3.7.67 \(\int \frac {e^{4 x}}{\sqrt {16+e^{8 x}}} \, dx\) [667]

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 = 0.01, 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 = {2281, 221} \begin {gather*} \frac {1}{4} \sinh ^{-1}\left (\frac {e^{4 x}}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4*x)/Sqrt[16 + E^(8*x)],x]

[Out]

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

Rule 221

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 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^{4 x}}{\sqrt {16+e^{8 x}}} \, dx &=\frac {1}{4} \text {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 = 0.03, size = 22, normalized size = 1.57 \begin {gather*} \frac {1}{4} \tanh ^{-1}\left (\frac {e^{4 x}}{\sqrt {16+e^{8 x}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[E^(4*x)/Sqrt[16 + E^(8*x)]]/4

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{4 x}}{\sqrt {16+{\mathrm e}^{8 x}}}\, dx\]

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 = 0.72, size = 9, normalized size = 0.64 \begin {gather*} \frac {1}{4} \, \operatorname {arsinh}\left (\frac {1}{4} \, e^{\left (4 \, x\right )}\right ) \end {gather*}

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 = 0.36, size = 18, normalized size = 1.29 \begin {gather*} -\frac {1}{4} \, \log \left (\sqrt {e^{\left (8 \, x\right )} + 16} - e^{\left (4 \, x\right )}\right ) \end {gather*}

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 = 0.43, size = 8, normalized size = 0.57 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {e^{4 x}}{4} \right )}}{4} \end {gather*}

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 = 5.10, size = 18, normalized size = 1.29 \begin {gather*} -\frac {1}{4} \, \log \left (\sqrt {e^{\left (8 \, x\right )} + 16} - e^{\left (4 \, x\right )}\right ) \end {gather*}

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

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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