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]

1/4*arcsinh(1/4*exp(4*x))

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Rubi [A]  time = 0.02, antiderivative size = 14, 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, 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 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]

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^{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*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \[ \frac {1}{4} \sinh ^{-1}\left (\frac {e^{4 x}}{4}\right ) \]

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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fricas [A]  time = 0.39, size = 18, normalized size = 1.29 \[ -\frac {1}{4} \, \log \left (\sqrt {e^{\left (8 \, x\right )} + 16} - e^{\left (4 \, x\right )}\right ) \]

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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giac [A]  time = 0.21, size = 18, normalized size = 1.29 \[ -\frac {1}{4} \, \log \left (\sqrt {e^{\left (8 \, x\right )} + 16} - e^{\left (4 \, x\right )}\right ) \]

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))

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{4 x}}{\sqrt {{\mathrm e}^{8 x}+16}}\, dx \]

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 = 1.25, size = 9, normalized size = 0.64 \[ \frac {1}{4} \, \operatorname {arsinh}\left (\frac {1}{4} \, e^{\left (4 \, x\right )}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {{\mathrm {e}}^{4\,x}}{\sqrt {{\mathrm {e}}^{8\,x}+16}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x)/(exp(8*x) + 16)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.89, size = 8, normalized size = 0.57 \[ \frac {\operatorname {asinh}{\left (\frac {e^{4 x}}{4} \right )}}{4} \]

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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