3.19 \(\int \frac{1}{\sqrt{b \text{sech}(c+d x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0220029, 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, 2639} \[ -\frac{2 i E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*Sech[c + d*x]],x]

[Out]

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

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

Mathematica [A]  time = 0.0295799, size = 42, normalized size = 1. \[ -\frac{2 i E\left (\left .\frac{1}{2} i (c+d x)\right |2\right )}{d \sqrt{\cosh (c+d x)} \sqrt{b \text{sech}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*Sech[c + d*x]],x]

[Out]

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

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Maple [B]  time = 0.141, size = 244, normalized size = 5.8 \begin{align*}{\frac{\sqrt{2}}{d}{\frac{1}{\sqrt{{\frac{b{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{dx+c}} \right ) ^{2}+1}}}}}}+{\frac{\sqrt{2}}{d \left ( \left ({{\rm e}^{dx+c}} \right ) ^{2}+1 \right ) } \left ( -2\,{\frac{b \left ({{\rm e}^{dx+c}} \right ) ^{2}+b}{b\sqrt{{{\rm e}^{dx+c}} \left ( b \left ({{\rm e}^{dx+c}} \right ) ^{2}+b \right ) }}}+{i\sqrt{2}\sqrt{-i \left ({{\rm e}^{dx+c}}+i \right ) }\sqrt{i \left ({{\rm e}^{dx+c}}-i \right ) }\sqrt{i{{\rm e}^{dx+c}}} \left ( -2\,i{\it EllipticE} \left ( \sqrt{-i \left ({{\rm e}^{dx+c}}+i \right ) },{\frac{\sqrt{2}}{2}} \right ) +i{\it EllipticF} \left ( \sqrt{-i \left ({{\rm e}^{dx+c}}+i \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{b \left ({{\rm e}^{dx+c}} \right ) ^{3}+b{{\rm e}^{dx+c}}}}}} \right ) \sqrt{b{{\rm e}^{dx+c}} \left ( \left ({{\rm e}^{dx+c}} \right ) ^{2}+1 \right ) }{\frac{1}{\sqrt{{\frac{b{{\rm e}^{dx+c}}}{ \left ({{\rm e}^{dx+c}} \right ) ^{2}+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*2^(1/2)/(b*exp(d*x+c)/(exp(d*x+c)^2+1))^(1/2)+1/d*(-2*(b*exp(d*x+c)^2+b)/b/(exp(d*x+c)*(b*exp(d*x+c)^2+b))
^(1/2)+I*(-I*(exp(d*x+c)+I))^(1/2)*2^(1/2)*(I*(exp(d*x+c)-I))^(1/2)*(I*exp(d*x+c))^(1/2)/(b*exp(d*x+c)^3+b*exp
(d*x+c))^(1/2)*(-2*I*EllipticE((-I*(exp(d*x+c)+I))^(1/2),1/2*2^(1/2))+I*EllipticF((-I*(exp(d*x+c)+I))^(1/2),1/
2*2^(1/2))))*2^(1/2)/(b*exp(d*x+c)/(exp(d*x+c)^2+1))^(1/2)*(b*exp(d*x+c)*(exp(d*x+c)^2+1))^(1/2)/(exp(d*x+c)^2
+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b*sech(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/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 (\frac{\sqrt{b \operatorname{sech}\left (d x + c\right )}}{b \operatorname{sech}\left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sech(d*x + c))/(b*sech(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(b*sech(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b*sech(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(b*sech(d*x + c)), x)