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

Optimal. Leaf size=23 \[ -e^x+e^x \left (4+e^{\frac {e^8 x}{2}}\right )+x \]

________________________________________________________________________________________

Rubi [B]  time = 0.41, antiderivative size = 50, normalized size of antiderivative = 2.17, number of steps used = 5, number of rules used = 2, integrand size = 97, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6688, 2194} \begin {gather*} x+3 e^x+\frac {2 e^{\frac {1}{2} \left (2+e^8\right ) x}}{2+e^8}+\frac {e^{\frac {1}{2} \left (2+e^8\right ) x+8}}{2+e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*E^x + (2*E^(((2 + E^8)*x)/2))/(2 + E^8) + E^(8 + ((2 + E^8)*x)/2)/(2 + E^8) + 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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 19, normalized size = 0.83 \begin {gather*} 3 e^x+e^{\frac {1}{2} \left (2+e^8\right ) x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*E^x + E^(((2 + E^8)*x)/2) + x

________________________________________________________________________________________

fricas [A]  time = 1.18, size = 44, normalized size = 1.91 \begin {gather*} {\left ({\left (x + 3 \, e^{x}\right )} e^{\left (\log \relax (2) - 8\right )} + e^{\left ({\left ({\left (\log \relax (2) - 8\right )} e^{\left (\log \relax (2) - 8\right )} + x\right )} e^{\left (-\log \relax (2) + 8\right )} + x\right )}\right )} e^{\left (-\log \relax (2) + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x + 3*e^x)*e^(log(2) - 8) + e^(((log(2) - 8)*e^(log(2) - 8) + x)*e^(-log(2) + 8) + x))*e^(-log(2) + 8)

________________________________________________________________________________________

giac [A]  time = 0.40, size = 14, normalized size = 0.61 \begin {gather*} x + e^{\left (\frac {1}{2} \, x e^{8} + x\right )} + 3 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^(1/2*x*e^8 + x) + 3*e^x

________________________________________________________________________________________

maple [A]  time = 0.34, size = 15, normalized size = 0.65

method	result	size
risch	\({\mathrm e}^{\frac {x \left (2+{\mathrm e}^{8}\right )}{2}}+x +3 \,{\mathrm e}^{x}\)	\(15\)
default	\(x +{\mathrm e}^{\ln \left ({\mathrm e}^{\frac {x \,{\mathrm e}^{8}}{2}}+4\right )+x}-{\mathrm e}^{x}\)	\(23\)
norman	\(x +\frac {{\mathrm e}^{-8} \left (2+{\mathrm e}^{8}\right ) {\mathrm e}^{x} {\mathrm e}^{\frac {x \,{\mathrm e}^{8}}{2}}}{2 \,{\mathrm e}^{-8}+1}+3 \,{\mathrm e}^{x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(1/2*x*(2+exp(8)))+x+3*exp(x)

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 41, normalized size = 1.78 \begin {gather*} {\left (e^{\left (\frac {1}{2} \, x e^{8} + 16 \, e^{\left (-8\right )} + 8\right )} + 3 \, e^{\left (16 \, e^{\left (-8\right )} + 8\right )}\right )} e^{\left ({\left (x e^{8} - 16\right )} e^{\left (-8\right )} - 8\right )} + x - 16 \, e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^(1/2*x*e^8 + 16*e^(-8) + 8) + 3*e^(16*e^(-8) + 8))*e^((x*e^8 - 16)*e^(-8) - 8) + x - 16*e^(-8)

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 14, normalized size = 0.61 \begin {gather*} x+{\mathrm {e}}^{x+\frac {x\,{\mathrm {e}}^8}{2}}+3\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(x + (x*exp(8))/2) + 3*exp(x)

________________________________________________________________________________________

sympy [A]  time = 0.43, size = 17, normalized size = 0.74 \begin {gather*} x + e^{x} e^{\frac {x e^{8}}{2}} + 3 e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(x)*exp(x*exp(8)/2) + 3*exp(x)

________________________________________________________________________________________