\(\int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx\) [18]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}+\frac {2 i \sqrt {b \text {csch}(c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right ) \sqrt {i \sinh (c+d x)}}{3 b^2 d} \] Output:

2/3*cosh(d*x+c)/b/d/(b*csch(d*x+c))^(1/2)+2/3*I*(b*csch(d*x+c))^(1/2)*Inve 
rseJacobiAM(1/2*I*c-1/4*Pi+1/2*I*d*x,2^(1/2))*(I*sinh(d*x+c))^(1/2)/b^2/d
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\frac {\text {csch}^2(c+d x) \left (-2 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i c+\pi -2 i d x),2\right ) \sqrt {i \sinh (c+d x)}+\sinh (2 (c+d x))\right )}{3 d (b \text {csch}(c+d x))^{3/2}} \] Input:

Integrate[(b*Csch[c + d*x])^(-3/2),x]
 

Output:

(Csch[c + d*x]^2*((-2*I)*EllipticF[((-2*I)*c + Pi - (2*I)*d*x)/4, 2]*Sqrt[ 
I*Sinh[c + d*x]] + Sinh[2*(c + d*x)]))/(3*d*(b*Csch[c + d*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4256, 3042, 4258, 3042, 3120}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(i b \csc (i c+i d x))^{3/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\int \sqrt {b \text {csch}(c+d x)}dx}{3 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\int \sqrt {i b \csc (i c+i d x)}dx}{3 b^2}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)} \int \frac {1}{\sqrt {i \sinh (c+d x)}}dx}{3 b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}-\frac {\sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)} \int \frac {1}{\sqrt {\sin (i c+i d x)}}dx}{3 b^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {2 \cosh (c+d x)}{3 b d \sqrt {b \text {csch}(c+d x)}}+\frac {2 i \sqrt {i \sinh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right ),2\right ) \sqrt {b \text {csch}(c+d x)}}{3 b^2 d}\)

Input:

Int[(b*Csch[c + d*x])^(-3/2),x]
 

Output:

(2*Cosh[c + d*x])/(3*b*d*Sqrt[b*Csch[c + d*x]]) + (((2*I)/3)*Sqrt[b*Csch[c 
 + d*x]]*EllipticF[(I*c - Pi/2 + I*d*x)/2, 2]*Sqrt[I*Sinh[c + d*x]])/(b^2* 
d)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3120

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

rule 4256

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* 
n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
 

Maple [F]

\[\int \frac {1}{\left (\operatorname {csch}\left (d x +c \right ) b \right )^{\frac {3}{2}}}d x\]

Input:

int(1/(csch(d*x+c)*b)^(3/2),x)
 

Output:

int(1/(csch(d*x+c)*b)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (68) = 136\).

Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.58 \[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=-\frac {4 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} - 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1}}}{6 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}} \] Input:

integrate(1/(b*csch(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

-1/6*(4*sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d* 
x + c)^2)*sqrt(b)*weierstrassPInverse(4, 0, cosh(d*x + c) + sinh(d*x + c)) 
 - sqrt(2)*(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d*x 
 + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^ 
4 - 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^2*d*cosh(d*x + c)^2 + 2 
*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2)
 

Sympy [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (b \operatorname {csch}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(b*csch(d*x+c))**(3/2),x)
 

Output:

Integral((b*csch(c + d*x))**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(b*csch(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*csch(d*x + c))^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{\left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(b*csch(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate((b*csch(d*x + c))^(-3/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(1/(b/sinh(c + d*x))^(3/2),x)
 

Output:

int(1/(b/sinh(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{(b \text {csch}(c+d x))^{3/2}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\mathrm {csch}\left (d x +c \right )}}{\mathrm {csch}\left (d x +c \right )^{2}}d x \right )}{b^{2}} \] Input:

int(1/(b*csch(d*x+c))^(3/2),x)
 

Output:

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