3.18 \(\int \sqrt{b \text{sech}(c+d x)} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 i \sqrt{\cosh (c+d x)} \text{EllipticF}\left (\frac{1}{2} i (c+d x),2\right ) \sqrt{b \text{sech}(c+d x)}}{d} \]

[Out]

((-2*I)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/d

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Rubi [A]  time = 0.0215461, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3771, 2641} \[ -\frac{2 i \sqrt{\cosh (c+d x)} F\left (\left .\frac{1}{2} i (c+d x)\right |2\right ) \sqrt{b \text{sech}(c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Sech[c + d*x]],x]

[Out]

((-2*I)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/d

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{b \text{sech}(c+d x)} \, dx &=\left (\sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}\right ) \int \frac{1}{\sqrt{\cosh (c+d x)}} \, dx\\ &=-\frac{2 i \sqrt{\cosh (c+d x)} F\left (\left .\frac{1}{2} i (c+d x)\right |2\right ) \sqrt{b \text{sech}(c+d x)}}{d}\\ \end{align*}

Mathematica [A]  time = 0.0214048, size = 42, normalized size = 1. \[ -\frac{2 i \sqrt{\cosh (c+d x)} \text{EllipticF}\left (\frac{1}{2} i (c+d x),2\right ) \sqrt{b \text{sech}(c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Sech[c + d*x]],x]

[Out]

((-2*I)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/d

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Maple [F]  time = 0.138, size = 0, normalized size = 0. \begin{align*} \int \sqrt{b{\rm sech} \left (dx+c\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*sech(d*x+c))^(1/2),x)

[Out]

int((b*sech(d*x+c))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sech(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sech(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sech(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sech(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sech(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(b*sech(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sech(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sech(d*x + c)), x)