3.12 \(\int \frac{1}{\sqrt{\text{sech}(a+b x)}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]

[Out]

((-2*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b

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Rubi [A]  time = 0.0198356, antiderivative size = 40, 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 = {3771, 2639} \[ -\frac{2 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sech[a + b*x]],x]

[Out]

((-2*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b

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 2639

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

Rubi steps

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

Mathematica [A]  time = 0.0342797, size = 40, normalized size = 1. \[ -\frac{2 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Sech[a + b*x]],x]

[Out]

((-2*I)*EllipticE[(I/2)*(a + b*x), 2])/(b*Sqrt[Cosh[a + b*x]]*Sqrt[Sech[a + b*x]])

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Maple [B]  time = 0.245, size = 135, normalized size = 3.4 \begin{align*} -2\,{\frac{\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) }{\sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sinh \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1
/2*a)^2+1)^(1/2)*EllipticE(cosh(1/2*b*x+1/2*a),2^(1/2))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/
sinh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{sech}\left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sech(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\operatorname{sech}\left (b x + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(sech(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{sech}{\left (a + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(sech(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{sech}\left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(sech(b*x + a)), x)