3.1.27 \(\int \frac {e^{2 x}}{A+B e^{4 x}} \, dx\) [27]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 31, 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, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {B} e^{2 x}}{\sqrt {A}}\right )}{2 \sqrt {A} \sqrt {B}} \end {gather*}

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/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^{2 x}}{A+B e^{4 x}} \, dx &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {B} e^{2 x}}{\sqrt {A}}\right )}{2 \sqrt {A} \sqrt {B}} \end {gather*}

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 = 0.03, size = 20, normalized size = 0.65

method	result	size
default	\(\frac {\arctan \left (\frac {B \,{\mathrm e}^{2 x}}{\sqrt {A B}}\right )}{2 \sqrt {A B}}\)	\(20\)
risch	\(-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {A}{\sqrt {-A B}}\right )}{4 \sqrt {-A B}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {A}{\sqrt {-A B}}\right )}{4 \sqrt {-A B}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.02, size = 19, normalized size = 0.61 \begin {gather*} \frac {\arctan \left (\frac {B e^{\left (2 \, x\right )}}{\sqrt {A B}}\right )}{2 \, \sqrt {A B}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.79, size = 76, normalized size = 2.45 \begin {gather*} \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 {gather*}

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 = 0.06, size = 22, normalized size = 0.71 \begin {gather*} \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 {gather*}

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 = 0.46, size = 19, normalized size = 0.61 \begin {gather*} \frac {\arctan \left (\frac {B e^{\left (2 \, x\right )}}{\sqrt {A B}}\right )}{2 \, \sqrt {A B}} \end {gather*}

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)

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Mupad [B]
time = 0.23, size = 19, normalized size = 0.61 \begin {gather*} \frac {\mathrm {atan}\left (\frac {B\,{\mathrm {e}}^{2\,x}}{\sqrt {A\,B}}\right )}{2\,\sqrt {A\,B}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/(A + B*exp(4*x)),x)

[Out]

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

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