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

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

[Out]

-arcsin(exp(-x))

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Rubi [A]  time = 0.03, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [B]  time = 0.02, 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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fricas [B]  time = 0.41, size = 18, normalized size = 2.25 \[ 2 \, \arctan \left ({\left (\sqrt {-e^{\left (-2 \, x\right )} + 1} - 1\right )} e^{x}\right ) \]

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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giac [A]  time = 0.20, size = 14, normalized size = 1.75 \[ -\arctan \relax (i) + \arctan \left (\sqrt {e^{\left (2 \, x\right )} - 1}\right ) \]

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))

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maple [B]  time = 0.05, size = 37, normalized size = 4.62 \[ -\frac {\sqrt {{\mathrm e}^{2 x}-1}\, \arctan \left (\frac {1}{\sqrt {{\mathrm e}^{2 x}-1}}\right ) {\mathrm e}^{-x}}{\sqrt {\left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-2 x}}} \]

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 [A]  time = 2.29, size = 14, normalized size = 1.75 \[ \arctan \left (\sqrt {-e^{\left (-2 \, x\right )} + 1} e^{x}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.12 \[ \int \frac {{\mathrm {e}}^{-x}}{\sqrt {1-{\mathrm {e}}^{-2\,x}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)/(1 - exp(-2*x))^(1/2),x)

[Out]

int(exp(-x)/(1 - exp(-2*x))^(1/2), x)

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sympy [A]  time = 0.99, size = 7, normalized size = 0.88 \[ - \operatorname {asin}{\left (e^{- x} \right )} \]

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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