3.649 \(\int \frac {e^{2 x}}{1+e^{4 x}} \, dx\)

Optimal. Leaf size=10 \[ \frac {1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

[Out]

1/2*arctan(exp(2*x))

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Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, 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, 203} \[ \frac {1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*x)/(1 + E^(4*x)),x]

[Out]

ArcTan[E^(2*x)]/2

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (e^{2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)/(1 + E^(4*x)),x]

[Out]

ArcTan[E^(2*x)]/2

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fricas [A]  time = 0.39, size = 7, normalized size = 0.70 \[ \frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(e^(2*x))

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giac [A]  time = 0.21, size = 7, normalized size = 0.70 \[ \frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(e^(2*x))

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maple [A]  time = 0.03, size = 8, normalized size = 0.80 \[ \frac {\arctan \left ({\mathrm e}^{2 x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(exp(x)^2)

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maxima [A]  time = 2.11, size = 7, normalized size = 0.70 \[ \frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(e^(2*x))

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mupad [B]  time = 3.37, size = 7, normalized size = 0.70 \[ \frac {\mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/(exp(4*x) + 1),x)

[Out]

atan(exp(2*x))/2

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sympy [B]  time = 0.11, size = 17, normalized size = 1.70 \[ \operatorname {RootSum} {\left (16 z^{2} + 1, \left (i \mapsto i \log {\left (4 i + e^{2 x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16*_z**2 + 1, Lambda(_i, _i*log(4*_i + exp(2*x))))

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