3.23 \(\int (b \text{sech}(c+d x))^n \, dx\)

Optimal. Leaf size=75 \[ -\frac{b \sinh (c+d x) (b \text{sech}(c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cosh ^2(c+d x)\right )}{d (1-n) \sqrt{-\sinh ^2(c+d x)}} \]

[Out]

-((b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cosh[c + d*x]^2]*(b*Sech[c + d*x])^(-1 + n)*Sinh[c + d*x])/(
d*(1 - n)*Sqrt[-Sinh[c + d*x]^2]))

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Rubi [A]  time = 0.0365971, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3772, 2643} \[ -\frac{b \sinh (c+d x) (b \text{sech}(c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cosh ^2(c+d x)\right )}{d (1-n) \sqrt{-\sinh ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sech[c + d*x])^n,x]

[Out]

-((b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cosh[c + d*x]^2]*(b*Sech[c + d*x])^(-1 + n)*Sinh[c + d*x])/(
d*(1 - n)*Sqrt[-Sinh[c + d*x]^2]))

Rule 3772

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

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (b \text{sech}(c+d x))^n \, dx &=\left (\frac{\cosh (c+d x)}{b}\right )^n (b \text{sech}(c+d x))^n \int \left (\frac{\cosh (c+d x)}{b}\right )^{-n} \, dx\\ &=-\frac{\cosh (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cosh ^2(c+d x)\right ) (b \text{sech}(c+d x))^n \sinh (c+d x)}{d (1-n) \sqrt{-\sinh ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0622459, size = 60, normalized size = 0.8 \[ -\frac{\sqrt{\tanh ^2(c+d x)} \coth (c+d x) (b \text{sech}(c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\text{sech}^2(c+d x)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sech[c + d*x])^n,x]

[Out]

-((Coth[c + d*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sech[c + d*x]^2]*(b*Sech[c + d*x])^n*Sqrt[Tanh[c + d*x
]^2])/(d*n))

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm sech} \left (dx+c\right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{sech}\left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sech(d*x + c))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \operatorname{sech}\left (d x + c\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sech(d*x + c))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sech(d*x+c))**n,x)

[Out]

Integral((b*sech(c + d*x))**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \operatorname{sech}\left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sech(d*x + c))^n, x)