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

Optimal. Leaf size=8 \[ -\sin ^{-1}\left (e^{-x}\right ) \]

[Out]

-ArcSin[E^(-x)]

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Rubi [A]  time = 0.0284385, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2249, 216} \[ -\sin ^{-1}\left (e^{-x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(E^x*Sqrt[1 - E^(-2*x)]),x]

[Out]

-ArcSin[E^(-x)]

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{e^{-x}}{\sqrt{1-e^{-2 x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,e^{-x}\right )\\ &=-\sin ^{-1}\left (e^{-x}\right )\\ \end{align*}

Mathematica [B]  time = 0.017202, size = 42, normalized size = 5.25 \[ \frac{e^{-x} \sqrt{e^{2 x}-1} \tan ^{-1}\left (\sqrt{e^{2 x}-1}\right )}{\sqrt{1-e^{-2 x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.073, size = 37, normalized size = 4.6 \begin{align*} -{\frac{1}{{{\rm e}^{x}}}\sqrt{ \left ({{\rm e}^{x}} \right ) ^{2}-1}\arctan \left ({\frac{1}{\sqrt{ \left ({{\rm e}^{x}} \right ) ^{2}-1}}} \right ){\frac{1}{\sqrt{{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}-1}{ \left ({{\rm e}^{x}} \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((exp(x)^2-1)/exp(x)^2)^(1/2)/exp(x)*(exp(x)^2-1)^(1/2)*arctan(1/(exp(x)^2-1)^(1/2))

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Maxima [B]  time = 1.45803, size = 19, normalized size = 2.38 \begin{align*} \arctan \left (\sqrt{-e^{\left (-2 \, x\right )} + 1} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(-e^(-2*x) + 1)*e^x)

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Fricas [B]  time = 0.755627, size = 55, normalized size = 6.88 \begin{align*} 2 \, \arctan \left ({\left (\sqrt{-e^{\left (-2 \, x\right )} + 1} - 1\right )} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.894799, size = 7, normalized size = 0.88 \begin{align*} - \operatorname{asin}{\left (e^{- x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-asin(exp(-x))

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Giac [B]  time = 1.3274, size = 19, normalized size = 2.38 \begin{align*} -\arctan \left (i\right ) + \arctan \left (\sqrt{e^{\left (2 \, x\right )} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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