Integrand size = 22, antiderivative size = 157 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^n}{a^2 e n}-\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d e n}+\frac {b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text {sech}\left (c+d x^n\right )\right )} \]
(e*x)^n/a^2/e/n-2*b*(2*a^2-b^2)*(e*x)^n*arctan((a-b)^(1/2)*tanh(1/2*c+1/2* d*x^n)/(a+b)^(1/2))/a^2/(a-b)^(3/2)/(a+b)^(3/2)/d/e/n/(x^n)+b^2*(e*x)^n*ta nh(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(a+b*sech(c+d*x^n))
Time = 1.27 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.48 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {x^{-n} (e x)^n \left (a \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \arctan \left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \cosh \left (c+d x^n\right )+b \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \arctan \left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+a b \sqrt {a^2-b^2} \sinh \left (c+d x^n\right )\right )\right )}{a^2 (a-b) (a+b) \sqrt {a^2-b^2} d e n \left (b+a \cosh \left (c+d x^n\right )\right )} \]
((e*x)^n*(a*((a^2 - b^2)^(3/2)*(c + d*x^n) + (4*a^2*b - 2*b^3)*ArcTan[((-a + b)*Tanh[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Cosh[c + d*x^n] + b*((a^2 - b ^2)^(3/2)*(c + d*x^n) + (4*a^2*b - 2*b^3)*ArcTan[((-a + b)*Tanh[(c + d*x^n )/2])/Sqrt[a^2 - b^2]] + a*b*Sqrt[a^2 - b^2]*Sinh[c + d*x^n])))/(a^2*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d*e*n*x^n*(b + a*Cosh[c + d*x^n]))
Time = 0.83 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5963, 5959, 3042, 4272, 25, 3042, 4407, 3042, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {(e x)^{n-1}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx\) |
| \(\Big \downarrow \) 5963 |
| \(\displaystyle \frac {x^{-n} (e x)^n \int \frac {x^{n-1}}{\left (a+b \text {sech}\left (d x^n+c\right )\right )^2}dx}{e}\) |
| \(\Big \downarrow \) 5959 |
| \(\displaystyle \frac {x^{-n} (e x)^n \int \frac {1}{\left (a+b \text {sech}\left (d x^n+c\right )\right )^2}dx^n}{e n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{-n} (e x)^n \int \frac {1}{\left (a+b \csc \left (i d x^n+i c+\frac {\pi }{2}\right )\right )^2}dx^n}{e n}\) |
| \(\Big \downarrow \) 4272 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}-\frac {\int -\frac {a^2-b \text {sech}\left (d x^n+c\right ) a-b^2}{a+b \text {sech}\left (d x^n+c\right )}dx^n}{a \left (a^2-b^2\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\int \frac {a^2-b \text {sech}\left (d x^n+c\right ) a-b^2}{a+b \text {sech}\left (d x^n+c\right )}dx^n}{a \left (a^2-b^2\right )}+\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}+\frac {\int \frac {a^2-b \csc \left (i d x^n+i c+\frac {\pi }{2}\right ) a-b^2}{a+b \csc \left (i d x^n+i c+\frac {\pi }{2}\right )}dx^n}{a \left (a^2-b^2\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\frac {\left (a^2-b^2\right ) x^n}{a}-\frac {b \left (2 a^2-b^2\right ) \int \frac {\text {sech}\left (d x^n+c\right )}{a+b \text {sech}\left (d x^n+c\right )}dx^n}{a}}{a \left (a^2-b^2\right )}+\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}+\frac {\frac {\left (a^2-b^2\right ) x^n}{a}-\frac {b \left (2 a^2-b^2\right ) \int \frac {\csc \left (i d x^n+i c+\frac {\pi }{2}\right )}{a+b \csc \left (i d x^n+i c+\frac {\pi }{2}\right )}dx^n}{a}}{a \left (a^2-b^2\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\frac {\left (a^2-b^2\right ) x^n}{a}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{\frac {a \cosh \left (d x^n+c\right )}{b}+1}dx^n}{a}}{a \left (a^2-b^2\right )}+\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}+\frac {\frac {\left (a^2-b^2\right ) x^n}{a}-\frac {\left (2 a^2-b^2\right ) \int \frac {1}{\frac {a \sin \left (i d x^n+i c+\frac {\pi }{2}\right )}{b}+1}dx^n}{a}}{a \left (a^2-b^2\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}+\frac {\frac {\left (a^2-b^2\right ) x^n}{a}+\frac {2 i \left (2 a^2-b^2\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) x^{2 n}+\frac {a+b}{b}}d\left (i \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )\right )}{a d}}{a \left (a^2-b^2\right )}\right )}{e n}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\frac {\left (a^2-b^2\right ) x^n}{a}-\frac {2 b \left (2 a^2-b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a \left (a^2-b^2\right )}+\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}\right )}{e n}\) |
((e*x)^n*((((a^2 - b^2)*x^n)/a - (2*b*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Ta nh[(c + d*x^n)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d))/(a*(a^2 - b^2)) + (b^2*Tanh[c + d*x^n])/(a*(a^2 - b^2)*d*(a + b*Sech[c + d*x^n]))))/ (e*n*x^n)
3.1.82.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(n + 1)*(a^2 - b^2)) Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x ], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ erQ[2*n]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m* (a + b*Sech[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.45 (sec) , antiderivative size = 491, normalized size of antiderivative = 3.13
| method | result | size |
| risch | \(\frac {x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{a^{2} n}-\frac {2 b^{2} {\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}} x \left (b \,{\mathrm e}^{c +d \,x^{n}}+a \right ) x^{-n}}{a^{2} \left (a^{2}-b^{2}\right ) d n \left (2 b \,{\mathrm e}^{c +d \,x^{n}}+a \,{\mathrm e}^{2 c +2 d \,x^{n}}+a \right )}-\frac {2 b \left (2 a^{2}-b^{2}\right ) {\mathrm e}^{-\frac {i \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi n \operatorname {csgn}\left (i e x \right )^{3}}{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi }{2}} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi }{2}} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi }{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e x \right )^{3} \pi }{2}} e^{n} {\mathrm e}^{c} \arctan \left (\frac {2 a \,{\mathrm e}^{2 c +d \,x^{n}}+2 \,{\mathrm e}^{c} b}{2 \sqrt {a^{2} {\mathrm e}^{2 c}-{\mathrm e}^{2 c} b^{2}}}\right )}{a^{2} \left (a^{2}-b^{2}\right ) n e d \sqrt {a^{2} {\mathrm e}^{2 c}-{\mathrm e}^{2 c} b^{2}}}\) | \(491\) |
1/a^2/n*x*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn(I*e )*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e) +2*ln(x)))-2*b^2*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*c sgn(I*e)*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+ 2*ln(e)+2*ln(x)))*x*(b*exp(c+d*x^n)+a)/a^2/(a^2-b^2)/d/n/(x^n)/(2*b*exp(c+ d*x^n)+a*exp(2*c+2*d*x^n)+a)-2*b/a^2*(2*a^2-b^2)/(a^2-b^2)/n*exp(-1/2*I*Pi *n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*n*csgn(I*e)*csgn(I*e*x)^2 )*exp(1/2*I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-1/2*I*Pi*n*csgn(I*e*x)^3)*e xp(1/2*I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*csgn(I*e)*csgn( I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1/2*I*Pi*csgn(I*e*x)^ 3)*e^n/e*exp(c)/d/(a^2*exp(2*c)-exp(2*c)*b^2)^(1/2)*arctan(1/2*(2*a*exp(2* c+d*x^n)+2*exp(c)*b)/(a^2*exp(2*c)-exp(2*c)*b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1758 vs. \(2 (148) = 296\).
Time = 0.32 (sec) , antiderivative size = 3547, normalized size of antiderivative = 22.59 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]
[((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 - 2*a^3* b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a *b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*lo g(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ( (a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 - 2 *a^3*b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^ 2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^ 2 + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) - ( a^2*b^3 - b^5)*cosh((n - 1)*log(e)) - (a^2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5)*d*cosh(n*log(x)))*sinh((n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5) *d*cosh((n - 1)*log(e)) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh((n - 1)*log(e)) )*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 2*(a^3*b ^2 - a*b^4)*cosh((n - 1)*log(e)) - (((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*co sh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*sinh((n - 1)*log(e )))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^3*b - a*b^3)*s qrt(-a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)* sinh((n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2 *a^3*b - a*b^3)*sqrt(-a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^...
\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{n - 1}}{\left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
-2*(2*a^2*b*e^n*e^c - b^3*e^n*e^c)*integrate(e^(d*x^n + n*log(x))/((a^5*e* e^(2*c) - a^3*b^2*e*e^(2*c))*x*e^(2*d*x^n) + 2*(a^4*b*e*e^c - a^2*b^3*e*e^ c)*x*e^(d*x^n) + (a^5*e - a^3*b^2*e)*x), x) - (2*a*b^2*e^n - (a^3*d*e^n - a*b^2*d*e^n)*x^n - (a^3*d*e^n*e^(2*c) - a*b^2*d*e^n*e^(2*c))*e^(2*d*x^n + n*log(x)) + 2*(b^3*e^n*e^c - (a^2*b*d*e^n*e^c - b^3*d*e^n*e^c)*x^n)*e^(d*x ^n))/(a^5*d*e*n - a^3*b^2*d*e*n + (a^5*d*e*n*e^(2*c) - a^3*b^2*d*e*n*e^(2* c))*e^(2*d*x^n) + 2*(a^4*b*d*e*n*e^c - a^2*b^3*d*e*n*e^c)*e^(d*x^n))
\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{n-1}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]