3.82.86 \(\int \frac {-2+e^{3+x+2 x^2+x^4-x^3 \log (x)} (2+8 x-2 x^2+8 x^3-6 x^2 \log (x))}{e^{6+2 x+4 x^2+2 x^4-2 x^3 \log (x)}-2 e^{3+x+2 x^2+x^4-x^3 \log (x)} x+x^2} \, dx\)

Optimal. Leaf size=30 \[ -3+\frac {2}{-e^{3+x-x^2 (-2+x (-x+\log (x)))}+x} \]

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Rubi [F]  time = 2.44, 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 {-2+e^{3+x+2 x^2+x^4-x^3 \log (x)} \left (2+8 x-2 x^2+8 x^3-6 x^2 \log (x)\right )}{e^{6+2 x+4 x^2+2 x^4-2 x^3 \log (x)}-2 e^{3+x+2 x^2+x^4-x^3 \log (x)} x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 + E^(3 + x + 2*x^2 + x^4 - x^3*Log[x])*(2 + 8*x - 2*x^2 + 8*x^3 - 6*x^2*Log[x]))/(E^(6 + 2*x + 4*x^2 +
2*x^4 - 2*x^3*Log[x]) - 2*E^(3 + x + 2*x^2 + x^4 - x^3*Log[x])*x + x^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 32, normalized size = 1.07 \begin {gather*} -\frac {2 x^{x^3}}{e^{3+x+2 x^2+x^4}-x^{1+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^(3 + x + 2*x^2 + x^4 - x^3*Log[x])*(2 + 8*x - 2*x^2 + 8*x^3 - 6*x^2*Log[x]))/(E^(6 + 2*x + 4
*x^2 + 2*x^4 - 2*x^3*Log[x]) - 2*E^(3 + x + 2*x^2 + x^4 - x^3*Log[x])*x + x^2),x]

[Out]

(-2*x^x^3)/(E^(3 + x + 2*x^2 + x^4) - x^(1 + x^3))

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fricas [A]  time = 0.94, size = 27, normalized size = 0.90 \begin {gather*} \frac {2}{x - e^{\left (x^{4} - x^{3} \log \relax (x) + 2 \, x^{2} + x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 3.98, size = 27, normalized size = 0.90 \begin {gather*} \frac {2}{x - e^{\left (x^{4} - x^{3} \log \relax (x) + 2 \, x^{2} + x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 28, normalized size = 0.93

method	result	size
risch	\(\frac {2}{x -x^{-x^{3}} {\mathrm e}^{x^{4}+2 x^{2}+x +3}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^2*ln(x)+8*x^3-2*x^2+8*x+2)*exp(-x^3*ln(x)+x^4+2*x^2+x+3)-2)/(exp(-x^3*ln(x)+x^4+2*x^2+x+3)^2-2*x*ex
p(-x^3*ln(x)+x^4+2*x^2+x+3)+x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 31, normalized size = 1.03 \begin {gather*} \frac {2 \, x^{\left (x^{3}\right )}}{x x^{\left (x^{3}\right )} - e^{\left (x^{4} + 2 \, x^{2} + x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^(x^3)/(x*x^(x^3) - e^(x^4 + 2*x^2 + x + 3))

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mupad [B]  time = 5.43, size = 29, normalized size = 0.97 \begin {gather*} \frac {2}{x-\frac {{\mathrm {e}}^{x^4}\,{\mathrm {e}}^3\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^x}{x^{x^3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - x^3*log(x) + 2*x^2 + x^4 + 3)*(8*x - 6*x^2*log(x) - 2*x^2 + 8*x^3 + 2) - 2)/(exp(2*x - 2*x^3*log(
x) + 4*x^2 + 2*x^4 + 6) - 2*x*exp(x - x^3*log(x) + 2*x^2 + x^4 + 3) + x^2),x)

[Out]

2/(x - (exp(x^4)*exp(3)*exp(2*x^2)*exp(x))/x^(x^3))

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sympy [A]  time = 0.33, size = 24, normalized size = 0.80 \begin {gather*} - \frac {2}{- x + e^{x^{4} - x^{3} \log {\relax (x )} + 2 x^{2} + x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**2*ln(x)+8*x**3-2*x**2+8*x+2)*exp(-x**3*ln(x)+x**4+2*x**2+x+3)-2)/(exp(-x**3*ln(x)+x**4+2*x**
2+x+3)**2-2*x*exp(-x**3*ln(x)+x**4+2*x**2+x+3)+x**2),x)

[Out]

-2/(-x + exp(x**4 - x**3*log(x) + 2*x**2 + x + 3))

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