3.84.31 \(\int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x))}{e^4} \, dx\)

Optimal. Leaf size=22 \[ 13+4 x+e^{-4+e^{\left (2+e^x\right )^4+x}} x \]

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Rubi [B]  time = 0.18, antiderivative size = 100, normalized size of antiderivative = 4.55, number of steps used = 3, number of rules used = 2, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {12, 2288} \begin {gather*} \frac {\left (32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x+x\right ) \exp \left (e^{x+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+16}-4\right )}{32 e^x+48 e^{2 x}+24 e^{3 x}+4 e^{4 x}+1}+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^4 + E^E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x)*(1 + E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(
3*x) + E^(4*x) + x)*(x + 32*E^x*x + 48*E^(2*x)*x + 24*E^(3*x)*x + 4*E^(4*x)*x)))/E^4,x]

[Out]

4*x + (E^(-4 + E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x))*(x + 32*E^x*x + 48*E^(2*x)*x + 24*E^(3*
x)*x + 4*E^(4*x)*x))/(1 + 32*E^x + 48*E^(2*x) + 24*E^(3*x) + 4*E^(4*x))

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 {\int \left (4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )\right ) \, dx}{e^4}\\ &=4 x+\frac {\int e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right ) \, dx}{e^4}\\ &=4 x+\frac {\exp \left (-4+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}\right ) \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )}{1+32 e^x+48 e^{2 x}+24 e^{3 x}+4 e^{4 x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 39, normalized size = 1.77 \begin {gather*} 4 x+e^{-4+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^4 + E^E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x)*(1 + E^(16 + 32*E^x + 24*E^(2*x) +
8*E^(3*x) + E^(4*x) + x)*(x + 32*E^x*x + 48*E^(2*x)*x + 24*E^(3*x)*x + 4*E^(4*x)*x)))/E^4,x]

[Out]

4*x + E^(-4 + E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x))*x

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fricas [A]  time = 0.77, size = 36, normalized size = 1.64 \begin {gather*} {\left (4 \, x e^{4} + x e^{\left (e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )}\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*ex
p(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x, algorithm="fricas")

[Out]

(4*x*e^4 + x*e^(e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16)))*e^(-4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left ({\left ({\left (4 \, x e^{\left (4 \, x\right )} + 24 \, x e^{\left (3 \, x\right )} + 48 \, x e^{\left (2 \, x\right )} + 32 \, x e^{x} + x\right )} e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )} + 1\right )} e^{\left (e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )}\right )} + 4 \, e^{4}\right )} e^{\left (-4\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*ex
p(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x, algorithm="giac")

[Out]

integrate((((4*x*e^(4*x) + 24*x*e^(3*x) + 48*x*e^(2*x) + 32*x*e^x + x)*e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x)
+ 32*e^x + 16) + 1)*e^(e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16)) + 4*e^4)*e^(-4), x)

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maple [A]  time = 0.15, size = 34, normalized size = 1.55

method	result	size
risch	\(4 x +x \,{\mathrm e}^{-4+{\mathrm e}^{{\mathrm e}^{4 x}+8 \,{\mathrm e}^{3 x}+24 \,{\mathrm e}^{2 x}+32 \,{\mathrm e}^{x}+x +16}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x
+16)+1)*exp(exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x,method=_RETURNVERBOSE)

[Out]

4*x+x*exp(-4+exp(exp(4*x)+8*exp(3*x)+24*exp(2*x)+32*exp(x)+x+16))

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maxima [A]  time = 0.46, size = 36, normalized size = 1.64 \begin {gather*} {\left (4 \, x e^{4} + x e^{\left (e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )}\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*ex
p(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x, algorithm="maxima")

[Out]

(4*x*e^4 + x*e^(e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16)))*e^(-4)

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mupad [B]  time = 0.29, size = 39, normalized size = 1.77 \begin {gather*} x\,{\mathrm {e}}^{-4}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{8\,{\mathrm {e}}^{3\,x}}\,{\mathrm {e}}^{24\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,{\mathrm {e}}^{32\,{\mathrm {e}}^x}\,{\mathrm {e}}^x}+4\,{\mathrm {e}}^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(4*exp(4) + exp(exp(x + 24*exp(2*x) + 8*exp(3*x) + exp(4*x) + 32*exp(x) + 16))*(exp(x + 24*exp(2*x
) + 8*exp(3*x) + exp(4*x) + 32*exp(x) + 16)*(x + 48*x*exp(2*x) + 24*x*exp(3*x) + 4*x*exp(4*x) + 32*x*exp(x)) +
1)),x)

[Out]

x*exp(-4)*(exp(exp(8*exp(3*x))*exp(24*exp(2*x))*exp(16)*exp(exp(4*x))*exp(32*exp(x))*exp(x)) + 4*exp(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((((4*x*exp(x)**4+24*x*exp(x)**3+48*x*exp(x)**2+32*exp(x)*x+x)*exp(exp(x)**4+8*exp(x)**3+24*exp(x)**2
+32*exp(x)+x+16)+1)*exp(exp(exp(x)**4+8*exp(x)**3+24*exp(x)**2+32*exp(x)+x+16))+4*exp(4))/exp(4),x)

[Out]

Timed out

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