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

Optimal result

Integrand size = 6, antiderivative size = 11 \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3855} \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b} \]

[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*} \text {integral}& = \frac {\arctan (\sinh (a+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{b} \]

[In]

Integrate[Sech[a + b*x],x]

[Out]

ArcTan[Sinh[a + b*x]]/b

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 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	\(\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{b}\)	\(34\)
parallelrisch	\(-\frac {i \left (\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-i\right )-\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i\right )\right )}{b}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \text {sech}(a+b x) \, dx=\frac {2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).

Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \text {sech}(a+b x) \, dx=\begin {cases} \frac {2 \operatorname {atan}{\left (\tanh {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\x \operatorname {sech}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*atan(tanh(a/2 + b*x/2))/b, Ne(b, 0)), (x*sech(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \text {sech}(a+b x) \, dx=\frac {\arctan \left (\sinh \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

arctan(sinh(b*x + a))/b

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \text {sech}(a+b x) \, dx=\frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} \]

[In]

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

[Out]

2*arctan(e^(b*x + a))/b

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \text {sech}(a+b x) \, dx=\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}} \]

[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)