\right )} \log \relax (x) + \log \relax (x)^{2} - 120 \, x + 100}\right )}}{216 \, x^{3} + 6 \, {\left (3 \, x - 5\right )} \log \relax (x)^{2} - \log \relax (x)^{3} - 1080 \, x^{2} - 12 \, {\left (9 \, x^{2} - 30 \, x + 25\right )} \log \relax (x) + 1800 \, x - 1000}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)^3+(36*x^2-60*x)*log(x)^2+((4*x^3-8*x^2)*exp(1)^2-216*x^3+720*x^2-600*x)*log(x)+(-12*x^3
+32*x^2-16*x)*exp(1)^2+432*x^4-2160*x^3+3600*x^2-2000*x)/(log(x)^3+(-18*x+30)*log(x)^2+(108*x^2-360*x+300)*log
(x)-216*x^3+1080*x^2-1800*x+1000)/exp((x^2-4*x+4)*exp(1)^2/(log(x)^2+(-12*x+20)*log(x)+36*x^2-120*x+100))^2,x,
algorithm="maxima")

[Out]

-2*integrate((216*x^4 - x*log(x)^3 - 1080*x^3 + 6*(3*x^2 - 5*x)*log(x)^2 + 1800*x^2 - 2*(3*x^3 - 8*x^2 + 4*x)*
e^2 - 2*(54*x^3 - 180*x^2 - (x^3 - 2*x^2)*e^2 + 150*x)*log(x) - 1000*x)*e^(-2*(x^2 - 4*x + 4)*e^2/(36*x^2 - 4*
(3*x - 5)*log(x) + log(x)^2 - 120*x + 100))/(216*x^3 + 6*(3*x - 5)*log(x)^2 - log(x)^3 - 1080*x^2 - 12*(9*x^2
- 30*x + 25)*log(x) + 1800*x - 1000), x)

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mupad [B]  time = 3.53, size = 50, normalized size = 1.67 \begin {gather*} -x^2\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^2\,x^2-8\,{\mathrm {e}}^2\,x+8\,{\mathrm {e}}^2}{36\,x^2-12\,x\,\ln \relax (x)-120\,x+{\ln \relax (x)}^2+20\,\ln \relax (x)+100}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*exp(2)*(x^2 - 4*x + 4))/(log(x)^2 - 120*x - log(x)*(12*x - 20) + 36*x^2 + 100))*(2000*x + log(x)
^2*(60*x - 36*x^2) + 2*x*log(x)^3 + exp(2)*(16*x - 32*x^2 + 12*x^3) + log(x)*(600*x + exp(2)*(8*x^2 - 4*x^3) -
720*x^2 + 216*x^3) - 3600*x^2 + 2160*x^3 - 432*x^4))/(log(x)^3 - 1800*x + log(x)*(108*x^2 - 360*x + 300) + 10
80*x^2 - 216*x^3 - log(x)^2*(18*x - 30) + 1000),x)

[Out]

-x^2*exp(-(8*exp(2) - 8*x*exp(2) + 2*x^2*exp(2))/(20*log(x) - 120*x + log(x)^2 - 12*x*log(x) + 36*x^2 + 100))

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sympy [A]  time = 42.88, size = 42, normalized size = 1.40 \begin {gather*} - x^{2} e^{- \frac {2 \left (x^{2} - 4 x + 4\right ) e^{2}}{36 x^{2} - 120 x + \left (20 - 12 x\right ) \log {\relax (x )} + \log {\relax (x )}^{2} + 100}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(x)**3+(36*x**2-60*x)*ln(x)**2+((4*x**3-8*x**2)*exp(1)**2-216*x**3+720*x**2-600*x)*ln(x)+(-1
2*x**3+32*x**2-16*x)*exp(1)**2+432*x**4-2160*x**3+3600*x**2-2000*x)/(ln(x)**3+(-18*x+30)*ln(x)**2+(108*x**2-36
0*x+300)*ln(x)-216*x**3+1080*x**2-1800*x+1000)/exp((x**2-4*x+4)*exp(1)**2/(ln(x)**2+(-12*x+20)*ln(x)+36*x**2-1
20*x+100))**2,x)

[Out]

-x**2*exp(-2*(x**2 - 4*x + 4)*exp(2)/(36*x**2 - 120*x + (20 - 12*x)*log(x) + log(x)**2 + 100))

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