3.96.57 \(\int e^{-3+2 x+e^{e^4} x^2} (-3-6 x-6 e^{e^4} x^2) \, dx\)

Optimal. Leaf size=20 \[ 3-3 e^{-3+x \left (2+e^{e^4} x\right )} x \]

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Rubi [A]  time = 0.06, antiderivative size = 40, normalized size of antiderivative = 2.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2288} \begin {gather*} -\frac {3 e^{e^{e^4} x^2+2 x-3} \left (e^{e^4} x^2+x\right )}{e^{e^4} x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-3 + 2*x + E^E^4*x^2)*(-3 - 6*x - 6*E^E^4*x^2),x]

[Out]

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

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 {3 e^{-3+2 x+e^{e^4} x^2} \left (x+e^{e^4} x^2\right )}{1+e^{e^4} x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} -3 e^{-3+2 x+e^{e^4} x^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-3 + 2*x + E^E^4*x^2)*(-3 - 6*x - 6*E^E^4*x^2),x]

[Out]

-3*E^(-3 + 2*x + E^E^4*x^2)*x

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fricas [A]  time = 0.53, size = 17, normalized size = 0.85 \begin {gather*} -3 \, e^{\left (x^{2} e^{\left (e^{4}\right )} + 2 \, x + \log \relax (x) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*exp(exp(4))-6*x-3)*exp(x^2*exp(exp(4))+2*x)/x/exp(3-log(x)),x, algorithm="fricas")

[Out]

-3*e^(x^2*e^(e^4) + 2*x + log(x) - 3)

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giac [A]  time = 0.15, size = 17, normalized size = 0.85 \begin {gather*} -3 \, e^{\left (x^{2} e^{\left (e^{4}\right )} + 2 \, x + \log \relax (x) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*exp(exp(4))-6*x-3)*exp(x^2*exp(exp(4))+2*x)/x/exp(3-log(x)),x, algorithm="giac")

[Out]

-3*e^(x^2*e^(e^4) + 2*x + log(x) - 3)

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maple [A]  time = 0.08, size = 17, normalized size = 0.85

method	result	size
risch	\(-3 x \,{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2 x -3}\)	\(17\)
norman	\(-3 x \,{\mathrm e}^{-3} {\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2 x}\)	\(20\)
gosper	\(-3 x \,{\mathrm e}^{-3} {\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2 x}\)	\(24\)
default	\(\frac {3 i \sqrt {\pi }\, {\mathrm e}^{-3-{\mathrm e}^{-{\mathrm e}^{4}}} {\mathrm e}^{-\frac {{\mathrm e}^{4}}{2}} \erf \left (i {\mathrm e}^{\frac {{\mathrm e}^{4}}{2}} x +i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{2}}\right )}{2}-3 \,{\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2 x -3}-3 i {\mathrm e}^{-\frac {3 \,{\mathrm e}^{4}}{2}} \sqrt {\pi }\, {\mathrm e}^{-3-{\mathrm e}^{-{\mathrm e}^{4}}} \erf \left (i {\mathrm e}^{\frac {{\mathrm e}^{4}}{2}} x +i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{2}}\right )-3 \,{\mathrm e}^{-{\mathrm e}^{4}} x \,{\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2 x -3+{\mathrm e}^{4}}+6 \,{\mathrm e}^{-{\mathrm e}^{4}} \left (\frac {{\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{x^{2} {\mathrm e}^{{\mathrm e}^{4}}+2 x -3+{\mathrm e}^{4}}}{2}+\frac {i {\mathrm e}^{-\frac {3 \,{\mathrm e}^{4}}{2}} \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{4}-3-{\mathrm e}^{-{\mathrm e}^{4}}} \erf \left (i {\mathrm e}^{\frac {{\mathrm e}^{4}}{2}} x +i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{2}}\right )}{2}\right )-\frac {3 i {\mathrm e}^{-\frac {3 \,{\mathrm e}^{4}}{2}} \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{4}-3-{\mathrm e}^{-{\mathrm e}^{4}}} \erf \left (i {\mathrm e}^{\frac {{\mathrm e}^{4}}{2}} x +i {\mathrm e}^{-\frac {{\mathrm e}^{4}}{2}}\right )}{2}\)	\(239\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2*exp(exp(4))-6*x-3)*exp(x^2*exp(exp(4))+2*x)/x/exp(3-ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.52, size = 285, normalized size = 14.25 \begin {gather*} 3 \, {\left (\frac {{\left (x e^{\left (e^{4}\right )} + 1\right )}^{3} e^{\left (-\frac {5}{2} \, e^{4}\right )} \Gamma \left (\frac {3}{2}, -{\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )}\right )}{\left (-{\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (x e^{\left (e^{4}\right )} + 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )}}\right ) - 1\right )} e^{\left (-\frac {5}{2} \, e^{4}\right )}}{\sqrt {-{\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )}}} + 2 \, e^{\left ({\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )} - \frac {3}{2} \, e^{4}\right )}\right )} e^{\left (\frac {1}{2} \, e^{4} - e^{\left (-e^{4}\right )} - 3\right )} + 3 \, {\left (\frac {\sqrt {\pi } {\left (x e^{\left (e^{4}\right )} + 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )}}\right ) - 1\right )} e^{\left (-\frac {3}{2} \, e^{4}\right )}}{\sqrt {-{\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )}}} - e^{\left ({\left (x e^{\left (e^{4}\right )} + 1\right )}^{2} e^{\left (-e^{4}\right )} - \frac {1}{2} \, e^{4}\right )}\right )} e^{\left (-\frac {1}{2} \, e^{4} - e^{\left (-e^{4}\right )} - 3\right )} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (x \sqrt {-e^{\left (e^{4}\right )}} - \frac {1}{\sqrt {-e^{\left (e^{4}\right )}}}\right ) e^{\left (-e^{\left (-e^{4}\right )} - 3\right )}}{2 \, \sqrt {-e^{\left (e^{4}\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*exp(exp(4))-6*x-3)*exp(x^2*exp(exp(4))+2*x)/x/exp(3-log(x)),x, algorithm="maxima")

[Out]

3*((x*e^(e^4) + 1)^3*e^(-5/2*e^4)*gamma(3/2, -(x*e^(e^4) + 1)^2*e^(-e^4))/(-(x*e^(e^4) + 1)^2*e^(-e^4))^(3/2)
- sqrt(pi)*(x*e^(e^4) + 1)*(erf(sqrt(-(x*e^(e^4) + 1)^2*e^(-e^4))) - 1)*e^(-5/2*e^4)/sqrt(-(x*e^(e^4) + 1)^2*e
^(-e^4)) + 2*e^((x*e^(e^4) + 1)^2*e^(-e^4) - 3/2*e^4))*e^(1/2*e^4 - e^(-e^4) - 3) + 3*(sqrt(pi)*(x*e^(e^4) + 1
)*(erf(sqrt(-(x*e^(e^4) + 1)^2*e^(-e^4))) - 1)*e^(-3/2*e^4)/sqrt(-(x*e^(e^4) + 1)^2*e^(-e^4)) - e^((x*e^(e^4)
+ 1)^2*e^(-e^4) - 1/2*e^4))*e^(-1/2*e^4 - e^(-e^4) - 3) - 3/2*sqrt(pi)*erf(x*sqrt(-e^(e^4)) - 1/sqrt(-e^(e^4))
)*e^(-e^(-e^4) - 3)/sqrt(-e^(e^4))

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mupad [B]  time = 8.05, size = 17, normalized size = 0.85 \begin {gather*} -3\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{{\mathrm {e}}^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x + x^2*exp(exp(4)))*exp(log(x) - 3)*(6*x + 6*x^2*exp(exp(4)) + 3))/x,x)

[Out]

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

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sympy [A]  time = 0.13, size = 20, normalized size = 1.00 \begin {gather*} - \frac {3 x e^{x^{2} e^{e^{4}} + 2 x}}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2*exp(exp(4))-6*x-3)*exp(x**2*exp(exp(4))+2*x)/x/exp(3-ln(x)),x)

[Out]

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

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