3.4.52 \(\int \frac {e^{-\frac {20 e^{e^x}}{x}} (-20 x+x^3-3 x^5+e^{e^x} (400+20 x^2-20 x^4+e^x (-400 x-20 x^3+20 x^5)))}{2 x^3} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{2} e^{-\frac {20 e^{e^x}}{x}} \left (\frac {4}{x}+x\right ) \left (5-x^2\right ) \]

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Rubi [B]  time = 0.25, antiderivative size = 74, normalized size of antiderivative = 2.47, number of steps used = 2, number of rules used = 2, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {12, 2288} \begin {gather*} \frac {e^{e^x-\frac {20 e^{e^x}}{x}} \left (-x^4+x^2-e^x \left (-x^5+x^3+20 x\right )+20\right )}{2 \left (\frac {e^{e^x}}{x^2}-\frac {e^{x+e^x}}{x}\right ) x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20*x + x^3 - 3*x^5 + E^E^x*(400 + 20*x^2 - 20*x^4 + E^x*(-400*x - 20*x^3 + 20*x^5)))/(2*E^((20*E^E^x)/x)
*x^3),x]

[Out]

(E^(E^x - (20*E^E^x)/x)*(20 + x^2 - x^4 - E^x*(20*x + x^3 - x^5)))/(2*(E^E^x/x^2 - E^(E^x + x)/x)*x^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{-\frac {20 e^{e^x}}{x}} \left (-20 x+x^3-3 x^5+e^{e^x} \left (400+20 x^2-20 x^4+e^x \left (-400 x-20 x^3+20 x^5\right )\right )\right )}{x^3} \, dx\\ &=\frac {e^{e^x-\frac {20 e^{e^x}}{x}} \left (20+x^2-x^4-e^x \left (20 x+x^3-x^5\right )\right )}{2 \left (\frac {e^{e^x}}{x^2}-\frac {e^{e^x+x}}{x}\right ) x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 29, normalized size = 0.97 \begin {gather*} \frac {e^{-\frac {20 e^{e^x}}{x}} \left (20+x^2-x^4\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*x + x^3 - 3*x^5 + E^E^x*(400 + 20*x^2 - 20*x^4 + E^x*(-400*x - 20*x^3 + 20*x^5)))/(2*E^((20*E^E
^x)/x)*x^3),x]

[Out]

(20 + x^2 - x^4)/(2*E^((20*E^E^x)/x)*x)

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fricas [A]  time = 0.48, size = 24, normalized size = 0.80 \begin {gather*} -\frac {{\left (x^{4} - x^{2} - 20\right )} e^{\left (-\frac {20 \, e^{\left (e^{x}\right )}}{x}\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((20*x^5-20*x^3-400*x)*exp(x)-20*x^4+20*x^2+400)*exp(exp(x))-3*x^5+x^3-20*x)/x^3/exp(20*exp(exp
(x))/x),x, algorithm="fricas")

[Out]

-1/2*(x^4 - x^2 - 20)*e^(-20*e^(e^x)/x)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (3 \, x^{5} - x^{3} + 20 \, {\left (x^{4} - x^{2} - {\left (x^{5} - x^{3} - 20 \, x\right )} e^{x} - 20\right )} e^{\left (e^{x}\right )} + 20 \, x\right )} e^{\left (-\frac {20 \, e^{\left (e^{x}\right )}}{x}\right )}}{2 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((20*x^5-20*x^3-400*x)*exp(x)-20*x^4+20*x^2+400)*exp(exp(x))-3*x^5+x^3-20*x)/x^3/exp(20*exp(exp
(x))/x),x, algorithm="giac")

[Out]

integrate(-1/2*(3*x^5 - x^3 + 20*(x^4 - x^2 - (x^5 - x^3 - 20*x)*e^x - 20)*e^(e^x) + 20*x)*e^(-20*e^(e^x)/x)/x
^3, x)

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maple [A]  time = 0.07, size = 25, normalized size = 0.83

method	result	size
risch	\(-\frac {\left (x^{4}-x^{2}-20\right ) {\mathrm e}^{-\frac {20 \,{\mathrm e}^{{\mathrm e}^{x}}}{x}}}{2 x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((20*x^5-20*x^3-400*x)*exp(x)-20*x^4+20*x^2+400)*exp(exp(x))-3*x^5+x^3-20*x)/x^3/exp(20*exp(exp(x))/x
),x,method=_RETURNVERBOSE)

[Out]

-1/2*(x^4-x^2-20)/x*exp(-20*exp(exp(x))/x)

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maxima [A]  time = 0.60, size = 24, normalized size = 0.80 \begin {gather*} -\frac {{\left (x^{4} - x^{2} - 20\right )} e^{\left (-\frac {20 \, e^{\left (e^{x}\right )}}{x}\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((20*x^5-20*x^3-400*x)*exp(x)-20*x^4+20*x^2+400)*exp(exp(x))-3*x^5+x^3-20*x)/x^3/exp(20*exp(exp
(x))/x),x, algorithm="maxima")

[Out]

-1/2*(x^4 - x^2 - 20)*e^(-20*e^(e^x)/x)/x

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mupad [B]  time = 0.47, size = 24, normalized size = 0.80 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {20\,{\mathrm {e}}^{{\mathrm {e}}^x}}{x}}\,\left (-x^4+x^2+20\right )}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(20*exp(exp(x)))/x)*(10*x - (exp(exp(x))*(20*x^2 - 20*x^4 - exp(x)*(400*x + 20*x^3 - 20*x^5) + 400)
)/2 - x^3/2 + (3*x^5)/2))/x^3,x)

[Out]

(exp(-(20*exp(exp(x)))/x)*(x^2 - x^4 + 20))/(2*x)

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sympy [A]  time = 0.28, size = 20, normalized size = 0.67 \begin {gather*} \frac {\left (- x^{4} + x^{2} + 20\right ) e^{- \frac {20 e^{e^{x}}}{x}}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((20*x**5-20*x**3-400*x)*exp(x)-20*x**4+20*x**2+400)*exp(exp(x))-3*x**5+x**3-20*x)/x**3/exp(20*
exp(exp(x))/x),x)

[Out]

(-x**4 + x**2 + 20)*exp(-20*exp(exp(x))/x)/(2*x)

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