3.334 \(\int \frac{1}{-e^x+e^{3 x}} \, dx\)

Optimal. Leaf size=12 \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]

[Out]

E^(-x) - ArcTanh[E^x]

_______________________________________________________________________________________

Rubi [A]  time = 0.024791, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-E^x + E^(3*x))^(-1),x]

[Out]

E^(-x) - ArcTanh[E^x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.73425, size = 10, normalized size = 0.83 \[ - \operatorname{atanh}{\left (e^{- x} \right )} + e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-exp(x)+exp(3*x)),x)

[Out]

-atanh(exp(-x)) + exp(-x)

_______________________________________________________________________________________

Mathematica [B]  time = 0.021508, size = 32, normalized size = 2.67 \[ e^{-x}+\frac{1}{2} \log \left (1-e^{-x}\right )-\frac{1}{2} \log \left (e^{-x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-E^x + E^(3*x))^(-1),x]

[Out]

E^(-x) + Log[1 - E^(-x)]/2 - Log[1 + E^(-x)]/2

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 20, normalized size = 1.7 \[ -{\frac{\ln \left ( 1+{{\rm e}^{x}} \right ) }{2}}+ \left ({{\rm e}^{x}} \right ) ^{-1}+{\frac{\ln \left ( -1+{{\rm e}^{x}} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-exp(x)+exp(3*x)),x)

[Out]

-1/2*ln(1+exp(x))+1/exp(x)+1/2*ln(-1+exp(x))

_______________________________________________________________________________________

Maxima [A]  time = 1.34042, size = 26, normalized size = 2.17 \[ e^{\left (-x\right )} - \frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e^(3*x) - e^x),x, algorithm="maxima")

[Out]

e^(-x) - 1/2*log(e^x + 1) + 1/2*log(e^x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.216942, size = 34, normalized size = 2.83 \[ -\frac{1}{2} \,{\left (e^{x} \log \left (e^{x} + 1\right ) - e^{x} \log \left (e^{x} - 1\right ) - 2\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e^(3*x) - e^x),x, algorithm="fricas")

[Out]

-1/2*(e^x*log(e^x + 1) - e^x*log(e^x - 1) - 2)*e^(-x)

_______________________________________________________________________________________

Sympy [A]  time = 0.114666, size = 20, normalized size = 1.67 \[ \frac{\log{\left (e^{x} - 1 \right )}}{2} - \frac{\log{\left (e^{x} + 1 \right )}}{2} + e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-exp(x)+exp(3*x)),x)

[Out]

log(exp(x) - 1)/2 - log(exp(x) + 1)/2 + exp(-x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219618, size = 27, normalized size = 2.25 \[ e^{\left (-x\right )} - \frac{1}{2} \,{\rm ln}\left (e^{x} + 1\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e^(3*x) - e^x),x, algorithm="giac")

[Out]

e^(-x) - 1/2*ln(e^x + 1) + 1/2*ln(abs(e^x - 1))