3.49.63 \(\int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x) \, dx\)

Optimal. Leaf size=31 \[ e^{25+e^2-x+e^{2 e^4} \left (-e^{e^3}+2 x\right )^2} \]

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Rubi [A]  time = 0.44, antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2244, 2236} \begin {gather*} \exp \left (4 e^{2 e^4} x^2-\left (1+4 e^{e^3+2 e^4}\right ) x+e^{2 e^3 (1+e)}+e^2+25\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(25 + E^2 + E^(2*E^3*(1 + E)) - (1 + 4*E^(E^3 + 2*E^4))*x + 4*E^(2*E^4)*x^2)

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \exp \left (25+e^2+e^{2 e^3 (1+e)}+\left (-1-4 e^{e^3+2 e^4}\right ) x+4 e^{2 e^4} x^2\right ) \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx\\ &=\exp \left (25+e^2+e^{2 e^3 (1+e)}-\left (1+4 e^{e^3+2 e^4}\right ) x+4 e^{2 e^4} x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 47, normalized size = 1.52 \begin {gather*} e^{25+e^2+e^{2 e^3 (1+e)}-\left (1+4 e^{e^3+2 e^4}\right ) x+4 e^{2 e^4} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(25 + E^2 + E^(2*E^3*(1 + E)) - (1 + 4*E^(E^3 + 2*E^4))*x + 4*E^(2*E^4)*x^2)

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fricas [B]  time = 0.58, size = 68, normalized size = 2.19 \begin {gather*} e^{\left ({\left (4 \, x^{2} e^{\left (4 \, e^{4} + 2 \, e^{3}\right )} - {\left (4 \, x e^{\left (2 \, e^{4} + e^{3}\right )} + x - e^{2} - 25\right )} e^{\left (2 \, e^{4} + 2 \, e^{3}\right )} + e^{\left (4 \, e^{4} + 4 \, e^{3}\right )}\right )} e^{\left (-2 \, e^{4} - 2 \, e^{3}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(2)^2)^2*exp(exp(3))+8*x*exp(exp(2)^2)^2-1)*exp(exp(exp(2)^2)^2*exp(exp(3))^2-4*x*exp(exp
(2)^2)^2*exp(exp(3))+4*x^2*exp(exp(2)^2)^2+exp(2)-x+25),x, algorithm="fricas")

[Out]

e^((4*x^2*e^(4*e^4 + 2*e^3) - (4*x*e^(2*e^4 + e^3) + x - e^2 - 25)*e^(2*e^4 + 2*e^3) + e^(4*e^4 + 4*e^3))*e^(-
2*e^4 - 2*e^3))

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giac [A]  time = 0.31, size = 39, normalized size = 1.26 \begin {gather*} e^{\left (4 \, x^{2} e^{\left (2 \, e^{4}\right )} - 4 \, x e^{\left (2 \, e^{4} + e^{3}\right )} - x + e^{2} + e^{\left (2 \, e^{4} + 2 \, e^{3}\right )} + 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(4*x^2*e^(2*e^4) - 4*x*e^(2*e^4 + e^3) - x + e^2 + e^(2*e^4 + 2*e^3) + 25)

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maple [A]  time = 0.06, size = 40, normalized size = 1.29

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}+{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\)	\(40\)
gosper	\({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\)	\(47\)
derivativedivides	\({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\)	\(47\)
default	\({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\)	\(47\)
norman	\({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(2*exp(4)+2*exp(3))-4*x*exp(2*exp(4)+exp(3))+4*x^2*exp(2*exp(4))+exp(2)-x+25)

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maxima [A]  time = 0.36, size = 39, normalized size = 1.26 \begin {gather*} e^{\left (4 \, x^{2} e^{\left (2 \, e^{4}\right )} - 4 \, x e^{\left (2 \, e^{4} + e^{3}\right )} - x + e^{2} + e^{\left (2 \, e^{4} + 2 \, e^{3}\right )} + 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(2)^2)^2*exp(exp(3))+8*x*exp(exp(2)^2)^2-1)*exp(exp(exp(2)^2)^2*exp(exp(3))^2-4*x*exp(exp
(2)^2)^2*exp(exp(3))+4*x^2*exp(exp(2)^2)^2+exp(2)-x+25),x, algorithm="maxima")

[Out]

e^(4*x^2*e^(2*e^4) - 4*x*e^(2*e^4 + e^3) - x + e^2 + e^(2*e^4 + 2*e^3) + 25)

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mupad [B]  time = 0.31, size = 45, normalized size = 1.45 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}}\,{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{{\mathrm {e}}^3}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}}\,{\mathrm {e}}^{{\mathrm {e}}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(2*exp(3))*exp(2*exp(4)))*exp(-4*x*exp(2*exp(4))*exp(exp(3)))*exp(-x)*exp(25)*exp(4*x^2*exp(2*exp(4)))*
exp(exp(2))

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sympy [A]  time = 0.15, size = 48, normalized size = 1.55 \begin {gather*} e^{4 x^{2} e^{2 e^{4}} - 4 x e^{e^{3}} e^{2 e^{4}} - x + e^{2} + 25 + e^{2 e^{3}} e^{2 e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(exp(2)**2)**2*exp(exp(3))+8*x*exp(exp(2)**2)**2-1)*exp(exp(exp(2)**2)**2*exp(exp(3))**2-4*x*
exp(exp(2)**2)**2*exp(exp(3))+4*x**2*exp(exp(2)**2)**2+exp(2)-x+25),x)

[Out]

exp(4*x**2*exp(2*exp(4)) - 4*x*exp(exp(3))*exp(2*exp(4)) - x + exp(2) + 25 + exp(2*exp(3))*exp(2*exp(4)))

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