Integrand size = 10, antiderivative size = 69 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {a \text {sech}^3(x)}+\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \]
-10/21*I*a*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sin h(1/2*x),2^(1/2))*(a*sech(x)^3)^(1/2)+10/21*a*sinh(x)*(a*sech(x)^3)^(1/2)+ 2/7*a*sech(x)*(a*sech(x)^3)^(1/2)*tanh(x)
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\frac {2}{21} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \left (-5 i \cosh ^{\frac {5}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+5 \cosh (x) \sinh (x)+3 \tanh (x)\right ) \]
(2*a*Sech[x]*Sqrt[a*Sech[x]^3]*((-5*I)*Cosh[x]^(5/2)*EllipticF[(I/2)*x, 2] + 5*Cosh[x]*Sinh[x] + 3*Tanh[x]))/21
Time = 0.43 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4611, 3042, 4255, 3042, 4255, 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.
| \(\displaystyle \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sec (i x)^3\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \int \text {sech}^{\frac {9}{2}}(x)dx}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \int \csc \left (i x+\frac {\pi }{2}\right )^{9/2}dx}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {5}{7} \int \text {sech}^{\frac {5}{2}}(x)dx+\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)+\frac {5}{7} \int \csc \left (i x+\frac {\pi }{2}\right )^{5/2}dx\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\text {sech}(x)}dx+\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)\right )+\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)+\frac {1}{3} \int \sqrt {\csc \left (i x+\frac {\pi }{2}\right )}dx\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\cosh (x)}}dx+\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)\right )+\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)+\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)-\frac {2}{3} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\) |
(a*Sqrt[a*Sech[x]^3]*((2*Sech[x]^(7/2)*Sinh[x])/7 + (5*(((-2*I)/3)*Sqrt[Co sh[x]]*EllipticF[(I/2)*x, 2]*Sqrt[Sech[x]] + (2*Sech[x]^(3/2)*Sinh[x])/3)) /7))/Sech[x]^(3/2)
3.1.40.3.1 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[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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]
\[\int \left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {3}{2}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 391, normalized size of antiderivative = 5.67 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\frac {2 \, {\left (5 \, \sqrt {2} {\left (a \cosh \left (x\right )^{6} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + a \sinh \left (x\right )^{6} + 3 \, a \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a \cosh \left (x\right )^{4} + 6 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a \cosh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (5 \, a \cosh \left (x\right )^{6} + 30 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + 5 \, a \sinh \left (x\right )^{6} + 17 \, a \cosh \left (x\right )^{4} + {\left (75 \, a \cosh \left (x\right )^{2} + 17 \, a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (25 \, a \cosh \left (x\right )^{3} + 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )^{2} + {\left (75 \, a \cosh \left (x\right )^{4} + 102 \, a \cosh \left (x\right )^{2} - 17 \, a\right )} \sinh \left (x\right )^{2} + 2 \, {\left (15 \, a \cosh \left (x\right )^{5} + 34 \, a \cosh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\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}}\right )}}{21 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
2/21*(5*sqrt(2)*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*c osh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x ))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh( x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(a)*we ierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + sqrt(2)*(5*a*cosh(x)^6 + 30* a*cosh(x)*sinh(x)^5 + 5*a*sinh(x)^6 + 17*a*cosh(x)^4 + (75*a*cosh(x)^2 + 1 7*a)*sinh(x)^4 + 4*(25*a*cosh(x)^3 + 17*a*cosh(x))*sinh(x)^3 - 17*a*cosh(x )^2 + (75*a*cosh(x)^4 + 102*a*cosh(x)^2 - 17*a)*sinh(x)^2 + 2*(15*a*cosh(x )^5 + 34*a*cosh(x)^3 - 17*a*cosh(x))*sinh(x) - 5*a)*sqrt((a*cosh(x) + a*si nh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(cosh(x)^6 + 6*co sh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1 )*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)
\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int \left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int { \left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int { \left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{3/2} \,d x \]