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

Optimal. Leaf size=15 \[ a x+\frac{b \tanh (c+d x)}{d} \]

[Out]

a*x + (b*Tanh[c + d*x])/d

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Rubi [A]  time = 0.0132529, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3767, 8} \[ a x+\frac{b \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sech[c + d*x]^2,x]

[Out]

a*x + (b*Tanh[c + d*x])/d

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

Mathematica [A]  time = 0.0027872, size = 15, normalized size = 1. \[ a x+\frac{b \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sech[c + d*x]^2,x]

[Out]

a*x + (b*Tanh[c + d*x])/d

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Maple [A]  time = 0.007, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\tanh \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*tanh(d*x+c)/d

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Maxima [A]  time = 1.15253, size = 31, normalized size = 2.07 \begin{align*} a x + \frac{2 \, b}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 2*b/(d*(e^(-2*d*x - 2*c) + 1))

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Fricas [B]  time = 1.99825, size = 88, normalized size = 5.87 \begin{align*} \frac{{\left (a d x - b\right )} \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(a + b*sech(c + d*x)**2, x)

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Giac [A]  time = 1.14616, size = 31, normalized size = 2.07 \begin{align*} a x - \frac{2 \, b}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x - 2*b/(d*(e^(2*d*x + 2*c) + 1))