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(d*x+c))/d

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ a x+\frac {b \tan ^{-1}(\sinh (c+d x))}{d} \]

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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fricas [A]  time = 0.40, size = 26, normalized size = 1.62 \[ \frac {a d x + 2 \, b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d} \]

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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giac [A]  time = 0.11, size = 17, normalized size = 1.06 \[ a x + \frac {2 \, b \arctan \left (e^{\left (d x + c\right )}\right )}{d} \]

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

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maple [A]  time = 0.02, size = 17, normalized size = 1.06 \[ a x +\frac {b \arctan \left (\sinh \left (d x +c \right )\right )}{d} \]

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 = 1.18, size = 16, normalized size = 1.00 \[ a x + \frac {b \arctan \left (\sinh \left (d x + c\right )\right )}{d} \]

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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mupad [B]  time = 1.30, size = 38, normalized size = 2.38 \[ a\,x+\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b/cosh(c + d*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}{\left (c + d x \right )}\right )\, dx \]

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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