3.90.45 \(\int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x (-3 x+x^2-x^3)))}{16+8 x+9 x^2+2 x^3+x^4} \, dx\) [8945]
Optimal. Leaf size=29 \[ 3+4 e^{-e^{5+x+\frac {e^x}{4 \left (4+x+x^2\right )}}} x \]
[Out]
3+4*x/exp(exp(x+exp(x)/(4*x^2+4*x+16)+5))
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(156\) vs. \(2(29)=58\).
time = 5.48, antiderivative size = 156, normalized size of antiderivative = 5.38, number of steps
used = 1, number of rules used = 1, integrand size = 152, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {2326}
\begin {gather*} \frac {4 \left (4 x^5+8 x^4+36 x^3+32 x^2+e^x \left (x^3-x^2+3 x\right )+64 x\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^4+2 x^3+9 x^2+8 x+16\right ) \left (\frac {12 x^2+48 x+e^x+36}{x^2+x+4}-\frac {(2 x+1) \left (4 x^3+24 x^2+36 x+e^x+80\right )}{\left (x^2+x+4\right )^2}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(64 + 32*x + 36*x^2 + 8*x^3 + 4*x^4 + E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4*x + 4*x^2))*(-64*x - 3
2*x^2 - 36*x^3 - 8*x^4 - 4*x^5 + E^x*(-3*x + x^2 - x^3)))/(E^E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4*x +
4*x^2))*(16 + 8*x + 9*x^2 + 2*x^3 + x^4)),x]
[Out]
(4*(64*x + 32*x^2 + 36*x^3 + 8*x^4 + 4*x^5 + E^x*(3*x - x^2 + x^3)))/(E^E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/
(4*(4 + x + x^2)))*(16 + 8*x + 9*x^2 + 2*x^3 + x^4)*((36 + E^x + 48*x + 12*x^2)/(4 + x + x^2) - ((1 + 2*x)*(80
+ E^x + 36*x + 24*x^2 + 4*x^3))/(4 + x + x^2)^2))
Rule 2326
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 {4 \exp \left (-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{4 \left (4+x+x^2\right )}}\right ) \left (64 x+32 x^2+36 x^3+8 x^4+4 x^5+e^x \left (3 x-x^2+x^3\right )\right )}{\left (16+8 x+9 x^2+2 x^3+x^4\right ) \left (\frac {36+e^x+48 x+12 x^2}{4+x+x^2}-\frac {(1+2 x) \left (80+e^x+36 x+24 x^2+4 x^3\right )}{\left (4+x+x^2\right )^2}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.38, size = 27, normalized size = 0.93 \begin {gather*} 4 e^{-e^{5+x+\frac {e^x}{4 \left (4+x+x^2\right )}}} x \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(64 + 32*x + 36*x^2 + 8*x^3 + 4*x^4 + E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4*x + 4*x^2))*(-64
*x - 32*x^2 - 36*x^3 - 8*x^4 - 4*x^5 + E^x*(-3*x + x^2 - x^3)))/(E^E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 +
4*x + 4*x^2))*(16 + 8*x + 9*x^2 + 2*x^3 + x^4)),x]
[Out]
(4*x)/E^E^(5 + x + E^x/(4*(4 + x + x^2)))
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Maple [A]
time = 0.59, size = 35, normalized size = 1.21
| | |
| method |
result |
size |
| | |
| risch |
\(4 x \,{\mathrm e}^{-{\mathrm e}^{\frac {{\mathrm e}^{x}+4 x^{3}+24 x^{2}+36 x +80}{4 x^{2}+4 x +16}}}\) |
\(35\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))+
4*x^4+8*x^3+36*x^2+32*x+64)/(x^4+2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16))),x,
method=_RETURNVERBOSE)
[Out]
4*x*exp(-exp(1/4*(exp(x)+4*x^3+24*x^2+36*x+80)/(x^2+x+4)))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x
+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4+2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16
))),x, algorithm="maxima")
[Out]
4*((x^3 - x^2 + 3*x)*e^(2*x) + 4*(x^5 + 2*x^4 + 9*x^3 + 8*x^2 + 16*x)*e^x)*e^(-e^(x + 1/4*e^x/(x^2 + x + 4) +
5))/((x^2 - x + 3)*e^(2*x) + 4*(x^4 + 2*x^3 + 9*x^2 + 8*x + 16)*e^x) - integrate(-4*((x^4 - 2*x^3 + 7*x^2 - 6*
x + 9)*e^(3*x) - (4*x^7 + 48*x^5 + 28*x^4 + 156*x^3 + 84*x^2 + (x^4 - 2*x^3 + 7*x^2 - 6*x + 9)*e^x + 64*x + 19
2)*e^(2*x) + 8*(x^6 + x^5 + 10*x^4 + 5*x^3 + 35*x^2 + 8*x + 48)*e^(2*x) - 4*(4*x^8 + 16*x^7 + 88*x^6 + 208*x^5
+ 580*x^4 + 832*x^3 + 1408*x^2 - (x^7 - 2*x^6 + 10*x^5 - 13*x^4 + 29*x^3 - 49*x^2 - 48)*e^x + 1024*x + 1024)*
e^x + 16*(x^8 + 4*x^7 + 22*x^6 + 52*x^5 + 145*x^4 + 208*x^3 + 352*x^2 + 256*x + 256)*e^x)*e^(-e^(x + 1/4*e^x/(
x^2 + x + 4) + 5))/((x^4 - 2*x^3 + 7*x^2 - 6*x + 9)*e^(3*x) + 8*(x^6 + x^5 + 10*x^4 + 5*x^3 + 35*x^2 + 8*x + 4
8)*e^(2*x) + 16*(x^8 + 4*x^7 + 22*x^6 + 52*x^5 + 145*x^4 + 208*x^3 + 352*x^2 + 256*x + 256)*e^x), x)
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Fricas [A]
time = 0.40, size = 34, normalized size = 1.17 \begin {gather*} 4 \, x e^{\left (-e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x
+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4+2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16
))),x, algorithm="fricas")
[Out]
4*x*e^(-e^(1/4*(4*x^3 + 24*x^2 + 36*x + e^x + 80)/(x^2 + x + 4)))
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-x**3+x**2-3*x)*exp(x)-4*x**5-8*x**4-36*x**3-32*x**2-64*x)*exp((exp(x)+4*x**3+24*x**2+36*x+80)/(4
*x**2+4*x+16))+4*x**4+8*x**3+36*x**2+32*x+64)/(x**4+2*x**3+9*x**2+8*x+16)/exp(exp((exp(x)+4*x**3+24*x**2+36*x+
80)/(4*x**2+4*x+16))),x)
[Out]
Timed out
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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((((-x^3+x^2-3*x)*exp(x)-4*x^5-8*x^4-36*x^3-32*x^2-64*x)*exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x
+16))+4*x^4+8*x^3+36*x^2+32*x+64)/(x^4+2*x^3+9*x^2+8*x+16)/exp(exp((exp(x)+4*x^3+24*x^2+36*x+80)/(4*x^2+4*x+16
))),x, algorithm="giac")
[Out]
integrate((4*x^4 + 8*x^3 + 36*x^2 - (4*x^5 + 8*x^4 + 36*x^3 + 32*x^2 + (x^3 - x^2 + 3*x)*e^x + 64*x)*e^(1/4*(4
*x^3 + 24*x^2 + 36*x + e^x + 80)/(x^2 + x + 4)) + 32*x + 64)*e^(-e^(1/4*(4*x^3 + 24*x^2 + 36*x + e^x + 80)/(x^
2 + x + 4)))/(x^4 + 2*x^3 + 9*x^2 + 8*x + 16), x)
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Mupad [B]
time = 0.51, size = 72, normalized size = 2.48 \begin {gather*} 4\,x\,{\mathrm {e}}^{-{\mathrm {e}}^{\frac {9\,x}{x^2+x+4}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{4\,x^2+4\,x+16}}\,{\mathrm {e}}^{\frac {x^3}{x^2+x+4}}\,{\mathrm {e}}^{\frac {6\,x^2}{x^2+x+4}}\,{\mathrm {e}}^{\frac {20}{x^2+x+4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(-exp((36*x + exp(x) + 24*x^2 + 4*x^3 + 80)/(4*x + 4*x^2 + 16)))*(32*x - exp((36*x + exp(x) + 24*x^2 +
4*x^3 + 80)/(4*x + 4*x^2 + 16))*(64*x + exp(x)*(3*x - x^2 + x^3) + 32*x^2 + 36*x^3 + 8*x^4 + 4*x^5) + 36*x^2
+ 8*x^3 + 4*x^4 + 64))/(8*x + 9*x^2 + 2*x^3 + x^4 + 16),x)
[Out]
4*x*exp(-exp((9*x)/(x + x^2 + 4))*exp(exp(x)/(4*x + 4*x^2 + 16))*exp(x^3/(x + x^2 + 4))*exp((6*x^2)/(x + x^2 +
4))*exp(20/(x + x^2 + 4)))
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