\(\int \frac {(e x)^{-1+n}}{(a+b \text {sech}(c+d x^n))^2} \, dx\) [89]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^n}{a^2 e n}-\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d e n}+\frac {b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text {sech}\left (c+d x^n\right )\right )} \] Output:

(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))
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.48 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {x^{-n} (e x)^n \left (a \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \arctan \left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \cosh \left (c+d x^n\right )+b \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \arctan \left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+a b \sqrt {a^2-b^2} \sinh \left (c+d x^n\right )\right )\right )}{a^2 (a-b) (a+b) \sqrt {a^2-b^2} d e n \left (b+a \cosh \left (c+d x^n\right )\right )} \] Input:

Integrate[(e*x)^(-1 + n)/(a + b*Sech[c + d*x^n])^2,x]
 

Output:

((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]))
 

Rubi [A] (verified)

Time = 1.41 (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

\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\frac {\left (a^2-b^2\right ) x^n}{a}-\frac {2 b \left (2 a^2-b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a \left (a^2-b^2\right )}+\frac {b^2 \tanh \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^n\right )\right )}\right )}{e n}\)

Input:

Int[(e*x)^(-1 + n)/(a + b*Sech[c + d*x^n])^2,x]
 

Output:

((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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

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]
 

rule 4272

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]
 

rule 4318

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]
 

rule 4407

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]
 

rule 5959

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]
 

rule 5963

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 8.80 (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 \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-i \pi \operatorname {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{a^{2} n}-\frac {2 b^{2} {\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-i \pi \operatorname {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \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 (a \,{\mathrm e}^{2 c +2 d \,x^{n}}+2 b \,{\mathrm e}^{c +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 \pi \,\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 \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i e x \right )^{3}}{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}}}\)	\(491\)

Input:

int((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x,method=_RETURNVERBOSE)
 

Output:

1/a^2/n*x*exp(1/2*(-1+n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi*csgn( 
I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln(x) 
+2*ln(e)))-2*b^2*exp(1/2*(-1+n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*P 
i*csgn(I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+ 
2*ln(x)+2*ln(e)))*x*(b*exp(c+d*x^n)+a)/a^2/(a^2-b^2)/d/n/(x^n)/(a*exp(2*c+ 
2*d*x^n)+2*b*exp(c+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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1758 vs. \(2 (148) = 296\).

Time = 0.16 (sec) , antiderivative size = 3547, normalized size of antiderivative = 22.59 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{n - 1}}{\left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \] Input:

integrate((e*x)**(-1+n)/(a+b*sech(c+d*x**n))**2,x)
 

Output:

Integral((e*x)**(n - 1)/(a + b*sech(c + d*x**n))**2, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x, algorithm="maxima")
 

Output:

-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))
 

Giac [F]

\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x, algorithm="giac")
 

Output:

integrate((e*x)^(n - 1)/(b*sech(d*x^n + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{n-1}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2} \,d x \] Input:

int((e*x)^(n - 1)/(a + b/cosh(c + d*x^n))^2,x)
 

Output:

int((e*x)^(n - 1)/(a + b/cosh(c + d*x^n))^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 638, normalized size of antiderivative = 4.06 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {e^{n} \left (-4 e^{2 x^{n} d +2 c} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3} b +2 e^{2 x^{n} d +2 c} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a +b}{\sqrt {a^{2}-b^{2}}}\right ) a \,b^{3}-8 e^{x^{n} d +c} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{2} b^{2}+4 e^{x^{n} d +c} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a +b}{\sqrt {a^{2}-b^{2}}}\right ) b^{4}-4 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3} b +2 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a +b}{\sqrt {a^{2}-b^{2}}}\right ) a \,b^{3}+x^{n} e^{2 x^{n} d +2 c} a^{5} d -2 x^{n} e^{2 x^{n} d +2 c} a^{3} b^{2} d +x^{n} e^{2 x^{n} d +2 c} a \,b^{4} d +e^{2 x^{n} d +2 c} a^{3} b^{2}-e^{2 x^{n} d +2 c} a \,b^{4}+2 x^{n} e^{x^{n} d +c} a^{4} b d -4 x^{n} e^{x^{n} d +c} a^{2} b^{3} d +2 x^{n} e^{x^{n} d +c} b^{5} d +x^{n} a^{5} d -2 x^{n} a^{3} b^{2} d +x^{n} a \,b^{4} d -a^{3} b^{2}+a \,b^{4}\right )}{a^{2} d e n \left (e^{2 x^{n} d +2 c} a^{5}-2 e^{2 x^{n} d +2 c} a^{3} b^{2}+e^{2 x^{n} d +2 c} a \,b^{4}+2 e^{x^{n} d +c} a^{4} b -4 e^{x^{n} d +c} a^{2} b^{3}+2 e^{x^{n} d +c} b^{5}+a^{5}-2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:

int((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x)
 

Output:

(e**n*( - 4*e**(2*x**n*d + 2*c)*sqrt(a**2 - b**2)*atan((e**(x**n*d + c)*a 
+ b)/sqrt(a**2 - b**2))*a**3*b + 2*e**(2*x**n*d + 2*c)*sqrt(a**2 - b**2)*a 
tan((e**(x**n*d + c)*a + b)/sqrt(a**2 - b**2))*a*b**3 - 8*e**(x**n*d + c)* 
sqrt(a**2 - b**2)*atan((e**(x**n*d + c)*a + b)/sqrt(a**2 - b**2))*a**2*b** 
2 + 4*e**(x**n*d + c)*sqrt(a**2 - b**2)*atan((e**(x**n*d + c)*a + b)/sqrt( 
a**2 - b**2))*b**4 - 4*sqrt(a**2 - b**2)*atan((e**(x**n*d + c)*a + b)/sqrt 
(a**2 - b**2))*a**3*b + 2*sqrt(a**2 - b**2)*atan((e**(x**n*d + c)*a + b)/s 
qrt(a**2 - b**2))*a*b**3 + x**n*e**(2*x**n*d + 2*c)*a**5*d - 2*x**n*e**(2* 
x**n*d + 2*c)*a**3*b**2*d + x**n*e**(2*x**n*d + 2*c)*a*b**4*d + e**(2*x**n 
*d + 2*c)*a**3*b**2 - e**(2*x**n*d + 2*c)*a*b**4 + 2*x**n*e**(x**n*d + c)* 
a**4*b*d - 4*x**n*e**(x**n*d + c)*a**2*b**3*d + 2*x**n*e**(x**n*d + c)*b** 
5*d + x**n*a**5*d - 2*x**n*a**3*b**2*d + x**n*a*b**4*d - a**3*b**2 + a*b** 
4))/(a**2*d*e*n*(e**(2*x**n*d + 2*c)*a**5 - 2*e**(2*x**n*d + 2*c)*a**3*b** 
2 + e**(2*x**n*d + 2*c)*a*b**4 + 2*e**(x**n*d + c)*a**4*b - 4*e**(x**n*d + 
 c)*a**2*b**3 + 2*e**(x**n*d + c)*b**5 + a**5 - 2*a**3*b**2 + a*b**4))