3.2.69 \(\int \frac {e^{1-e^{x^2} x+2 x^2} (x+2 x^3)}{(1-e^{x^2} x)^2} \, dx\) [169]

Optimal. Leaf size=25 \[ -\frac {e^{1-e^{x^2} x}}{-1+e^{x^2} x} \]

[Out]

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

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Rubi [F]
time = 0.66, 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^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(1 - E^x^2*x + 2*x^2)*(x + 2*x^3))/(1 - E^x^2*x)^2,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {e^{1-e^{x^2} x+2 x^2} \left (x+2 x^3\right )}{\left (1-e^{x^2} x\right )^2} \, dx &=\int \frac {e^{1-e^{x^2} x+2 x^2} x \left (1+2 x^2\right )}{\left (1-e^{x^2} x\right )^2} \, dx\\ &=\int \left (\frac {e^{1-e^{x^2} x+2 x^2} x}{\left (-1+e^{x^2} x\right )^2}+\frac {2 e^{1-e^{x^2} x+2 x^2} x^3}{\left (-1+e^{x^2} x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{1-e^{x^2} x+2 x^2} x^3}{\left (-1+e^{x^2} x\right )^2} \, dx+\int \frac {e^{1-e^{x^2} x+2 x^2} x}{\left (-1+e^{x^2} x\right )^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 25, normalized size = 1.00 \begin {gather*} -\frac {e^{1-e^{x^2} x}}{-1+e^{x^2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 - E^x^2*x + 2*x^2)*(x + 2*x^3))/(1 - E^x^2*x)^2,x]

[Out]

-(E^(1 - E^x^2*x)/(-1 + E^x^2*x))

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Mathics [A]
time = 2.21, size = 25, normalized size = 1.00 \begin {gather*} -\frac {E^{1-x E^{x^2}}}{-1+x E^{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(2*x^3+x)*(E^(x^2))^2*E^(1-x*E^(x^2))/(1-x*E^(x^2))^2,x]')

[Out]

-E ^ (1 - x E ^ x ^ 2) / (-1 + x E ^ x ^ 2)

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Maple [A]
time = 0.05, size = 23, normalized size = 0.92

method	result	size
risch	\(-\frac {{\mathrm e}^{1-{\mathrm e}^{x^{2}} x}}{-1+{\mathrm e}^{x^{2}} x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 22, normalized size = 0.88 \begin {gather*} -\frac {e^{\left (-x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (x^{2}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x, algorithm="maxima")

[Out]

-e^(-x*e^(x^2) + 1)/(x*e^(x^2) - 1)

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Fricas [A]
time = 0.31, size = 36, normalized size = 1.44 \begin {gather*} -\frac {e^{\left (2 \, x^{2} - x e^{\left (x^{2}\right )} + 1\right )}}{x e^{\left (3 \, x^{2}\right )} - e^{\left (2 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x, algorithm="fricas")

[Out]

-e^(2*x^2 - x*e^(x^2) + 1)/(x*e^(3*x^2) - e^(2*x^2))

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Sympy [A]
time = 0.12, size = 31, normalized size = 1.24 \begin {gather*} - \frac {e^{2 x^{2} - x e^{x^{2}} + 1}}{x e^{3 x^{2}} - e^{2 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1-exp(x**2)*x+2*x**2)*(2*x**3+x)/(1-exp(x**2)*x)**2,x)

[Out]

-exp(2*x**2 - x*exp(x**2) + 1)/(x*exp(3*x**2) - exp(2*x**2))

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Giac [A]
time = 0.01, size = 35, normalized size = 1.40 \begin {gather*} -\frac {\mathrm {e} \mathrm {e}^{2 x^{2}-x \mathrm {e}^{x^{2}}}}{x \left (\mathrm {e}^{x^{2}}\right )^{3}-\left (\mathrm {e}^{x^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1-exp(x^2)*x+2*x^2)*(2*x^3+x)/(1-exp(x^2)*x)^2,x)

[Out]

-e^(-e^(x^2)*x + 2*x^2 + 1)/(e^(3*x^2)*x - e^(2*x^2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x^2-x\,{\mathrm {e}}^{x^2}+1}\,\left (2\,x^3+x\right )}{{\left (x\,{\mathrm {e}}^{x^2}-1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x^2 - x*exp(x^2) + 1)*(x + 2*x^3))/(x*exp(x^2) - 1)^2,x)

[Out]

int((exp(2*x^2 - x*exp(x^2) + 1)*(x + 2*x^3))/(x*exp(x^2) - 1)^2, x)

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