\(\int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx\) [10]

Optimal result

Integrand size = 10, antiderivative size = 54 \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3856, 2719} \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]

[In]

Int[1/Sqrt[Csch[a + b*x]],x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=\frac {2 \sqrt {\text {csch}(a+b x)} E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i (a+b x)\right )\right |2\right ) \sqrt {i \sinh (a+b x)}}{b} \]

[In]

Integrate[1/Sqrt[Csch[a + b*x]],x]

[Out]

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

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.00

method	result	size
default	\(\frac {\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\)	\(108\)
risch	\(\frac {\sqrt {2}}{b \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}}-\frac {\left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \operatorname {EllipticE}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}}{b \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, \left ({\mathrm e}^{2 b x +2 a}-1\right )}\)	\(210\)

[In]

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

[Out]

(-I*(sinh(b*x+a)+I))^(1/2)*2^(1/2)*(-I*(-sinh(b*x+a)+I))^(1/2)*(I*sinh(b*x+a))^(1/2)*(2*EllipticE((1-I*sinh(b*
x+a))^(1/2),1/2*2^(1/2))-EllipticF((1-I*sinh(b*x+a))^(1/2),1/2*2^(1/2)))/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.78 \[ \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}} + 2 \, {\left (\sqrt {2} \cosh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )} \]

[In]

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

[Out]

-(sqrt(2)*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*sqrt((cosh(b*x + a) + sinh(b
*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)) + 2*(sqrt(2)*cosh(b*x + a) +
sqrt(2)*sinh(b*x + a))*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(b*x + a) + sinh(b*x + a))))/(b*co
sh(b*x + a) + b*sinh(b*x + a))

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(csch(a + b*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/sqrt(csch(b*x + a)), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/sqrt(csch(b*x + a)), x)

Mupad [F(-1)]

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

[In]

int(1/(1/sinh(a + b*x))^(1/2),x)

[Out]

int(1/(1/sinh(a + b*x))^(1/2), x)