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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0218582, 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, 2641} \[ -\frac{2 i \sqrt{i \sinh (c+d x)} F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \text{csch}(c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*Csch[c + d*x]],x]

[Out]

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

Mathematica [A]  time = 0.0333764, size = 54, normalized size = 0.96 \[ \frac{2 i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (\frac{\pi }{2}-i (c+d x)\right ),2\right ) \sqrt{b \text{csch}(c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*Csch[c + d*x]],x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.129, size = 0, normalized size = 0. \begin{align*} \int \sqrt{b{\rm csch} \left (dx+c\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

integrate(sqrt(b*csch(d*x + c)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*csch(d*x + c)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*csch(c + d*x)), x)

________________________________________________________________________________________

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

[Out]

integrate(sqrt(b*csch(d*x + c)), x)