Integrand size = 12, antiderivative size = 104 \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=-\frac {10 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{21 b^4 d}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {10 \sinh (c+d x)}{21 b^3 d \sqrt {b \text {sech}(c+d x)}} \]
2/7*sinh(d*x+c)/b/d/(b*sech(d*x+c))^(5/2)+10/21*sinh(d*x+c)/b^3/d/(b*sech( d*x+c))^(1/2)-10/21*I*(cosh(1/2*d*x+1/2*c)^2)^(1/2)/cosh(1/2*d*x+1/2*c)*El lipticF(I*sinh(1/2*d*x+1/2*c),2^(1/2))*cosh(d*x+c)^(1/2)*(b*sech(d*x+c))^( 1/2)/b^4/d
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\frac {\sqrt {b \text {sech}(c+d x)} \left (-40 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right )+26 \sinh (2 (c+d x))+3 \sinh (4 (c+d x))\right )}{84 b^4 d} \]
(Sqrt[b*Sech[c + d*x]]*((-40*I)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d *x), 2] + 26*Sinh[2*(c + d*x)] + 3*Sinh[4*(c + d*x)]))/(84*b^4*d)
Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4256, 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.
| \(\displaystyle \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {5 \int \frac {1}{(b \text {sech}(c+d x))^{3/2}}dx}{7 b^2}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {5 \int \frac {1}{\left (b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{7 b^2}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {5 \left (\frac {\int \sqrt {b \text {sech}(c+d x)}dx}{3 b^2}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}\right )}{7 b^2}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {5 \left (\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\int \sqrt {b \csc \left (i c+i d x+\frac {\pi }{2}\right )}dx}{3 b^2}\right )}{7 b^2}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {5 \left (\frac {\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)} \int \frac {1}{\sqrt {\cosh (c+d x)}}dx}{3 b^2}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}\right )}{7 b^2}+\frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {5 \left (\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)} \int \frac {1}{\sqrt {\sin \left (i c+i d x+\frac {\pi }{2}\right )}}dx}{3 b^2}\right )}{7 b^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 \sinh (c+d x)}{7 b d (b \text {sech}(c+d x))^{5/2}}+\frac {5 \left (\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}-\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d}\right )}{7 b^2}\) |
(2*Sinh[c + d*x])/(7*b*d*(b*Sech[c + d*x])^(5/2)) + (5*((((-2*I)/3)*Sqrt[C osh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/(b^2*d) + (2*Sinh[c + d*x])/(3*b*d*Sqrt[b*Sech[c + d*x]])))/(7*b^2)
3.1.22.3.1 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]
\[\int \frac {1}{\left (b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 483, normalized size of antiderivative = 4.64 \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\frac {80 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (3 \, \cosh \left (d x + c\right )^{8} + 24 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, \sinh \left (d x + c\right )^{8} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{2} + 13\right )} \sinh \left (d x + c\right )^{6} + 26 \, \cosh \left (d x + c\right )^{6} + 12 \, {\left (14 \, \cosh \left (d x + c\right )^{3} + 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 30 \, {\left (7 \, \cosh \left (d x + c\right )^{4} + 13 \, \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (21 \, \cosh \left (d x + c\right )^{5} + 65 \, \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{6} + 195 \, \cosh \left (d x + c\right )^{4} - 13\right )} \sinh \left (d x + c\right )^{2} - 26 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (6 \, \cosh \left (d x + c\right )^{7} + 39 \, \cosh \left (d x + c\right )^{5} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 3\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{168 \, {\left (b^{4} d \cosh \left (d x + c\right )^{4} + 4 \, b^{4} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{4} d \sinh \left (d x + c\right )^{4}\right )}} \]
1/168*(80*sqrt(2)*(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*c osh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d* x + c)^4)*sqrt(b)*weierstrassPInverse(-4, 0, cosh(d*x + c) + sinh(d*x + c) ) + sqrt(2)*(3*cosh(d*x + c)^8 + 24*cosh(d*x + c)*sinh(d*x + c)^7 + 3*sinh (d*x + c)^8 + 2*(42*cosh(d*x + c)^2 + 13)*sinh(d*x + c)^6 + 26*cosh(d*x + c)^6 + 12*(14*cosh(d*x + c)^3 + 13*cosh(d*x + c))*sinh(d*x + c)^5 + 30*(7* cosh(d*x + c)^4 + 13*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(21*cosh(d*x + c )^5 + 65*cosh(d*x + c)^3)*sinh(d*x + c)^3 + 2*(42*cosh(d*x + c)^6 + 195*co sh(d*x + c)^4 - 13)*sinh(d*x + c)^2 - 26*cosh(d*x + c)^2 + 4*(6*cosh(d*x + c)^7 + 39*cosh(d*x + c)^5 - 13*cosh(d*x + c))*sinh(d*x + c) - 3)*sqrt((b* cosh(d*x + c) + b*sinh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d *x + c) + sinh(d*x + c)^2 + 1)))/(b^4*d*cosh(d*x + c)^4 + 4*b^4*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^4*d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^4*d* cosh(d*x + c)*sinh(d*x + c)^3 + b^4*d*sinh(d*x + c)^4)
\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int \frac {1}{\left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int { \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{7/2}} \,d x \]