\(\int \frac {1}{(a \text {csch}^3(x))^{3/2}} \, dx\) [40]

Optimal result

Integrand size = 10, antiderivative size = 89 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=-\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}} \] Output:

-14/45*cosh(x)/a/(a*csch(x)^3)^(1/2)+14/15*I*csch(x)*EllipticE(cos(1/4*Pi+ 
1/2*I*x),2^(1/2))/a/(a*csch(x)^3)^(1/2)/(I*sinh(x))^(1/2)+2/9*cosh(x)*sinh 
(x)^2/a/(a*csch(x)^3)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx=\frac {-33 \cosh (x)+5 \cosh (3 x)+84 \text {csch}^2(x) E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}}{90 a \sqrt {a \text {csch}^3(x)}} \] Input:

Integrate[(a*Csch[x]^3)^(-3/2),x]
 

Output:

(-33*Cosh[x] + 5*Cosh[3*x] + 84*Csch[x]^2*EllipticE[(Pi - (2*I)*x)/4, 2]*S 
qrt[I*Sinh[x]])/(90*a*Sqrt[a*Csch[x]^3])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4611, 3042, 4256, 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.

\(\displaystyle \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (i a \sec \left (\frac {\pi }{2}+i x\right )^3\right )^{3/2}}dx\)

\(\Big \downarrow \) 4611

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \int \frac {1}{(i \text {csch}(x))^{9/2}}dx}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \int \frac {1}{(-\csc (i x))^{9/2}}dx}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 4256

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \int \frac {1}{(i \text {csch}(x))^{5/2}}dx-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \int \frac {1}{(-\csc (i x))^{5/2}}dx-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 4256

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\sqrt {i \text {csch}(x)}}dx-\frac {2 i \cosh (x)}{5 (i \text {csch}(x))^{3/2}}\right )-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\sqrt {-\csc (i x)}}dx-\frac {2 i \cosh (x)}{5 (i \text {csch}(x))^{3/2}}\right )-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 4258

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \left (\frac {3 \int \sqrt {i \sinh (x)}dx}{5 \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-\frac {2 i \cosh (x)}{5 (i \text {csch}(x))^{3/2}}\right )-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \left (\frac {3 \int \sqrt {\sin (i x)}dx}{5 \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-\frac {2 i \cosh (x)}{5 (i \text {csch}(x))^{3/2}}\right )-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {i (i \text {csch}(x))^{3/2} \left (\frac {7}{9} \left (\frac {6 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{5 \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-\frac {2 i \cosh (x)}{5 (i \text {csch}(x))^{3/2}}\right )-\frac {2 i \cosh (x)}{9 (i \text {csch}(x))^{7/2}}\right )}{a \sqrt {a \text {csch}^3(x)}}\)

Input:

Int[(a*Csch[x]^3)^(-3/2),x]
 

Output:

((-I)*(I*Csch[x])^(3/2)*((((-2*I)/9)*Cosh[x])/(I*Csch[x])^(7/2) + (7*((((- 
2*I)/5)*Cosh[x])/(I*Csch[x])^(3/2) + (((6*I)/5)*EllipticE[Pi/4 - (I/2)*x, 
2])/(Sqrt[I*Csch[x]]*Sqrt[I*Sinh[x]])))/9))/(a*Sqrt[a*Csch[x]^3])
 

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]
 

rule 4611

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ 
IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart 
[p]))   Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] 
&&  !IntegerQ[p]
 

Maple [F]

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

Input:

int(1/(a*csch(x)^3)^(3/2),x)
 

Output:

int(1/(a*csch(x)^3)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (68) = 136\).

Time = 0.08 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.57 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(a*csch(x)^3)^(3/2),x, algorithm="fricas")
 

Output:

-1/720*(672*sqrt(2)*(cosh(x)^5 + 5*cosh(x)^4*sinh(x) + 10*cosh(x)^3*sinh(x 
)^2 + 10*cosh(x)^2*sinh(x)^3 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5)*sqrt(a)*we 
ierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(x) + sinh(x))) - sqrt(2 
)*(5*cosh(x)^10 + 50*cosh(x)*sinh(x)^9 + 5*sinh(x)^10 + (225*cosh(x)^2 - 4 
3)*sinh(x)^8 - 43*cosh(x)^8 + 8*(75*cosh(x)^3 - 43*cosh(x))*sinh(x)^7 + 2* 
(525*cosh(x)^4 - 602*cosh(x)^2 - 149)*sinh(x)^6 - 298*cosh(x)^6 + 4*(315*c 
osh(x)^5 - 602*cosh(x)^3 - 447*cosh(x))*sinh(x)^5 + 2*(525*cosh(x)^6 - 150 
5*cosh(x)^4 - 2235*cosh(x)^2 + 187)*sinh(x)^4 + 374*cosh(x)^4 + 8*(75*cosh 
(x)^7 - 301*cosh(x)^5 - 745*cosh(x)^3 + 187*cosh(x))*sinh(x)^3 + (225*cosh 
(x)^8 - 1204*cosh(x)^6 - 4470*cosh(x)^4 + 2244*cosh(x)^2 - 43)*sinh(x)^2 - 
 43*cosh(x)^2 + 2*(25*cosh(x)^9 - 172*cosh(x)^7 - 894*cosh(x)^5 + 748*cosh 
(x)^3 - 43*cosh(x))*sinh(x) + 5)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 
 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)))/(a^2*cosh(x)^5 + 5*a^2*cosh(x)^4*sin 
h(x) + 10*a^2*cosh(x)^3*sinh(x)^2 + 10*a^2*cosh(x)^2*sinh(x)^3 + 5*a^2*cos 
h(x)*sinh(x)^4 + a^2*sinh(x)^5)
 

Sympy [F]

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

integrate(1/(a*csch(x)**3)**(3/2),x)
 

Output:

Integral((a*csch(x)**3)**(-3/2), x)
 

Maxima [F]

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

integrate(1/(a*csch(x)^3)^(3/2),x, algorithm="maxima")
 

Output:

integrate((a*csch(x)^3)^(-3/2), x)
 

Giac [F]

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

integrate(1/(a*csch(x)^3)^(3/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((a*csch(x)^3)^(-3/2), x)
 

Mupad [F(-1)]

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

int(1/(a/sinh(x)^3)^(3/2),x)
 

Output:

int(1/(a/sinh(x)^3)^(3/2), x)
 

Reduce [F]

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

int(1/(a*csch(x)^3)^(3/2),x)
 

Output:

(sqrt(a)*int(sqrt(csch(x))/csch(x)**5,x))/a**2