Integrand size = 10, antiderivative size = 62 \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{3 \sqrt {a \text {csch}^3(x)}} \]
2/3*coth(x)/(a*csch(x)^3)^(1/2)-2/3*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*csch(x)^3)^(1/2)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\frac {2 \left (\coth (x)+\frac {\text {csch}(x) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right )}{3 \sqrt {a \text {csch}^3(x)}} \]
(2*(Coth[x] + (Csch[x]*EllipticF[(Pi - (2*I)*x)/4, 2])/Sqrt[I*Sinh[x]]))/( 3*Sqrt[a*Csch[x]^3])
Time = 0.39 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.31, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4611, 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.
\(\displaystyle \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {i a \sec \left (\frac {\pi }{2}+i x\right )^3}}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \int \frac {1}{(i \text {csch}(x))^{3/2}}dx}{\sqrt {a \text {csch}^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \int \frac {1}{(-\csc (i x))^{3/2}}dx}{\sqrt {a \text {csch}^3(x)}}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \left (\frac {1}{3} \int \sqrt {i \text {csch}(x)}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )}{\sqrt {a \text {csch}^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \left (\frac {1}{3} \int \sqrt {-\csc (i x)}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )}{\sqrt {a \text {csch}^3(x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \left (\frac {1}{3} \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)} \int \frac {1}{\sqrt {i \sinh (x)}}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )}{\sqrt {a \text {csch}^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \left (\frac {1}{3} \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)} \int \frac {1}{\sqrt {\sin (i x)}}dx-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )}{\sqrt {a \text {csch}^3(x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {(i \text {csch}(x))^{3/2} \left (\frac {2}{3} i \sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right )-\frac {2 i \cosh (x)}{3 \sqrt {i \text {csch}(x)}}\right )}{\sqrt {a \text {csch}^3(x)}}\) |
((I*Csch[x])^(3/2)*((((-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]]))/Sqrt[a*Csch[x]^3 ]
3.1.39.3.1 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]
\[\int \frac {1}{\sqrt {a \operatorname {csch}\left (x \right )^{3}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=-\frac {4 \, \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - \sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}}{6 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2}\right )}} \]
-1/6*(4*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(a)*weiers trassPInverse(4, 0, cosh(x) + sinh(x)) - sqrt(2)*(cosh(x)^4 + 4*cosh(x)^3* sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 - 1)*sqr t((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)) )/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2)
\[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\int \frac {1}{\sqrt {a \operatorname {csch}^{3}{\left (x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \operatorname {csch}\left (x\right )^{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \operatorname {csch}\left (x\right )^{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\mathrm {sinh}\left (x\right )}^3}}} \,d x \]