3.1.1 \(\int \text {sech}(a+b x) \, dx\) [1]

Optimal. Leaf size=11 \[ \frac {\text {ArcTan}(\sinh (a+b x))}{b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {3855} \begin {gather*} \frac {\text {ArcTan}(\sinh (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x],x]

[Out]

ArcTan[Sinh[a + b*x]]/b

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

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {\text {ArcTan}(\sinh (a+b x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x],x]

[Out]

ArcTan[Sinh[a + b*x]]/b

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Maple [A]
time = 0.24, size = 12, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\arctan \left (\sinh \left (b x +a \right )\right )}{b}\)	\(12\)
default	\(\frac {\arctan \left (\sinh \left (b x +a \right )\right )}{b}\)	\(12\)
risch	\(\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{b}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 11, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\sinh \left (b x + a\right )\right )}{b} \end {gather*}

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

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Fricas [A]
time = 0.35, size = 19, normalized size = 1.73 \begin {gather*} \frac {2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {sech}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(a + b*x), x)

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Giac [A]
time = 0.39, size = 12, normalized size = 1.09 \begin {gather*} \frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \end {gather*}

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

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Mupad [B]
time = 0.08, size = 23, normalized size = 2.09 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(a + b*x),x)

[Out]

(2*atan((exp(b*x)*exp(a)*(b^2)^(1/2))/b))/(b^2)^(1/2)

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