Integrand size = 10, antiderivative size = 135 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=-\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}} \]
-26/77*coth(x)/a^2/(a*csch(x)^3)^(1/2)+78/385*cosh(x)*sinh(x)/a^2/(a*csch( x)^3)^(1/2)-26/165*cosh(x)*sinh(x)^3/a^2/(a*csch(x)^3)^(1/2)+2/15*cosh(x)* sinh(x)^5/a^2/(a*csch(x)^3)^(1/2)+26/77*I*csch(x)^2*(sin(1/4*Pi+1/2*I*x)^2 )^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2))*(I*sinh (x))^(1/2)/a^2/(a*csch(x)^3)^(1/2)
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \text {csch}^3(x)} \sinh (x) \left (24960 i \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right ) \sqrt {i \sinh (x)}-19122 \sinh (2 x)+4406 \sinh (4 x)-826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \]
(Sqrt[a*Csch[x]^3]*Sinh[x]*((24960*I)*EllipticF[(Pi - (2*I)*x)/4, 2]*Sqrt[ I*Sinh[x]] - 19122*Sinh[2*x] + 4406*Sinh[4*x] - 826*Sinh[6*x] + 77*Sinh[8* x]))/(73920*a^3)
Time = 0.67 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.14, 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, 4256, 3042, 4256, 3042, 4256, 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.
| \(\displaystyle \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (i a \sec \left (\frac {\pi }{2}+i x\right )^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \int \frac {1}{(i \text {csch}(x))^{15/2}}dx}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \int \frac {1}{(-\csc (i x))^{15/2}}dx}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \int \frac {1}{(i \text {csch}(x))^{11/2}}dx-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \int \frac {1}{(-\csc (i x))^{11/2}}dx-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \int \frac {1}{(i \text {csch}(x))^{7/2}}dx-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \int \frac {1}{(-\csc (i x))^{7/2}}dx-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \frac {1}{(i \text {csch}(x))^{3/2}}dx-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \frac {1}{(-\csc (i x))^{3/2}}dx-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {i \text {csch}(x)}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {-\csc (i x)}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)} \int \frac {1}{\sqrt {i \sinh (x)}}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)} \int \frac {1}{\sqrt {\sin (i x)}}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {(i \text {csch}(x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {2}{3} i \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right )-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )-\frac {2 i \cosh (x)}{7 (i \text {csch}(x))^{5/2}}\right )-\frac {2 i \cosh (x)}{11 (i \text {csch}(x))^{9/2}}\right )-\frac {2 i \cosh (x)}{15 (i \text {csch}(x))^{13/2}}\right )}{a^2 \sqrt {a \text {csch}^3(x)}}\) |
-(((I*Csch[x])^(3/2)*((((-2*I)/15)*Cosh[x])/(I*Csch[x])^(13/2) + (13*((((- 2*I)/11)*Cosh[x])/(I*Csch[x])^(9/2) + (9*((((-2*I)/7)*Cosh[x])/(I*Csch[x]) ^(5/2) + (5*((((-2*I)/3)*Cosh[x])/Sqrt[I*Csch[x]] + ((2*I)/3)*Sqrt[I*Csch[ x]]*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]))/7))/11))/15))/(a^2*Sqrt [a*Csch[x]^3]))
3.1.41.3.1 Defintions of rubi rules used
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]
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]
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]
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]
\[\int \frac {1}{\left (a \operatorname {csch}\left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.32 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/147840*(49920*sqrt(2)*(cosh(x)^8 + 8*cosh(x)^7*sinh(x) + 28*cosh(x)^6*si nh(x)^2 + 56*cosh(x)^5*sinh(x)^3 + 70*cosh(x)^4*sinh(x)^4 + 56*cosh(x)^3*s inh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8)*sqrt( a)*weierstrassPInverse(4, 0, cosh(x) + sinh(x)) + sqrt(2)*(77*cosh(x)^16 + 1232*cosh(x)*sinh(x)^15 + 77*sinh(x)^16 + 14*(660*cosh(x)^2 - 59)*sinh(x) ^14 - 826*cosh(x)^14 + 196*(220*cosh(x)^3 - 59*cosh(x))*sinh(x)^13 + 2*(70 070*cosh(x)^4 - 37583*cosh(x)^2 + 2203)*sinh(x)^12 + 4406*cosh(x)^12 + 8*( 42042*cosh(x)^5 - 37583*cosh(x)^3 + 6609*cosh(x))*sinh(x)^11 + 2*(308308*c osh(x)^6 - 413413*cosh(x)^4 + 145398*cosh(x)^2 - 9561)*sinh(x)^10 - 19122* cosh(x)^10 + 4*(220220*cosh(x)^7 - 413413*cosh(x)^5 + 242330*cosh(x)^3 - 4 7805*cosh(x))*sinh(x)^9 + 6*(165165*cosh(x)^8 - 413413*cosh(x)^6 + 363495* cosh(x)^4 - 143415*cosh(x)^2)*sinh(x)^8 + 16*(55055*cosh(x)^9 - 177177*cos h(x)^7 + 218097*cosh(x)^5 - 143415*cosh(x)^3)*sinh(x)^7 + 2*(308308*cosh(x )^10 - 1240239*cosh(x)^8 + 2035572*cosh(x)^6 - 2007810*cosh(x)^4 + 9561)*s inh(x)^6 + 19122*cosh(x)^6 + 4*(84084*cosh(x)^11 - 413413*cosh(x)^9 + 8723 88*cosh(x)^7 - 1204686*cosh(x)^5 + 28683*cosh(x))*sinh(x)^5 + 2*(70070*cos h(x)^12 - 413413*cosh(x)^10 + 1090485*cosh(x)^8 - 2007810*cosh(x)^6 + 1434 15*cosh(x)^2 - 2203)*sinh(x)^4 - 4406*cosh(x)^4 + 8*(5390*cosh(x)^13 - 375 83*cosh(x)^11 + 121165*cosh(x)^9 - 286830*cosh(x)^7 + 47805*cosh(x)^3 - 22 03*cosh(x))*sinh(x)^3 + 2*(4620*cosh(x)^14 - 37583*cosh(x)^12 + 145398*...
\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^3}\right )}^{5/2}} \,d x \]