\(\int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx\) [17]

Optimal result

Integrand size = 12, antiderivative size = 56 \[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \]

[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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3856, 2719} \[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)}} \]

[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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3856

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*} \text {integral}& = \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] (verified)

Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=\frac {2 i E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right )}{d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}} \]

[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]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(79)=158\).

Time = 0.50 (sec) , antiderivative size = 227, normalized size of antiderivative = 4.05

method	result	size
risch	\(\frac {\sqrt {2}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{{\mathrm e}^{2 d x +2 c}-1}}}-\frac {\left (\frac {2 b \,{\mathrm e}^{2 d x +2 c}-2 b}{b \sqrt {{\mathrm e}^{d x +c} \left (b \,{\mathrm e}^{2 d x +2 c}-b \right )}}-\frac {\sqrt {{\mathrm e}^{d x +c}+1}\, \sqrt {-2 \,{\mathrm e}^{d x +c}+2}\, \sqrt {-{\mathrm e}^{d x +c}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 d x +3 c} b -{\mathrm e}^{d x +c} b}}\right ) \sqrt {2}\, \sqrt {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}}{d \sqrt {\frac {b \,{\mathrm e}^{d x +c}}{{\mathrm e}^{2 d x +2 c}-1}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right )}\)	\(227\)

[In]

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

[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*exp(d*x+c)+2)^(1/2)*(-exp(d*x+c))^(1/2)/(exp(d*x+c)^3*b-exp(d*x+c)*b)^(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)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.75 \[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {b} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}}}{b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right )} \]

[In]

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

[Out]

-(2*sqrt(2)*sqrt(b)*(cosh(d*x + c) + sinh(d*x + c))*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(d*x +
c) + sinh(d*x + c))) + sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt((
b*cosh(d*x + c) + b*sinh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)))/(
b*d*cosh(d*x + c) + b*d*sinh(d*x + c))

Sympy [F]

\[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {b \operatorname {csch}{\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(b*csch(c + d*x)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {csch}\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(b*csch(d*x + c)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {csch}\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(b*csch(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]

[In]

int(1/(b/sinh(c + d*x))^(1/2),x)

[Out]

int(1/(b/sinh(c + d*x))^(1/2), x)