3.89.95 \(\int (1+e^{-25+e^{-1-e^x+2 x}-9 x-x^2} (-9+e^{-1-e^x+2 x} (2-e^x)-2 x)) \, dx\)

Optimal. Leaf size=25 \[ e^{e^{-1-e^x+2 x}+x-(5+x)^2}+x \]

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Rubi [A]  time = 0.31, antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6706} \begin {gather*} e^{-x^2-9 x+e^{2 x-e^x-1}-25}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(-25 + E^(-1 - E^x + 2*x) - 9*x - x^2)*(-9 + E^(-1 - E^x + 2*x)*(2 - E^x) - 2*x),x]

[Out]

E^(-25 + E^(-1 - E^x + 2*x) - 9*x - x^2) + 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} &=x+\int e^{-25+e^{-1-e^x+2 x}-9 x-x^2} \left (-9+e^{-1-e^x+2 x} \left (2-e^x\right )-2 x\right ) \, dx\\ &=e^{-25+e^{-1-e^x+2 x}-9 x-x^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.83, size = 26, normalized size = 1.04 \begin {gather*} e^{-25+e^{-1-e^x+2 x}-9 x-x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(-25 + E^(-1 - E^x + 2*x) - 9*x - x^2)*(-9 + E^(-1 - E^x + 2*x)*(2 - E^x) - 2*x),x]

[Out]

E^(-25 + E^(-1 - E^x + 2*x) - 9*x - x^2) + x

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fricas [A]  time = 0.60, size = 23, normalized size = 0.92 \begin {gather*} x + e^{\left (-x^{2} - 9 \, x + e^{\left (2 \, x - e^{x} - 1\right )} - 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+2)*exp(-exp(x)+2*x-1)-2*x-9)*exp(exp(-exp(x)+2*x-1)-x^2-9*x-25)+1,x, algorithm="fricas")

[Out]

x + e^(-x^2 - 9*x + e^(2*x - e^x - 1) - 25)

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giac [A]  time = 0.20, size = 23, normalized size = 0.92 \begin {gather*} x + e^{\left (-x^{2} - 9 \, x + e^{\left (2 \, x - e^{x} - 1\right )} - 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+2)*exp(-exp(x)+2*x-1)-2*x-9)*exp(exp(-exp(x)+2*x-1)-x^2-9*x-25)+1,x, algorithm="giac")

[Out]

x + e^(-x^2 - 9*x + e^(2*x - e^x - 1) - 25)

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maple [A]  time = 0.04, size = 24, normalized size = 0.96

method	result	size
default	\(x +{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x}+2 x -1}-x^{2}-9 x -25}\)	\(24\)
norman	\(x +{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x}+2 x -1}-x^{2}-9 x -25}\)	\(24\)
risch	\(x +{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x}+2 x -1}-x^{2}-9 x -25}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(x)+2)*exp(-exp(x)+2*x-1)-2*x-9)*exp(exp(-exp(x)+2*x-1)-x^2-9*x-25)+1,x,method=_RETURNVERBOSE)

[Out]

x+exp(exp(-exp(x)+2*x-1)-x^2-9*x-25)

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maxima [A]  time = 0.63, size = 23, normalized size = 0.92 \begin {gather*} x + e^{\left (-x^{2} - 9 \, x + e^{\left (2 \, x - e^{x} - 1\right )} - 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+2)*exp(-exp(x)+2*x-1)-2*x-9)*exp(exp(-exp(x)+2*x-1)-x^2-9*x-25)+1,x, algorithm="maxima")

[Out]

x + e^(-x^2 - 9*x + e^(2*x - e^x - 1) - 25)

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mupad [B]  time = 0.16, size = 28, normalized size = 1.12 \begin {gather*} x+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-9\,x}\,{\mathrm {e}}^{-25}\,{\mathrm {e}}^{-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - exp(exp(2*x - exp(x) - 1) - 9*x - x^2 - 25)*(2*x + exp(2*x - exp(x) - 1)*(exp(x) - 2) + 9),x)

[Out]

x + exp(exp(2*x)*exp(-1)*exp(-exp(x)))*exp(-9*x)*exp(-25)*exp(-x^2)

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sympy [A]  time = 0.31, size = 20, normalized size = 0.80 \begin {gather*} x + e^{- x^{2} - 9 x + e^{2 x - e^{x} - 1} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+2)*exp(-exp(x)+2*x-1)-2*x-9)*exp(exp(-exp(x)+2*x-1)-x**2-9*x-25)+1,x)

[Out]

x + exp(-x**2 - 9*x + exp(2*x - exp(x) - 1) - 25)

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