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

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

[Out]

-(Sqrt[1 + E^(2*x)]/E^x)

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Rubi [A]  time = 0.0255804, antiderivative size = 18, normalized size of antiderivative = 1., 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 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 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]

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*}

Mathematica [A]  time = 0.0121807, size = 18, normalized size = 1. \[ -e^{-x} \sqrt{e^{2 x}+1} \]

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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Maple [A]  time = 0.06, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{{{\rm e}^{x}}}\sqrt{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+exp(2*x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.999682, size = 19, normalized size = 1.06 \begin{align*} -\sqrt{e^{\left (2 \, x\right )} + 1} e^{\left (-x\right )} \end{align*}

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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Fricas [A]  time = 0.919925, size = 28, normalized size = 1.56 \begin{align*} -\sqrt{e^{\left (-2 \, x\right )} + 1} \end{align*}

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

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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Giac [A]  time = 1.31232, size = 28, normalized size = 1.56 \begin{align*} \frac{2}{{\left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right )}^{2} - 1} \end{align*}

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)