\(\int \frac {1}{(a \sinh ^3(x))^{5/2}} \, dx\) [151]

Optimal result

Integrand size = 10, antiderivative size = 135 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \] Output:

-154/585*coth(x)/a^2/(a*sinh(x)^3)^(1/2)+22/117*coth(x)*csch(x)^2/a^2/(a*s 
inh(x)^3)^(1/2)-2/13*coth(x)*csch(x)^4/a^2/(a*sinh(x)^3)^(1/2)+154/195*cos 
h(x)*sinh(x)/a^2/(a*sinh(x)^3)^(1/2)-154/195*I*EllipticE(cos(1/4*Pi+1/2*I* 
x),2^(1/2))*sinh(x)^2/a^2/(I*sinh(x))^(1/2)/(a*sinh(x)^3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\frac {-2 \coth (x) \left (77-55 \text {csch}^2(x)+45 \text {csch}^4(x)\right )+462 i E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) (i \sinh (x))^{3/2}+462 \cosh (x) \sinh (x)}{585 a^2 \sqrt {a \sinh ^3(x)}} \] Input:

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

Output:

(-2*Coth[x]*(77 - 55*Csch[x]^2 + 45*Csch[x]^4) + (462*I)*EllipticE[(Pi - ( 
2*I)*x)/4, 2]*(I*Sinh[x])^(3/2) + 462*Cosh[x]*Sinh[x])/(585*a^2*Sqrt[a*Sin 
h[x]^3])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.87, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 3686, 3042, 3116, 3042, 3116, 3042, 3116, 3042, 3116, 3042, 3121, 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.

\(\displaystyle \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (i a \sin (i x)^3\right )^{5/2}}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {15}{2}}(x)}dx}{a^2 \sqrt {a \sinh ^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{(-i \sin (i x))^{15/2}}dx}{a^2 \sqrt {a \sinh ^3(x)}}\)

\(\Big \downarrow \) 3116

\(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \left (-\frac {11}{13} \int \frac {1}{\sinh ^{\frac {11}{2}}(x)}dx-\frac {2 \cosh (x)}{13 \sinh ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sinh ^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \left (-\frac {2 \cosh (x)}{13 \sinh ^{\frac {13}{2}}(x)}-\frac {11}{13} \int \frac {1}{(-i \sin (i x))^{11/2}}dx\right )}{a^2 \sqrt {a \sinh ^3(x)}}\)

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 3042

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

rule 3116

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

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 3121

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

rule 3686

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ 
n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Si 
n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] 
 && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / 
; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
 

Maple [F]

\[\int \frac {1}{\left (a \sinh \left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]

Input:

int(1/(a*sinh(x)^3)^(5/2),x)
 

Output:

int(1/(a*sinh(x)^3)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (106) = 212\).

Time = 0.13 (sec) , antiderivative size = 1477, normalized size of antiderivative = 10.94 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*sinh(x)^3)^(5/2),x, algorithm="fricas")
 

Output:

4/585*(231*sqrt(1/2)*(cosh(x)^14 + 14*cosh(x)*sinh(x)^13 + sinh(x)^14 + 7* 
(13*cosh(x)^2 - 1)*sinh(x)^12 - 7*cosh(x)^12 + 28*(13*cosh(x)^3 - 3*cosh(x 
))*sinh(x)^11 + 7*(143*cosh(x)^4 - 66*cosh(x)^2 + 3)*sinh(x)^10 + 21*cosh( 
x)^10 + 14*(143*cosh(x)^5 - 110*cosh(x)^3 + 15*cosh(x))*sinh(x)^9 + 7*(429 
*cosh(x)^6 - 495*cosh(x)^4 + 135*cosh(x)^2 - 5)*sinh(x)^8 - 35*cosh(x)^8 + 
 8*(429*cosh(x)^7 - 693*cosh(x)^5 + 315*cosh(x)^3 - 35*cosh(x))*sinh(x)^7 
+ 7*(429*cosh(x)^8 - 924*cosh(x)^6 + 630*cosh(x)^4 - 140*cosh(x)^2 + 5)*si 
nh(x)^6 + 35*cosh(x)^6 + 14*(143*cosh(x)^9 - 396*cosh(x)^7 + 378*cosh(x)^5 
 - 140*cosh(x)^3 + 15*cosh(x))*sinh(x)^5 + 7*(143*cosh(x)^10 - 495*cosh(x) 
^8 + 630*cosh(x)^6 - 350*cosh(x)^4 + 75*cosh(x)^2 - 3)*sinh(x)^4 - 21*cosh 
(x)^4 + 28*(13*cosh(x)^11 - 55*cosh(x)^9 + 90*cosh(x)^7 - 70*cosh(x)^5 + 2 
5*cosh(x)^3 - 3*cosh(x))*sinh(x)^3 + 7*(13*cosh(x)^12 - 66*cosh(x)^10 + 13 
5*cosh(x)^8 - 140*cosh(x)^6 + 75*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x)^2 + 
 7*cosh(x)^2 + 14*(cosh(x)^13 - 6*cosh(x)^11 + 15*cosh(x)^9 - 20*cosh(x)^7 
 + 15*cosh(x)^5 - 6*cosh(x)^3 + cosh(x))*sinh(x) - 1)*sqrt(a)*weierstrassZ 
eta(4, 0, weierstrassPInverse(4, 0, cosh(x) + sinh(x))) + (231*cosh(x)^14 
+ 3234*cosh(x)*sinh(x)^13 + 231*sinh(x)^14 + 77*(273*cosh(x)^2 - 20)*sinh( 
x)^12 - 1540*cosh(x)^12 + 924*(91*cosh(x)^3 - 20*cosh(x))*sinh(x)^11 + 11* 
(21021*cosh(x)^4 - 9240*cosh(x)^2 + 397)*sinh(x)^10 + 4367*cosh(x)^10 + 22 
*(21021*cosh(x)^5 - 15400*cosh(x)^3 + 1985*cosh(x))*sinh(x)^9 + (693693...
 

Sympy [F]

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

integrate(1/(a*sinh(x)**3)**(5/2),x)
 

Output:

Integral((a*sinh(x)**3)**(-5/2), x)
 

Maxima [F]

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

integrate(1/(a*sinh(x)^3)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(a*sinh(x)^3)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{5/2}} \,d x \] Input:

int(1/(a*sinh(x)^3)^(5/2),x)
 

Output:

int(1/(a*sinh(x)^3)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sinh \left (x \right )}}{\sinh \left (x \right )^{8}}d x \right )}{a^{3}} \] Input:

int(1/(a*sinh(x)^3)^(5/2),x)
 

Output:

(sqrt(a)*int(sqrt(sinh(x))/sinh(x)**8,x))/a**3