\(\int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx\) [12]

Optimal result

Integrand size = 10, antiderivative size = 80 \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{5 b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \] Output:

2/5*cosh(b*x+a)/b/csch(b*x+a)^(3/2)-6/5*I*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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \left (\cosh (a+b x)-3 \text {csch}^2(a+b x) E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}\right )}{5 b \text {csch}^{\frac {3}{2}}(a+b x)} \] Input:

Integrate[Csch[a + b*x]^(-5/2),x]
 

Output:

(2*(Cosh[a + b*x] - 3*Csch[a + b*x]^2*EllipticE[((-2*I)*a + Pi - (2*I)*b*x 
)/4, 2]*Sqrt[I*Sinh[a + b*x]]))/(5*b*Csch[a + b*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4256, 3042, 4258, 3042, 3119}

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.

\(\displaystyle \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(i \csc (i a+i b x))^{5/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {3}{5} \int \frac {1}{\sqrt {\text {csch}(a+b x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {3}{5} \int \frac {1}{\sqrt {i \csc (i a+i b x)}}dx\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {3 \int \sqrt {i \sinh (a+b x)}dx}{5 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}-\frac {3 \int \sqrt {\sin (i a+i b x)}dx}{5 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 \cosh (a+b x)}{5 b \text {csch}^{\frac {3}{2}}(a+b x)}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\)

Input:

Int[Csch[a + b*x]^(-5/2),x]
 

Output:

(2*Cosh[a + b*x])/(5*b*Csch[a + b*x]^(3/2)) + (((6*I)/5)*EllipticE[(I*a - 
Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])
 

Defintions of rubi rules used

rule 3042

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

rule 3119

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 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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (63 ) = 126\).

Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.05

method	result	size
default	\(\frac {-\frac {6 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 \cosh \left (b x +a \right )^{4}}{5}-\frac {2 \cosh \left (b x +a \right )^{2}}{5}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\)	\(164\)

Input:

int(1/csch(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(-6/5*(1-I*sinh(b*x+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+ 
a))^(1/2)*EllipticE((1-I*sinh(b*x+a))^(1/2),1/2*2^(1/2))+3/5*(1-I*sinh(b*x 
+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a))^(1/2)*EllipticF 
((1-I*sinh(b*x+a))^(1/2),1/2*2^(1/2))+2/5*cosh(b*x+a)^4-2/5*cosh(b*x+a)^2) 
/cosh(b*x+a)/sinh(b*x+a)^(1/2)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (59) = 118\).

Time = 0.10 (sec) , antiderivative size = 370, normalized size of antiderivative = 4.62 \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} + 11\right )} \sinh \left (b x + a\right )^{4} + 11 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 11 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} + 66 \, \cosh \left (b x + a\right )^{2} - 13\right )} \sinh \left (b x + a\right )^{2} - 13 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} + 22 \, \cosh \left (b x + a\right )^{3} - 13 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}} + 24 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{20 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \] Input:

integrate(1/csch(b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

1/20*(sqrt(2)*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b* 
x + a)^6 + (15*cosh(b*x + a)^2 + 11)*sinh(b*x + a)^4 + 11*cosh(b*x + a)^4 
+ 4*(5*cosh(b*x + a)^3 + 11*cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x 
+ a)^4 + 66*cosh(b*x + a)^2 - 13)*sinh(b*x + a)^2 - 13*cosh(b*x + a)^2 + 2 
*(3*cosh(b*x + a)^5 + 22*cosh(b*x + a)^3 - 13*cosh(b*x + a))*sinh(b*x + a) 
 + 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)) + 24*(sqrt(2)*cosh(b*x + a)^3 + 
3*sqrt(2)*cosh(b*x + a)^2*sinh(b*x + a) + 3*sqrt(2)*cosh(b*x + a)*sinh(b*x 
 + a)^2 + sqrt(2)*sinh(b*x + a)^3)*weierstrassZeta(4, 0, weierstrassPInver 
se(4, 0, cosh(b*x + a) + sinh(b*x + a))))/(b*cosh(b*x + a)^3 + 3*b*cosh(b* 
x + a)^2*sinh(b*x + a) + 3*b*cosh(b*x + a)*sinh(b*x + a)^2 + b*sinh(b*x + 
a)^3)
 

Sympy [F]

\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{\operatorname {csch}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \] Input:

integrate(1/csch(b*x+a)**(5/2),x)
 

Output:

Integral(csch(a + b*x)**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/csch(b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(csch(b*x + a)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {csch}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/csch(b*x+a)^(5/2),x, algorithm="giac")
 

Output:

integrate(csch(b*x + a)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \] Input:

int(1/(1/sinh(a + b*x))^(5/2),x)
 

Output:

int(1/(1/sinh(a + b*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\text {csch}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (b x +a \right )}}{\mathrm {csch}\left (b x +a \right )^{3}}d x \] Input:

int(1/csch(b*x+a)^(5/2),x)
 

Output:

int(sqrt(csch(a + b*x))/csch(a + b*x)**3,x)