3.31.25 \(\int \frac {e^{-\frac {-16 x^3-12 x^4+8 x^5-x^6+(16 x+32 x^2-8 x^3) \log (x)-16 \log ^2(x)}{4 x}} (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+(32-32 x^2+16 x^3) \log (x)-16 \log ^2(x))}{20 x} \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{5} e^{-x+\frac {\left (x \left (-1-2 x+\frac {x^2}{2}\right )+2 \log (x)\right )^2}{x}} x \]

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Rubi [F]  time = 9.26, 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 {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{20 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12*x - 32*x^2 + 40*x^3 + 36*x^4 - 32*x^5 + 5*x^6 + (32 - 32*x^2 + 16*x^3)*Log[x] - 16*Log[x]^2)/(20*E^((
-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x))*x),x]

[Out]

(-3*Defer[Int][E^(-1/4*(-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/x), x])
/5 - (8*Defer[Int][x/E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x))
, x])/5 + 2*Defer[Int][x^2/E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/
(4*x)), x] + (9*Defer[Int][x^3/E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]
^2)/(4*x)), x])/5 - (8*Defer[Int][x^4/E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16
*Log[x]^2)/(4*x)), x])/5 + Defer[Int][x^5/E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x]
- 16*Log[x]^2)/(4*x)), x]/4 + (8*Defer[Int][Log[x]/(E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^
3)*Log[x] - 16*Log[x]^2)/(4*x))*x), x])/5 - (8*Defer[Int][(x*Log[x])/E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*
x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x)), x])/5 + (4*Defer[Int][(x^2*Log[x])/E^((-16*x^3 - 12*x^4 + 8*
x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x)), x])/5 - (4*Defer[Int][Log[x]^2/(E^((-16*x^3
- 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x))*x), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{20} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (-12 x-32 x^2+40 x^3+36 x^4-32 x^5+5 x^6+\left (32-32 x^2+16 x^3\right ) \log (x)-16 \log ^2(x)\right )}{x} \, dx\\ &=\frac {1}{20} \int \left (-12 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right )-32 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x+40 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2+36 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3-32 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4+5 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5+\frac {16 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (2-2 x^2+x^3\right ) \log (x)}{x}-\frac {16 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x}\right ) \, dx\\ &=\frac {1}{4} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5 \, dx-\frac {3}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \, dx+\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \left (2-2 x^2+x^3\right ) \log (x)}{x} \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x} \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4 \, dx+\frac {9}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3 \, dx+2 \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \, dx\\ &=\frac {1}{4} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5 \, dx-\frac {3}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x} \, dx+\frac {4}{5} \int \left (\frac {2 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log (x)}{x}-2 \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \log (x)+\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \log (x)\right ) \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4 \, dx+\frac {9}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3 \, dx+2 \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \, dx\\ &=\frac {1}{4} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^5 \, dx-\frac {3}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \, dx+\frac {4}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \log (x) \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log ^2(x)}{x} \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^4 \, dx+\frac {8}{5} \int \frac {\exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) \log (x)}{x} \, dx-\frac {8}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x \log (x) \, dx+\frac {9}{5} \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^3 \, dx+2 \int \exp \left (-\frac {-16 x^3-12 x^4+8 x^5-x^6+\left (16 x+32 x^2-8 x^3\right ) \log (x)-16 \log ^2(x)}{4 x}\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 48, normalized size = 1.33 \begin {gather*} \frac {1}{5} e^{4 x^2+3 x^3-2 x^4+\frac {x^5}{4}+\frac {4 \log ^2(x)}{x}} x^{-3+2 (-4+x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x - 32*x^2 + 40*x^3 + 36*x^4 - 32*x^5 + 5*x^6 + (32 - 32*x^2 + 16*x^3)*Log[x] - 16*Log[x]^2)/(2
0*E^((-16*x^3 - 12*x^4 + 8*x^5 - x^6 + (16*x + 32*x^2 - 8*x^3)*Log[x] - 16*Log[x]^2)/(4*x))*x),x]

[Out]

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

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fricas [A]  time = 0.95, size = 50, normalized size = 1.39 \begin {gather*} \frac {1}{5} \, x e^{\left (\frac {x^{6} - 8 \, x^{5} + 12 \, x^{4} + 16 \, x^{3} + 8 \, {\left (x^{3} - 4 \, x^{2} - 2 \, x\right )} \log \relax (x) + 16 \, \log \relax (x)^{2}}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*
log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x^6+8*x^5-12*x^4-16*x^3)/x),x, algorithm="fricas")

[Out]

1/5*x*e^(1/4*(x^6 - 8*x^5 + 12*x^4 + 16*x^3 + 8*(x^3 - 4*x^2 - 2*x)*log(x) + 16*log(x)^2)/x)

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giac [A]  time = 0.59, size = 53, normalized size = 1.47 \begin {gather*} \frac {1}{5} \, x e^{\left (\frac {x^{6} - 8 \, x^{5} + 12 \, x^{4} + 8 \, x^{3} \log \relax (x) + 16 \, x^{3} - 32 \, x^{2} \log \relax (x) - 16 \, x \log \relax (x) + 16 \, \log \relax (x)^{2}}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*
log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x^6+8*x^5-12*x^4-16*x^3)/x),x, algorithm="giac")

[Out]

1/5*x*e^(1/4*(x^6 - 8*x^5 + 12*x^4 + 8*x^3*log(x) + 16*x^3 - 32*x^2*log(x) - 16*x*log(x) + 16*log(x)^2)/x)

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maple [A]  time = 0.02, size = 39, normalized size = 1.08

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (x^{3}-4 x^{2}+4 \ln \relax (x )\right ) \left (x^{3}-4 x^{2}+4 \ln \relax (x )-4 x \right )}{4 x}}}{5}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/20*(-16*ln(x)^2+(16*x^3-32*x^2+32)*ln(x)+5*x^6-32*x^5+36*x^4+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*ln(x)^2+
(-8*x^3+32*x^2+16*x)*ln(x)-x^6+8*x^5-12*x^4-16*x^3)/x),x,method=_RETURNVERBOSE)

[Out]

1/5*x*exp(1/4*(x^3-4*x^2+4*ln(x))*(x^3-4*x^2+4*ln(x)-4*x)/x)

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maxima [A]  time = 0.83, size = 48, normalized size = 1.33 \begin {gather*} \frac {e^{\left (\frac {1}{4} \, x^{5} - 2 \, x^{4} + 3 \, x^{3} + 2 \, x^{2} \log \relax (x) + 4 \, x^{2} - 8 \, x \log \relax (x) + \frac {4 \, \log \relax (x)^{2}}{x}\right )}}{5 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-16*log(x)^2+(16*x^3-32*x^2+32)*log(x)+5*x^6-32*x^5+36*x^4+40*x^3-32*x^2-12*x)/x/exp(1/4*(-16*
log(x)^2+(-8*x^3+32*x^2+16*x)*log(x)-x^6+8*x^5-12*x^4-16*x^3)/x),x, algorithm="maxima")

[Out]

1/5*e^(1/4*x^5 - 2*x^4 + 3*x^3 + 2*x^2*log(x) + 4*x^2 - 8*x*log(x) + 4*log(x)^2/x)/x^3

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mupad [B]  time = 1.92, size = 48, normalized size = 1.33 \begin {gather*} \frac {x^{2\,x^2-8\,x-3}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{3\,x^3}\,{\mathrm {e}}^{-2\,x^4}\,{\mathrm {e}}^{\frac {x^5}{4}}\,{\mathrm {e}}^{\frac {4\,{\ln \relax (x)}^2}{x}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((4*log(x)^2 + 4*x^3 + 3*x^4 - 2*x^5 + x^6/4 - (log(x)*(16*x + 32*x^2 - 8*x^3))/4)/x)*((3*x)/5 - (log
(x)*(16*x^3 - 32*x^2 + 32))/20 + (4*log(x)^2)/5 + (8*x^2)/5 - 2*x^3 - (9*x^4)/5 + (8*x^5)/5 - x^6/4))/x,x)

[Out]

(x^(2*x^2 - 8*x - 3)*exp(4*x^2)*exp(3*x^3)*exp(-2*x^4)*exp(x^5/4)*exp((4*log(x)^2)/x))/5

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sympy [A]  time = 0.50, size = 51, normalized size = 1.42 \begin {gather*} \frac {x e^{- \frac {- \frac {x^{6}}{4} + 2 x^{5} - 3 x^{4} - 4 x^{3} + \frac {\left (- 8 x^{3} + 32 x^{2} + 16 x\right ) \log {\relax (x )}}{4} - 4 \log {\relax (x )}^{2}}{x}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-16*ln(x)**2+(16*x**3-32*x**2+32)*ln(x)+5*x**6-32*x**5+36*x**4+40*x**3-32*x**2-12*x)/x/exp(1/4
*(-16*ln(x)**2+(-8*x**3+32*x**2+16*x)*ln(x)-x**6+8*x**5-12*x**4-16*x**3)/x),x)

[Out]

x*exp(-(-x**6/4 + 2*x**5 - 3*x**4 - 4*x**3 + (-8*x**3 + 32*x**2 + 16*x)*log(x)/4 - 4*log(x)**2)/x)/5

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