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

Optimal. Leaf size=40 \[ \frac{(2 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0265989, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4046, 3770} \[ \frac{(2 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]*(a + b*Sech[c + d*x]^2),x]

[Out]

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

Rule 4046

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> -Simp[(C*Cot[
e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x]
/; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

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

Mathematica [A]  time = 0.0207111, size = 48, normalized size = 1.2 \[ \frac{a \tan ^{-1}(\sinh (c+d x))}{d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]*(a + b*Sech[c + d*x]^2),x]

[Out]

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

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Maple [A]  time = 0.02, size = 45, normalized size = 1.1 \begin{align*} 2\,{\frac{a\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*a*arctan(exp(d*x+c))+1/2*b*sech(d*x+c)*tanh(d*x+c)/d+1/d*b*arctan(exp(d*x+c))

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Maxima [B]  time = 1.72274, size = 109, normalized size = 2.72 \begin{align*} -b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac{a \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(arctan(e^(-d*x - c))/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))
) + a*arctan(sinh(d*x + c))/d

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Fricas [B]  time = 2.09404, size = 882, normalized size = 22.05 \begin{align*} \frac{b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} +{\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \,{\left (2 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (2 \, a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} +{\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \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 ) - b \cosh \left (d x + c\right ) +{\left (3 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + ((2*a + b)*cosh(d*x + c)^4 + 4*(2
*a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a + b)*sinh(d*x + c)^4 + 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*(2*a +
b)*cosh(d*x + c)^2 + 2*a + b)*sinh(d*x + c)^2 + 4*((2*a + b)*cosh(d*x + c)^3 + (2*a + b)*cosh(d*x + c))*sinh(d
*x + c) + 2*a + b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - b*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 - b)*sinh(d*
x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh(d*x + c)^4 + 2*d*cosh(d*x + c)^2 + 2*(
3*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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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 ) \operatorname{sech}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.26995, size = 115, normalized size = 2.88 \begin{align*} \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )}{\left (2 \, a + b\right )}}{4 \, d} + \frac{b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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