3.90.45 \(\int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x (-3 x+x^2-x^3)))}{16+8 x+9 x^2+2 x^3+x^4} \, dx\)

Optimal. Leaf size=29 \[ 3+4 e^{-e^{5+x+\frac {e^x}{4 \left (4+x+x^2\right )}}} x \]

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Rubi [B]  time = 5.40, 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 = {2288} \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 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 {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 = 0.42, 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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fricas [A]  time = 0.57, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{4} + 8 \, x^{3} + 36 \, x^{2} - {\left (4 \, x^{5} + 8 \, x^{4} + 36 \, x^{3} + 32 \, x^{2} + {\left (x^{3} - x^{2} + 3 \, x\right )} e^{x} + 64 \, x\right )} e^{\left (\frac {4 \, x^{3} + 24 \, x^{2} + 36 \, x + e^{x} + 80}{4 \, {\left (x^{2} + x + 4\right )}}\right )} + 32 \, x + 64\right )} 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 )}}{x^{4} + 2 \, x^{3} + 9 \, x^{2} + 8 \, x + 16}\,{d x} \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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maple [A]  time = 0.21, 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*} \frac {4 \, {\left ({\left (x^{3} - x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{5} + 2 \, x^{4} + 9 \, x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{x}\right )} e^{\left (-e^{\left (x + \frac {e^{x}}{4 \, {\left (x^{2} + x + 4\right )}} + 5\right )}\right )}}{{\left (x^{2} - x + 3\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} + 2 \, x^{3} + 9 \, x^{2} + 8 \, x + 16\right )} e^{x}} - \int -\frac {4 \, {\left ({\left (x^{4} - 2 \, x^{3} + 7 \, x^{2} - 6 \, x + 9\right )} e^{\left (3 \, x\right )} - {\left (4 \, x^{7} + 48 \, x^{5} + 28 \, x^{4} + 156 \, x^{3} + 84 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + 7 \, x^{2} - 6 \, x + 9\right )} e^{x} + 64 \, x + 192\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{6} + x^{5} + 10 \, x^{4} + 5 \, x^{3} + 35 \, x^{2} + 8 \, x + 48\right )} e^{\left (2 \, x\right )} - 4 \, {\left (4 \, x^{8} + 16 \, x^{7} + 88 \, x^{6} + 208 \, x^{5} + 580 \, x^{4} + 832 \, x^{3} + 1408 \, x^{2} - {\left (x^{7} - 2 \, x^{6} + 10 \, x^{5} - 13 \, x^{4} + 29 \, x^{3} - 49 \, x^{2} - 48\right )} e^{x} + 1024 \, x + 1024\right )} e^{x} + 16 \, {\left (x^{8} + 4 \, x^{7} + 22 \, x^{6} + 52 \, x^{5} + 145 \, x^{4} + 208 \, x^{3} + 352 \, x^{2} + 256 \, x + 256\right )} e^{x}\right )} e^{\left (-e^{\left (x + \frac {e^{x}}{4 \, {\left (x^{2} + x + 4\right )}} + 5\right )}\right )}}{{\left (x^{4} - 2 \, x^{3} + 7 \, x^{2} - 6 \, x + 9\right )} e^{\left (3 \, x\right )} + 8 \, {\left (x^{6} + x^{5} + 10 \, x^{4} + 5 \, x^{3} + 35 \, x^{2} + 8 \, x + 48\right )} e^{\left (2 \, x\right )} + 16 \, {\left (x^{8} + 4 \, x^{7} + 22 \, x^{6} + 52 \, x^{5} + 145 \, x^{4} + 208 \, x^{3} + 352 \, x^{2} + 256 \, x + 256\right )} e^{x}}\,{d x} \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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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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sympy [F(-1)]  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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