3.4.86 \(\int \frac {e^{\frac {x}{2 e^x+x}} (-4 e^{2 x}-x^2+e^x (-2 x-2 x^2))}{4 e^{2 x} x^2+4 e^x x^3+x^4} \, dx\) [386]

Optimal. Leaf size=17 \[ \frac {e^{\frac {x}{2 e^x+x}}}{x} \]

[Out]

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

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Rubi [F]
time = 1.50, 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 {x}{2 e^x+x}} \left (-4 e^{2 x}-x^2+e^x \left (-2 x-2 x^2\right )\right )}{4 e^{2 x} x^2+4 e^x x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 17, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {x}{2 e^x+x}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(x/(2*E^x + x))/x

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Maple [A]
time = 0.10, size = 16, normalized size = 0.94

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x}{2 \,{\mathrm e}^{x}+x}}}{x}\)	\(16\)
norman	\(\frac {x \,{\mathrm e}^{\frac {x}{2 \,{\mathrm e}^{x}+x}}+2 \,{\mathrm e}^{x} {\mathrm e}^{\frac {x}{2 \,{\mathrm e}^{x}+x}}}{x \left (2 \,{\mathrm e}^{x}+x \right )}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} \frac {e^{\left (\frac {x}{x + 2 \, e^{x}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x/(x + 2*e^x))/x

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Sympy [A]
time = 0.11, size = 10, normalized size = 0.59 \begin {gather*} \frac {e^{\frac {x}{x + 2 e^{x}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x/(x + 2*exp(x)))/x

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Giac [A]
time = 0.41, size = 15, normalized size = 0.88 \begin {gather*} \frac {e^{\left (\frac {x}{x + 2 \, e^{x}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x/(x + 2*e^x))/x

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Mupad [B]
time = 0.77, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x}{x+2\,{\mathrm {e}}^x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x/(x + 2*exp(x)))/x

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