Integrand size = 10, antiderivative size = 66 \[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=-\frac {6 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{5 b}+\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)} \]
2/5*sinh(b*x+a)/b/sech(b*x+a)^(3/2)-6/5*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/co sh(1/2*a+1/2*b*x)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))*cosh(b*x+a)^(1/ 2)*sech(b*x+a)^(1/2)/b
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (-12 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+\sinh (a+b x)+\sinh (3 (a+b x))\right )}{10 b} \]
(Sqrt[Sech[a + b*x]]*((-12*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x ), 2] + Sinh[a + b*x] + Sinh[3*(a + b*x)]))/(10*b)
Time = 0.32 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4256, 3042, 4258, 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.
\(\displaystyle \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (i a+i b x+\frac {\pi }{2}\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {3}{5} \int \frac {1}{\sqrt {\text {sech}(a+b x)}}dx+\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (i a+i b x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {3}{5} \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\cosh (a+b x)}dx+\frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}+\frac {3}{5} \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 \sinh (a+b x)}{5 b \text {sech}^{\frac {3}{2}}(a+b x)}-\frac {6 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}\) |
(((-6*I)/5)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b + (2*Sinh[a + b*x])/(5*b*Sech[a + b*x]^(3/2))
3.1.14.3.1 Defintions of rubi rules used
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[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(187\) vs. \(2(82)=164\).
Time = 1.92 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.85
method | result | size |
default | \(\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (8 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}-16 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}+10 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}-3 \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\) | \(188\) |
2/5*((-1+2*cosh(1/2*b*x+1/2*a)^2)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(8*cosh(1/2 *b*x+1/2*a)^7-16*cosh(1/2*b*x+1/2*a)^5+10*cosh(1/2*b*x+1/2*a)^3-3*(-sinh(1 /2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticE(cosh(1 /2*b*x+1/2*a),2^(1/2))-2*cosh(1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sin h(1/2*b*x+1/2*a)^2)^(1/2)/sinh(1/2*b*x+1/2*a)/(-1+2*cosh(1/2*b*x+1/2*a)^2) ^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 370, normalized size of antiderivative = 5.61 \[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} - 11\right )} \sinh \left (b x + a\right )^{4} - 11 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 11 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} - 66 \, \cosh \left (b x + a\right )^{2} - 13\right )} \sinh \left (b x + a\right )^{2} - 13 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} - 22 \, \cosh \left (b x + a\right )^{3} - 13 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} - 24 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{20 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \]
1/20*(sqrt(2)*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b* x + a)^6 + (15*cosh(b*x + a)^2 - 11)*sinh(b*x + a)^4 - 11*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - 11*cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 - 66*cosh(b*x + a)^2 - 13)*sinh(b*x + a)^2 - 13*cosh(b*x + a)^2 + 2 *(3*cosh(b*x + a)^5 - 22*cosh(b*x + a)^3 - 13*cosh(b*x + a))*sinh(b*x + a) - 1)*sqrt((cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)) - 24*(sqrt(2)*cosh(b*x + a)^3 + 3*sqrt(2)*cosh(b*x + a)^2*sinh(b*x + a) + 3*sqrt(2)*cosh(b*x + a)*sinh(b*x + a)^2 + sqrt(2)*sinh(b*x + a)^3)*weierstrassZeta(-4, 0, weierstrassPInve rse(-4, 0, cosh(b*x + a) + sinh(b*x + a))))/(b*cosh(b*x + a)^3 + 3*b*cosh( b*x + a)^2*sinh(b*x + a) + 3*b*cosh(b*x + a)*sinh(b*x + a)^2 + b*sinh(b*x + a)^3)
\[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{\operatorname {sech}^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\operatorname {sech}\left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\text {sech}^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{5/2}} \,d x \]