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

Optimal. Leaf size=56 \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0226316, antiderivative size = 56, 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} \[ -\frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*Csch[c + d*x]],x]

[Out]

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

Mathematica [A]  time = 0.0445771, size = 52, normalized size = 0.93 \[ \frac{2 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )}{d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*Csch[c + d*x]],x]

[Out]

((2*I)*EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2])/(d*Sqrt[b*Csch[c + d*x]]*Sqrt[I*Sinh[c + d*x]])

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Maple [B]  time = 0.138, size = 227, normalized size = 4.1 \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 ) }}}-{\sqrt{{{\rm e}^{dx+c}}+1}\sqrt{2-2\,{{\rm e}^{dx+c}}}\sqrt{-{{\rm e}^{dx+c}}} \left ( -2\,{\it EllipticE} \left ( \sqrt{{{\rm e}^{dx+c}}+1},1/2\,\sqrt{2} \right ) +{\it EllipticF} \left ( \sqrt{{{\rm e}^{dx+c}}+1},{\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*csch(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)-(exp(d*x+c)+1)^(1/2)*(2-2*exp(d*x+c))^(1/2)*(-exp(d*x+c))^(1/2)/(b*exp(d*x+c)^3-b*exp(d*x+c))^(1/2)*(-2*
EllipticE((exp(d*x+c)+1)^(1/2),1/2*2^(1/2))+EllipticF((exp(d*x+c)+1)^(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{csch}\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*csch(d*x + c))/(b*csch(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{csch}{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(b*csch(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{csch}\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*csch(d*x + c)), x)