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

Optimal. Leaf size=14 \[ \sinh ^{-1}\left (\frac {1+2 e^x}{\sqrt {3}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 633, 221} \begin {gather*} \sinh ^{-1}\left (\frac {2 e^x+1}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {e^x}{\sqrt {1+e^x+e^{2 x}}} \, dx &=\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*}

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Mathematica [A]
time = 0.06, size = 26, normalized size = 1.86 \begin {gather*} -\log \left (-1-2 e^x+2 \sqrt {1+e^x+e^{2 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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)]]

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Maple [A]
time = 0.02, size = 11, normalized size = 0.79

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [A]
time = 0.49, size = 12, normalized size = 0.86 \begin {gather*} \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [A]
time = 0.46, size = 21, normalized size = 1.50 \begin {gather*} -\log \left (2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x} + 1} - 2 \, e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{x}}{\sqrt {e^{2 x} + e^{x} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x)/sqrt(exp(2*x) + exp(x) + 1), x)

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Giac [A]
time = 4.24, size = 21, normalized size = 1.50 \begin {gather*} -\log \left (2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x} + 1} - 2 \, e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 3.64, size = 15, normalized size = 1.07 \begin {gather*} \ln \left ({\mathrm {e}}^x+\sqrt {{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x+1}+\frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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