3.16.74 \(\int e^{77-e^{16/x}+e^{2 x}-108 x+54 x^2-12 x^3+x^4+e^x (19-12 x+2 x^2)} (16 e^{16/x}+2 x-108 x^2+2 e^{2 x} x^2+108 x^3-36 x^4+4 x^5+e^x (7 x^2-8 x^3+2 x^4)) \, dx\)

Optimal. Leaf size=33 \[ e^{-4-e^{16/x}+e^x+\left (e^x+(3-x)^2\right )^2} x^2 \]

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Rubi [B]  time = 12.16, antiderivative size = 171, normalized size of antiderivative = 5.18, number of steps used = 1, number of rules used = 1, integrand size = 112, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2288} \begin {gather*} -\frac {\left (4 x^5-36 x^4+108 x^3+2 e^{2 x} x^2-108 x^2+e^x \left (2 x^4-8 x^3+7 x^2\right )+16 e^{16/x}\right ) \exp \left (x^4-12 x^3+54 x^2+e^x \left (2 x^2-12 x+19\right )-108 x-e^{16/x}+e^{2 x}+77\right )}{-4 x^3+36 x^2-e^x \left (2 x^2-12 x+19\right )-\frac {16 e^{16/x}}{x^2}-108 x-2 e^{2 x}+4 e^x (3-x)+108} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(77 - E^(16/x) + E^(2*x) - 108*x + 54*x^2 - 12*x^3 + x^4 + E^x*(19 - 12*x + 2*x^2))*(16*E^(16/x) + 2*x -
108*x^2 + 2*E^(2*x)*x^2 + 108*x^3 - 36*x^4 + 4*x^5 + E^x*(7*x^2 - 8*x^3 + 2*x^4)),x]

[Out]

-((E^(77 - E^(16/x) + E^(2*x) - 108*x + 54*x^2 - 12*x^3 + x^4 + E^x*(19 - 12*x + 2*x^2))*(16*E^(16/x) - 108*x^
2 + 2*E^(2*x)*x^2 + 108*x^3 - 36*x^4 + 4*x^5 + E^x*(7*x^2 - 8*x^3 + 2*x^4)))/(108 - 2*E^(2*x) + 4*E^x*(3 - x)
- (16*E^(16/x))/x^2 - 108*x + 36*x^2 - 4*x^3 - E^x*(19 - 12*x + 2*x^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 {\exp \left (77-e^{16/x}+e^{2 x}-108 x+54 x^2-12 x^3+x^4+e^x \left (19-12 x+2 x^2\right )\right ) \left (16 e^{16/x}-108 x^2+2 e^{2 x} x^2+108 x^3-36 x^4+4 x^5+e^x \left (7 x^2-8 x^3+2 x^4\right )\right )}{108-2 e^{2 x}+4 e^x (3-x)-\frac {16 e^{16/x}}{x^2}-108 x+36 x^2-4 x^3-e^x \left (19-12 x+2 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 52, normalized size = 1.58 \begin {gather*} e^{77-e^{16/x}+e^{2 x}-108 x+54 x^2-12 x^3+x^4+e^x \left (19-12 x+2 x^2\right )} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(77 - E^(16/x) + E^(2*x) - 108*x + 54*x^2 - 12*x^3 + x^4 + E^x*(19 - 12*x + 2*x^2))*(16*E^(16/x) +
2*x - 108*x^2 + 2*E^(2*x)*x^2 + 108*x^3 - 36*x^4 + 4*x^5 + E^x*(7*x^2 - 8*x^3 + 2*x^4)),x]

[Out]

E^(77 - E^(16/x) + E^(2*x) - 108*x + 54*x^2 - 12*x^3 + x^4 + E^x*(19 - 12*x + 2*x^2))*x^2

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fricas [A]  time = 0.82, size = 48, normalized size = 1.45 \begin {gather*} x^{2} e^{\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} + {\left (2 \, x^{2} - 12 \, x + 19\right )} e^{x} - 108 \, x + e^{\left (2 \, x\right )} - e^{\frac {16}{x}} + 77\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*x^2+(2*x^4-8*x^3+7*x^2)*exp(x)+16*exp(16/x)+4*x^5-36*x^4+108*x^3-108*x^2+2*x)*exp(exp(x)
^2+(2*x^2-12*x+19)*exp(x)-exp(16/x)+x^4-12*x^3+54*x^2-108*x+77),x, algorithm="fricas")

[Out]

x^2*e^(x^4 - 12*x^3 + 54*x^2 + (2*x^2 - 12*x + 19)*e^x - 108*x + e^(2*x) - e^(16/x) + 77)

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giac [B]  time = 0.57, size = 72, normalized size = 2.18 \begin {gather*} x^{2} e^{\left (\frac {x^{5} - 12 \, x^{4} + 2 \, x^{3} e^{x} + 54 \, x^{3} - 12 \, x^{2} e^{x} - 108 \, x^{2} + x e^{\left (2 \, x\right )} + 19 \, x e^{x} - x e^{\frac {16}{x}} + 77 \, x + 16}{x} - \frac {16}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*x^2+(2*x^4-8*x^3+7*x^2)*exp(x)+16*exp(16/x)+4*x^5-36*x^4+108*x^3-108*x^2+2*x)*exp(exp(x)
^2+(2*x^2-12*x+19)*exp(x)-exp(16/x)+x^4-12*x^3+54*x^2-108*x+77),x, algorithm="giac")

[Out]

x^2*e^((x^5 - 12*x^4 + 2*x^3*e^x + 54*x^3 - 12*x^2*e^x - 108*x^2 + x*e^(2*x) + 19*x*e^x - x*e^(16/x) + 77*x +
16)/x - 16/x)

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maple [A]  time = 0.27, size = 52, normalized size = 1.58

method	result	size
risch	\(x^{2} {\mathrm e}^{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x^{2}-12 \,{\mathrm e}^{x} x +19 \,{\mathrm e}^{x}-{\mathrm e}^{\frac {16}{x}}+x^{4}-12 x^{3}+54 x^{2}-108 x +77}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)^2*x^2+(2*x^4-8*x^3+7*x^2)*exp(x)+16*exp(16/x)+4*x^5-36*x^4+108*x^3-108*x^2+2*x)*exp(exp(x)^2+(2*
x^2-12*x+19)*exp(x)-exp(16/x)+x^4-12*x^3+54*x^2-108*x+77),x,method=_RETURNVERBOSE)

[Out]

x^2*exp(exp(2*x)+2*exp(x)*x^2-12*exp(x)*x+19*exp(x)-exp(16/x)+x^4-12*x^3+54*x^2-108*x+77)

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maxima [A]  time = 0.93, size = 51, normalized size = 1.55 \begin {gather*} x^{2} e^{\left (x^{4} - 12 \, x^{3} + 2 \, x^{2} e^{x} + 54 \, x^{2} - 12 \, x e^{x} - 108 \, x + e^{\left (2 \, x\right )} + 19 \, e^{x} - e^{\frac {16}{x}} + 77\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^2*x^2+(2*x^4-8*x^3+7*x^2)*exp(x)+16*exp(16/x)+4*x^5-36*x^4+108*x^3-108*x^2+2*x)*exp(exp(x)
^2+(2*x^2-12*x+19)*exp(x)-exp(16/x)+x^4-12*x^3+54*x^2-108*x+77),x, algorithm="maxima")

[Out]

x^2*e^(x^4 - 12*x^3 + 2*x^2*e^x + 54*x^2 - 12*x*e^x - 108*x + e^(2*x) + 19*e^x - e^(16/x) + 77)

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mupad [B]  time = 1.36, size = 59, normalized size = 1.79 \begin {gather*} x^2\,{\mathrm {e}}^{-12\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-108\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-{\mathrm {e}}^{16/x}}\,{\mathrm {e}}^{77}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-12\,x^3}\,{\mathrm {e}}^{54\,x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{19\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(2*x) - 108*x - exp(16/x) + exp(x)*(2*x^2 - 12*x + 19) + 54*x^2 - 12*x^3 + x^4 + 77)*(2*x + 16*exp(
16/x) + exp(x)*(7*x^2 - 8*x^3 + 2*x^4) + 2*x^2*exp(2*x) - 108*x^2 + 108*x^3 - 36*x^4 + 4*x^5),x)

[Out]

x^2*exp(-12*x*exp(x))*exp(-108*x)*exp(x^4)*exp(-exp(16/x))*exp(77)*exp(2*x^2*exp(x))*exp(-12*x^3)*exp(54*x^2)*
exp(exp(2*x))*exp(19*exp(x))

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sympy [A]  time = 1.98, size = 46, normalized size = 1.39 \begin {gather*} x^{2} e^{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + \left (2 x^{2} - 12 x + 19\right ) e^{x} - e^{\frac {16}{x}} + e^{2 x} + 77} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)**2*x**2+(2*x**4-8*x**3+7*x**2)*exp(x)+16*exp(16/x)+4*x**5-36*x**4+108*x**3-108*x**2+2*x)*e
xp(exp(x)**2+(2*x**2-12*x+19)*exp(x)-exp(16/x)+x**4-12*x**3+54*x**2-108*x+77),x)

[Out]

x**2*exp(x**4 - 12*x**3 + 54*x**2 - 108*x + (2*x**2 - 12*x + 19)*exp(x) - exp(16/x) + exp(2*x) + 77)

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