3.637 \(\int \frac{e^{-x}+e^x}{-e^{-x}+e^x} \, dx\)

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

[Out]

Log[E^(-x) - E^x]

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Rubi [A]  time = 0.0824308, antiderivative size = 14, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \log \left (1-e^{2 x}\right )-x \]

Antiderivative was successfully verified.

[In]  Int[(E^(-x) + E^x)/(-E^(-x) + E^x),x]

[Out]

-x + Log[1 - E^(2*x)]

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Rubi in Sympy [A]  time = 44.3662, size = 15, normalized size = 1.25 \[ \log{\left (- e^{2 x} + 1 \right )} - \frac{\log{\left (e^{2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(-exp(2*x) + 1) - log(exp(2*x))/2

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Mathematica [A]  time = 0.00504645, size = 14, normalized size = 1.17 \[ \log \left (1-e^{2 x}\right )-x \]

Antiderivative was successfully verified.

[In]  Integrate[(E^(-x) + E^x)/(-E^(-x) + E^x),x]

[Out]

-x + Log[1 - E^(2*x)]

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Maple [A]  time = 0.014, size = 17, normalized size = 1.4 \[ \ln \left ( 1+{{\rm e}^{x}} \right ) +\ln \left ( -1+{{\rm e}^{x}} \right ) -\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((exp(-x)+exp(x))/(-1/exp(x)+exp(x)),x)

[Out]

ln(1+exp(x))+ln(-1+exp(x))-ln(exp(x))

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Maxima [A]  time = 0.780558, size = 14, normalized size = 1.17 \[ \log \left (e^{\left (-x\right )} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(e^(-x) - e^x)

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Fricas [A]  time = 0.23904, size = 15, normalized size = 1.25 \[ -x + \log \left (e^{\left (2 \, x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x + log(e^(2*x) - 1)

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Sympy [A]  time = 0.078101, size = 8, normalized size = 0.67 \[ - x + \log{\left (e^{2 x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x + log(exp(2*x) - 1)

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GIAC/XCAS [A]  time = 0.237516, size = 16, normalized size = 1.33 \[ -x +{\rm ln}\left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x + ln(abs(e^(2*x) - 1))