\(\int (a \text {csch}^3(x))^{5/2} \, dx\) [36]

Optimal result

Integrand size = 10, antiderivative size = 135 \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)-\frac {154 i a^2 \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 \sqrt {i \sinh (x)}} \] Output:

-154/585*a^2*coth(x)*(a*csch(x)^3)^(1/2)+22/117*a^2*coth(x)*csch(x)^2*(a*c 
sch(x)^3)^(1/2)-2/13*a^2*coth(x)*csch(x)^4*(a*csch(x)^3)^(1/2)+154/195*a^2 
*cosh(x)*(a*csch(x)^3)^(1/2)*sinh(x)-154/195*I*a^2*(a*csch(x)^3)^(1/2)*Ell 
ipticE(cos(1/4*Pi+1/2*I*x),2^(1/2))*sinh(x)^2/(I*sinh(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.50 \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=-\frac {2}{585} a^2 \sqrt {a \text {csch}^3(x)} \left (-231 \cosh (x)+\coth (x) \text {csch}(x) \left (77-55 \text {csch}^2(x)+45 \text {csch}^4(x)\right )+231 E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}\right ) \sinh (x) \] Input:

Integrate[(a*Csch[x]^3)^(5/2),x]
 

Output:

(-2*a^2*Sqrt[a*Csch[x]^3]*(-231*Cosh[x] + Coth[x]*Csch[x]*(77 - 55*Csch[x] 
^2 + 45*Csch[x]^4) + 231*EllipticE[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]])*S 
inh[x])/585
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 4611, 3042, 4255, 3042, 4255, 3042, 4255, 3042, 4255, 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.

\(\displaystyle \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (i a \sec \left (\frac {\pi }{2}+i x\right )^3\right )^{5/2}dx\)

\(\Big \downarrow \) 4611

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \int (i \text {csch}(x))^{15/2}dx}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \int (-\csc (i x))^{15/2}dx}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 4255

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \int (i \text {csch}(x))^{11/2}dx-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \int (-\csc (i x))^{11/2}dx-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 4255

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \int (i \text {csch}(x))^{7/2}dx-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \int (-\csc (i x))^{7/2}dx-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 4255

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int (i \text {csch}(x))^{3/2}dx-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int (-\csc (i x))^{3/2}dx-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 4255

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-\int \frac {1}{\sqrt {i \text {csch}(x)}}dx-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-\int \frac {1}{\sqrt {-\csc (i x)}}dx-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 4258

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-\frac {\int \sqrt {i \sinh (x)}dx}{\sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-\frac {\int \sqrt {\sin (i x)}dx}{\sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {a^2 \sqrt {a \text {csch}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-2 i \cosh (x) \sqrt {i \text {csch}(x)}-\frac {2 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}\right )-\frac {2}{5} i \cosh (x) (i \text {csch}(x))^{5/2}\right )-\frac {2}{9} i \cosh (x) (i \text {csch}(x))^{9/2}\right )-\frac {2}{13} i \cosh (x) (i \text {csch}(x))^{13/2}\right )}{(i \text {csch}(x))^{3/2}}\)

Input:

Int[(a*Csch[x]^3)^(5/2),x]
 

Output:

-((a^2*Sqrt[a*Csch[x]^3]*(((-2*I)/13)*Cosh[x]*(I*Csch[x])^(13/2) + (11*((( 
-2*I)/9)*Cosh[x]*(I*Csch[x])^(9/2) + (7*(((-2*I)/5)*Cosh[x]*(I*Csch[x])^(5 
/2) + (3*((-2*I)*Cosh[x]*Sqrt[I*Csch[x]] - ((2*I)*EllipticE[Pi/4 - (I/2)*x 
, 2])/(Sqrt[I*Csch[x]]*Sqrt[I*Sinh[x]])))/5))/9))/13))/(I*Csch[x])^(3/2))
 

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 4255

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

\[\int \left (a \operatorname {csch}\left (x \right )^{3}\right )^{\frac {5}{2}}d x\]

Input:

int((a*csch(x)^3)^(5/2),x)
 

Output:

int((a*csch(x)^3)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (106) = 212\).

Time = 0.11 (sec) , antiderivative size = 1389, normalized size of antiderivative = 10.29 \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a*csch(x)^3)^(5/2),x, algorithm="fricas")
 

Output:

2/585*(231*sqrt(2)*(a^2*cosh(x)^12 + 12*a^2*cosh(x)*sinh(x)^11 + a^2*sinh( 
x)^12 - 6*a^2*cosh(x)^10 + 6*(11*a^2*cosh(x)^2 - a^2)*sinh(x)^10 + 15*a^2* 
cosh(x)^8 + 20*(11*a^2*cosh(x)^3 - 3*a^2*cosh(x))*sinh(x)^9 + 15*(33*a^2*c 
osh(x)^4 - 18*a^2*cosh(x)^2 + a^2)*sinh(x)^8 - 20*a^2*cosh(x)^6 + 24*(33*a 
^2*cosh(x)^5 - 30*a^2*cosh(x)^3 + 5*a^2*cosh(x))*sinh(x)^7 + 4*(231*a^2*co 
sh(x)^6 - 315*a^2*cosh(x)^4 + 105*a^2*cosh(x)^2 - 5*a^2)*sinh(x)^6 + 15*a^ 
2*cosh(x)^4 + 24*(33*a^2*cosh(x)^7 - 63*a^2*cosh(x)^5 + 35*a^2*cosh(x)^3 - 
 5*a^2*cosh(x))*sinh(x)^5 + 15*(33*a^2*cosh(x)^8 - 84*a^2*cosh(x)^6 + 70*a 
^2*cosh(x)^4 - 20*a^2*cosh(x)^2 + a^2)*sinh(x)^4 - 6*a^2*cosh(x)^2 + 20*(1 
1*a^2*cosh(x)^9 - 36*a^2*cosh(x)^7 + 42*a^2*cosh(x)^5 - 20*a^2*cosh(x)^3 + 
 3*a^2*cosh(x))*sinh(x)^3 + 6*(11*a^2*cosh(x)^10 - 45*a^2*cosh(x)^8 + 70*a 
^2*cosh(x)^6 - 50*a^2*cosh(x)^4 + 15*a^2*cosh(x)^2 - a^2)*sinh(x)^2 + a^2 
+ 12*(a^2*cosh(x)^11 - 5*a^2*cosh(x)^9 + 10*a^2*cosh(x)^7 - 10*a^2*cosh(x) 
^5 + 5*a^2*cosh(x)^3 - a^2*cosh(x))*sinh(x))*sqrt(a)*weierstrassZeta(4, 0, 
 weierstrassPInverse(4, 0, cosh(x) + sinh(x))) + sqrt(2)*(231*a^2*cosh(x)^ 
13 + 3003*a^2*cosh(x)*sinh(x)^12 + 231*a^2*sinh(x)^13 - 1540*a^2*cosh(x)^1 
1 + 154*(117*a^2*cosh(x)^2 - 10*a^2)*sinh(x)^11 + 4367*a^2*cosh(x)^9 + 169 
4*(39*a^2*cosh(x)^3 - 10*a^2*cosh(x))*sinh(x)^10 + 11*(15015*a^2*cosh(x)^4 
 - 7700*a^2*cosh(x)^2 + 397*a^2)*sinh(x)^9 - 6808*a^2*cosh(x)^7 + 33*(9009 
*a^2*cosh(x)^5 - 7700*a^2*cosh(x)^3 + 1191*a^2*cosh(x))*sinh(x)^8 + 4*(...
 

Sympy [F]

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

integrate((a*csch(x)**3)**(5/2),x)
 

Output:

Integral((a*csch(x)**3)**(5/2), x)
 

Maxima [F]

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

integrate((a*csch(x)^3)^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*csch(x)^3)^(5/2), x)
 

Giac [F]

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

integrate((a*csch(x)^3)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((a*csch(x)^3)^(5/2), x)
 

Mupad [F(-1)]

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

int((a/sinh(x)^3)^(5/2),x)
 

Output:

int((a/sinh(x)^3)^(5/2), x)
 

Reduce [F]

\[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\mathrm {csch}\left (x \right )}\, \mathrm {csch}\left (x \right )^{7}d x \right ) a^{2} \] Input:

int((a*csch(x)^3)^(5/2),x)
 

Output:

sqrt(a)*int(sqrt(csch(x))*csch(x)**7,x)*a**2