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

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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, 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 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]

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 \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*}

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Mathematica [A]  time = 0.05, 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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fricas [F]  time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {csch}\left (d x + c\right )}}{b \operatorname {csch}\left (d x + c\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {csch}\left (d x + c\right )}}\,{d x} \]

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)

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maple [B]  time = 0.33, size = 227, normalized size = 4.05 \[ \frac {\sqrt {2}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{{\mathrm e}^{2 d x +2 c}-1}}}-\frac {\left (\frac {2 b \,{\mathrm e}^{2 d x +2 c}-2 b}{b \sqrt {{\mathrm e}^{d x +c} \left (b \,{\mathrm e}^{2 d x +2 c}-b \right )}}-\frac {\sqrt {{\mathrm e}^{d x +c}+1}\, \sqrt {2-2 \,{\mathrm e}^{d x +c}}\, \sqrt {-{\mathrm e}^{d x +c}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 d x +3 c} b -b \,{\mathrm e}^{d x +c}}}\right ) \sqrt {2}\, \sqrt {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{{\mathrm e}^{2 d x +2 c}-1}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right )} \]

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)/(exp(d*x+c)^3*b-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {csch}\left (d x + c\right )}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/sinh(c + d*x))^(1/2),x)

[Out]

int(1/(b/sinh(c + d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \operatorname {csch}{\left (c + d x \right )}}}\, dx \]

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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