3.1.90 \(\int \frac {1}{2} (-e^{e^4}+2 e^{8+4 x}+2 x) \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{4} \left (e^{4 (2+x)}+\left (e^{e^4}-x\right )^2+x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 2194} \begin {gather*} \frac {x^2}{2}-\frac {e^{e^4} x}{2}+\frac {1}{4} e^{4 x+8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^E^4 + 2*E^(8 + 4*x) + 2*x)/2,x]

[Out]

E^(8 + 4*x)/4 - (E^E^4*x)/2 + x^2/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 29, normalized size = 1.12 \begin {gather*} \frac {1}{4} e^{8+4 x}-\frac {e^{e^4} x}{2}+\frac {x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^E^4 + 2*E^(8 + 4*x) + 2*x)/2,x]

[Out]

E^(8 + 4*x)/4 - (E^E^4*x)/2 + x^2/2

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, x e^{\left (e^{4}\right )} + \frac {1}{4} \, e^{\left (4 \, x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*x*e^(e^4) + 1/4*e^(4*x + 8)

________________________________________________________________________________________

giac [A]  time = 0.29, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, x e^{\left (e^{4}\right )} + \frac {1}{4} \, e^{\left (4 \, x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*x*e^(e^4) + 1/4*e^(4*x + 8)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 21, normalized size = 0.81

method	result	size
default	\(\frac {x^{2}}{2}-\frac {x \,{\mathrm e}^{{\mathrm e}^{4}}}{2}+\frac {{\mathrm e}^{4 x +8}}{4}\)	\(21\)
norman	\(\frac {x^{2}}{2}-\frac {x \,{\mathrm e}^{{\mathrm e}^{4}}}{2}+\frac {{\mathrm e}^{4 x +8}}{4}\)	\(21\)
risch	\(\frac {x^{2}}{2}-\frac {x \,{\mathrm e}^{{\mathrm e}^{4}}}{2}+\frac {{\mathrm e}^{4 x +8}}{4}\)	\(21\)
derivativedivides	\(-2 x -4+\frac {\left (4 x +8\right )^{2}}{32}+\frac {{\mathrm e}^{4 x +8}}{4}-\frac {{\mathrm e}^{{\mathrm e}^{4}} \left (4 x +8\right )}{8}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2-1/2*x*exp(exp(4))+1/4*exp(4*x+8)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, x e^{\left (e^{4}\right )} + \frac {1}{4} \, e^{\left (4 \, x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*x*e^(e^4) + 1/4*e^(4*x + 8)

________________________________________________________________________________________

mupad [B]  time = 0.27, size = 20, normalized size = 0.77 \begin {gather*} \frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^8}{4}-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^4}}{2}+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x + exp(4*x + 8) - exp(exp(4))/2,x)

[Out]

(exp(4*x)*exp(8))/4 - (x*exp(exp(4)))/2 + x^2/2

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 20, normalized size = 0.77 \begin {gather*} \frac {x^{2}}{2} - \frac {x e^{e^{4}}}{2} + \frac {e^{4 x + 8}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - x*exp(exp(4))/2 + exp(4*x + 8)/4

________________________________________________________________________________________