3.27 \(\int \frac{e^{2 x}}{A+B e^{4 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]

[Out]

ArcTan[(Sqrt[B]*E^(2*x))/Sqrt[A]]/(2*Sqrt[A]*Sqrt[B])

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Rubi [A]  time = 0.0347668, antiderivative size = 31, 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, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*x)/(A + B*E^(4*x)),x]

[Out]

ArcTan[(Sqrt[B]*E^(2*x))/Sqrt[A]]/(2*Sqrt[A]*Sqrt[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]

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.008076, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{B} e^{2 x}}{\sqrt{A}}\right )}{2 \sqrt{A} \sqrt{B}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)/(A + B*E^(4*x)),x]

[Out]

ArcTan[(Sqrt[B]*E^(2*x))/Sqrt[A]]/(2*Sqrt[A]*Sqrt[B])

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Maple [A]  time = 0.007, size = 20, normalized size = 0.7 \begin{align*}{\frac{1}{2}\arctan \left ({B \left ({{\rm e}^{x}} \right ) ^{2}{\frac{1}{\sqrt{AB}}}} \right ){\frac{1}{\sqrt{AB}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/(A+B*exp(4*x)),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93193, size = 186, normalized size = 6. \begin{align*} \left [-\frac{\sqrt{-A B} \log \left (\frac{B e^{\left (4 \, x\right )} - 2 \, \sqrt{-A B} e^{\left (2 \, x\right )} - A}{B e^{\left (4 \, x\right )} + A}\right )}{4 \, A B}, -\frac{\sqrt{A B} \arctan \left (\frac{\sqrt{A B} e^{\left (-2 \, x\right )}}{B}\right )}{2 \, A B}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*sqrt(-A*B)*log((B*e^(4*x) - 2*sqrt(-A*B)*e^(2*x) - A)/(B*e^(4*x) + A))/(A*B), -1/2*sqrt(A*B)*arctan(sqrt
(A*B)*e^(-2*x)/B)/(A*B)]

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Sympy [A]  time = 0.150645, size = 22, normalized size = 0.71 \begin{align*} \operatorname{RootSum}{\left (16 z^{2} A B + 1, \left ( i \mapsto i \log{\left (4 i A + e^{2 x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(A+B*exp(4*x)),x)

[Out]

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

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Giac [A]  time = 1.08468, size = 26, normalized size = 0.84 \begin{align*} \frac{\arctan \left (\frac{B e^{\left (2 \, x\right )}}{\sqrt{A B}}\right )}{2 \, \sqrt{A B}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(B*e^(2*x)/sqrt(A*B))/sqrt(A*B)