3.574 \(\int \text{sech}(x) \, dx\)

Optimal. Leaf size=3 \[ \tan ^{-1}(\sinh (x)) \]

[Out]

ArcTan[Sinh[x]]

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Rubi [A]  time = 0.0032907, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3770} \[ \tan ^{-1}(\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x],x]

[Out]

ArcTan[Sinh[x]]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \text{sech}(x) \, dx &=\tan ^{-1}(\sinh (x))\\ \end{align*}

Mathematica [B]  time = 0.0039811, size = 9, normalized size = 3. \[ 2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x],x]

[Out]

2*ArcTan[Tanh[x/2]]

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Maple [A]  time = 0., size = 4, normalized size = 1.3 \begin{align*} \arctan \left ( \sinh \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x),x)

[Out]

arctan(sinh(x))

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Maxima [A]  time = 0.927777, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\sinh \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x),x, algorithm="maxima")

[Out]

arctan(sinh(x))

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Fricas [B]  time = 2.08164, size = 39, normalized size = 13. \begin{align*} 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x),x, algorithm="fricas")

[Out]

2*arctan(cosh(x) + sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x),x)

[Out]

Integral(sech(x), x)

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Giac [A]  time = 1.09022, size = 7, normalized size = 2.33 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x),x, algorithm="giac")

[Out]

2*arctan(e^x)