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

Optimal. Leaf size=33 \[ a^2 x+\frac{2 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \tanh (c+d x)}{d} \]

[Out]

a^2*x + (2*a*b*ArcTan[Sinh[c + d*x]])/d + (b^2*Tanh[c + d*x])/d

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Rubi [A]  time = 0.0280215, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3773, 3770, 3767, 8} \[ a^2 x+\frac{2 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b^2 \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x])^2,x]

[Out]

a^2*x + (2*a*b*ArcTan[Sinh[c + d*x]])/d + (b^2*Tanh[c + d*x])/d

Rule 3773

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Dist[2*a*b, Int[Csc[c + d*x], x],
x] + Dist[b^2, Int[Csc[c + d*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A]  time = 0.063079, size = 32, normalized size = 0.97 \[ \frac{a \left (a d x+2 b \tan ^{-1}(\sinh (c+d x))\right )+b^2 \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sech[c + d*x])^2,x]

[Out]

(a*(a*d*x + 2*b*ArcTan[Sinh[c + d*x]]) + b^2*Tanh[c + d*x])/d

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Maple [A]  time = 0.01, size = 42, normalized size = 1.3 \begin{align*}{a}^{2}x+{\frac{{b}^{2}\tanh \left ( dx+c \right ) }{d}}+4\,{\frac{ab\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{{a}^{2}c}{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^2*x+b^2*tanh(d*x+c)/d+4/d*a*b*arctan(exp(d*x+c))+1/d*a^2*c

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Maxima [A]  time = 1.0838, size = 55, normalized size = 1.67 \begin{align*} a^{2} x + \frac{2 \, a b \arctan \left (\sinh \left (d x + c\right )\right )}{d} + \frac{2 \, b^{2}}{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^2*x + 2*a*b*arctan(sinh(d*x + c))/d + 2*b^2/(d*(e^(-2*d*x - 2*c) + 1))

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Fricas [B]  time = 2.43496, size = 428, normalized size = 12.97 \begin{align*} \frac{a^{2} d x \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d x \sinh \left (d x + c\right )^{2} + a^{2} d x - 2 \, b^{2} + 4 \,{\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + a b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \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^2*d*x*cosh(d*x + c)^2 + 2*a^2*d*x*cosh(d*x + c)*sinh(d*x + c) + a^2*d*x*sinh(d*x + c)^2 + a^2*d*x - 2*b^2 +
4*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + a*b)*arctan(cosh(d*x + c)
+ sinh(d*x + c)))/(d*cosh(d*x + c)^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^2 + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}{\left (c + d x \right )}\right )^{2}\, 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 = 1.16557, size = 65, normalized size = 1.97 \begin{align*} \frac{{\left (d x + c\right )} a^{2}}{d} + \frac{4 \, a b \arctan \left (e^{\left (d x + c\right )}\right )}{d} - \frac{2 \, b^{2}}{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]

(d*x + c)*a^2/d + 4*a*b*arctan(e^(d*x + c))/d - 2*b^2/(d*(e^(2*d*x + 2*c) + 1))