3.77.71 \(\int \frac {-3 x^3-x^4+e^x (-2 x+x^2)+e^{x^2} (e^2 (-12 x^2-4 x^3)+e^{2+x} (-8+8 x^3))}{4 e^{2+x+x^2} x^2+e^x x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {2}{x}+e^{-x} (4+x)+\log \left (4 e^{2+x^2}+x\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.45, 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 {-3 x^3-x^4+e^x \left (-2 x+x^2\right )+e^{x^2} \left (e^2 \left (-12 x^2-4 x^3\right )+e^{2+x} \left (-8+8 x^3\right )\right )}{4 e^{2+x+x^2} x^2+e^x x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x^3 - x^4 + E^x*(-2*x + x^2) + E^x^2*(E^2*(-12*x^2 - 4*x^3) + E^(2 + x)*(-8 + 8*x^3)))/(4*E^(2 + x + x
^2)*x^2 + E^x*x^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-3 x^3-x^4+e^x \left (-2 x+x^2\right )+e^{x^2} \left (e^2 \left (-12 x^2-4 x^3\right )+e^{2+x} \left (-8+8 x^3\right )\right )\right )}{x^2 \left (4 e^{2+x^2}+x\right )} \, dx\\ &=\int \left (-\frac {-1+2 x^2}{4 e^{2+x^2}+x}+\frac {e^{-x} \left (-2 e^x-3 x^2-x^3+2 e^x x^3\right )}{x^2}\right ) \, dx\\ &=-\int \frac {-1+2 x^2}{4 e^{2+x^2}+x} \, dx+\int \frac {e^{-x} \left (-2 e^x-3 x^2-x^3+2 e^x x^3\right )}{x^2} \, dx\\ &=\int \left (-\frac {2}{x^2}+2 x-e^{-x} (3+x)\right ) \, dx-\int \left (-\frac {1}{4 e^{2+x^2}+x}+\frac {2 x^2}{4 e^{2+x^2}+x}\right ) \, dx\\ &=\frac {2}{x}+x^2-2 \int \frac {x^2}{4 e^{2+x^2}+x} \, dx-\int e^{-x} (3+x) \, dx+\int \frac {1}{4 e^{2+x^2}+x} \, dx\\ &=\frac {2}{x}+x^2+e^{-x} (3+x)-2 \int \frac {x^2}{4 e^{2+x^2}+x} \, dx-\int e^{-x} \, dx+\int \frac {1}{4 e^{2+x^2}+x} \, dx\\ &=e^{-x}+\frac {2}{x}+x^2+e^{-x} (3+x)-2 \int \frac {x^2}{4 e^{2+x^2}+x} \, dx+\int \frac {1}{4 e^{2+x^2}+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.63, size = 27, normalized size = 1.00 \begin {gather*} \frac {2}{x}+e^{-x} (4+x)+\log \left (4 e^{2+x^2}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^3 - x^4 + E^x*(-2*x + x^2) + E^x^2*(E^2*(-12*x^2 - 4*x^3) + E^(2 + x)*(-8 + 8*x^3)))/(4*E^(2 +
x + x^2)*x^2 + E^x*x^3),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.38, size = 59, normalized size = 2.19 \begin {gather*} \frac {{\left (x e^{\left (x^{2} + x + 2\right )} \log \left (x + 4 \, e^{\left (x^{2} + 2\right )}\right ) + {\left (x^{2} + 4 \, x\right )} e^{\left (x^{2} + 2\right )} + 2 \, e^{\left (x^{2} + x + 2\right )}\right )} e^{\left (-x^{2} - x - 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3-8)*exp(2)*exp(x)+(-4*x^3-12*x^2)*exp(2))*exp(x^2)+(x^2-2*x)*exp(x)-x^4-3*x^3)/(4*x^2*exp(2)
*exp(x)*exp(x^2)+exp(x)*x^3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.21, size = 48, normalized size = 1.78 \begin {gather*} -\frac {{\left (x^{2} e^{x} - x e^{x} \log \left (x e^{x} + 4 \, e^{\left (x^{2} + x + 2\right )}\right ) - x^{2} - 4 \, x - 2 \, e^{x}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3-8)*exp(2)*exp(x)+(-4*x^3-12*x^2)*exp(2))*exp(x^2)+(x^2-2*x)*exp(x)-x^4-3*x^3)/(4*x^2*exp(2)
*exp(x)*exp(x^2)+exp(x)*x^3),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.11, size = 26, normalized size = 0.96

method	result	size
risch	\(\frac {2}{x}+\left (4+x \right ) {\mathrm e}^{-x}+\ln \left ({\mathrm e}^{x^{2}}+\frac {x \,{\mathrm e}^{-2}}{4}\right )\)	\(26\)
norman	\(\frac {\left (x^{2}+4 x +2 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}+\ln \left (x +4 \,{\mathrm e}^{x^{2}} {\mathrm e}^{2}\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^3-8)*exp(2)*exp(x)+(-4*x^3-12*x^2)*exp(2))*exp(x^2)+(x^2-2*x)*exp(x)-x^4-3*x^3)/(4*x^2*exp(2)*exp(x
)*exp(x^2)+exp(x)*x^3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 34, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{2} + 4 \, x\right )} e^{\left (-x\right )} + 2}{x} + \log \left (\frac {1}{4} \, {\left (x + 4 \, e^{\left (x^{2} + 2\right )}\right )} e^{\left (-2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3-8)*exp(2)*exp(x)+(-4*x^3-12*x^2)*exp(2))*exp(x^2)+(x^2-2*x)*exp(x)-x^4-3*x^3)/(4*x^2*exp(2)
*exp(x)*exp(x^2)+exp(x)*x^3),x, algorithm="maxima")

[Out]

((x^2 + 4*x)*e^(-x) + 2)/x + log(1/4*(x + 4*e^(x^2 + 2))*e^(-2))

________________________________________________________________________________________

mupad [B]  time = 0.25, size = 29, normalized size = 1.07 \begin {gather*} 4\,{\mathrm {e}}^{-x}+\ln \left ({\mathrm {e}}^{x^2}+\frac {x\,{\mathrm {e}}^{-2}}{4}\right )+x\,{\mathrm {e}}^{-x}+\frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(2*x - x^2) + exp(x^2)*(exp(2)*(12*x^2 + 4*x^3) - exp(2)*exp(x)*(8*x^3 - 8)) + 3*x^3 + x^4)/(x^3*
exp(x) + 4*x^2*exp(x^2)*exp(2)*exp(x)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 22, normalized size = 0.81 \begin {gather*} \left (x + 4\right ) e^{- x} + \log {\left (\frac {x}{4 e^{2}} + e^{x^{2}} \right )} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**3-8)*exp(2)*exp(x)+(-4*x**3-12*x**2)*exp(2))*exp(x**2)+(x**2-2*x)*exp(x)-x**4-3*x**3)/(4*x**
2*exp(2)*exp(x)*exp(x**2)+exp(x)*x**3),x)

[Out]

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

________________________________________________________________________________________