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)

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Rubi [A]  time = 0.0514737, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} e^{m x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} m} \]

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)

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Rubi in Sympy [A]  time = 5.50959, size = 29, normalized size = 0.94 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt{b} e^{- m x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b/exp(m*x)+a*exp(m*x)),x)

[Out]

-atan(sqrt(b)*exp(-m*x)/sqrt(a))/(sqrt(a)*sqrt(b)*m)

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Mathematica [A]  time = 0.0136105, 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)

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Maple [A]  time = 0.006, size = 22, normalized size = 0.7 \[{\frac{1}{m}\arctan \left ({a{{\rm e}^{mx}}{\frac{1}{\sqrt{ba}}}} \right ){\frac{1}{\sqrt{ba}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*e^(m*x) + b*e^(-m*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222215, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a b e^{\left (m x\right )} + \sqrt{-a b}{\left (a e^{\left (2 \, m x\right )} - b\right )}}{a e^{\left (2 \, m x\right )} + b}\right )}{2 \, \sqrt{-a b} m}, -\frac{\arctan \left (\frac{b e^{\left (-m x\right )}}{\sqrt{a b}}\right )}{\sqrt{a b} m}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*e^(m*x) + b*e^(-m*x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.190262, size = 26, normalized size = 0.84 \[ \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (- 2 i a + e^{- m x} \right )} \right )\right )}}{m} \]

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

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GIAC/XCAS [A]  time = 0.20011, size = 28, normalized size = 0.9 \[ \frac{\arctan \left (\frac{a e^{\left (m x\right )}}{\sqrt{a b}}\right )}{\sqrt{a b} m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*e^(m*x) + b*e^(-m*x)),x, algorithm="giac")

[Out]

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