3.18.9 \(\int \frac {e^{\frac {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}} (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+(-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7) \log (x))}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx\)

Optimal. Leaf size=31 \[ 1-e^{\frac {x^4 \left (-x+x \left (\frac {1}{x}+x\right )\right )^2}{(x-\log (x))^2}} \]

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Rubi [F]  time = 7.01, 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 {x^4-2 x^5+3 x^6-2 x^7+x^8}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (2 x^3-2 x^4+8 x^6-8 x^7+6 x^8+\left (-4 x^3+10 x^4-18 x^5+14 x^6-8 x^7\right ) \log (x)\right )}{-x^3+3 x^2 \log (x)-3 x \log ^2(x)+\log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((x^4 - 2*x^5 + 3*x^6 - 2*x^7 + x^8)/(x^2 - 2*x*Log[x] + Log[x]^2))*(2*x^3 - 2*x^4 + 8*x^6 - 8*x^7 + 6*
x^8 + (-4*x^3 + 10*x^4 - 18*x^5 + 14*x^6 - 8*x^7)*Log[x]))/(-x^3 + 3*x^2*Log[x] - 3*x*Log[x]^2 + Log[x]^3),x]

[Out]

-2*Defer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^3)/(x - Log[x])^3, x] + 6*Defer[Int][(E^((x^4*(1 - x
+ x^2)^2)/(x - Log[x])^2)*x^4)/(x - Log[x])^3, x] - 10*Defer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x
^5)/(x - Log[x])^3, x] + 10*Defer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^6)/(x - Log[x])^3, x] - 6*D
efer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^7)/(x - Log[x])^3, x] + 2*Defer[Int][(E^((x^4*(1 - x + x
^2)^2)/(x - Log[x])^2)*x^8)/(x - Log[x])^3, x] - 4*Defer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^3)/(
x - Log[x])^2, x] + 10*Defer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^4)/(x - Log[x])^2, x] - 18*Defer
[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^5)/(x - Log[x])^2, x] + 14*Defer[Int][(E^((x^4*(1 - x + x^2)
^2)/(x - Log[x])^2)*x^6)/(x - Log[x])^2, x] - 8*Defer[Int][(E^((x^4*(1 - x + x^2)^2)/(x - Log[x])^2)*x^7)/(x -
Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (1-x+x^2\right ) \left (-1+x^2-3 x^3-\left (-2+3 x-4 x^2\right ) \log (x)\right )}{(x-\log (x))^3} \, dx\\ &=2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (1-x+x^2\right ) \left (-1+x^2-3 x^3-\left (-2+3 x-4 x^2\right ) \log (x)\right )}{(x-\log (x))^3} \, dx\\ &=2 \int \left (\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} (-1+x) x^3 \left (1-x+x^2\right )^2}{(x-\log (x))^3}-\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (2-5 x+9 x^2-7 x^3+4 x^4\right )}{(x-\log (x))^2}\right ) \, dx\\ &=2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} (-1+x) x^3 \left (1-x+x^2\right )^2}{(x-\log (x))^3} \, dx-2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3 \left (2-5 x+9 x^2-7 x^3+4 x^4\right )}{(x-\log (x))^2} \, dx\\ &=2 \int \left (-\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^3}+\frac {3 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^3}-\frac {5 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^3}+\frac {5 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^3}-\frac {3 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^3}+\frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^8}{(x-\log (x))^3}\right ) \, dx-2 \int \left (\frac {2 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^2}-\frac {5 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^2}+\frac {9 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^2}-\frac {7 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^2}+\frac {4 e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^3} \, dx\right )+2 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^8}{(x-\log (x))^3} \, dx-4 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^3}{(x-\log (x))^2} \, dx+6 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^3} \, dx-6 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^3} \, dx-8 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^7}{(x-\log (x))^2} \, dx-10 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^3} \, dx+10 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^3} \, dx+10 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^4}{(x-\log (x))^2} \, dx+14 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^6}{(x-\log (x))^2} \, dx-18 \int \frac {e^{\frac {x^4 \left (1-x+x^2\right )^2}{(x-\log (x))^2}} x^5}{(x-\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.48, size = 26, normalized size = 0.84 \begin {gather*} -e^{\frac {x^4 \left (1-x+x^2\right )^2}{(-x+\log (x))^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x^4 - 2*x^5 + 3*x^6 - 2*x^7 + x^8)/(x^2 - 2*x*Log[x] + Log[x]^2))*(2*x^3 - 2*x^4 + 8*x^6 - 8*x^
7 + 6*x^8 + (-4*x^3 + 10*x^4 - 18*x^5 + 14*x^6 - 8*x^7)*Log[x]))/(-x^3 + 3*x^2*Log[x] - 3*x*Log[x]^2 + Log[x]^
3),x]

[Out]

-E^((x^4*(1 - x + x^2)^2)/(-x + Log[x])^2)

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fricas [A]  time = 0.91, size = 41, normalized size = 1.32 \begin {gather*} -e^{\left (\frac {x^{8} - 2 \, x^{7} + 3 \, x^{6} - 2 \, x^{5} + x^{4}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^7+14*x^6-18*x^5+10*x^4-4*x^3)*log(x)+6*x^8-8*x^7+8*x^6-2*x^4+2*x^3)*exp((x^8-2*x^7+3*x^6-2*x^
5+x^4)/(log(x)^2-2*x*log(x)+x^2))/(log(x)^3-3*x*log(x)^2+3*x^2*log(x)-x^3),x, algorithm="fricas")

[Out]

-e^((x^8 - 2*x^7 + 3*x^6 - 2*x^5 + x^4)/(x^2 - 2*x*log(x) + log(x)^2))

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giac [B]  time = 0.51, size = 102, normalized size = 3.29 \begin {gather*} -e^{\left (\frac {x^{8}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} - \frac {2 \, x^{7}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} + \frac {3 \, x^{6}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} - \frac {2 \, x^{5}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} + \frac {x^{4}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^7+14*x^6-18*x^5+10*x^4-4*x^3)*log(x)+6*x^8-8*x^7+8*x^6-2*x^4+2*x^3)*exp((x^8-2*x^7+3*x^6-2*x^
5+x^4)/(log(x)^2-2*x*log(x)+x^2))/(log(x)^3-3*x*log(x)^2+3*x^2*log(x)-x^3),x, algorithm="giac")

[Out]

-e^(x^8/(x^2 - 2*x*log(x) + log(x)^2) - 2*x^7/(x^2 - 2*x*log(x) + log(x)^2) + 3*x^6/(x^2 - 2*x*log(x) + log(x)
^2) - 2*x^5/(x^2 - 2*x*log(x) + log(x)^2) + x^4/(x^2 - 2*x*log(x) + log(x)^2))

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maple [A]  time = 0.03, size = 26, normalized size = 0.84

method	result	size
risch	\(-{\mathrm e}^{\frac {x^{4} \left (x^{2}-x +1\right )^{2}}{\left (\ln \relax (x )-x \right )^{2}}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^7+14*x^6-18*x^5+10*x^4-4*x^3)*ln(x)+6*x^8-8*x^7+8*x^6-2*x^4+2*x^3)*exp((x^8-2*x^7+3*x^6-2*x^5+x^4)/
(ln(x)^2-2*x*ln(x)+x^2))/(ln(x)^3-3*x*ln(x)^2+3*x^2*ln(x)-x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 2.17, size = 343, normalized size = 11.06 \begin {gather*} -e^{\left (\frac {\log \relax (x)^{8}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} + x^{6} + 2 \, x^{5} \log \relax (x) + 3 \, x^{4} \log \relax (x)^{2} + 4 \, x^{3} \log \relax (x)^{3} + 5 \, x^{2} \log \relax (x)^{4} + 6 \, x \log \relax (x)^{5} + 7 \, \log \relax (x)^{6} - \frac {2 \, \log \relax (x)^{7}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} + \frac {8 \, \log \relax (x)^{7}}{x - \log \relax (x)} - 2 \, x^{5} - 4 \, x^{4} \log \relax (x) - 6 \, x^{3} \log \relax (x)^{2} - 8 \, x^{2} \log \relax (x)^{3} - 10 \, x \log \relax (x)^{4} - 12 \, \log \relax (x)^{5} + \frac {3 \, \log \relax (x)^{6}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} - \frac {14 \, \log \relax (x)^{6}}{x - \log \relax (x)} + 3 \, x^{4} + 6 \, x^{3} \log \relax (x) + 9 \, x^{2} \log \relax (x)^{2} + 12 \, x \log \relax (x)^{3} + 15 \, \log \relax (x)^{4} - \frac {2 \, \log \relax (x)^{5}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} + \frac {18 \, \log \relax (x)^{5}}{x - \log \relax (x)} - 2 \, x^{3} - 4 \, x^{2} \log \relax (x) - 6 \, x \log \relax (x)^{2} - 8 \, \log \relax (x)^{3} + \frac {\log \relax (x)^{4}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} - \frac {10 \, \log \relax (x)^{4}}{x - \log \relax (x)} + x^{2} + 2 \, x \log \relax (x) + 3 \, \log \relax (x)^{2} + \frac {4 \, \log \relax (x)^{3}}{x - \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^7+14*x^6-18*x^5+10*x^4-4*x^3)*log(x)+6*x^8-8*x^7+8*x^6-2*x^4+2*x^3)*exp((x^8-2*x^7+3*x^6-2*x^
5+x^4)/(log(x)^2-2*x*log(x)+x^2))/(log(x)^3-3*x*log(x)^2+3*x^2*log(x)-x^3),x, algorithm="maxima")

[Out]

-e^(log(x)^8/(x^2 - 2*x*log(x) + log(x)^2) + x^6 + 2*x^5*log(x) + 3*x^4*log(x)^2 + 4*x^3*log(x)^3 + 5*x^2*log(
x)^4 + 6*x*log(x)^5 + 7*log(x)^6 - 2*log(x)^7/(x^2 - 2*x*log(x) + log(x)^2) + 8*log(x)^7/(x - log(x)) - 2*x^5
- 4*x^4*log(x) - 6*x^3*log(x)^2 - 8*x^2*log(x)^3 - 10*x*log(x)^4 - 12*log(x)^5 + 3*log(x)^6/(x^2 - 2*x*log(x)
+ log(x)^2) - 14*log(x)^6/(x - log(x)) + 3*x^4 + 6*x^3*log(x) + 9*x^2*log(x)^2 + 12*x*log(x)^3 + 15*log(x)^4 -
2*log(x)^5/(x^2 - 2*x*log(x) + log(x)^2) + 18*log(x)^5/(x - log(x)) - 2*x^3 - 4*x^2*log(x) - 6*x*log(x)^2 - 8
*log(x)^3 + log(x)^4/(x^2 - 2*x*log(x) + log(x)^2) - 10*log(x)^4/(x - log(x)) + x^2 + 2*x*log(x) + 3*log(x)^2
+ 4*log(x)^3/(x - log(x)))

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mupad [B]  time = 1.31, size = 105, normalized size = 3.39 \begin {gather*} -{\mathrm {e}}^{\frac {x^4}{x^2-2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{-\frac {2\,x^5}{x^2-2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {x^8}{x^2-2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{\frac {3\,x^6}{x^2-2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}}\,{\mathrm {e}}^{-\frac {2\,x^7}{x^2-2\,x\,\ln \relax (x)+{\ln \relax (x)}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x^4 - 2*x^5 + 3*x^6 - 2*x^7 + x^8)/(log(x)^2 - 2*x*log(x) + x^2))*(log(x)*(4*x^3 - 10*x^4 + 18*x^5 -
14*x^6 + 8*x^7) - 2*x^3 + 2*x^4 - 8*x^6 + 8*x^7 - 6*x^8))/(3*x*log(x)^2 - 3*x^2*log(x) - log(x)^3 + x^3),x)

[Out]

-exp(x^4/(log(x)^2 - 2*x*log(x) + x^2))*exp(-(2*x^5)/(log(x)^2 - 2*x*log(x) + x^2))*exp(x^8/(log(x)^2 - 2*x*lo
g(x) + x^2))*exp((3*x^6)/(log(x)^2 - 2*x*log(x) + x^2))*exp(-(2*x^7)/(log(x)^2 - 2*x*log(x) + x^2))

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sympy [A]  time = 0.63, size = 39, normalized size = 1.26 \begin {gather*} - e^{\frac {x^{8} - 2 x^{7} + 3 x^{6} - 2 x^{5} + x^{4}}{x^{2} - 2 x \log {\relax (x )} + \log {\relax (x )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**7+14*x**6-18*x**5+10*x**4-4*x**3)*ln(x)+6*x**8-8*x**7+8*x**6-2*x**4+2*x**3)*exp((x**8-2*x**7
+3*x**6-2*x**5+x**4)/(ln(x)**2-2*x*ln(x)+x**2))/(ln(x)**3-3*x*ln(x)**2+3*x**2*ln(x)-x**3),x)

[Out]

-exp((x**8 - 2*x**7 + 3*x**6 - 2*x**5 + x**4)/(x**2 - 2*x*log(x) + log(x)**2))

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