3.16.71 \(\int \frac {e^{\frac {-x^5+e^x (-2 x^3+x^4)}{-4-20 x^4-5 x^5+e^x (-40 x^2+10 x^3+5 x^4)}} (20 x^4+20 x^8+e^{2 x} (80 x^4-80 x^5+20 x^6)+e^x (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7))}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8)+e^x (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9)} \, dx\)
Optimal. Leaf size=33 \[ e^{\frac {x}{x+4 (5+x)+\frac {4}{x^2 \left (e^x (2-x)+x^2\right )}}} \]
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Rubi [F] time = 74.84, 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 {\exp \left (\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}\right ) \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(E^((-x^5 + E^x*(-2*x^3 + x^4))/(-4 - 20*x^4 - 5*x^5 + E^x*(-40*x^2 + 10*x^3 + 5*x^4)))*(20*x^4 + 20*x^8 +
E^(2*x)*(80*x^4 - 80*x^5 + 20*x^6) + E^x*(24*x^2 - 8*x^3 - 4*x^4 + 80*x^6 - 40*x^7)))/(16 + 160*x^4 + 40*x^5
+ 400*x^8 + 200*x^9 + 25*x^10 + E^(2*x)*(1600*x^4 - 800*x^5 - 300*x^6 + 100*x^7 + 25*x^8) + E^x*(320*x^2 - 80*
x^3 - 40*x^4 + 1600*x^6 - 300*x^8 - 50*x^9)),x]
[Out]
(4*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + x)^2), x]
)/5 - (336*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(-4 -
40*E^x*x^2 + 10*E^x*x^3 - 20*x^4 + 5*E^x*x^4 - 5*x^5)^2), x])/5 - (4*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)
/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(-4 - 40*E^x*x^2 + 10*E^x*x^3 - 20*x^4 + 5*E^x*x^4 - 5*x^5
)), x])/5 - (1936*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))
*(-2 + x)*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)^2), x])/15 - 32*Defer[Int][x/(E^((E^x*(-2
+ x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*
E^x*x^4 + 5*x^5)^2), x] - 16*Defer[Int][x^2/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 +
2*x + x^2)))*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)^2), x] - 8*Defer[Int][x^3/(E^((E^x*(-
2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5
*E^x*x^4 + 5*x^5)^2), x] + 4*Defer[Int][x^4/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 +
2*x + x^2)))*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)^2), x] - 4*Defer[Int][x^5/(E^((E^x*(-
2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5
*E^x*x^4 + 5*x^5)^2), x] + (64*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 +
2*x + x^2)))*(4 + x)^2*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)^2), x])/5 + (16*Defer[Int][
1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + x)*(4 + 40*E^x*x^2 - 10
*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)^2), x])/15 + (4*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 +
5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(-2 + x)*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)), x]
)/15 - (32*Defer[Int][1/(E^((E^x*(-2 + x)*x^3 - x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + x
)^2*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E^x*x^4 + 5*x^5)), x])/5 - (4*Defer[Int][1/(E^((E^x*(-2 + x)*x^3
- x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))*(4 + x)*(4 + 40*E^x*x^2 - 10*E^x*x^3 + 20*x^4 - 5*E
^x*x^4 + 5*x^5)), x])/15
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) x^2 \left (5 e^{2 x} (-2+x)^2 x^2-e^x \left (-6+2 x+x^2-20 x^4+10 x^5\right )+5 \left (x^2+x^6\right )\right )}{\left (4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )\right )^2} \, dx\\ &=4 \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) x^2 \left (5 e^{2 x} (-2+x)^2 x^2-e^x \left (-6+2 x+x^2-20 x^4+10 x^5\right )+5 \left (x^2+x^6\right )\right )}{\left (4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )\right )^2} \, dx\\ &=4 \int \left (\frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right )}{5 (4+x)^2}+\frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-8-6 x+6 x^2+x^3\right )}{5 (-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )}-\frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-64-8 x+24 x^2+4 x^3+320 x^4-80 x^5-20 x^6+25 x^7+5 x^8\right )}{5 (-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}\right ) \, dx\\ &=\frac {4}{5} \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right )}{(4+x)^2} \, dx+\frac {4}{5} \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-8-6 x+6 x^2+x^3\right )}{(-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )} \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-64-8 x+24 x^2+4 x^3+320 x^4-80 x^5-20 x^6+25 x^7+5 x^8\right )}{(-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.27, size = 48, normalized size = 1.45 \begin {gather*} e^{\frac {-e^x (-2+x) x^3+x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^((-x^5 + E^x*(-2*x^3 + x^4))/(-4 - 20*x^4 - 5*x^5 + E^x*(-40*x^2 + 10*x^3 + 5*x^4)))*(20*x^4 + 20
*x^8 + E^(2*x)*(80*x^4 - 80*x^5 + 20*x^6) + E^x*(24*x^2 - 8*x^3 - 4*x^4 + 80*x^6 - 40*x^7)))/(16 + 160*x^4 + 4
0*x^5 + 400*x^8 + 200*x^9 + 25*x^10 + E^(2*x)*(1600*x^4 - 800*x^5 - 300*x^6 + 100*x^7 + 25*x^8) + E^x*(320*x^2
- 80*x^3 - 40*x^4 + 1600*x^6 - 300*x^8 - 50*x^9)),x]
[Out]
E^((-(E^x*(-2 + x)*x^3) + x^5)/(4 + 20*x^4 + 5*x^5 - 5*E^x*x^2*(-8 + 2*x + x^2)))
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fricas [A] time = 0.68, size = 51, normalized size = 1.55 \begin {gather*} e^{\left (\frac {x^{5} - {\left (x^{4} - 2 \, x^{3}\right )} e^{x}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((20*x^6-80*x^5+80*x^4)*exp(x)^2+(-40*x^7+80*x^6-4*x^4-8*x^3+24*x^2)*exp(x)+20*x^8+20*x^4)*exp(((x^4
-2*x^3)*exp(x)-x^5)/((5*x^4+10*x^3-40*x^2)*exp(x)-5*x^5-20*x^4-4))/((25*x^8+100*x^7-300*x^6-800*x^5+1600*x^4)*
exp(x)^2+(-50*x^9-300*x^8+1600*x^6-40*x^4-80*x^3+320*x^2)*exp(x)+25*x^10+200*x^9+400*x^8+40*x^5+160*x^4+16),x,
algorithm="fricas")
[Out]
e^((x^5 - (x^4 - 2*x^3)*e^x)/(5*x^5 + 20*x^4 - 5*(x^4 + 2*x^3 - 8*x^2)*e^x + 4))
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giac [A] time = 4.22, size = 55, normalized size = 1.67 \begin {gather*} e^{\left (\frac {x^{5} - x^{4} e^{x} + 2 \, x^{3} e^{x}}{5 \, x^{5} - 5 \, x^{4} e^{x} + 20 \, x^{4} - 10 \, x^{3} e^{x} + 40 \, x^{2} e^{x} + 4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((20*x^6-80*x^5+80*x^4)*exp(x)^2+(-40*x^7+80*x^6-4*x^4-8*x^3+24*x^2)*exp(x)+20*x^8+20*x^4)*exp(((x^4
-2*x^3)*exp(x)-x^5)/((5*x^4+10*x^3-40*x^2)*exp(x)-5*x^5-20*x^4-4))/((25*x^8+100*x^7-300*x^6-800*x^5+1600*x^4)*
exp(x)^2+(-50*x^9-300*x^8+1600*x^6-40*x^4-80*x^3+320*x^2)*exp(x)+25*x^10+200*x^9+400*x^8+40*x^5+160*x^4+16),x,
algorithm="giac")
[Out]
e^((x^5 - x^4*e^x + 2*x^3*e^x)/(5*x^5 - 5*x^4*e^x + 20*x^4 - 10*x^3*e^x + 40*x^2*e^x + 4))
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maple [A] time = 0.21, size = 55, normalized size = 1.67
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\({\mathrm e}^{\frac {x^{3} \left ({\mathrm e}^{x} x -x^{2}-2 \,{\mathrm e}^{x}\right )}{5 \,{\mathrm e}^{x} x^{4}-5 x^{5}+10 \,{\mathrm e}^{x} x^{3}-20 x^{4}-40 \,{\mathrm e}^{x} x^{2}-4}}\) |
\(55\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((20*x^6-80*x^5+80*x^4)*exp(x)^2+(-40*x^7+80*x^6-4*x^4-8*x^3+24*x^2)*exp(x)+20*x^8+20*x^4)*exp(((x^4-2*x^3
)*exp(x)-x^5)/((5*x^4+10*x^3-40*x^2)*exp(x)-5*x^5-20*x^4-4))/((25*x^8+100*x^7-300*x^6-800*x^5+1600*x^4)*exp(x)
^2+(-50*x^9-300*x^8+1600*x^6-40*x^4-80*x^3+320*x^2)*exp(x)+25*x^10+200*x^9+400*x^8+40*x^5+160*x^4+16),x,method
=_RETURNVERBOSE)
[Out]
exp(x^3*(exp(x)*x-x^2-2*exp(x))/(5*exp(x)*x^4-5*x^5+10*exp(x)*x^3-20*x^4-40*exp(x)*x^2-4))
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maxima [B] time = 1.66, size = 152, normalized size = 4.61 \begin {gather*} e^{\left (-\frac {4 \, x^{4}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4} + \frac {4 \, x^{3} e^{x}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4} - \frac {8 \, x^{2} e^{x}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4} - \frac {4}{5 \, {\left (5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4\right )}} + \frac {1}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((20*x^6-80*x^5+80*x^4)*exp(x)^2+(-40*x^7+80*x^6-4*x^4-8*x^3+24*x^2)*exp(x)+20*x^8+20*x^4)*exp(((x^4
-2*x^3)*exp(x)-x^5)/((5*x^4+10*x^3-40*x^2)*exp(x)-5*x^5-20*x^4-4))/((25*x^8+100*x^7-300*x^6-800*x^5+1600*x^4)*
exp(x)^2+(-50*x^9-300*x^8+1600*x^6-40*x^4-80*x^3+320*x^2)*exp(x)+25*x^10+200*x^9+400*x^8+40*x^5+160*x^4+16),x,
algorithm="maxima")
[Out]
e^(-4*x^4/(5*x^5 + 20*x^4 - 5*(x^4 + 2*x^3 - 8*x^2)*e^x + 4) + 4*x^3*e^x/(5*x^5 + 20*x^4 - 5*(x^4 + 2*x^3 - 8*
x^2)*e^x + 4) - 8*x^2*e^x/(5*x^5 + 20*x^4 - 5*(x^4 + 2*x^3 - 8*x^2)*e^x + 4) - 4/5/(5*x^5 + 20*x^4 - 5*(x^4 +
2*x^3 - 8*x^2)*e^x + 4) + 1/5)
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mupad [B] time = 1.50, size = 127, normalized size = 3.85 \begin {gather*} {\mathrm {e}}^{\frac {x^5}{40\,x^2\,{\mathrm {e}}^x-10\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+20\,x^4+5\,x^5+4}}\,{\mathrm {e}}^{\frac {2\,x^3\,{\mathrm {e}}^x}{40\,x^2\,{\mathrm {e}}^x-10\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+20\,x^4+5\,x^5+4}}\,{\mathrm {e}}^{-\frac {x^4\,{\mathrm {e}}^x}{40\,x^2\,{\mathrm {e}}^x-10\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+20\,x^4+5\,x^5+4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp((exp(x)*(2*x^3 - x^4) + x^5)/(20*x^4 - exp(x)*(10*x^3 - 40*x^2 + 5*x^4) + 5*x^5 + 4))*(exp(2*x)*(80*x
^4 - 80*x^5 + 20*x^6) - exp(x)*(8*x^3 - 24*x^2 + 4*x^4 - 80*x^6 + 40*x^7) + 20*x^4 + 20*x^8))/(exp(2*x)*(1600*
x^4 - 800*x^5 - 300*x^6 + 100*x^7 + 25*x^8) + 160*x^4 + 40*x^5 + 400*x^8 + 200*x^9 + 25*x^10 - exp(x)*(80*x^3
- 320*x^2 + 40*x^4 - 1600*x^6 + 300*x^8 + 50*x^9) + 16),x)
[Out]
exp(x^5/(40*x^2*exp(x) - 10*x^3*exp(x) - 5*x^4*exp(x) + 20*x^4 + 5*x^5 + 4))*exp((2*x^3*exp(x))/(40*x^2*exp(x)
- 10*x^3*exp(x) - 5*x^4*exp(x) + 20*x^4 + 5*x^5 + 4))*exp(-(x^4*exp(x))/(40*x^2*exp(x) - 10*x^3*exp(x) - 5*x^
4*exp(x) + 20*x^4 + 5*x^5 + 4))
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sympy [B] time = 1.33, size = 46, normalized size = 1.39 \begin {gather*} e^{\frac {- x^{5} + \left (x^{4} - 2 x^{3}\right ) e^{x}}{- 5 x^{5} - 20 x^{4} + \left (5 x^{4} + 10 x^{3} - 40 x^{2}\right ) e^{x} - 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((20*x**6-80*x**5+80*x**4)*exp(x)**2+(-40*x**7+80*x**6-4*x**4-8*x**3+24*x**2)*exp(x)+20*x**8+20*x**4
)*exp(((x**4-2*x**3)*exp(x)-x**5)/((5*x**4+10*x**3-40*x**2)*exp(x)-5*x**5-20*x**4-4))/((25*x**8+100*x**7-300*x
**6-800*x**5+1600*x**4)*exp(x)**2+(-50*x**9-300*x**8+1600*x**6-40*x**4-80*x**3+320*x**2)*exp(x)+25*x**10+200*x
**9+400*x**8+40*x**5+160*x**4+16),x)
[Out]
exp((-x**5 + (x**4 - 2*x**3)*exp(x))/(-5*x**5 - 20*x**4 + (5*x**4 + 10*x**3 - 40*x**2)*exp(x) - 4))
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