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

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

[Out]

2*I*(sin(1/2*I*c+1/4*Pi+1/2*I*d*x)^2)^(1/2)/sin(1/2*I*c+1/4*Pi+1/2*I*d*x)*EllipticF(cos(1/2*I*c+1/4*Pi+1/2*I*d
*x),2^(1/2))*(b*csch(d*x+c))^(1/2)*(I*sinh(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.04, size = 54, normalized size = 0.96 \[ \frac {2 i \sqrt {i \sinh (c+d x)} F\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (c+d x)\right )\right |2\right ) \sqrt {b \text {csch}(c+d x)}}{d} \]

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

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \operatorname {csch}\left (d x + c\right )}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \operatorname {csch}\left (d x + c\right )}\,{d x} \]

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)

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maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,\mathrm {csch}\left (d x +c \right )}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \operatorname {csch}\left (d x + c\right )}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/sinh(c + d*x))^(1/2),x)

[Out]

int((b/sinh(c + d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

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)

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