\(\int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx\) [643]

Optimal result

Integrand size = 18, antiderivative size = 14 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=\text {arcsinh}\left (\frac {1+2 e^x}{\sqrt {3}}\right ) \]

[Out]

arcsinh(1/3*(1+2*exp(x))*3^(1/2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 633, 221} \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=\text {arcsinh}\left (\frac {2 e^x+1}{\sqrt {3}}\right ) \]

[In]

Int[E^x/Sqrt[1 + E^x + E^(2*x)],x]

[Out]

ArcSinh[(1 + 2*E^x)/Sqrt[3]]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,e^x\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 e^x\right )}{\sqrt {3}} \\ & = \sinh ^{-1}\left (\frac {1+2 e^x}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=-\log \left (-1-2 e^x+2 \sqrt {1+e^x+e^{2 x}}\right ) \]

[In]

Integrate[E^x/Sqrt[1 + E^x + E^(2*x)],x]

[Out]

-Log[-1 - 2*E^x + 2*Sqrt[1 + E^x + E^(2*x)]]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79

method	result	size
default	\(\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left ({\mathrm e}^{x}+\frac {1}{2}\right )}{3}\right )\)	\(11\)

[In]

int(exp(x)/(1+exp(x)+exp(2*x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsinh(2/3*3^(1/2)*(exp(x)+1/2))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=-\log \left (2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x} + 1} - 2 \, e^{x} - 1\right ) \]

[In]

integrate(exp(x)/(1+exp(x)+exp(2*x))^(1/2),x, algorithm="fricas")

[Out]

-log(2*sqrt(e^(2*x) + e^x + 1) - 2*e^x - 1)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=\operatorname {asinh}{\left (\frac {2 \sqrt {3} \left (e^{x} + \frac {1}{2}\right )}{3} \right )} \]

[In]

integrate(exp(x)/(1+exp(x)+exp(2*x))**(1/2),x)

[Out]

asinh(2*sqrt(3)*(exp(x) + 1/2)/3)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=\operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) \]

[In]

integrate(exp(x)/(1+exp(x)+exp(2*x))^(1/2),x, algorithm="maxima")

[Out]

arcsinh(1/3*sqrt(3)*(2*e^x + 1))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=-\log \left (2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x} + 1} - 2 \, e^{x} - 1\right ) \]

[In]

integrate(exp(x)/(1+exp(x)+exp(2*x))^(1/2),x, algorithm="giac")

[Out]

-log(2*sqrt(e^(2*x) + e^x + 1) - 2*e^x - 1)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx=\ln \left ({\mathrm {e}}^x+\sqrt {{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x+1}+\frac {1}{2}\right ) \]

[In]

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

[Out]

log(exp(x) + (exp(2*x) + exp(x) + 1)^(1/2) + 1/2)