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]

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 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])

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]]]

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*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{5}} e^{m x}\right )}{\sqrt {10} m} \]

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))

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fricas [B]  time = 0.72, size = 40, normalized size = 1.67 \[ \frac {\sqrt {10} \log \left (-\frac {2 \, \sqrt {10} e^{\left (m x\right )} - 2 \, e^{\left (2 \, m x\right )} - 5}{2 \, e^{\left (2 \, m x\right )} - 5}\right )}{20 \, m} \]

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

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giac [B]  time = 1.07, size = 38, normalized size = 1.58 \[ -\frac {\sqrt {10} \log \left (\frac {1}{2} \, \sqrt {10} + e^{\left (m x\right )}\right ) - \sqrt {10} \log \left ({\left | -\frac {1}{2} \, \sqrt {10} + e^{\left (m x\right )} \right |}\right )}{20 \, m} \]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.79 \[ -\frac {\sqrt {10}\, \arctanh \left (\frac {\sqrt {10}\, {\mathrm e}^{m x}}{5}\right )}{10 m} \]

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)

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maxima [A]  time = 1.40, size = 35, normalized size = 1.46 \[ \frac {\sqrt {10} \log \left (-\frac {\sqrt {10} - 5 \, e^{\left (-m x\right )}}{\sqrt {10} + 5 \, e^{\left (-m x\right )}}\right )}{20 \, m} \]

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

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mupad [B]  time = 0.46, size = 18, normalized size = 0.75 \[ -\frac {\sqrt {10}\,\mathrm {atanh}\left (\frac {\sqrt {10}\,{\mathrm {e}}^{m\,x}}{5}\right )}{10\,m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*exp(m*x) - 5*exp(-m*x)),x)

[Out]

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

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sympy [A]  time = 0.15, size = 19, normalized size = 0.79 \[ \frac {\operatorname {RootSum} {\left (40 z^{2} - 1, \left (i \mapsto i \log {\left (- 10 i + e^{m x} \right )} \right )\right )}}{m} \]

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(-10*_i + exp(m*x))))/m

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