3.5 \(\int \text{sech}^5(a+b x) \, dx\)

Optimal. Leaf size=55 \[ \frac{3 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac{\tanh (a+b x) \text{sech}^3(a+b x)}{4 b}+\frac{3 \tanh (a+b x) \text{sech}(a+b x)}{8 b} \]

[Out]

(3*ArcTan[Sinh[a + b*x]])/(8*b) + (3*Sech[a + b*x]*Tanh[a + b*x])/(8*b) + (Sech[a + b*x]^3*Tanh[a + b*x])/(4*b
)

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Rubi [A]  time = 0.027885, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3770} \[ \frac{3 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac{\tanh (a+b x) \text{sech}^3(a+b x)}{4 b}+\frac{3 \tanh (a+b x) \text{sech}(a+b x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^5,x]

[Out]

(3*ArcTan[Sinh[a + b*x]])/(8*b) + (3*Sech[a + b*x]*Tanh[a + b*x])/(8*b) + (Sech[a + b*x]^3*Tanh[a + b*x])/(4*b
)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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}^5(a+b x) \, dx &=\frac{\text{sech}^3(a+b x) \tanh (a+b x)}{4 b}+\frac{3}{4} \int \text{sech}^3(a+b x) \, dx\\ &=\frac{3 \text{sech}(a+b x) \tanh (a+b x)}{8 b}+\frac{\text{sech}^3(a+b x) \tanh (a+b x)}{4 b}+\frac{3}{8} \int \text{sech}(a+b x) \, dx\\ &=\frac{3 \tan ^{-1}(\sinh (a+b x))}{8 b}+\frac{3 \text{sech}(a+b x) \tanh (a+b x)}{8 b}+\frac{\text{sech}^3(a+b x) \tanh (a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.0405043, size = 47, normalized size = 0.85 \[ \frac{3 \tan ^{-1}(\sinh (a+b x))+2 \tanh (a+b x) \text{sech}^3(a+b x)+3 \tanh (a+b x) \text{sech}(a+b x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^5,x]

[Out]

(3*ArcTan[Sinh[a + b*x]] + 3*Sech[a + b*x]*Tanh[a + b*x] + 2*Sech[a + b*x]^3*Tanh[a + b*x])/(8*b)

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Maple [A]  time = 0.01, size = 50, normalized size = 0.9 \begin{align*}{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}\tanh \left ( bx+a \right ) }{4\,b}}+{\frac{3\,{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{8\,b}}+{\frac{3\,\arctan \left ({{\rm e}^{bx+a}} \right ) }{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^5,x)

[Out]

1/4*sech(b*x+a)^3*tanh(b*x+a)/b+3/8*sech(b*x+a)*tanh(b*x+a)/b+3/4*arctan(exp(b*x+a))/b

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Maxima [B]  time = 1.51204, size = 151, normalized size = 2.75 \begin{align*} -\frac{3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac{3 \, e^{\left (-b x - a\right )} + 11 \, e^{\left (-3 \, b x - 3 \, a\right )} - 11 \, e^{\left (-5 \, b x - 5 \, a\right )} - 3 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^5,x, algorithm="maxima")

[Out]

-3/4*arctan(e^(-b*x - a))/b + 1/4*(3*e^(-b*x - a) + 11*e^(-3*b*x - 3*a) - 11*e^(-5*b*x - 5*a) - 3*e^(-7*b*x -
7*a))/(b*(4*e^(-2*b*x - 2*a) + 6*e^(-4*b*x - 4*a) + 4*e^(-6*b*x - 6*a) + e^(-8*b*x - 8*a) + 1))

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Fricas [B]  time = 2.14876, size = 2263, normalized size = 41.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^5,x, algorithm="fricas")

[Out]

1/4*(3*cosh(b*x + a)^7 + 21*cosh(b*x + a)*sinh(b*x + a)^6 + 3*sinh(b*x + a)^7 + (63*cosh(b*x + a)^2 + 11)*sinh
(b*x + a)^5 + 11*cosh(b*x + a)^5 + 5*(21*cosh(b*x + a)^3 + 11*cosh(b*x + a))*sinh(b*x + a)^4 + (105*cosh(b*x +
a)^4 + 110*cosh(b*x + a)^2 - 11)*sinh(b*x + a)^3 - 11*cosh(b*x + a)^3 + (63*cosh(b*x + a)^5 + 110*cosh(b*x +
a)^3 - 33*cosh(b*x + a))*sinh(b*x + a)^2 + 3*(cosh(b*x + a)^8 + 8*cosh(b*x + a)*sinh(b*x + a)^7 + sinh(b*x + a
)^8 + 4*(7*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^6 + 4*cosh(b*x + a)^6 + 8*(7*cosh(b*x + a)^3 + 3*cosh(b*x + a))*
sinh(b*x + a)^5 + 2*(35*cosh(b*x + a)^4 + 30*cosh(b*x + a)^2 + 3)*sinh(b*x + a)^4 + 6*cosh(b*x + a)^4 + 8*(7*c
osh(b*x + a)^5 + 10*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*cosh(b*x + a)^6 + 15*cosh(b*x +
a)^4 + 9*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 4*cosh(b*x + a)^2 + 8*(cosh(b*x + a)^7 + 3*cosh(b*x + a)^5 + 3
*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) + 1)*arctan(cosh(b*x + a) + sinh(b*x + a)) + (21*cosh(b*x + a)
^6 + 55*cosh(b*x + a)^4 - 33*cosh(b*x + a)^2 - 3)*sinh(b*x + a) - 3*cosh(b*x + a))/(b*cosh(b*x + a)^8 + 8*b*co
sh(b*x + a)*sinh(b*x + a)^7 + b*sinh(b*x + a)^8 + 4*b*cosh(b*x + a)^6 + 4*(7*b*cosh(b*x + a)^2 + b)*sinh(b*x +
a)^6 + 8*(7*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a))*sinh(b*x + a)^5 + 6*b*cosh(b*x + a)^4 + 2*(35*b*cosh(b*x +
a)^4 + 30*b*cosh(b*x + a)^2 + 3*b)*sinh(b*x + a)^4 + 8*(7*b*cosh(b*x + a)^5 + 10*b*cosh(b*x + a)^3 + 3*b*cosh
(b*x + a))*sinh(b*x + a)^3 + 4*b*cosh(b*x + a)^2 + 4*(7*b*cosh(b*x + a)^6 + 15*b*cosh(b*x + a)^4 + 9*b*cosh(b*
x + a)^2 + b)*sinh(b*x + a)^2 + 8*(b*cosh(b*x + a)^7 + 3*b*cosh(b*x + a)^5 + 3*b*cosh(b*x + a)^3 + b*cosh(b*x
+ a))*sinh(b*x + a) + b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{5}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**5,x)

[Out]

Integral(sech(a + b*x)**5, x)

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Giac [B]  time = 1.1439, size = 140, normalized size = 2.55 \begin{align*} \frac{3 \,{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )\right )}}{16 \, b} + \frac{3 \,{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 20 \, e^{\left (b x + a\right )} - 20 \, e^{\left (-b x - a\right )}}{4 \,{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^5,x, algorithm="giac")

[Out]

3/16*(pi + 2*arctan(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)))/b + 1/4*(3*(e^(b*x + a) - e^(-b*x - a))^3 + 20*e^
(b*x + a) - 20*e^(-b*x - a))/(((e^(b*x + a) - e^(-b*x - a))^2 + 4)^2*b)