3.1 \(\int \text{sech}(a+b x) \, dx\)

Optimal. Leaf size=11 \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b} \]

[Out]

ArcTan[Sinh[a + b*x]]/b

________________________________________________________________________________________

Rubi [A]  time = 0.0050338, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3770} \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x],x]

[Out]

ArcTan[Sinh[a + b*x]]/b

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}(a+b x) \, dx &=\frac{\tan ^{-1}(\sinh (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0020975, size = 11, normalized size = 1. \[ \frac{\tan ^{-1}(\sinh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x],x]

[Out]

ArcTan[Sinh[a + b*x]]/b

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 12, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \sinh \left ( bx+a \right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a),x)

[Out]

arctan(sinh(b*x+a))/b

________________________________________________________________________________________

Maxima [A]  time = 1.0037, size = 15, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\sinh \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a),x, algorithm="maxima")

[Out]

arctan(sinh(b*x + a))/b

________________________________________________________________________________________

Fricas [A]  time = 2.00671, size = 58, normalized size = 5.27 \begin{align*} \frac{2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a),x, algorithm="fricas")

[Out]

2*arctan(cosh(b*x + a) + sinh(b*x + a))/b

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a),x)

[Out]

Integral(sech(a + b*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.09724, size = 16, normalized size = 1.45 \begin{align*} \frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a),x, algorithm="giac")

[Out]

2*arctan(e^(b*x + a))/b