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

Optimal. Leaf size=4 \[ \tan ^{-1}\left (e^x\right ) \]

[Out]

ArcTan[E^x]

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Rubi [A]  time = 0.0171728, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(E^(-x) + E^x)^(-1),x]

[Out]

ArcTan[E^x]

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Rubi in Sympy [A]  time = 6.33744, size = 7, normalized size = 1.75 \[ - \operatorname{atan}{\left (e^{- x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(exp(-x))

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Mathematica [A]  time = 0.00479911, size = 4, normalized size = 1. \[ \tan ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[E^x]

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Maple [A]  time = 0.004, size = 4, normalized size = 1. \[ \arctan \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(exp(-x)+exp(x)),x)

[Out]

arctan(exp(x))

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Maxima [A]  time = 0.851387, size = 9, normalized size = 2.25 \[ -\arctan \left (e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan(e^(-x))

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Fricas [A]  time = 0.241353, size = 4, normalized size = 1. \[ \arctan \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(e^x)

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Sympy [A]  time = 0.092894, size = 15, normalized size = 3.75 \[ \operatorname{RootSum}{\left (4 z^{2} + 1, \left ( i \mapsto i \log{\left (2 i + e^{x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(exp(-x)+exp(x)),x)

[Out]

RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))

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GIAC/XCAS [A]  time = 0.240204, size = 4, normalized size = 1. \[ \arctan \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(e^x)