3.2 \(\int \frac{1}{-5 e^{-m x}+2 e^{m x}} \, dx\)

Optimal. Leaf size=24 \[ -\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m} \]

[Out]

-(ArcTanh[Sqrt[2/5]*E^(m*x)]/(Sqrt[10]*m))

________________________________________________________________________________________

Rubi [A]  time = 0.0178917, antiderivative size = 24, 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, 207} \[ -\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m} \]

Antiderivative was successfully verified.

[In]

Int[(-5/E^(m*x) + 2*E^(m*x))^(-1),x]

[Out]

-(ArcTanh[Sqrt[2/5]*E^(m*x)]/(Sqrt[10]*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 207

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

Rubi steps

\begin{align*} \int \frac{1}{-5 e^{-m x}+2 e^{m x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{-5+2 x^2} \, dx,x,e^{m x}\right )}{m}\\ &=-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m}\\ \end{align*}

Mathematica [A]  time = 0.0096272, size = 24, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{5}} e^{m x}\right )}{\sqrt{10} m} \]

Antiderivative was successfully verified.

[In]

Integrate[(-5/E^(m*x) + 2*E^(m*x))^(-1),x]

[Out]

-(ArcTanh[Sqrt[2/5]*E^(m*x)]/(Sqrt[10]*m))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 19, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{10}}{10\,m}{\it Artanh} \left ({\frac{{{\rm e}^{mx}}\sqrt{10}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-5/exp(m*x)+2*exp(m*x)),x)

[Out]

-1/10*arctanh(1/5*exp(m*x)*10^(1/2))/m*10^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.40219, size = 47, normalized size = 1.96 \begin{align*} \frac{\sqrt{10} \log \left (-\frac{\sqrt{10} - 5 \, e^{\left (-m x\right )}}{\sqrt{10} + 5 \, e^{\left (-m x\right )}}\right )}{20 \, m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5/exp(m*x)+2*exp(m*x)),x, algorithm="maxima")

[Out]

1/20*sqrt(10)*log(-(sqrt(10) - 5*e^(-m*x))/(sqrt(10) + 5*e^(-m*x)))/m

________________________________________________________________________________________

Fricas [B]  time = 1.92609, size = 108, normalized size = 4.5 \begin{align*} \frac{\sqrt{10} \log \left (-\frac{2 \, \sqrt{10} e^{\left (m x\right )} - 2 \, e^{\left (2 \, m x\right )} - 5}{2 \, e^{\left (2 \, m x\right )} - 5}\right )}{20 \, m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5/exp(m*x)+2*exp(m*x)),x, algorithm="fricas")

[Out]

1/20*sqrt(10)*log(-(2*sqrt(10)*e^(m*x) - 2*e^(2*m*x) - 5)/(2*e^(2*m*x) - 5))/m

________________________________________________________________________________________

Sympy [A]  time = 0.13892, size = 20, normalized size = 0.83 \begin{align*} \frac{\operatorname{RootSum}{\left (40 z^{2} - 1, \left ( i \mapsto i \log{\left (- 4 i + e^{- m x} \right )} \right )\right )}}{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5/exp(m*x)+2*exp(m*x)),x)

[Out]

RootSum(40*_z**2 - 1, Lambda(_i, _i*log(-4*_i + exp(-m*x))))/m

________________________________________________________________________________________

Giac [B]  time = 1.10253, size = 51, normalized size = 2.12 \begin{align*} -\frac{\sqrt{10} \log \left (\frac{1}{2} \, \sqrt{10} + e^{\left (m x\right )}\right ) - \sqrt{10} \log \left ({\left | -\frac{1}{2} \, \sqrt{10} + e^{\left (m x\right )} \right |}\right )}{20 \, m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5/exp(m*x)+2*exp(m*x)),x, algorithm="giac")

[Out]

-1/20*(sqrt(10)*log(1/2*sqrt(10) + e^(m*x)) - sqrt(10)*log(abs(-1/2*sqrt(10) + e^(m*x))))/m