3.74.61 \(\int \frac {1}{27} e^{-25+e^{3+2 x}-e^{\frac {1}{54} (-162+x^2)}+x} (27+54 e^{3+2 x}-e^{\frac {1}{54} (-162+x^2)} x) \, dx\)

Optimal. Leaf size=25 \[ e^{-25+e^{3+2 x}-e^{-3+\frac {x^2}{54}}+x} \]

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Rubi [A]  time = 0.24, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {12, 6706} \begin {gather*} e^{-e^{\frac {1}{54} \left (x^2-162\right )}+x+e^{2 x+3}-25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-25 + E^(3 + 2*x) - E^((-162 + x^2)/54) + x)*(27 + 54*E^(3 + 2*x) - E^((-162 + x^2)/54)*x))/27,x]

[Out]

E^(-25 + E^(3 + 2*x) - E^((-162 + x^2)/54) + 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 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} &=\frac {1}{27} \int e^{-25+e^{3+2 x}-e^{\frac {1}{54} \left (-162+x^2\right )}+x} \left (27+54 e^{3+2 x}-e^{\frac {1}{54} \left (-162+x^2\right )} x\right ) \, dx\\ &=e^{-25+e^{3+2 x}-e^{\frac {1}{54} \left (-162+x^2\right )}+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 25, normalized size = 1.00 \begin {gather*} e^{-25+e^{3+2 x}-e^{-3+\frac {x^2}{54}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-25 + E^(3 + 2*x) - E^((-162 + x^2)/54) + x)*(27 + 54*E^(3 + 2*x) - E^((-162 + x^2)/54)*x))/27,x
]

[Out]

E^(-25 + E^(3 + 2*x) - E^(-3 + x^2/54) + x)

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fricas [A]  time = 0.65, size = 20, normalized size = 0.80 \begin {gather*} e^{\left (x - e^{\left (\frac {1}{54} \, x^{2} - 3\right )} + e^{\left (2 \, x + 3\right )} - 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-x*exp(1/54*x^2-3)+54*exp(2*x+3)+27)*exp(-exp(1/54*x^2-3)+exp(2*x+3)+x-25),x, algorithm="frica
s")

[Out]

e^(x - e^(1/54*x^2 - 3) + e^(2*x + 3) - 25)

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giac [A]  time = 0.22, size = 20, normalized size = 0.80 \begin {gather*} e^{\left (x - e^{\left (\frac {1}{54} \, x^{2} - 3\right )} + e^{\left (2 \, x + 3\right )} - 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-x*exp(1/54*x^2-3)+54*exp(2*x+3)+27)*exp(-exp(1/54*x^2-3)+exp(2*x+3)+x-25),x, algorithm="giac"
)

[Out]

e^(x - e^(1/54*x^2 - 3) + e^(2*x + 3) - 25)

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maple [A]  time = 0.12, size = 21, normalized size = 0.84

method	result	size
norman	\({\mathrm e}^{-{\mathrm e}^{\frac {x^{2}}{54}-3}+{\mathrm e}^{2 x +3}+x -25}\)	\(21\)
risch	\({\mathrm e}^{-{\mathrm e}^{\frac {x^{2}}{54}-3}+{\mathrm e}^{2 x +3}+x -25}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/27*(-x*exp(1/54*x^2-3)+54*exp(2*x+3)+27)*exp(-exp(1/54*x^2-3)+exp(2*x+3)+x-25),x,method=_RETURNVERBOSE)

[Out]

exp(-exp(1/54*x^2-3)+exp(2*x+3)+x-25)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.80 \begin {gather*} e^{\left (x - e^{\left (\frac {1}{54} \, x^{2} - 3\right )} + e^{\left (2 \, x + 3\right )} - 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-x*exp(1/54*x^2-3)+54*exp(2*x+3)+27)*exp(-exp(1/54*x^2-3)+exp(2*x+3)+x-25),x, algorithm="maxim
a")

[Out]

e^(x - e^(1/54*x^2 - 3) + e^(2*x + 3) - 25)

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mupad [B]  time = 4.59, size = 24, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-25}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-{\mathrm {e}}^{-3}\,{\mathrm {e}}^{\frac {x^2}{54}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + exp(2*x + 3) - exp(x^2/54 - 3) - 25)*(54*exp(2*x + 3) - x*exp(x^2/54 - 3) + 27))/27,x)

[Out]

exp(-25)*exp(exp(2*x)*exp(3))*exp(-exp(-3)*exp(x^2/54))*exp(x)

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sympy [A]  time = 0.29, size = 19, normalized size = 0.76 \begin {gather*} e^{x + e^{2 x + 3} - e^{\frac {x^{2}}{54} - 3} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*(-x*exp(1/54*x**2-3)+54*exp(2*x+3)+27)*exp(-exp(1/54*x**2-3)+exp(2*x+3)+x-25),x)

[Out]

exp(x + exp(2*x + 3) - exp(x**2/54 - 3) - 25)

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