3.90 \(\int (a+b \text{sech}(c+d x)) \, dx\)

Optimal. Leaf size=16 \[ a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

[Out]

a*x + (b*ArcTan[Sinh[c + d*x]])/d

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Rubi [A]  time = 0.009225, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3770} \[ a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sech[c + d*x],x]

[Out]

a*x + (b*ArcTan[Sinh[c + d*x]])/d

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

Mathematica [A]  time = 0.0017844, size = 16, normalized size = 1. \[ a x+\frac{b \tan ^{-1}(\sinh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sech[c + d*x],x]

[Out]

a*x + (b*ArcTan[Sinh[c + d*x]])/d

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Maple [A]  time = 0.003, size = 17, normalized size = 1.1 \begin{align*} ax+{\frac{b\arctan \left ( \sinh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21941, size = 22, normalized size = 1.38 \begin{align*} a x + \frac{b \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*arctan(sinh(d*x + c))/d

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Fricas [A]  time = 2.31228, size = 74, normalized size = 4.62 \begin{align*} \frac{a d x + 2 \, b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*sech(c + d*x), x)

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Giac [A]  time = 1.13199, size = 23, normalized size = 1.44 \begin{align*} a x + \frac{2 \, b \arctan \left (e^{\left (d x + c\right )}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 2*b*arctan(e^(d*x + c))/d