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

Optimal. Leaf size=54 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]

[Out]

((-2*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])

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Rubi [A]  time = 0.0201483, antiderivative size = 54, 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} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Csch[a + b*x]],x]

[Out]

((-2*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*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{\text{csch}(a+b x)}} \, dx &=\frac{\int \sqrt{i \sinh (a+b x)} \, dx}{\sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ &=-\frac{2 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right )}{b \sqrt{\text{csch}(a+b x)} \sqrt{i \sinh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0339411, size = 50, normalized size = 0.93 \[ \frac{2 \sqrt{i \sinh (a+b x)} \sqrt{\text{csch}(a+b x)} E\left (\left .\frac{1}{2} \left (\frac{\pi }{2}-i (a+b x)\right )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Csch[a + b*x]],x]

[Out]

(2*Sqrt[Csch[a + b*x]]*EllipticE[(Pi/2 - I*(a + b*x))/2, 2]*Sqrt[I*Sinh[a + b*x]])/b

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Maple [A]  time = 0.152, size = 108, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{\cosh \left ( bx+a \right ) b}\sqrt{-i \left ( i+\sinh \left ( bx+a \right ) \right ) }\sqrt{-i \left ( -\sinh \left ( bx+a \right ) +i \right ) }\sqrt{i\sinh \left ( bx+a \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csch(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{csch}\left (b x + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(csch(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(csch(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csch(b*x + a)), x)