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

Optimal. Leaf size=717 \[ -\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cosh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 d e n \sqrt {b^2-a^2}}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a^2 d e n \sqrt {b^2-a^2}}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^n\right )+b\right )}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {(e x)^{2 n}}{2 a^2 e n} \]

[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]  time = 1.22, antiderivative size = 717, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5440, 5436, 4191, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac {2 b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cosh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 d e n \sqrt {b^2-a^2}}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a^2 d e n \sqrt {b^2-a^2}}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {b^2 x^{-n} (e x)^{2 n} \sinh \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cosh \left (c+d x^n\right )+b\right )}+\frac {(e x)^{2 n}}{2 a^2 e n} \]

Antiderivative was successfully verified.

[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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2264

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 2279

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 2391

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 2668

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 3320

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))/(E^(I*Pi*(k - 1/2))*(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))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3324

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 4191

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 5436

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 5440

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*} \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 ) \operatorname {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 ) \operatorname {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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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*}

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Mathematica [A]  time = 30.82, size = 542, normalized size = 0.76 \[ \frac {x^{-2 n} (e x)^{2 n} \text {sech}^2\left (c+d x^n\right ) \left (a \cosh \left (c+d x^n\right )+b\right ) \left (\frac {2 b \left (a \cosh \left (c+d x^n\right )+b\right ) \left (\frac {2 b e^{2 c} d x^n}{e^{2 c}+1}-\frac {\left (2 a^2-b^2\right ) \text {Li}_2\left (\frac {a e^{d x^n+c}}{\sqrt {b^2-a^2}-b}\right )+\left (b^2-2 a^2\right ) \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )+2 a^2 d x^n \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )-b^2 d x^n \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {b^2-a^2}}+1\right )-2 a^2 d x^n \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )+b^2 d x^n \log \left (\frac {a e^{c+d x^n}}{\sqrt {b^2-a^2}+b}+1\right )+b \sqrt {b^2-a^2} \log \left (a e^{2 \left (c+d x^n\right )}+a+2 b e^{c+d x^n}\right )}{\sqrt {b^2-a^2}}\right )}{a^2-b^2}+\frac {2 b^2 d \tanh (c) x^n \left (a \cosh \left (c+d x^n\right )+b\right )}{b^2-a^2}+\frac {d x^n \left (a \cosh \left (c+d x^n\right )+b\right ) \left (d \left (a^2-b^2\right ) x^n+2 b^2 \tanh (c)\right )}{(a-b) (a+b)}+\frac {2 b^2 d \text {sech}(c) x^n \left (b \sinh (c)-a \sinh \left (d x^n\right )\right )}{(b-a) (a+b)}\right )}{2 a^2 d^2 e n \left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \]

Antiderivative was successfully verified.

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

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fricas [B]  time = 0.56, size = 9020, normalized size = 12.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*co
sh((2*n - 1)*log(e))*cosh(n*log(x))^2 + 4*(a^3*b^2 - a*b^4)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*(a^3*b
^2 - a*b^4)*c*cosh((2*n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 - 2*a^3*b^
2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))^2 + 4*
(a^3*b^2 - a*b^4)*d*cosh(n*log(x)) + 4*(a^3*b^2 - a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 - 2*a^3*b^2 + a*b
^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 2*(a^3*b^2 - a*b^4)*d*cosh((2*n - 1)*log(e)) + ((a^5 - 2*a^3*b
^2 + a*b^4)*d^2*cosh(n*log(x)) + 2*(a^3*b^2 - a*b^4)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*
log(x)) + d*sinh(n*log(x)) + c)^2 + 4*(a^3*b^2 - a*b^4)*c*cosh((2*n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*
d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 + 4*(a^3*b^2 - a*b^4)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*
(a^3*b^2 - a*b^4)*c*cosh((2*n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 - 2*
a^3*b^2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))^
2 + 4*(a^3*b^2 - a*b^4)*d*cosh(n*log(x)) + 4*(a^3*b^2 - a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 - 2*a^3*b^2
 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 2*(a^3*b^2 - a*b^4)*d*cosh((2*n - 1)*log(e)) + ((a^5 - 2
*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x)) + 2*(a^3*b^2 - a*b^4)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*c
osh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 - 2*a^3
*b^2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*l
og(e))*cosh(n*log(x))^2 + 2*(a^2*b^3 - b^5)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*(a^2*b^3 - b^5)*c*cosh
((2*n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e)) + (a^4*b - 2*a^2*b^3 + b^5)*d^2*sin
h((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh(n*log(x))^2 + 2*(a^2*b^3 - b^5)*d*
cosh(n*log(x)) + 4*(a^2*b^3 - b^5)*c)*sinh((2*n - 1)*log(e)) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)
*log(e))*cosh(n*log(x)) + (a^2*b^3 - b^5)*d*cosh((2*n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh(n*log
(x)) + (a^2*b^3 - b^5)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c
) + 2*(((2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b
^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^4*b - a^2*b^3)*sqrt(-
(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*s
inh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log
(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)) + 2*((2*a^3*b^2 - a*b^4)*sqrt(-(a^2 -
 b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 - a*b^4)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*
cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*((2*a^3*b^2 - a*b^4)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))
+ (2*a^3*b^2 - a*b^4)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a
^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*
log(x)) + d*sinh(n*log(x)) + c))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*dilog(-((a*sqrt(-(a^2 - b^2)/a
^2) + b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt(-(a^2 - b^2)/a^2) + b)*sinh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c) + a)/a + 1) - 2*(((2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) +
(2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c)^2 + ((2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b
^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2*a^4*b - a^2*b^3)*sqrt(-(
a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)) + 2
*((2*a^3*b^2 - a*b^4)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 - a*b^4)*sqrt(-(a^2 - b^2)/a^
2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*((2*a^3*b^2 - a*b^4)*sqrt(-(a^2 -
 b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 - a*b^4)*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)) + ((2*a^
4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*sqrt(-(a^2 - b^2)/a^2)*sinh
((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c))*dilog(((a*sqrt(-(a^2 - b^2)/a^2) - b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt(-(a^2 - b^2)
/a^2) - b)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - a)/a + 1) - 2*(((a^3*b^2 - a*b^4 - (2*a^4*b - a^2*b
^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 - a*b^4 - (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b
^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^3*b^2 - a*b^4 - (2*a^4
*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 - a*b^4 - (2*a^4*b - a^2*b^3)*c*sqrt
(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((a^2*b^3 - b^
5 - (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^2*b^3 - b^5 - (2*a^3*b^2 - a*b^4
)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^3*b^2 -
 a*b^4 - (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + 2*(((a^3*b^2 - a*b^4 - (2*a^4*
b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 - a*b^4 - (2*a^4*b - a^2*b^3)*c*sqrt(
-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^2*b^3 - b^5 - (2
*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^2*b^3 - b^5 - (2*a^3*b^2 - a*b^4)*c*sq
rt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^3*b^2 - a*b^4
 - (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*log(2*a*cosh(d*cosh(n*log(x)) + d*sin
h(n*log(x)) + c) + 2*a*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 2*(
((a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 - a*b^4 +
(2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x))
 + c)^2 + ((a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2
- a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh
(n*log(x)) + c)^2 + 2*((a^2*b^3 - b^5 + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) +
 (a^2*b^3 - b^5 + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c) + (a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(
e)) + 2*(((a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 -
 a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(
n*log(x)) + c) + (a^2*b^3 - b^5 + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^2*
b^3 - b^5 + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c) + (a^3*b^2 - a*b^4 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)))*l
og(2*a*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*a*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 2*a
*sqrt(-(a^2 - b^2)/a^2) + 2*b) + 2*((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n
*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-
(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*
n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a
^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))
+ (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) +
 d*sinh(n*log(x)) + c)^2 + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x))
 + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b
^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a
^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2
*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((2*a^3*b^2 - a*b^4)*d*sq
rt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2)*cosh
((2*n - 1)*log(e)) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^3*b^2 - a*b^4)*c*sqrt
(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(
e)) + (2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x
)) + d*sinh(n*log(x)) + c) + 2*((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log
(x)) + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2
 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n -
1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 -
 b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2
*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*s
inh(n*log(x)) + c) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^3*b^2 - a*b^4)*c*sqrt
(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(
e)) + (2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x
)) + d*sinh(n*log(x)) + c) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^2*b^3
)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n -
 1)*log(e)) + (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*log(((a*sqr
t(-(a^2 - b^2)/a^2) + b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt(-(a^2 - b^2)/a^2) + b)*sinh(d
*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + a)/a) - 2*((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n -
1)*log(e))*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b -
a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b
^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^
2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((
2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d
*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e
))*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)
*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)
) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 -
b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((2*a^3*
b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2
 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^3*b^
2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*co
sh((2*n - 1)*log(e)) + (2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*co
sh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*l
og(e))*cosh(n*log(x)) + (2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*
b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/
a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b - a^2*b^
3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh((2*n
- 1)*log(e)) + (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (2*a^3*b^
2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*co
sh((2*n - 1)*log(e)) + (2*a^3*b^2 - a*b^4)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*si
nh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*cosh(n*log(x)) + (
2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)
/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b - a^2*b^3)*d*sqrt(-(a^2 - b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log
(x)))*log(-((a*sqrt(-(a^2 - b^2)/a^2) - b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt(-(a^2 - b^2
)/a^2) - b)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - a)/a) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh((2*n
 - 1)*log(e))*cosh(n*log(x))^2 + 2*(a^2*b^3 - b^5)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*(a^2*b^3 - b^5)
*c*cosh((2*n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e)) + (a^4*b - 2*a^2*b^3 + b^5)*
d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*lo
g(x))^2 + 4*(a^3*b^2 - a*b^4)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*(a^3*b^2 - a*b^4)*c*cosh((2*n - 1)*l
og(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d^2*sinh((2*n - 1)*
log(e)))*sinh(n*log(x))^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))^2 + 4*(a^3*b^2 - a*b^4)*d*cosh(n*log
(x)) + 4*(a^3*b^2 - a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))
*cosh(n*log(x)) + 2*(a^3*b^2 - a*b^4)*d*cosh((2*n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))
 + 2*(a^3*b^2 - a*b^4)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c
) + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh(n*log(x))^2 + 2*(a^2*b^3 - b^5)*d*cosh(n*log(x)) + 4*(a^2*b^3 - b^5)*c
)*sinh((2*n - 1)*log(e)) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (a^2*b^3 -
 b^5)*d*cosh((2*n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5)*d^2*cosh(n*log(x)) + (a^2*b^3 - b^5)*d)*sinh((2*n
- 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*c
osh(n*log(x))^2 + 4*(a^3*b^2 - a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n -
 1)*log(e))*cosh(n*log(x)) + (a^5 - 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))*sinh((2*n - 1)*log(e)))*sinh(n*log(x
)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (a^7 - 2*a^5*b^2 + a^
3*b^4)*d^2*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2*n*cosh(d*co
sh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*n + 2*((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*
n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2*n)*sinh(d*cosh(n*log(x)) +
 d*sinh(n*log(x)) + c))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{2 \, n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 1.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{-1+2 n}}{\left (a +b \,\mathrm {sech}\left (c +d \,x^{n}\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {4 \, a b^{2} e^{2 \, n} x^{n} - {\left (a^{3} d e^{2 \, n} - a b^{2} d e^{2 \, n}\right )} x^{2 \, n} - {\left (a^{3} d e^{2 \, n} e^{\left (2 \, c\right )} - a b^{2} d e^{2 \, n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + 2 \, n \log \relax (x)\right )} + 2 \, {\left (2 \, b^{3} e^{2 \, n} e^{\left (n \log \relax (x) + c\right )} - {\left (a^{2} b d e^{2 \, n} e^{c} - b^{3} d e^{2 \, n} e^{c}\right )} x^{2 \, n}\right )} e^{\left (d x^{n}\right )}}{2 \, {\left (a^{5} d e n - a^{3} b^{2} d e n + {\left (a^{5} d e n e^{\left (2 \, c\right )} - a^{3} b^{2} d e n e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n}\right )} + 2 \, {\left (a^{4} b d e n e^{c} - a^{2} b^{3} d e n e^{c}\right )} e^{\left (d x^{n}\right )}\right )}} - \int -\frac {2 \, {\left (a b^{2} e^{2 \, n} x^{n} + {\left (b^{3} e^{2 \, n} e^{\left (n \log \relax (x) + c\right )} - {\left (2 \, a^{2} b d e^{2 \, n} e^{c} - b^{3} d e^{2 \, n} e^{c}\right )} x^{2 \, n}\right )} e^{\left (d x^{n}\right )}\right )}}{{\left (a^{5} d e e^{\left (2 \, c\right )} - a^{3} b^{2} d e e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x^{n}\right )} + 2 \, {\left (a^{4} b d e e^{c} - a^{2} b^{3} d e e^{c}\right )} x e^{\left (d x^{n}\right )} + {\left (a^{5} d e - a^{3} b^{2} d e\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{2\,n-1}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{2 n - 1}}{\left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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