3.625 \(\int e^x \sqrt{1-e^{2 x}} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{2} e^x \sqrt{1-e^{2 x}}+\frac{1}{2} \sin ^{-1}\left (e^x\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0448223, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{2} e^x \sqrt{1-e^{2 x}}+\frac{1}{2} \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x*Sqrt[1 - E^(2*x)],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.0512, size = 20, normalized size = 0.69 \[ \frac{\sqrt{- e^{2 x} + 1} e^{x}}{2} + \frac{\operatorname{asin}{\left (e^{x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-exp(2*x) + 1)*exp(x)/2 + asin(exp(x))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.021292, size = 26, normalized size = 0.9 \[ \frac{1}{2} \left (e^x \sqrt{1-e^{2 x}}+\sin ^{-1}\left (e^x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 21, normalized size = 0.7 \[{\frac{{{\rm e}^{x}}}{2}\sqrt{1- \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{\arcsin \left ({{\rm e}^{x}} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*exp(x)*(1-exp(x)^2)^(1/2)+1/2*arcsin(exp(x))

_______________________________________________________________________________________

Maxima [A]  time = 0.854537, size = 27, normalized size = 0.93 \[ \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 1} e^{x} + \frac{1}{2} \, \arcsin \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-e^(2*x) + 1)*e^x + 1/2*arcsin(e^x)

_______________________________________________________________________________________

Fricas [A]  time = 0.246567, size = 124, normalized size = 4.28 \[ -\frac{2 \,{\left (2 \, \sqrt{-e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 2\right )} \arctan \left ({\left (\sqrt{-e^{\left (2 \, x\right )} + 1} - 1\right )} e^{\left (-x\right )}\right ) -{\left (e^{\left (3 \, x\right )} - 2 \, e^{x}\right )} \sqrt{-e^{\left (2 \, x\right )} + 1} + 2 \, e^{\left (3 \, x\right )} - 2 \, e^{x}}{2 \,{\left (2 \, \sqrt{-e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*(2*sqrt(-e^(2*x) + 1) + e^(2*x) - 2)*arctan((sqrt(-e^(2*x) + 1) - 1)*e^(
-x)) - (e^(3*x) - 2*e^x)*sqrt(-e^(2*x) + 1) + 2*e^(3*x) - 2*e^x)/(2*sqrt(-e^(2*x
) + 1) + e^(2*x) - 2)

_______________________________________________________________________________________

Sympy [A]  time = 1.54093, size = 24, normalized size = 0.83 \[ \begin{cases} \frac{\sqrt{- e^{2 x} + 1} e^{x}}{2} + \frac{\operatorname{asin}{\left (e^{x} \right )}}{2} & \text{for}\: e^{x} < 0 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*(1-exp(2*x))**(1/2),x)

[Out]

Piecewise((sqrt(-exp(2*x) + 1)*exp(x)/2 + asin(exp(x))/2, exp(x) < 0))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231957, size = 27, normalized size = 0.93 \[ \frac{1}{2} \, \sqrt{-e^{\left (2 \, x\right )} + 1} e^{x} + \frac{1}{2} \, \arcsin \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-e^(2*x) + 1)*e^x + 1/2*arcsin(e^x)