3.65.72 \(\int \frac {e^{e^{-4+2 x}} (1+8 x+e^{-4+2 x} (10-2 x-8 x^2))}{25-10 x-39 x^2+8 x^3+16 x^4} \, dx\)

Optimal. Leaf size=22 \[ \frac {e^{e^{-4+2 x}}}{5-x-4 x^2} \]

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Rubi [A]  time = 0.12, antiderivative size = 42, normalized size of antiderivative = 1.91, number of steps used = 1, number of rules used = 1, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2288} \begin {gather*} \frac {e^{e^{2 x-4}} \left (-4 x^2-x+5\right )}{16 x^4+8 x^3-39 x^2-10 x+25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^(-4 + 2*x)*(1 + 8*x + E^(-4 + 2*x)*(10 - 2*x - 8*x^2)))/(25 - 10*x - 39*x^2 + 8*x^3 + 16*x^4),x]

[Out]

(E^E^(-4 + 2*x)*(5 - x - 4*x^2))/(25 - 10*x - 39*x^2 + 8*x^3 + 16*x^4)

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 {e^{e^{-4+2 x}} \left (5-x-4 x^2\right )}{25-10 x-39 x^2+8 x^3+16 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 21, normalized size = 0.95 \begin {gather*} -\frac {e^{e^{-4+2 x}}}{-5+x+4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(-4 + 2*x)*(1 + 8*x + E^(-4 + 2*x)*(10 - 2*x - 8*x^2)))/(25 - 10*x - 39*x^2 + 8*x^3 + 16*x^4),x
]

[Out]

-(E^E^(-4 + 2*x)/(-5 + x + 4*x^2))

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fricas [A]  time = 0.57, size = 19, normalized size = 0.86 \begin {gather*} -\frac {e^{\left (e^{\left (2 \, x - 4\right )}\right )}}{4 \, x^{2} + x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-2*x+10)*exp(x-2)^2+8*x+1)*exp(exp(x-2)^2)/(16*x^4+8*x^3-39*x^2-10*x+25),x, algorithm="frica
s")

[Out]

-e^(e^(2*x - 4))/(4*x^2 + x - 5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, {\left (4 \, x^{2} + x - 5\right )} e^{\left (2 \, x - 4\right )} - 8 \, x - 1\right )} e^{\left (e^{\left (2 \, x - 4\right )}\right )}}{16 \, x^{4} + 8 \, x^{3} - 39 \, x^{2} - 10 \, x + 25}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-2*x+10)*exp(x-2)^2+8*x+1)*exp(exp(x-2)^2)/(16*x^4+8*x^3-39*x^2-10*x+25),x, algorithm="giac"
)

[Out]

integrate(-(2*(4*x^2 + x - 5)*e^(2*x - 4) - 8*x - 1)*e^(e^(2*x - 4))/(16*x^4 + 8*x^3 - 39*x^2 - 10*x + 25), x)

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maple [A]  time = 0.21, size = 20, normalized size = 0.91

method	result	size
norman	\(-\frac {{\mathrm e}^{{\mathrm e}^{2 x -4}}}{4 x^{2}+x -5}\)	\(20\)
risch	\(-\frac {{\mathrm e}^{{\mathrm e}^{2 x -4}}}{4 x^{2}+x -5}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2-2*x+10)*exp(x-2)^2+8*x+1)*exp(exp(x-2)^2)/(16*x^4+8*x^3-39*x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

-exp(exp(x-2)^2)/(4*x^2+x-5)

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maxima [A]  time = 0.45, size = 19, normalized size = 0.86 \begin {gather*} -\frac {e^{\left (e^{\left (2 \, x - 4\right )}\right )}}{4 \, x^{2} + x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2-2*x+10)*exp(x-2)^2+8*x+1)*exp(exp(x-2)^2)/(16*x^4+8*x^3-39*x^2-10*x+25),x, algorithm="maxim
a")

[Out]

-e^(e^(2*x - 4))/(4*x^2 + x - 5)

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mupad [B]  time = 4.20, size = 20, normalized size = 0.91 \begin {gather*} -\frac {{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4}}}{4\,x^2+x-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(2*x - 4))*(8*x - exp(2*x - 4)*(2*x + 8*x^2 - 10) + 1))/(8*x^3 - 39*x^2 - 10*x + 16*x^4 + 25),x)

[Out]

-exp(exp(2*x)*exp(-4))/(x + 4*x^2 - 5)

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sympy [A]  time = 0.18, size = 17, normalized size = 0.77 \begin {gather*} - \frac {e^{e^{2 x - 4}}}{4 x^{2} + x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2-2*x+10)*exp(x-2)**2+8*x+1)*exp(exp(x-2)**2)/(16*x**4+8*x**3-39*x**2-10*x+25),x)

[Out]

-exp(exp(2*x - 4))/(4*x**2 + x - 5)

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