3.75.83 \(\int \frac {380 x^4+100 e^{\frac {2 e^{2 x}}{x^2}} x^4-400 x^5+100 x^6+e^x (-40 x^2+80 x^3-20 x^4)+e^{\frac {e^{2 x}}{x^2}} (e^{3 x} (40-40 x)+400 x^4-200 x^5+e^{2 x} (-40 x+40 x^2)+e^x (-20 x^2+20 x^3))}{e^{2 x} x^2+x^4-20 x^5+110 x^6+25 e^{\frac {2 e^{2 x}}{x^2}} x^6-100 x^7+25 x^8+e^x (-2 x^3+20 x^4-10 x^5)+e^{\frac {e^{2 x}}{x^2}} (10 e^x x^4-10 x^5+100 x^6-50 x^7)} \, dx\)

Optimal. Leaf size=42 \[ \frac {4}{-x+\frac {-e^x+x}{5 \left (2 x+\left (e^{\frac {e^{2 x}}{x^2}}-x\right ) x\right )}} \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(380*x^4 + 100*E^((2*E^(2*x))/x^2)*x^4 - 400*x^5 + 100*x^6 + E^x*(-40*x^2 + 80*x^3 - 20*x^4) + E^(E^(2*x)/
x^2)*(E^(3*x)*(40 - 40*x) + 400*x^4 - 200*x^5 + E^(2*x)*(-40*x + 40*x^2) + E^x*(-20*x^2 + 20*x^3)))/(E^(2*x)*x
^2 + x^4 - 20*x^5 + 110*x^6 + 25*E^((2*E^(2*x))/x^2)*x^6 - 100*x^7 + 25*x^8 + E^x*(-2*x^3 + 20*x^4 - 10*x^5) +
E^(E^(2*x)/x^2)*(10*E^x*x^4 - 10*x^5 + 100*x^6 - 50*x^7)),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.18, size = 53, normalized size = 1.26 \begin {gather*} \frac {20 x \left (-2-e^{\frac {e^{2 x}}{x^2}}+x\right )}{e^x+5 e^{\frac {e^{2 x}}{x^2}} x^2+x \left (-1+10 x-5 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(380*x^4 + 100*E^((2*E^(2*x))/x^2)*x^4 - 400*x^5 + 100*x^6 + E^x*(-40*x^2 + 80*x^3 - 20*x^4) + E^(E^
(2*x)/x^2)*(E^(3*x)*(40 - 40*x) + 400*x^4 - 200*x^5 + E^(2*x)*(-40*x + 40*x^2) + E^x*(-20*x^2 + 20*x^3)))/(E^(
2*x)*x^2 + x^4 - 20*x^5 + 110*x^6 + 25*E^((2*E^(2*x))/x^2)*x^6 - 100*x^7 + 25*x^8 + E^x*(-2*x^3 + 20*x^4 - 10*
x^5) + E^(E^(2*x)/x^2)*(10*E^x*x^4 - 10*x^5 + 100*x^6 - 50*x^7)),x]

[Out]

(20*x*(-2 - E^(E^(2*x)/x^2) + x))/(E^x + 5*E^(E^(2*x)/x^2)*x^2 + x*(-1 + 10*x - 5*x^2))

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fricas [A]  time = 0.91, size = 53, normalized size = 1.26 \begin {gather*} -\frac {20 \, {\left (x^{2} - x e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}}\right )} - 2 \, x\right )}}{5 \, x^{3} - 5 \, x^{2} e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}}\right )} - 10 \, x^{2} + x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*x^4*exp(exp(x)^2/x^2)^2+((-40*x+40)*exp(x)^3+(40*x^2-40*x)*exp(x)^2+(20*x^3-20*x^2)*exp(x)-200*
x^5+400*x^4)*exp(exp(x)^2/x^2)+(-20*x^4+80*x^3-40*x^2)*exp(x)+100*x^6-400*x^5+380*x^4)/(25*x^6*exp(exp(x)^2/x^
2)^2+(10*exp(x)*x^4-50*x^7+100*x^6-10*x^5)*exp(exp(x)^2/x^2)+exp(x)^2*x^2+(-10*x^5+20*x^4-2*x^3)*exp(x)+25*x^8
-100*x^7+110*x^6-20*x^5+x^4),x, algorithm="fricas")

[Out]

-20*(x^2 - x*e^(e^(2*x)/x^2) - 2*x)/(5*x^3 - 5*x^2*e^(e^(2*x)/x^2) - 10*x^2 + x - e^x)

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giac [B]  time = 0.73, size = 708, normalized size = 16.86 \begin {gather*} -\frac {20 \, {\left (5 \, x^{8} e^{x} - 10 \, x^{7} e^{\left (3 \, x\right )} - 10 \, x^{7} e^{x} - 5 \, x^{7} e^{\left (\frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 50 \, x^{6} e^{\left (3 \, x\right )} - x^{6} e^{\left (2 \, x\right )} + 10 \, x^{6} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - x^{6} e^{x} - 82 \, x^{5} e^{\left (3 \, x\right )} + 4 \, x^{5} e^{\left (2 \, x\right )} - 30 \, x^{5} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + x^{5} e^{\left (x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 2 \, x^{5} e^{x} + x^{5} e^{\left (\frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 2 \, x^{4} e^{\left (4 \, x\right )} + 46 \, x^{4} e^{\left (3 \, x\right )} - 4 \, x^{4} e^{\left (2 \, x\right )} + 22 \, x^{4} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 2 \, x^{4} e^{\left (x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 6 \, x^{3} e^{\left (4 \, x\right )} - 4 \, x^{3} e^{\left (3 \, x\right )} - 2 \, x^{3} e^{\left (3 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 2 \, x^{3} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 4 \, x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (3 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )}\right )}}{25 \, x^{9} e^{x} - 50 \, x^{8} e^{\left (3 \, x\right )} - 50 \, x^{8} e^{x} - 25 \, x^{8} e^{\left (\frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 250 \, x^{7} e^{\left (3 \, x\right )} - 5 \, x^{7} e^{\left (2 \, x\right )} + 50 \, x^{7} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 420 \, x^{6} e^{\left (3 \, x\right )} + 15 \, x^{6} e^{\left (2 \, x\right )} - 150 \, x^{6} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 5 \, x^{6} e^{\left (x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 10 \, x^{6} e^{x} + 5 \, x^{6} e^{\left (\frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 20 \, x^{5} e^{\left (4 \, x\right )} + 260 \, x^{5} e^{\left (3 \, x\right )} - 21 \, x^{5} e^{\left (2 \, x\right )} + 110 \, x^{5} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 10 \, x^{5} e^{\left (x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - x^{5} e^{x} - 60 \, x^{4} e^{\left (4 \, x\right )} - 41 \, x^{4} e^{\left (3 \, x\right )} + 3 \, x^{4} e^{\left (2 \, x\right )} - 10 \, x^{4} e^{\left (3 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 10 \, x^{4} e^{\left (2 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} + 44 \, x^{3} e^{\left (4 \, x\right )} + 10 \, x^{3} e^{\left (3 \, x + \frac {x^{3} + e^{\left (2 \, x\right )}}{x^{2}}\right )} - 2 \, x^{2} e^{\left (5 \, x\right )} - 4 \, x^{2} e^{\left (4 \, x\right )} + 2 \, x e^{\left (5 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*x^4*exp(exp(x)^2/x^2)^2+((-40*x+40)*exp(x)^3+(40*x^2-40*x)*exp(x)^2+(20*x^3-20*x^2)*exp(x)-200*
x^5+400*x^4)*exp(exp(x)^2/x^2)+(-20*x^4+80*x^3-40*x^2)*exp(x)+100*x^6-400*x^5+380*x^4)/(25*x^6*exp(exp(x)^2/x^
2)^2+(10*exp(x)*x^4-50*x^7+100*x^6-10*x^5)*exp(exp(x)^2/x^2)+exp(x)^2*x^2+(-10*x^5+20*x^4-2*x^3)*exp(x)+25*x^8
-100*x^7+110*x^6-20*x^5+x^4),x, algorithm="giac")

[Out]

-20*(5*x^8*e^x - 10*x^7*e^(3*x) - 10*x^7*e^x - 5*x^7*e^((x^3 + e^(2*x))/x^2) + 50*x^6*e^(3*x) - x^6*e^(2*x) +
10*x^6*e^(2*x + (x^3 + e^(2*x))/x^2) - x^6*e^x - 82*x^5*e^(3*x) + 4*x^5*e^(2*x) - 30*x^5*e^(2*x + (x^3 + e^(2*
x))/x^2) + x^5*e^(x + (x^3 + e^(2*x))/x^2) + 2*x^5*e^x + x^5*e^((x^3 + e^(2*x))/x^2) + 2*x^4*e^(4*x) + 46*x^4*
e^(3*x) - 4*x^4*e^(2*x) + 22*x^4*e^(2*x + (x^3 + e^(2*x))/x^2) - 2*x^4*e^(x + (x^3 + e^(2*x))/x^2) - 6*x^3*e^(
4*x) - 4*x^3*e^(3*x) - 2*x^3*e^(3*x + (x^3 + e^(2*x))/x^2) - 2*x^3*e^(2*x + (x^3 + e^(2*x))/x^2) + 4*x^2*e^(4*
x) + 2*x^2*e^(3*x + (x^3 + e^(2*x))/x^2))/(25*x^9*e^x - 50*x^8*e^(3*x) - 50*x^8*e^x - 25*x^8*e^((x^3 + e^(2*x)
)/x^2) + 250*x^7*e^(3*x) - 5*x^7*e^(2*x) + 50*x^7*e^(2*x + (x^3 + e^(2*x))/x^2) - 420*x^6*e^(3*x) + 15*x^6*e^(
2*x) - 150*x^6*e^(2*x + (x^3 + e^(2*x))/x^2) + 5*x^6*e^(x + (x^3 + e^(2*x))/x^2) + 10*x^6*e^x + 5*x^6*e^((x^3
+ e^(2*x))/x^2) + 20*x^5*e^(4*x) + 260*x^5*e^(3*x) - 21*x^5*e^(2*x) + 110*x^5*e^(2*x + (x^3 + e^(2*x))/x^2) -
10*x^5*e^(x + (x^3 + e^(2*x))/x^2) - x^5*e^x - 60*x^4*e^(4*x) - 41*x^4*e^(3*x) + 3*x^4*e^(2*x) - 10*x^4*e^(3*x
+ (x^3 + e^(2*x))/x^2) - 10*x^4*e^(2*x + (x^3 + e^(2*x))/x^2) + 44*x^3*e^(4*x) + 10*x^3*e^(3*x + (x^3 + e^(2*
x))/x^2) - 2*x^2*e^(5*x) - 4*x^2*e^(4*x) + 2*x*e^(5*x))

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maple [A]  time = 0.07, size = 50, normalized size = 1.19

method	result	size
risch	\(-\frac {4}{x}+\frac {4 x -4 \,{\mathrm e}^{x}}{x \left (5 x^{3}-5 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{2 x}}{x^{2}}}-10 x^{2}+x -{\mathrm e}^{x}\right )}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x^4*exp(exp(x)^2/x^2)^2+((-40*x+40)*exp(x)^3+(40*x^2-40*x)*exp(x)^2+(20*x^3-20*x^2)*exp(x)-200*x^5+40
0*x^4)*exp(exp(x)^2/x^2)+(-20*x^4+80*x^3-40*x^2)*exp(x)+100*x^6-400*x^5+380*x^4)/(25*x^6*exp(exp(x)^2/x^2)^2+(
10*exp(x)*x^4-50*x^7+100*x^6-10*x^5)*exp(exp(x)^2/x^2)+exp(x)^2*x^2+(-10*x^5+20*x^4-2*x^3)*exp(x)+25*x^8-100*x
^7+110*x^6-20*x^5+x^4),x,method=_RETURNVERBOSE)

[Out]

-4/x+4*(x-exp(x))/x/(5*x^3-5*x^2*exp(exp(2*x)/x^2)-10*x^2+x-exp(x))

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maxima [A]  time = 0.47, size = 53, normalized size = 1.26 \begin {gather*} -\frac {20 \, {\left (x^{2} - x e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}}\right )} - 2 \, x\right )}}{5 \, x^{3} - 5 \, x^{2} e^{\left (\frac {e^{\left (2 \, x\right )}}{x^{2}}\right )} - 10 \, x^{2} + x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*x^4*exp(exp(x)^2/x^2)^2+((-40*x+40)*exp(x)^3+(40*x^2-40*x)*exp(x)^2+(20*x^3-20*x^2)*exp(x)-200*
x^5+400*x^4)*exp(exp(x)^2/x^2)+(-20*x^4+80*x^3-40*x^2)*exp(x)+100*x^6-400*x^5+380*x^4)/(25*x^6*exp(exp(x)^2/x^
2)^2+(10*exp(x)*x^4-50*x^7+100*x^6-10*x^5)*exp(exp(x)^2/x^2)+exp(x)^2*x^2+(-10*x^5+20*x^4-2*x^3)*exp(x)+25*x^8
-100*x^7+110*x^6-20*x^5+x^4),x, algorithm="maxima")

[Out]

-20*(x^2 - x*e^(e^(2*x)/x^2) - 2*x)/(5*x^3 - 5*x^2*e^(e^(2*x)/x^2) - 10*x^2 + x - e^x)

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mupad [B]  time = 4.62, size = 49, normalized size = 1.17 \begin {gather*} -\frac {20\,x\,\left ({\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}}{x^2}}-x+2\right )}{{\mathrm {e}}^x-x+5\,x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}}{x^2}}+10\,x^2-5\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x^4*exp((2*exp(2*x))/x^2) - exp(x)*(40*x^2 - 80*x^3 + 20*x^4) + 380*x^4 - 400*x^5 + 100*x^6 - exp(exp
(2*x)/x^2)*(exp(2*x)*(40*x - 40*x^2) + exp(x)*(20*x^2 - 20*x^3) + exp(3*x)*(40*x - 40) - 400*x^4 + 200*x^5))/(
25*x^6*exp((2*exp(2*x))/x^2) - exp(x)*(2*x^3 - 20*x^4 + 10*x^5) + exp(exp(2*x)/x^2)*(10*x^4*exp(x) - 10*x^5 +
100*x^6 - 50*x^7) + x^2*exp(2*x) + x^4 - 20*x^5 + 110*x^6 - 100*x^7 + 25*x^8),x)

[Out]

-(20*x*(exp(exp(2*x)/x^2) - x + 2))/(exp(x) - x + 5*x^2*exp(exp(2*x)/x^2) + 10*x^2 - 5*x^3)

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sympy [A]  time = 0.41, size = 44, normalized size = 1.05 \begin {gather*} \frac {- 4 x + 4 e^{x}}{- 5 x^{4} + 5 x^{3} e^{\frac {e^{2 x}}{x^{2}}} + 10 x^{3} - x^{2} + x e^{x}} - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*x**4*exp(exp(x)**2/x**2)**2+((-40*x+40)*exp(x)**3+(40*x**2-40*x)*exp(x)**2+(20*x**3-20*x**2)*ex
p(x)-200*x**5+400*x**4)*exp(exp(x)**2/x**2)+(-20*x**4+80*x**3-40*x**2)*exp(x)+100*x**6-400*x**5+380*x**4)/(25*
x**6*exp(exp(x)**2/x**2)**2+(10*exp(x)*x**4-50*x**7+100*x**6-10*x**5)*exp(exp(x)**2/x**2)+exp(x)**2*x**2+(-10*
x**5+20*x**4-2*x**3)*exp(x)+25*x**8-100*x**7+110*x**6-20*x**5+x**4),x)

[Out]

(-4*x + 4*exp(x))/(-5*x**4 + 5*x**3*exp(exp(2*x)/x**2) + 10*x**3 - x**2 + x*exp(x)) - 4/x

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