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

Optimal. Leaf size=4 \[ \sinh ^{-1}\left (e^x\right ) \]

[Out]

ArcSinh[E^x]

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Rubi [A]  time = 0.0214806, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2249, 215} \[ \sinh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x/Sqrt[1 + E^(2*x)],x]

[Out]

ArcSinh[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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 )\\ &=\sinh ^{-1}\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0037465, size = 4, normalized size = 1. \[ \sinh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/Sqrt[1 + E^(2*x)],x]

[Out]

ArcSinh[E^x]

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Maple [A]  time = 0.065, size = 4, normalized size = 1. \begin{align*}{\it Arcsinh} \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(exp(x))

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Maxima [A]  time = 1.45739, size = 4, normalized size = 1. \begin{align*} \operatorname{arsinh}\left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(e^x)

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Fricas [B]  time = 0.770909, size = 42, normalized size = 10.5 \begin{align*} -\log \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(exp(x)/(1+exp(2*x))^(1/2),x, algorithm="fricas")

[Out]

-log(sqrt(e^(2*x) + 1) - e^x)

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Sympy [A]  time = 0.620196, size = 3, normalized size = 0.75 \begin{align*} \operatorname{asinh}{\left (e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(exp(x))

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Giac [B]  time = 1.22555, size = 22, normalized size = 5.5 \begin{align*} -\log \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(exp(x)/(1+exp(2*x))^(1/2),x, algorithm="giac")

[Out]

-log(sqrt(e^(2*x) + 1) - e^x)