\(\int \frac {1}{2} e^{e^{\frac {1}{2} (14-2 e^4+2 e^{x^2}-x)}+\frac {1}{2} (14-2 e^4+2 e^{x^2}-x)} (-1+4 e^{x^2} x) \, dx\) [5979]

Optimal result

Integrand size = 61, antiderivative size = 21 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{e^{7-e^4+e^{x^2}-\frac {x}{2}}} \]

[Out]

exp(exp(exp(x^2)-exp(4)-1/2*x+7))

Rubi [F]

\[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=\int \frac {1}{2} \exp \left (e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )\right ) \left (-1+4 e^{x^2} x\right ) \, dx \]

[In]

Int[(E^(E^((14 - 2*E^4 + 2*E^x^2 - x)/2) + (14 - 2*E^4 + 2*E^x^2 - x)/2)*(-1 + 4*E^x^2*x))/2,x]

[Out]

-1/2*Defer[Int][E^(E^((14 - 2*E^4 + 2*E^x^2 - x)/2) + (14 - 2*E^4 + 2*E^x^2 - x)/2), x] + 2*Defer[Int][E^(E^((
14 - 2*E^4 + 2*E^x^2 - x)/2) + (14 - 2*E^4 + 2*E^x^2 - x)/2 + x^2)*x, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \exp \left (e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )\right ) \left (-1+4 e^{x^2} x\right ) \, dx \\ & = \frac {1}{2} \int \left (-\exp \left (e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )\right )+4 \exp \left (e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )+x^2\right ) x\right ) \, dx \\ & = -\left (\frac {1}{2} \int \exp \left (e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )\right ) \, dx\right )+2 \int \exp \left (e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )+x^2\right ) x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{e^{7-e^4+e^{x^2}-\frac {x}{2}}} \]

[In]

Integrate[(E^(E^((14 - 2*E^4 + 2*E^x^2 - x)/2) + (14 - 2*E^4 + 2*E^x^2 - x)/2)*(-1 + 4*E^x^2*x))/2,x]

[Out]

E^E^(7 - E^4 + E^x^2 - x/2)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

method	result	size
norman	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}}-{\mathrm e}^{4}-\frac {x}{2}+7}}\)	\(16\)
risch	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}}-{\mathrm e}^{4}-\frac {x}{2}+7}}\)	\(16\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}}-{\mathrm e}^{4}-\frac {x}{2}+7}}\)	\(16\)

[In]

int(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^2)-exp(4)-1/2*x+7)),x,method=_RETURNVERBOS
E)

[Out]

exp(exp(exp(x^2)-exp(4)-1/2*x+7))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{\left (e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + 7\right )}\right )} \]

[In]

integrate(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^2)-exp(4)-1/2*x+7)),x, algorithm="fr
icas")

[Out]

e^(e^(-1/2*x - e^4 + e^(x^2) + 7))

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{e^{- \frac {x}{2} + e^{x^{2}} - e^{4} + 7}} \]

[In]

integrate(1/2*(4*exp(x**2)*x-1)*exp(exp(x**2)-exp(4)-1/2*x+7)*exp(exp(exp(x**2)-exp(4)-1/2*x+7)),x)

[Out]

exp(exp(-x/2 + exp(x**2) - exp(4) + 7))

Maxima [A] (verification not implemented)

none

Time = 0.64 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=e^{\left (e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + 7\right )}\right )} \]

[In]

integrate(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^2)-exp(4)-1/2*x+7)),x, algorithm="ma
xima")

[Out]

e^(e^(-1/2*x - e^4 + e^(x^2) + 7))

Giac [F]

\[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx=\int { \frac {1}{2} \, {\left (4 \, x e^{\left (x^{2}\right )} - 1\right )} e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + e^{\left (-\frac {1}{2} \, x - e^{4} + e^{\left (x^{2}\right )} + 7\right )} + 7\right )} \,d x } \]

[In]

integrate(1/2*(4*exp(x^2)*x-1)*exp(exp(x^2)-exp(4)-1/2*x+7)*exp(exp(exp(x^2)-exp(4)-1/2*x+7)),x, algorithm="gi
ac")

[Out]

integrate(1/2*(4*x*e^(x^2) - 1)*e^(-1/2*x - e^4 + e^(x^2) + e^(-1/2*x - e^4 + e^(x^2) + 7) + 7), x)

Mupad [B] (verification not implemented)

Time = 11.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{2} e^{e^{\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )}+\frac {1}{2} \left (14-2 e^4+2 e^{x^2}-x\right )} \left (-1+4 e^{x^2} x\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^7\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}} \]

[In]

int((exp(exp(exp(x^2) - x/2 - exp(4) + 7))*exp(exp(x^2) - x/2 - exp(4) + 7)*(4*x*exp(x^2) - 1))/2,x)

[Out]

exp(exp(-exp(4))*exp(-x/2)*exp(7)*exp(exp(x^2)))