Integrand size = 10, antiderivative size = 135 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \]
-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*(sin(1/4*Pi+1/2*I*x)^2)^(1/ 2)/sin(1/4*Pi+1/2*I*x)*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)
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\frac {-2 \coth (x) \left (77-55 \text {csch}^2(x)+45 \text {csch}^4(x)\right )+462 i E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) (i \sinh (x))^{3/2}+462 \cosh (x) \sinh (x)}{585 a^2 \sqrt {a \sinh ^3(x)}} \]
(-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])
Time = 0.55 (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 |
| \(\displaystyle \frac {\sinh ^{\frac {3}{2}}(x) \left (-\frac {2 \cosh (x)}{13 \sinh ^{\frac {13}{2}}(x)}-\frac {11}{13} \left (-\frac {2 \cosh (x)}{9 \sinh ^{\frac {9}{2}}(x)}-\frac {7}{9} \left (-\frac {2 \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {3}{5} \left (-\frac {2 \cosh (x)}{\sqrt {\sinh (x)}}+\frac {2 i \sqrt {\sinh (x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)}}\right )\right )\right )\right )}{a^2 \sqrt {a \sinh ^3(x)}}\) |
(((-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])
3.2.51.3.1 Defintions of rubi rules used
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]
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]
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]
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]])
\[\int \frac {1}{\left (a \sinh \left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1676, normalized size of antiderivative = 12.41 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
2/585*(231*(sqrt(2)*cosh(x)^14 + 14*sqrt(2)*cosh(x)*sinh(x)^13 + sqrt(2)*s inh(x)^14 + 7*(13*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x)^12 - 7*sqrt(2)*cosh (x)^12 + 28*(13*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))*sinh(x)^11 + 7*(143 *sqrt(2)*cosh(x)^4 - 66*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^10 + 21*sqr t(2)*cosh(x)^10 + 14*(143*sqrt(2)*cosh(x)^5 - 110*sqrt(2)*cosh(x)^3 + 15*s qrt(2)*cosh(x))*sinh(x)^9 + 7*(429*sqrt(2)*cosh(x)^6 - 495*sqrt(2)*cosh(x) ^4 + 135*sqrt(2)*cosh(x)^2 - 5*sqrt(2))*sinh(x)^8 - 35*sqrt(2)*cosh(x)^8 + 8*(429*sqrt(2)*cosh(x)^7 - 693*sqrt(2)*cosh(x)^5 + 315*sqrt(2)*cosh(x)^3 - 35*sqrt(2)*cosh(x))*sinh(x)^7 + 7*(429*sqrt(2)*cosh(x)^8 - 924*sqrt(2)*c osh(x)^6 + 630*sqrt(2)*cosh(x)^4 - 140*sqrt(2)*cosh(x)^2 + 5*sqrt(2))*sinh (x)^6 + 35*sqrt(2)*cosh(x)^6 + 14*(143*sqrt(2)*cosh(x)^9 - 396*sqrt(2)*cos h(x)^7 + 378*sqrt(2)*cosh(x)^5 - 140*sqrt(2)*cosh(x)^3 + 15*sqrt(2)*cosh(x ))*sinh(x)^5 + 7*(143*sqrt(2)*cosh(x)^10 - 495*sqrt(2)*cosh(x)^8 + 630*sqr t(2)*cosh(x)^6 - 350*sqrt(2)*cosh(x)^4 + 75*sqrt(2)*cosh(x)^2 - 3*sqrt(2)) *sinh(x)^4 - 21*sqrt(2)*cosh(x)^4 + 28*(13*sqrt(2)*cosh(x)^11 - 55*sqrt(2) *cosh(x)^9 + 90*sqrt(2)*cosh(x)^7 - 70*sqrt(2)*cosh(x)^5 + 25*sqrt(2)*cosh (x)^3 - 3*sqrt(2)*cosh(x))*sinh(x)^3 + 7*(13*sqrt(2)*cosh(x)^12 - 66*sqrt( 2)*cosh(x)^10 + 135*sqrt(2)*cosh(x)^8 - 140*sqrt(2)*cosh(x)^6 + 75*sqrt(2) *cosh(x)^4 - 18*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 7*sqrt(2)*cosh(x) ^2 + 14*(sqrt(2)*cosh(x)^13 - 6*sqrt(2)*cosh(x)^11 + 15*sqrt(2)*cosh(x)...
\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sinh \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sinh \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{5/2}} \,d x \]