\(\int \frac {(e x)^{-1+2 n}}{(a+b \text {sech}(c+d x^n))^2} \, dx\) [83]

Optimal result

Integrand size = 24, antiderivative size = 717 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \cosh \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )} \]

[Out]

1/2*(e*x)^(2*n)/a^2/e/n-b^2*(e*x)^(2*n)*ln(b+a*cosh(c+d*x^n))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))+b^3*(e*x)^(2*n)*
ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)-b^3*(e*x)^(2*n)*ln(1+a*exp(c+d*x^n)
/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)+b^3*(e*x)^(2*n)*polylog(2,-a*exp(c+d*x^n)/(b-(-a^2+b^2
)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-b^3*(e*x)^(2*n)*polylog(2,-a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)
))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))+b^2*(e*x)^(2*n)*sinh(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(b+a*cosh(c+d*
x^n))-2*b*(e*x)^(2*n)*ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)+2*b*(e*x)^(2*
n)*ln(1+a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-2*b*(e*x)^(2*n)*polylog(2,-a*exp
(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)+2*b*(e*x)^(2*n)*polylog(2,-a*exp(c+d*x^
n)/(b+(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 717, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5548, 5544, 4276, 3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cosh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 d e n \sqrt {b^2-a^2}}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a^2 d e n \sqrt {b^2-a^2}}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^n\right )+b\right )}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {(e x)^{2 n}}{2 a^2 e n} \]

[In]

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

[Out]

(e*x)^(2*n)/(2*a^2*e*n) + (b^3*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2
)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]
*d*e*n*x^n) - (b^3*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*
n*x^n) + (2*b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n)
- (b^2*(e*x)^(2*n)*Log[b + a*Cosh[c + d*x^n]])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*n)*PolyLog[2,
-((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*Pol
yLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(2*n
)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (2*b*(e*
x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (b^
2*(e*x)^(2*n)*Sinh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cosh[c + d*x^n]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3401

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*(
(-I)*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 5544

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(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[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 5548

Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx}{e} \\ & = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(a+b \text {sech}(c+d x))^2} \, dx,x,x^n\right )}{e n} \\ & = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \cosh (c+d x))^2}-\frac {2 b x}{a^2 (b+a \cosh (c+d x))}\right ) \, dx,x,x^n\right )}{e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(b+a \cosh (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )}-\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )}-\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} e n}+\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} e n}-\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \cosh \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \cosh \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )}-\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}+\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}+\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \cosh \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )}+\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \cosh \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \cosh \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.46 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.65 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {x^{-2 n} (e x)^{2 n} \left (b+a \cosh \left (c+d x^n\right )\right ) \text {sech}^2\left (c+d x^n\right ) \left (\frac {4 b^2 d e^{2 c} x^n \left (b+a \cosh \left (c+d x^n\right )\right )}{\left (a^2-b^2\right ) \left (1+e^{2 c}\right )}+\frac {2 b \left (b+a \cosh \left (c+d x^n\right )\right ) \left (b \sqrt {-a^2+b^2} \log \left (a+2 b e^{c+d x^n}+a e^{2 \left (c+d x^n\right )}\right )+\left (2 a^2-b^2\right ) \left (d x^n \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{c+d x^n}}{-b+\sqrt {-a^2+b^2}}\right )\right )-\left (2 a^2-b^2\right ) \left (d x^n \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )+\operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )\right )\right )}{\left (-a^2+b^2\right )^{3/2}}+\frac {2 b^2 d x^n \text {sech}(c) \left (b \sinh (c)-a \sinh \left (d x^n\right )\right )}{(-a+b) (a+b)}+\frac {2 b^2 d x^n \left (b+a \cosh \left (c+d x^n\right )\right ) \tanh (c)}{-a^2+b^2}+\frac {d x^n \left (b+a \cosh \left (c+d x^n\right )\right ) \left (\left (a^2-b^2\right ) d x^n+2 b^2 \tanh (c)\right )}{(a-b) (a+b)}\right )}{2 a^2 d^2 e n \left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \]

[In]

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

[Out]

((e*x)^(2*n)*(b + a*Cosh[c + d*x^n])*Sech[c + d*x^n]^2*((4*b^2*d*E^(2*c)*x^n*(b + a*Cosh[c + d*x^n]))/((a^2 -
b^2)*(1 + E^(2*c))) + (2*b*(b + a*Cosh[c + d*x^n])*(b*Sqrt[-a^2 + b^2]*Log[a + 2*b*E^(c + d*x^n) + a*E^(2*(c +
d*x^n))] + (2*a^2 - b^2)*(d*x^n*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])] + PolyLog[2, (a*E^(c + d*x^
n))/(-b + Sqrt[-a^2 + b^2])]) - (2*a^2 - b^2)*(d*x^n*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])] + PolyL
og[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])))/(-a^2 + b^2)^(3/2) + (2*b^2*d*x^n*Sech[c]*(b*Sinh[c] - a
*Sinh[d*x^n]))/((-a + b)*(a + b)) + (2*b^2*d*x^n*(b + a*Cosh[c + d*x^n])*Tanh[c])/(-a^2 + b^2) + (d*x^n*(b + a
*Cosh[c + d*x^n])*((a^2 - b^2)*d*x^n + 2*b^2*Tanh[c]))/((a - b)*(a + b))))/(2*a^2*d^2*e*n*x^(2*n)*(a + b*Sech[
c + d*x^n])^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9020 vs. \(2 (681) = 1362\).

Time = 0.42 (sec) , antiderivative size = 9020, normalized size of antiderivative = 12.58 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/2*(4*a*b^2*e^(2*n)*x^n - (a^3*d*e^(2*n) - a*b^2*d*e^(2*n))*x^(2*n) - (a^3*d*e^(2*n)*e^(2*c) - a*b^2*d*e^(2*
n)*e^(2*c))*e^(2*d*x^n + 2*n*log(x)) + 2*(2*b^3*e^(2*n)*e^(n*log(x) + c) - (a^2*b*d*e^(2*n)*e^c - b^3*d*e^(2*n
)*e^c)*x^(2*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)) - integrate(-2*(a*b^2*e^(2*n)*x^n + (b^3*e^(2*n)*e^(n*l
og(x) + c) - (2*a^2*b*d*e^(2*n)*e^c - b^3*d*e^(2*n)*e^c)*x^(2*n))*e^(d*x^n))/((a^5*d*e*e^(2*c) - a^3*b^2*d*e*e
^(2*c))*x*e^(2*d*x^n) + 2*(a^4*b*d*e*e^c - a^2*b^3*d*e*e^c)*x*e^(d*x^n) + (a^5*d*e - a^3*b^2*d*e)*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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