3.100.33 \(\int \frac {e^{-x^2} (e^{3+\frac {e^{3-x^2}}{x+x^4}} (-1-2 x^2-4 x^3-2 x^5)+e^{x+x^2} (x^2+2 x^5+x^8))}{x^2+2 x^5+x^8} \, dx\) [9933]

Optimal. Leaf size=28 \[ -e^4+e^x+e^{\frac {e^{3-x^2}}{x+x^4}} \]

[Out]

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

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Rubi [F]
time = 7.67, 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^{-x^2} \left (e^{3+\frac {e^{3-x^2}}{x+x^4}} \left (-1-2 x^2-4 x^3-2 x^5\right )+e^{x+x^2} \left (x^2+2 x^5+x^8\right )\right )}{x^2+2 x^5+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3 + E^(3 - x^2)/(x + x^4))*(-1 - 2*x^2 - 4*x^3 - 2*x^5) + E^(x + x^2)*(x^2 + 2*x^5 + x^8))/(E^x^2*(x^2
+ 2*x^5 + x^8)),x]

[Out]

E^x + (4*Defer[Int][E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(1 + I*Sqrt[3] - 2*x)^2, x])/3 - (((4*I)/3)*Defer[Int]
[E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(1 + I*Sqrt[3] - 2*x), x])/Sqrt[3] - Defer[Int][E^(3 - x^2 + E^(3 - x^2)/
(x + x^4))/x^2, x] + Defer[Int][E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(1 + x)^2, x]/3 - (2*Defer[Int][E^(3 - x^2
+ E^(3 - x^2)/(x + x^4))/(1 + x), x])/3 + (2*(3 + I*Sqrt[3])*Defer[Int][E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(
-1 - I*Sqrt[3] + 2*x), x])/9 + (4*Defer[Int][E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(-1 + I*Sqrt[3] + 2*x)^2, x])
/3 - (((4*I)/3)*Defer[Int][E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(-1 + I*Sqrt[3] + 2*x), x])/Sqrt[3] + (2*(3 - I
*Sqrt[3])*Defer[Int][E^(3 - x^2 + E^(3 - x^2)/(x + x^4))/(-1 + I*Sqrt[3] + 2*x), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x^2} \left (e^{3+\frac {e^{3-x^2}}{x+x^4}} \left (-1-2 x^2-4 x^3-2 x^5\right )+e^{x+x^2} \left (x^2+2 x^5+x^8\right )\right )}{x^2 \left (1+2 x^3+x^6\right )} \, dx\\ &=\int \frac {e^{-x^2} \left (e^{3+\frac {e^{3-x^2}}{x+x^4}} \left (-1-2 x^2-4 x^3-2 x^5\right )+e^{x+x^2} \left (x^2+2 x^5+x^8\right )\right )}{x^2 \left (1+x^3\right )^2} \, dx\\ &=\int \left (e^x-\frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}} \left (1+2 x^2+4 x^3+2 x^5\right )}{x^2 \left (1+x^3\right )^2}\right ) \, dx\\ &=\int e^x \, dx-\int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}} \left (1+2 x^2+4 x^3+2 x^5\right )}{x^2 \left (1+x^3\right )^2} \, dx\\ &=e^x-\int \left (\frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{x^2}-\frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{3 (1+x)^2}+\frac {2 e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{3 (1+x)}+\frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{\left (1-x+x^2\right )^2}-\frac {2 e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}} (-1+x)}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=e^x+\frac {1}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{(1+x)^2} \, dx-\frac {2}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{1+x} \, dx+\frac {2}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}} (-1+x)}{1-x+x^2} \, dx-\int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{x^2} \, dx-\int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{\left (1-x+x^2\right )^2} \, dx\\ &=e^x+\frac {1}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{(1+x)^2} \, dx-\frac {2}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{1+x} \, dx+\frac {2}{3} \int \left (\frac {\left (1+\frac {i}{\sqrt {3}}\right ) e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{-1+i \sqrt {3}+2 x}\right ) \, dx-\int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{x^2} \, dx-\int \left (-\frac {4 e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {4 i e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {4 e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {4 i e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=e^x+\frac {1}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{(1+x)^2} \, dx-\frac {2}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{1+x} \, dx+\frac {4}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {(4 i) \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}-\frac {(4 i) \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{-1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}+\frac {1}{9} \left (2 \left (3-i \sqrt {3}\right )\right ) \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (3+i \sqrt {3}\right )\right ) \int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{-1-i \sqrt {3}+2 x} \, dx-\int \frac {e^{3-x^2+\frac {e^{3-x^2}}{x+x^4}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.34, size = 26, normalized size = 0.93 \begin {gather*} e^x+e^{\frac {e^{3-x^2}}{x \left (1+x^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 + E^(3 - x^2)/(x + x^4))*(-1 - 2*x^2 - 4*x^3 - 2*x^5) + E^(x + x^2)*(x^2 + 2*x^5 + x^8))/(E^x^
2*(x^2 + 2*x^5 + x^8)),x]

[Out]

E^x + E^(E^(3 - x^2)/(x*(1 + x^3)))

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Maple [A]
time = 0.14, size = 32, normalized size = 1.14

method	result	size
risch	\({\mathrm e}^{x}+{\mathrm e}^{\frac {{\mathrm e}^{-x^{2}+3}}{x \left (x +1\right ) \left (x^{2}-x +1\right )}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^5-4*x^3-2*x^2-1)*exp(3)*exp(exp(3)/(x^4+x)/exp(x^2))+(x^8+2*x^5+x^2)*exp(x)*exp(x^2))/(x^8+2*x^5+x^
2)/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (24) = 48\).
time = 0.53, size = 91, normalized size = 3.25 \begin {gather*} {\left (e^{\left (x + \frac {e^{\left (-x^{2} + 3\right )}}{3 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {2 \, x e^{\left (-x^{2} + 3\right )}}{3 \, {\left (x^{2} - x + 1\right )}} + \frac {e^{\left (-x^{2} + 3\right )}}{3 \, {\left (x^{2} - x + 1\right )}} + \frac {e^{\left (-x^{2} + 3\right )}}{x}\right )}\right )} e^{\left (-\frac {e^{\left (-x^{2} + 3\right )}}{3 \, {\left (x + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^5-4*x^3-2*x^2-1)*exp(3)*exp(exp(3)/(x^4+x)/exp(x^2))+(x^8+2*x^5+x^2)*exp(x)*exp(x^2))/(x^8+2*
x^5+x^2)/exp(x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.40, size = 34, normalized size = 1.21 \begin {gather*} {\left (e^{\left (x + 3\right )} + e^{\left (\frac {3 \, x^{4} + 3 \, x + e^{\left (-x^{2} + 3\right )}}{x^{4} + x}\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^5-4*x^3-2*x^2-1)*exp(3)*exp(exp(3)/(x^4+x)/exp(x^2))+(x^8+2*x^5+x^2)*exp(x)*exp(x^2))/(x^8+2*
x^5+x^2)/exp(x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.34, size = 17, normalized size = 0.61 \begin {gather*} e^{x} + e^{\frac {e^{3} e^{- x^{2}}}{x^{4} + x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**5-4*x**3-2*x**2-1)*exp(3)*exp(exp(3)/(x**4+x)/exp(x**2))+(x**8+2*x**5+x**2)*exp(x)*exp(x**2)
)/(x**8+2*x**5+x**2)/exp(x**2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^5-4*x^3-2*x^2-1)*exp(3)*exp(exp(3)/(x^4+x)/exp(x^2))+(x^8+2*x^5+x^2)*exp(x)*exp(x^2))/(x^8+2*
x^5+x^2)/exp(x^2),x, algorithm="giac")

[Out]

integrate(((x^8 + 2*x^5 + x^2)*e^(x^2 + x) - (2*x^5 + 4*x^3 + 2*x^2 + 1)*e^(e^(-x^2 + 3)/(x^4 + x) + 3))*e^(-x
^2)/(x^8 + 2*x^5 + x^2), x)

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Mupad [B]
time = 7.82, size = 20, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^3\,{\mathrm {e}}^{-x^2}}{x^4+x}}+{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x^2)*(exp(3)*exp((exp(3)*exp(-x^2))/(x + x^4))*(2*x^2 + 4*x^3 + 2*x^5 + 1) - exp(x^2)*exp(x)*(x^2 +
2*x^5 + x^8)))/(x^2 + 2*x^5 + x^8),x)

[Out]

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

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