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

Optimal. Leaf size=10 \[ \log \left (e^{-x}+e^x\right ) \]

[Out]

Log[E^(-x) + E^x]

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Rubi [A]  time = 0.073143, antiderivative size = 12, normalized size of antiderivative = 1.2, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \log \left (e^{2 x}+1\right )-x \]

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 = 42.0219, size = 15, normalized size = 1.5 \[ \log{\left (e^{2 x} + 1 \right )} - \frac{\log{\left (e^{2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(exp(2*x) + 1) - log(exp(2*x))/2

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Mathematica [A]  time = 0.00456616, size = 12, normalized size = 1.2 \[ \log \left (e^{2 x}+1\right )-x \]

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 = 0.011, size = 14, normalized size = 1.4 \[ \ln \left ( 1+ \left ({{\rm e}^{x}} \right ) ^{2} \right ) -\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(1+exp(x)^2)-ln(exp(x))

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Maxima [A]  time = 0.7584, size = 11, normalized size = 1.1 \[ \log \left (e^{\left (-x\right )} + e^{x}\right ) \]

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 = 0.449851, size = 15, normalized size = 1.5 \[ -x + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]

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 = 0.075641, size = 8, normalized size = 0.8 \[ - x + \log{\left (e^{2 x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x + log(exp(2*x) + 1)

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GIAC/XCAS [A]  time = 0.228538, size = 15, normalized size = 1.5 \[ -x +{\rm ln}\left (e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x + ln(e^(2*x) + 1)