3.8.5 \(\int \frac {e^{-x}}{\sqrt {1+e^{2 x}}} \, dx\) [705]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {2281, 197} \begin {gather*} -e^{-x} \sqrt {e^{2 x}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^x*Sqrt[1 + E^(2*x)]),x]

[Out]

-(Sqrt[1 + E^(2*x)]/E^x)

Rule 197

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 2281

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 &=-\text {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 = 2.33, size = 18, normalized size = 1.00 \begin {gather*} -e^{-x} \sqrt {1+e^{2 x}} \end {gather*}

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 = 0.02, size = 15, normalized size = 0.83

method	result	size
default	\(-{\mathrm e}^{-x} \sqrt {1+{\mathrm e}^{2 x}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 14, normalized size = 0.78 \begin {gather*} -\sqrt {e^{\left (2 \, x\right )} + 1} e^{\left (-x\right )} \end {gather*}

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 = 0.35, size = 10, normalized size = 0.56 \begin {gather*} -\sqrt {e^{\left (-2 \, x\right )} + 1} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{- x}}{\sqrt {e^{2 x} + 1}}\, dx \end {gather*}

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 = 6.28, size = 21, normalized size = 1.17 \begin {gather*} \frac {2}{{\left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right )}^{2} - 1} \end {gather*}

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

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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