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

Optimal. Leaf size=18 \[ -e^{-x} \sqrt {e^{2 x}+1} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2249, 191} \[ -e^{-x} \sqrt {e^{2 x}+1} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^x*Sqrt[1 + E^(2*x)]),x]

[Out]

-(Sqrt[1 + E^(2*x)]/E^x)

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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+\frac {1}{x^2}}} \, dx,x,e^{-x}\right )\\ &=-e^{-x} \sqrt {1+e^{2 x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -e^{-x} \sqrt {e^{2 x}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 + E^(2*x)]/E^x)

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fricas [A]  time = 0.39, size = 10, normalized size = 0.56 \[ -\sqrt {e^{\left (-2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(e^(-2*x) + 1)

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giac [A]  time = 0.24, size = 21, normalized size = 1.17 \[ \frac {2}{{\left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right )}^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((sqrt(e^(2*x) + 1) - e^x)^2 - 1)

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maple [A]  time = 0.04, size = 15, normalized size = 0.83 \[ -\sqrt {{\mathrm e}^{2 x}+1}\, {\mathrm e}^{-x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.99, size = 14, normalized size = 0.78 \[ -\sqrt {e^{\left (2 \, x\right )} + 1} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(e^(2*x) + 1)*e^(-x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {{\mathrm {e}}^{-x}}{\sqrt {{\mathrm {e}}^{2\,x}+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)/(exp(2*x) + 1)^(1/2),x)

[Out]

int(exp(-x)/(exp(2*x) + 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- x}}{\sqrt {e^{2 x} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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