3.46.7 \(\int \frac {e^{-6+x^2+2 e^{4/x} x^2+e^{8/x} x^2} (e^2 (-1+2 x^2)+e^{2+\frac {8}{x}} (-8 x+2 x^2)+e^{2+\frac {4}{x}} (-8 x+4 x^2))}{x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac {e^{-4+\left (x+e^{4/x} x\right )^2}}{x} \]

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Rubi [B]  time = 0.92, antiderivative size = 93, normalized size of antiderivative = 4.43, number of steps used = 2, number of rules used = 2, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6688, 2288} \begin {gather*} \frac {e^{\left (e^{4/x}+1\right )^2 x^2-4} \left (-x^2+2 e^{4/x} (2-x) x+e^{8/x} (4-x) x\right )}{x^2 \left (4 e^{4/x} \left (e^{4/x}+1\right )-\left (e^{4/x}+1\right )^2 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-6 + x^2 + 2*E^(4/x)*x^2 + E^(8/x)*x^2)*(E^2*(-1 + 2*x^2) + E^(2 + 8/x)*(-8*x + 2*x^2) + E^(2 + 4/x)*(
-8*x + 4*x^2)))/x^2,x]

[Out]

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.34, size = 23, normalized size = 1.10 \begin {gather*} \frac {e^{-4+\left (1+e^{4/x}\right )^2 x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 + x^2 + 2*E^(4/x)*x^2 + E^(8/x)*x^2)*(E^2*(-1 + 2*x^2) + E^(2 + 8/x)*(-8*x + 2*x^2) + E^(2 +
4/x)*(-8*x + 4*x^2)))/x^2,x]

[Out]

E^(-4 + (1 + E^(4/x))^2*x^2)/x

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fricas [B]  time = 0.73, size = 48, normalized size = 2.29 \begin {gather*} \frac {e^{\left ({\left (x^{2} e^{\left (\frac {4 \, {\left (x + 2\right )}}{x}\right )} + 2 \, x^{2} e^{\left (\frac {2 \, {\left (x + 2\right )}}{x} + 2\right )} + {\left (x^{2} - 6\right )} e^{4}\right )} e^{\left (-4\right )} + 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-8*x)*exp(2)*exp(4/x)^2+(4*x^2-8*x)*exp(2)*exp(4/x)+(2*x^2-1)*exp(2))/x^2/exp(-x^2*exp(4/x)^2
-2*x^2*exp(4/x)-x^2+6),x, algorithm="fricas")

[Out]

e^((x^2*e^(4*(x + 2)/x) + 2*x^2*e^(2*(x + 2)/x + 2) + (x^2 - 6)*e^4)*e^(-4) + 2)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (2 \, x^{2} - 1\right )} e^{2} + 2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (\frac {8}{x} + 2\right )} + 4 \, {\left (x^{2} - 2 \, x\right )} e^{\left (\frac {4}{x} + 2\right )}\right )} e^{\left (x^{2} e^{\frac {8}{x}} + 2 \, x^{2} e^{\frac {4}{x}} + x^{2} - 6\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-8*x)*exp(2)*exp(4/x)^2+(4*x^2-8*x)*exp(2)*exp(4/x)+(2*x^2-1)*exp(2))/x^2/exp(-x^2*exp(4/x)^2
-2*x^2*exp(4/x)-x^2+6),x, algorithm="giac")

[Out]

integrate(((2*x^2 - 1)*e^2 + 2*(x^2 - 4*x)*e^(8/x + 2) + 4*(x^2 - 2*x)*e^(4/x + 2))*e^(x^2*e^(8/x) + 2*x^2*e^(
4/x) + x^2 - 6)/x^2, x)

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maple [A]  time = 0.14, size = 32, normalized size = 1.52

method	result	size
risch	\(\frac {{\mathrm e}^{-4+x^{2} {\mathrm e}^{\frac {8}{x}}+2 x^{2} {\mathrm e}^{\frac {4}{x}}+x^{2}}}{x}\)	\(32\)
norman	\(\frac {{\mathrm e}^{2} {\mathrm e}^{x^{2} {\mathrm e}^{\frac {8}{x}}+2 x^{2} {\mathrm e}^{\frac {4}{x}}+x^{2}-6}}{x}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-8*x)*exp(2)*exp(4/x)^2+(4*x^2-8*x)*exp(2)*exp(4/x)+(2*x^2-1)*exp(2))/x^2/exp(-x^2*exp(4/x)^2-2*x^2
*exp(4/x)-x^2+6),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.54, size = 31, normalized size = 1.48 \begin {gather*} \frac {e^{\left (x^{2} e^{\frac {8}{x}} + 2 \, x^{2} e^{\frac {4}{x}} + x^{2} - 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-8*x)*exp(2)*exp(4/x)^2+(4*x^2-8*x)*exp(2)*exp(4/x)+(2*x^2-1)*exp(2))/x^2/exp(-x^2*exp(4/x)^2
-2*x^2*exp(4/x)-x^2+6),x, algorithm="maxima")

[Out]

e^(x^2*e^(8/x) + 2*x^2*e^(4/x) + x^2 - 4)/x

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mupad [B]  time = 3.35, size = 33, normalized size = 1.57 \begin {gather*} \frac {{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{4/x}}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{8/x}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x^2*exp(4/x) + x^2*exp(8/x) + x^2 - 6)*(exp(2)*exp(4/x)*(8*x - 4*x^2) - exp(2)*(2*x^2 - 1) + exp(2
)*exp(8/x)*(8*x - 2*x^2)))/x^2,x)

[Out]

(exp(2*x^2*exp(4/x))*exp(x^2*exp(8/x))*exp(x^2)*exp(-4))/x

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sympy [A]  time = 0.35, size = 29, normalized size = 1.38 \begin {gather*} \frac {e^{2} e^{x^{2} e^{\frac {8}{x}} + 2 x^{2} e^{\frac {4}{x}} + x^{2} - 6}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-8*x)*exp(2)*exp(4/x)**2+(4*x**2-8*x)*exp(2)*exp(4/x)+(2*x**2-1)*exp(2))/x**2/exp(-x**2*exp(
4/x)**2-2*x**2*exp(4/x)-x**2+6),x)

[Out]

exp(2)*exp(x**2*exp(8/x) + 2*x**2*exp(4/x) + x**2 - 6)/x

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