3.6.74 \(\int \text {sech}(x) \, dx\) [574]

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

[Out]

arctan(sinh(x))

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Rubi [A]
time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855} \begin {gather*} \text {ArcTan}(\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x],x]

[Out]

ArcTan[Sinh[x]]

Rule 3855

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(9\) vs. \(2(3)=6\).
time = 0.00, size = 9, normalized size = 3.00 \begin {gather*} 2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[x],x]

[Out]

2*ArcTan[Tanh[x/2]]

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Maple [A]
time = 0.01, size = 4, normalized size = 1.33

method	result	size
lookup	\(\arctan \left (\sinh \left (x \right )\right )\)	\(4\)
default	\(\arctan \left (\sinh \left (x \right )\right )\)	\(4\)
risch	\(i \ln \left ({\mathrm e}^{x}+i\right )-i \ln \left ({\mathrm e}^{x}-i\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x),x,method=_RETURNVERBOSE)

[Out]

arctan(sinh(x))

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Maxima [A]
time = 1.53, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\sinh \left (x\right )\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 8 vs. \(2 (3) = 6\).
time = 0.41, size = 8, normalized size = 2.67 \begin {gather*} 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {sech}{\left (x \right )}\, dx \end {gather*}

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 = 1.26, size = 5, normalized size = 1.67 \begin {gather*} 2 \, \arctan \left (e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(e^x)

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Mupad [B]
time = 0.02, size = 5, normalized size = 1.67 \begin {gather*} 2\,\mathrm {atan}\left ({\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(x),x)

[Out]

2*atan(exp(x))

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