3.83.95 \(\int \frac {-2+e^{1089+e^{2 x}+e^x (-66-2 x)+66 x+x^2} (66-130 x+62 x^2+2 x^3+e^{2 x} (2-4 x+2 x^2)+e^x (-68+134 x-64 x^2-2 x^3))}{1-2 x+x^2} \, dx\)

Optimal. Leaf size=24 \[ e^{\left (-33+e^x-x\right )^2}+\frac {3+x}{-2+2 x} \]

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Rubi [A]  time = 0.81, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {27, 6688, 6706} \begin {gather*} e^{\left (x-e^x+33\right )^2}-\frac {2}{1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + E^(1089 + E^(2*x) + E^x*(-66 - 2*x) + 66*x + x^2)*(66 - 130*x + 62*x^2 + 2*x^3 + E^(2*x)*(2 - 4*x +
2*x^2) + E^x*(-68 + 134*x - 64*x^2 - 2*x^3)))/(1 - 2*x + x^2),x]

[Out]

E^(33 - E^x + x)^2 - 2/(1 - x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+e^{1089+e^{2 x}+e^x (-66-2 x)+66 x+x^2} \left (66-130 x+62 x^2+2 x^3+e^{2 x} \left (2-4 x+2 x^2\right )+e^x \left (-68+134 x-64 x^2-2 x^3\right )\right )}{(-1+x)^2} \, dx\\ &=\int \left (2 e^{\left (33-e^x+x\right )^2} \left (-1+e^x\right ) \left (-33+e^x-x\right )-\frac {2}{(-1+x)^2}\right ) \, dx\\ &=-\frac {2}{1-x}+2 \int e^{\left (33-e^x+x\right )^2} \left (-1+e^x\right ) \left (-33+e^x-x\right ) \, dx\\ &=e^{\left (33-e^x+x\right )^2}-\frac {2}{1-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 31, normalized size = 1.29 \begin {gather*} e^{1089+e^{2 x}+66 x+x^2-2 e^x (33+x)}+\frac {2}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^(1089 + E^(2*x) + E^x*(-66 - 2*x) + 66*x + x^2)*(66 - 130*x + 62*x^2 + 2*x^3 + E^(2*x)*(2 -
4*x + 2*x^2) + E^x*(-68 + 134*x - 64*x^2 - 2*x^3)))/(1 - 2*x + x^2),x]

[Out]

E^(1089 + E^(2*x) + 66*x + x^2 - 2*E^x*(33 + x)) + 2/(-1 + x)

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fricas [A]  time = 0.83, size = 32, normalized size = 1.33 \begin {gather*} \frac {{\left (x - 1\right )} e^{\left (x^{2} - 2 \, {\left (x + 33\right )} e^{x} + 66 \, x + e^{\left (2 \, x\right )} + 1089\right )} + 2}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-4*x+2)*exp(x)^2+(-2*x^3-64*x^2+134*x-68)*exp(x)+2*x^3+62*x^2-130*x+66)*exp(exp(x)^2+(-2*x-6
6)*exp(x)+x^2+66*x+1089)-2)/(x^2-2*x+1),x, algorithm="fricas")

[Out]

((x - 1)*e^(x^2 - 2*(x + 33)*e^x + 66*x + e^(2*x) + 1089) + 2)/(x - 1)

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giac [B]  time = 0.29, size = 56, normalized size = 2.33 \begin {gather*} \frac {x e^{\left (x^{2} - 2 \, x e^{x} + 66 \, x + e^{\left (2 \, x\right )} - 66 \, e^{x} + 1089\right )} - e^{\left (x^{2} - 2 \, x e^{x} + 66 \, x + e^{\left (2 \, x\right )} - 66 \, e^{x} + 1089\right )} + 2}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-4*x+2)*exp(x)^2+(-2*x^3-64*x^2+134*x-68)*exp(x)+2*x^3+62*x^2-130*x+66)*exp(exp(x)^2+(-2*x-6
6)*exp(x)+x^2+66*x+1089)-2)/(x^2-2*x+1),x, algorithm="giac")

[Out]

(x*e^(x^2 - 2*x*e^x + 66*x + e^(2*x) - 66*e^x + 1089) - e^(x^2 - 2*x*e^x + 66*x + e^(2*x) - 66*e^x + 1089) + 2
)/(x - 1)

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maple [A]  time = 0.39, size = 31, normalized size = 1.29

method	result	size
risch	\(\frac {2}{x -1}+{\mathrm e}^{-2 \,{\mathrm e}^{x} x +x^{2}-66 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+66 x +1089}\)	\(31\)
norman	\(\frac {x \,{\mathrm e}^{{\mathrm e}^{2 x}+\left (-2 x -66\right ) {\mathrm e}^{x}+x^{2}+66 x +1089}-{\mathrm e}^{{\mathrm e}^{2 x}+\left (-2 x -66\right ) {\mathrm e}^{x}+x^{2}+66 x +1089}+2}{x -1}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-4*x+2)*exp(x)^2+(-2*x^3-64*x^2+134*x-68)*exp(x)+2*x^3+62*x^2-130*x+66)*exp(exp(x)^2+(-2*x-66)*exp
(x)+x^2+66*x+1089)-2)/(x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

2/(x-1)+exp(-2*exp(x)*x+x^2-66*exp(x)+exp(2*x)+66*x+1089)

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maxima [A]  time = 3.24, size = 30, normalized size = 1.25 \begin {gather*} \frac {2}{x - 1} + e^{\left (x^{2} - 2 \, x e^{x} + 66 \, x + e^{\left (2 \, x\right )} - 66 \, e^{x} + 1089\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-4*x+2)*exp(x)^2+(-2*x^3-64*x^2+134*x-68)*exp(x)+2*x^3+62*x^2-130*x+66)*exp(exp(x)^2+(-2*x-6
6)*exp(x)+x^2+66*x+1089)-2)/(x^2-2*x+1),x, algorithm="maxima")

[Out]

2/(x - 1) + e^(x^2 - 2*x*e^x + 66*x + e^(2*x) - 66*e^x + 1089)

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mupad [B]  time = 5.26, size = 35, normalized size = 1.46 \begin {gather*} \frac {2}{x-1}+{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{66\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{1089}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-66\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(66*x + exp(2*x) - exp(x)*(2*x + 66) + x^2 + 1089)*(exp(2*x)*(2*x^2 - 4*x + 2) - 130*x + 62*x^2 + 2*x^
3 - exp(x)*(64*x^2 - 134*x + 2*x^3 + 68) + 66) - 2)/(x^2 - 2*x + 1),x)

[Out]

2/(x - 1) + exp(-2*x*exp(x))*exp(66*x)*exp(x^2)*exp(1089)*exp(exp(2*x))*exp(-66*exp(x))

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sympy [A]  time = 0.41, size = 29, normalized size = 1.21 \begin {gather*} e^{x^{2} + 66 x + \left (- 2 x - 66\right ) e^{x} + e^{2 x} + 1089} + \frac {2}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-4*x+2)*exp(x)**2+(-2*x**3-64*x**2+134*x-68)*exp(x)+2*x**3+62*x**2-130*x+66)*exp(exp(x)**2+
(-2*x-66)*exp(x)+x**2+66*x+1089)-2)/(x**2-2*x+1),x)

[Out]

exp(x**2 + 66*x + (-2*x - 66)*exp(x) + exp(2*x) + 1089) + 2/(x - 1)

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