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

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

[Out]

ArcSinh[(1 + 2*E^x)/Sqrt[3]]

_______________________________________________________________________________________

Rubi [A]  time = 0.060464, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \sinh ^{-1}\left (\frac{2 e^x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcSinh[(1 + 2*E^x)/Sqrt[3]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.0855, size = 22, normalized size = 1.57 \[ \operatorname{atanh}{\left (\frac{2 e^{x} + 1}{2 \sqrt{e^{2 x} + e^{x} + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)/(1+exp(x)+exp(2*x))**(1/2),x)

[Out]

atanh((2*exp(x) + 1)/(2*sqrt(exp(2*x) + exp(x) + 1)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0184937, size = 14, normalized size = 1. \[ \sinh ^{-1}\left (\frac{2 e^x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcSinh[(1 + 2*E^x)/Sqrt[3]]

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 11, normalized size = 0.8 \[{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ({{\rm e}^{x}}+{\frac{1}{2}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.870914, size = 16, normalized size = 1.14 \[ \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{x} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/sqrt(e^(2*x) + e^x + 1),x, algorithm="maxima")

[Out]

arcsinh(1/3*sqrt(3)*(2*e^x + 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.255446, size = 28, normalized size = 2. \[ -\log \left (2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x} + 1} - 2 \, e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/sqrt(e^(2*x) + e^x + 1),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e^{x}}{\sqrt{e^{2 x} + e^{x} + 1}}\, dx \]

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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230663, size = 28, normalized size = 2. \[ -{\rm ln}\left (2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x} + 1} - 2 \, e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^x/sqrt(e^(2*x) + e^x + 1),x, algorithm="giac")

[Out]

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