3.2 \(\int \text{sech}^2(a+b x) \, dx\)

Optimal. Leaf size=10 \[ \frac{\tanh (a+b x)}{b} \]

[Out]

Tanh[a + b*x]/b

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Rubi [A]  time = 0.0100053, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767, 8} \[ \frac{\tanh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^2,x]

[Out]

Tanh[a + b*x]/b

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \text{sech}^2(a+b x) \, dx &=\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (a+b x))}{b}\\ &=\frac{\tanh (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0028304, size = 10, normalized size = 1. \[ \frac{\tanh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^2,x]

[Out]

Tanh[a + b*x]/b

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Maple [A]  time = 0.005, size = 11, normalized size = 1.1 \begin{align*}{\frac{\tanh \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^2,x)

[Out]

tanh(b*x+a)/b

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Maxima [A]  time = 0.991004, size = 24, normalized size = 2.4 \begin{align*} \frac{2}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2,x, algorithm="maxima")

[Out]

2/(b*(e^(-2*b*x - 2*a) + 1))

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Fricas [B]  time = 2.07124, size = 111, normalized size = 11.1 \begin{align*} -\frac{2}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2,x, algorithm="fricas")

[Out]

-2/(b*cosh(b*x + a)^2 + 2*b*cosh(b*x + a)*sinh(b*x + a) + b*sinh(b*x + a)^2 + b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**2,x)

[Out]

Integral(sech(a + b*x)**2, x)

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Giac [A]  time = 1.13855, size = 24, normalized size = 2.4 \begin{align*} -\frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^2,x, algorithm="giac")

[Out]

-2/(b*(e^(2*b*x + 2*a) + 1))