3.14.86 \(\int \frac {7 x^2+4 e^{e^{\frac {1+4 x-x^2}{x}}+\frac {1+4 x-x^2}{x}} (-2-2 x^2)}{x^2} \, dx\) [1386]

Optimal. Leaf size=23 \[ 1+5 x+2 \left (4 e^{e^{4+\frac {1}{x}-x}}+x\right ) \]

[Out]

2*exp(2*ln(2)+exp(1/x-x+4))+7*x+1

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Rubi [F]
time = 0.32, 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 {7 x^2+4 e^{e^{\frac {1+4 x-x^2}{x}}+\frac {1+4 x-x^2}{x}} \left (-2-2 x^2\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(7*x^2 + 4*E^(E^((1 + 4*x - x^2)/x) + (1 + 4*x - x^2)/x)*(-2 - 2*x^2))/x^2,x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (7-\frac {8 e^{4+e^{4+\frac {1}{x}-x}+\frac {1}{x}-x} \left (1+x^2\right )}{x^2}\right ) \, dx\\ &=7 x-8 \int \frac {e^{4+e^{4+\frac {1}{x}-x}+\frac {1}{x}-x} \left (1+x^2\right )}{x^2} \, dx\\ &=7 x-8 \int \left (e^{4+e^{4+\frac {1}{x}-x}+\frac {1}{x}-x}+\frac {e^{4+e^{4+\frac {1}{x}-x}+\frac {1}{x}-x}}{x^2}\right ) \, dx\\ &=7 x-8 \int e^{4+e^{4+\frac {1}{x}-x}+\frac {1}{x}-x} \, dx-8 \int \frac {e^{4+e^{4+\frac {1}{x}-x}+\frac {1}{x}-x}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.10, size = 18, normalized size = 0.78 \begin {gather*} 8 e^{e^{4+\frac {1}{x}-x}}+7 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7*x^2 + 4*E^(E^((1 + 4*x - x^2)/x) + (1 + 4*x - x^2)/x)*(-2 - 2*x^2))/x^2,x]

[Out]

8*E^E^(4 + x^(-1) - x) + 7*x

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Maple [A]
time = 0.31, size = 22, normalized size = 0.96

method	result	size
risch	\(7 x +8 \,{\mathrm e}^{{\mathrm e}^{-\frac {x^{2}-4 x -1}{x}}}\)	\(22\)
norman	\(\frac {7 x^{2}+2 x \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{2}+4 x +1}{x}}+2 \ln \left (2\right )}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-2)*exp((-x^2+4*x+1)/x)*exp(exp((-x^2+4*x+1)/x)+2*ln(2))+7*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

7*x+8*exp(exp(-(x^2-4*x-1)/x))

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Maxima [A]
time = 0.36, size = 16, normalized size = 0.70 \begin {gather*} 7 \, x + 8 \, e^{\left (e^{\left (-x + \frac {1}{x} + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-2)*exp((-x^2+4*x+1)/x)*exp(exp((-x^2+4*x+1)/x)+2*log(2))+7*x^2)/x^2,x, algorithm="maxima")

[Out]

7*x + 8*e^(e^(-x + 1/x + 4))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
time = 0.39, size = 70, normalized size = 3.04 \begin {gather*} {\left (7 \, x e^{\left (-\frac {x^{2} - 4 \, x - 1}{x}\right )} + 2 \, e^{\left (-\frac {x^{2} - x e^{\left (-\frac {x^{2} - 4 \, x - 1}{x}\right )} - 2 \, x \log \left (2\right ) - 4 \, x - 1}{x}\right )}\right )} e^{\left (\frac {x^{2} - 4 \, x - 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-2)*exp((-x^2+4*x+1)/x)*exp(exp((-x^2+4*x+1)/x)+2*log(2))+7*x^2)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.09, size = 17, normalized size = 0.74 \begin {gather*} 7 x + 8 e^{e^{\frac {- x^{2} + 4 x + 1}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-2)*exp((-x**2+4*x+1)/x)*exp(exp((-x**2+4*x+1)/x)+2*ln(2))+7*x**2)/x**2,x)

[Out]

7*x + 8*exp(exp((-x**2 + 4*x + 1)/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^2-2)*exp((-x^2+4*x+1)/x)*exp(exp((-x^2+4*x+1)/x)+2*log(2))+7*x^2)/x^2,x, algorithm="giac")

[Out]

integrate((7*x^2 - 2*(x^2 + 1)*e^(-(x^2 - 4*x - 1)/x + e^(-(x^2 - 4*x - 1)/x) + 2*log(2)))/x^2, x)

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Mupad [B]
time = 1.08, size = 18, normalized size = 0.78 \begin {gather*} 7\,x+8\,{\mathrm {e}}^{{\mathrm {e}}^{-x}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^2 - exp((4*x - x^2 + 1)/x)*exp(exp((4*x - x^2 + 1)/x) + 2*log(2))*(2*x^2 + 2))/x^2,x)

[Out]

7*x + 8*exp(exp(-x)*exp(1/x)*exp(4))

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