3.97.91 \(\int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} (-x^2+e^{2 x^2} (2 x-4 x^3))}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx\)

Optimal. Leaf size=27 \[ -2+e^{2 e^5+\frac {x^2}{e^{2 x^2}-x}} \]

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Rubi [F]  time = 8.01, 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 {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} \left (-x^2+e^{2 x^2} \left (2 x-4 x^3\right )\right )}{e^{4 x^2}-2 e^{2 x^2} x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x \left (-x-e^{2 x^2} \left (-2+4 x^2\right )\right )}{\left (e^{2 x^2}-x\right )^2} \, dx\\ &=\int \left (-\frac {2 e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x \left (-1+2 x^2\right )}{e^{2 x^2}-x}-\frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^2 \left (-1+4 x^2\right )}{\left (e^{2 x^2}-x\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x \left (-1+2 x^2\right )}{e^{2 x^2}-x} \, dx\right )-\int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^2 \left (-1+4 x^2\right )}{\left (e^{2 x^2}-x\right )^2} \, dx\\ &=-\left (2 \int \left (-\frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x}{e^{2 x^2}-x}+\frac {2 e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^3}{e^{2 x^2}-x}\right ) \, dx\right )-\int \left (-\frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^2}{\left (e^{2 x^2}-x\right )^2}+\frac {4 e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^4}{\left (e^{2 x^2}-x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x}{e^{2 x^2}-x} \, dx-4 \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^3}{e^{2 x^2}-x} \, dx-4 \int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^4}{\left (e^{2 x^2}-x\right )^2} \, dx+\int \frac {e^{\frac {2 e^{5+2 x^2}-2 e^5 x+x^2}{e^{2 x^2}-x}} x^2}{\left (e^{2 x^2}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.67, size = 25, normalized size = 0.93 \begin {gather*} e^{2 e^5+\frac {x^2}{e^{2 x^2}-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*E^5 + x^2/(E^(2*x^2) - x))

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fricas [A]  time = 0.65, size = 42, normalized size = 1.56 \begin {gather*} e^{\left (-\frac {x^{2} e^{5} - 2 \, x e^{10} + 2 \, e^{\left (2 \, x^{2} + 10\right )}}{x e^{5} - e^{\left (2 \, x^{2} + 5\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2
*x*exp(x^2)^2+x^2),x, algorithm="fricas")

[Out]

e^(-(x^2*e^5 - 2*x*e^10 + 2*e^(2*x^2 + 10))/(x*e^5 - e^(2*x^2 + 5)))

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giac [B]  time = 0.39, size = 58, normalized size = 2.15 \begin {gather*} e^{\left (-\frac {x^{2}}{x - e^{\left (2 \, x^{2}\right )}} + \frac {2 \, x e^{5}}{x - e^{\left (2 \, x^{2}\right )}} - \frac {2 \, e^{\left (2 \, x^{2} + 5\right )}}{x - e^{\left (2 \, x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2
*x*exp(x^2)^2+x^2),x, algorithm="giac")

[Out]

e^(-x^2/(x - e^(2*x^2)) + 2*x*e^5/(x - e^(2*x^2)) - 2*e^(2*x^2 + 5)/(x - e^(2*x^2)))

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maple [A]  time = 0.14, size = 36, normalized size = 1.33

method	result	size
risch	\({\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}-x^{2}-2 \,{\mathrm e}^{2 x^{2}+5}}{-{\mathrm e}^{2 x^{2}}+x}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2*x*exp
(x^2)^2+x^2),x,method=_RETURNVERBOSE)

[Out]

exp((2*x*exp(5)-x^2-2*exp(2*x^2+5))/(-exp(2*x^2)+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{2} + 2 \, {\left (2 \, x^{3} - x\right )} e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-\frac {x^{2} - 2 \, x e^{5} + 2 \, e^{\left (2 \, x^{2} + 5\right )}}{x - e^{\left (2 \, x^{2}\right )}}\right )}}{x^{2} - 2 \, x e^{\left (2 \, x^{2}\right )} + e^{\left (4 \, x^{2}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+2*x)*exp(x^2)^2-x^2)*exp((2*exp(5)*exp(x^2)^2-2*x*exp(5)+x^2)/(exp(x^2)^2-x))/(exp(x^2)^4-2
*x*exp(x^2)^2+x^2),x, algorithm="maxima")

[Out]

-integrate((x^2 + 2*(2*x^3 - x)*e^(2*x^2))*e^(-(x^2 - 2*x*e^5 + 2*e^(2*x^2 + 5))/(x - e^(2*x^2)))/(x^2 - 2*x*e
^(2*x^2) + e^(4*x^2)), x)

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mupad [B]  time = 6.01, size = 60, normalized size = 2.22 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{x-{\mathrm {e}}^{2\,x^2}}}\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^5}{x-{\mathrm {e}}^{2\,x^2}}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^5\,{\mathrm {e}}^{2\,x^2}}{x-{\mathrm {e}}^{2\,x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*exp(5)*exp(2*x^2) - 2*x*exp(5) + x^2)/(x - exp(2*x^2)))*(exp(2*x^2)*(2*x - 4*x^3) - x^2))/(exp(4*
x^2) - 2*x*exp(2*x^2) + x^2),x)

[Out]

exp(-x^2/(x - exp(2*x^2)))*exp((2*x*exp(5))/(x - exp(2*x^2)))*exp(-(2*exp(5)*exp(2*x^2))/(x - exp(2*x^2)))

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sympy [A]  time = 0.41, size = 31, normalized size = 1.15 \begin {gather*} e^{\frac {x^{2} - 2 x e^{5} + 2 e^{5} e^{2 x^{2}}}{- x + e^{2 x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3+2*x)*exp(x**2)**2-x**2)*exp((2*exp(5)*exp(x**2)**2-2*x*exp(5)+x**2)/(exp(x**2)**2-x))/(exp
(x**2)**4-2*x*exp(x**2)**2+x**2),x)

[Out]

exp((x**2 - 2*x*exp(5) + 2*exp(5)*exp(2*x**2))/(-x + exp(2*x**2)))

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