3.157 \(\int \frac{1}{b e^{-m x}+a e^{m x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]

[Out]

ArcTan[(Sqrt[a]*E^(m*x))/Sqrt[b]]/(Sqrt[a]*Sqrt[b]*m)

________________________________________________________________________________________

Rubi [A]  time = 0.0287608, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2282, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]

Antiderivative was successfully verified.

[In]

Int[(b/E^(m*x) + a*E^(m*x))^(-1),x]

[Out]

ArcTan[(Sqrt[a]*E^(m*x))/Sqrt[b]]/(Sqrt[a]*Sqrt[b]*m)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[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{1}{b e^{-m x}+a e^{m x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,e^{m x}\right )}{m}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m}\\ \end{align*}

Mathematica [A]  time = 0.0091206, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/E^(m*x) + a*E^(m*x))^(-1),x]

[Out]

ArcTan[(Sqrt[a]*E^(m*x))/Sqrt[b]]/(Sqrt[a]*Sqrt[b]*m)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 22, normalized size = 0.7 \begin{align*}{\frac{1}{m}\arctan \left ({a{{\rm e}^{mx}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/exp(m*x)+a*exp(m*x)),x)

[Out]

1/m/(a*b)^(1/2)*arctan(a*exp(m*x)/(a*b)^(1/2))

________________________________________________________________________________________

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(1/(b/exp(m*x)+a*exp(m*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.73, size = 192, normalized size = 6.19 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{a e^{\left (2 \, m x\right )} - 2 \, \sqrt{-a b} e^{\left (m x\right )} - b}{a e^{\left (2 \, m x\right )} + b}\right )}{2 \, a b m}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} e^{\left (-m x\right )}}{a}\right )}{a b m}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/exp(m*x)+a*exp(m*x)),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-a*b)*log((a*e^(2*m*x) - 2*sqrt(-a*b)*e^(m*x) - b)/(a*e^(2*m*x) + b))/(a*b*m), -sqrt(a*b)*arctan(sq
rt(a*b)*e^(-m*x)/a)/(a*b*m)]

________________________________________________________________________________________

Sympy [A]  time = 0.176954, size = 26, normalized size = 0.84 \begin{align*} \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (- 2 i a + e^{- m x} \right )} \right )\right )}}{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/exp(m*x)+a*exp(m*x)),x)

[Out]

RootSum(4*_z**2*a*b + 1, Lambda(_i, _i*log(-2*_i*a + exp(-m*x))))/m

________________________________________________________________________________________

Giac [A]  time = 1.09318, size = 28, normalized size = 0.9 \begin{align*} \frac{\arctan \left (\frac{a e^{\left (m x\right )}}{\sqrt{a b}}\right )}{\sqrt{a b} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/exp(m*x)+a*exp(m*x)),x, algorithm="giac")

[Out]

arctan(a*e^(m*x)/sqrt(a*b))/(sqrt(a*b)*m)