3.667 \(\int \frac{e^{4 x}}{\sqrt{16+e^{8 x}}} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{4} \sinh ^{-1}\left (\frac{e^{4 x}}{4}\right ) \]

[Out]

ArcSinh[E^(4*x)/4]/4

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Rubi [A]  time = 0.0234841, antiderivative size = 14, 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, 215} \[ \frac{1}{4} \sinh ^{-1}\left (\frac{e^{4 x}}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(4*x)/Sqrt[16 + E^(8*x)],x]

[Out]

ArcSinh[E^(4*x)/4]/4

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^{4 x}}{\sqrt{16+e^{8 x}}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{16+x^2}} \, dx,x,e^{4 x}\right )\\ &=\frac{1}{4} \sinh ^{-1}\left (\frac{e^{4 x}}{4}\right )\\ \end{align*}

Mathematica [A]  time = 0.0044799, size = 14, normalized size = 1. \[ \frac{1}{4} \sinh ^{-1}\left (\frac{e^{4 x}}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*x)/Sqrt[16 + E^(8*x)],x]

[Out]

ArcSinh[E^(4*x)/4]/4

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{4\,x}}{\frac{1}{\sqrt{16+{{\rm e}^{8\,x}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x)/(16+exp(8*x))^(1/2),x)

[Out]

int(exp(4*x)/(16+exp(8*x))^(1/2),x)

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Maxima [A]  time = 1.45731, size = 12, normalized size = 0.86 \begin{align*} \frac{1}{4} \, \operatorname{arsinh}\left (\frac{1}{4} \, e^{\left (4 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x)/(16+exp(8*x))^(1/2),x, algorithm="maxima")

[Out]

1/4*arcsinh(1/4*e^(4*x))

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Fricas [A]  time = 0.597388, size = 54, normalized size = 3.86 \begin{align*} -\frac{1}{4} \, \log \left (\sqrt{e^{\left (8 \, x\right )} + 16} - e^{\left (4 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x)/(16+exp(8*x))^(1/2),x, algorithm="fricas")

[Out]

-1/4*log(sqrt(e^(8*x) + 16) - e^(4*x))

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Sympy [A]  time = 0.946482, size = 8, normalized size = 0.57 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{e^{4 x}}{4} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x)/(16+exp(8*x))**(1/2),x)

[Out]

asinh(exp(4*x)/4)/4

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Giac [A]  time = 1.23226, size = 24, normalized size = 1.71 \begin{align*} -\frac{1}{4} \, \log \left (\sqrt{e^{\left (8 \, x\right )} + 16} - e^{\left (4 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4*x)/(16+exp(8*x))^(1/2),x, algorithm="giac")

[Out]

-1/4*log(sqrt(e^(8*x) + 16) - e^(4*x))