Integrand size = 10, antiderivative size = 48 \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \tanh (x)}{3 \sqrt {a \text {sech}^3(x)}} \]
-2/3*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))/ cosh(x)^(3/2)/(a*sech(x)^3)^(1/2)+2/3*tanh(x)/(a*sech(x)^3)^(1/2)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\frac {-\frac {2 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{\cosh ^{\frac {3}{2}}(x)}+2 \tanh (x)}{3 \sqrt {a \text {sech}^3(x)}} \]
Time = 0.36 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19, 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.
\(\displaystyle \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a \sec (i x)^3}}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)}dx}{\sqrt {a \text {sech}^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{3/2}}dx}{\sqrt {a \text {sech}^3(x)}}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {1}{3} \int \sqrt {\text {sech}(x)}dx+\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}\right )}{\sqrt {a \text {sech}^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \int \sqrt {\csc \left (i x+\frac {\pi }{2}\right )}dx\right )}{\sqrt {a \text {sech}^3(x)}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\cosh (x)}}dx+\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}\right )}{\sqrt {a \text {sech}^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )}}dx\right )}{\sqrt {a \text {sech}^3(x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {2}{3} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )\right )}{\sqrt {a \text {sech}^3(x)}}\) |
(Sech[x]^(3/2)*(((-2*I)/3)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2]*Sqrt[Sech[x ]] + (2*Sinh[x])/(3*Sqrt[Sech[x]])))/Sqrt[a*Sech[x]^3]
3.1.42.3.1 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]
\[\int \frac {1}{\sqrt {a \operatorname {sech}\left (x \right )^{3}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.62 \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\frac {4 \, \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{6 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2}\right )}} \]
1/6*(4*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(a)*weierst rassPInverse(-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)
\[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\int \frac {1}{\sqrt {a \operatorname {sech}^{3}{\left (x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \operatorname {sech}\left (x\right )^{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \operatorname {sech}\left (x\right )^{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^3}}} \,d x \]