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

Optimal. Leaf size=757 \[ \frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cos \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \sqrt{b^2-a^2}}-\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \sqrt{b^2-a^2}}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-n} (e x)^{2 n} \sin \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cos \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \]

[Out]

(e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2
+ b^2)^(3/2)*d*e*n*x^n) + ((2*I)*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sq
rt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a
^2 + b^2)^(3/2)*d*e*n*x^n) - ((2*I)*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(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*Cos[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^(I*(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^(I*(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^(I*(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^(I*(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)*Sin[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cos[c
+ d*x^n]))

________________________________________________________________________________________

Rubi [A]  time = 1.27256, antiderivative size = 757, normalized size of antiderivative = 1., 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 = {4208, 4204, 4191, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cos \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \sqrt{b^2-a^2}}-\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \sqrt{b^2-a^2}}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-n} (e x)^{2 n} \sin \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cos \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2
+ b^2)^(3/2)*d*e*n*x^n) + ((2*I)*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sq
rt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a
^2 + b^2)^(3/2)*d*e*n*x^n) - ((2*I)*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(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*Cos[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^(I*(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^(I*(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^(I*(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^(I*(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)*Sin[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cos[c
+ d*x^n]))

Rule 4208

Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(e^IntPart[m]*(e*x
)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sec[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]

Rule 4204

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[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 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 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 3321

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c
+ d*x)^m*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(
2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && 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 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 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 31

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

Rubi steps

\begin{align*} \int \frac{(e x)^{-1+2 n}}{\left (a+b \sec \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 \sec \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 \sec (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 \cos (c+d x))^2}-\frac{2 b x}{a^2 (b+a \cos (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 \cos (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 \cos (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} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \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^{i (c+d x)} x}{a+2 b e^{i (c+d x)}+a e^{2 i (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 \cos (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{\sin (c+d x)}{b+a \cos (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} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \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^{i (c+d x)} x}{a+2 b e^{i (c+d x)}+a e^{2 i (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^{i (c+d x)} x}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (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^{i (c+d x)} x}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (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 \cos \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 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{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 \cos \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} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \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^{i (c+d x)} x}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (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^{i (c+d x)} x}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (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 i b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{i (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 i b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{i (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{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{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 \cos \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} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \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^{i \left (c+d x^n\right )}\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^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^2 e n}-\frac{\left (i b^3 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{i (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 (i b^3 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 a e^{i (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{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{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 \cos \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^{i \left (c+d x^n\right )}}{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^{i \left (c+d x^n\right )}}{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} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \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^{i \left (c+d x^n\right )}\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^{i \left (c+d x^n\right )}\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{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d e n}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac{a e^{i \left (c+d x^n\right )}}{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 \cos \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^{i \left (c+d x^n\right )}}{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^{i \left (c+d x^n\right )}}{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^{i \left (c+d x^n\right )}}{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^{i \left (c+d x^n\right )}}{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} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}\\ \end{align*}

Mathematica [B]  time = 10.0344, size = 2450, normalized size = 3.24 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*b*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2]
)/Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(
b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sq
rt[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (Ar
cCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)
/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]
])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2
- b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + (-Ar
cCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(
a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2,
((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d
*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sq
rt[a^2 - b^2]*Tan[(c + d*x^n)/2]))]))*Sec[c + d*x^n]^2)/((a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2) + (
b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/
Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/
a)] - (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt
[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (ArcC
os[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2
])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]])
] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 -
b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + (-ArcC
os[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a
+ b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2, (
(b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x
^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt
[a^2 - b^2]*Tan[(c + d*x^n)/2]))]))*Sec[c + d*x^n]^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2) +
(x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c + d*x^n]^2*(a^2*d*x^n*Cos[c] - b^2*d*x^n*Cos[c] +
2*b^2*Sin[c]))/(2*a^2*(a - b)*(a + b)*d*n*(a + b*Sec[c + d*x^n])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])
) + (b^2*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c]*Sec[c + d*x^n]^2*(a*Cos[c]*Log[b + a*Cos
[c]*Cos[d*x^n] - a*Sin[c]*Sin[d*x^n]] + a*d*x^n*Sin[c] - ((2*I)*a*b*ArcTan[((-I)*a*Sin[c] - I*(-b + a*Cos[c])*
Tan[(d*x^n)/2])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]]*Sin[c])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]))/(
a*(a^2 - b^2)*d^2*n*(a + b*Sec[c + d*x^n])^2*(a^2*Cos[c]^2 + a^2*Sin[c]^2)) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*
(b + a*Cos[c + d*x^n])*Sec[c + d*x^n]^2*(b*Sin[c] - a*Sin[d*x^n]))/(a^2*(-a + b)*(a + b)*d*n*(a + b*Sec[c + d*
x^n])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^
2*Sec[c + d*x^n]^2*Tan[c])/(a^2*(-a^2 + b^2)*d*n*(a + b*Sec[c + d*x^n])^2) - ((2*I)*b^3*x^(1 - 2*n)*(e*x)^(-1
+ 2*n)*ArcTan[(b + a*Cos[c + d*x^n] + I*a*Sin[c + d*x^n])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x^n])^2*Sec[c + d*
x^n]^2*Tan[c])/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2)

________________________________________________________________________________________

Maple [C]  time = 0.332, size = 2819, normalized size = 3.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^2/n*x*exp(-1/2*(-1+2*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)*c
sgn(I*e*x)^2*Pi+I*csgn(I*e*x)^3*Pi-2*ln(x)-2*ln(e)))+2*I*b^2*x/a^2/(a^2-b^2)/d/n/(x^n)/(exp(2*I*(c+d*x^n))*a+2
*b*exp(I*(c+d*x^n))+a)*((-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(-1)^(-1/2*csgn(I*e)*csgn(I*e*x)^2)*(-1)^(-
1/2*csgn(I*x)*csgn(I*e*x)^2)/x/e*(x^n)^2*(e^n)^2*b*exp(1/2*I*Pi*csgn(I*e*x)^3)*exp(-I*Pi*n*csgn(I*e)*csgn(I*x)
*csgn(I*e*x))*exp(I*Pi*n*csgn(I*e)*csgn(I*e*x)^2)*exp(I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-I*Pi*n*csgn(I*e*x)^
3)*exp(I*x^n*d)*exp(I*c)+(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(-1)^(-1/2*csgn(I*e)*csgn(I*e*x)^2)*(-1)^(
-1/2*csgn(I*x)*csgn(I*e*x)^2)/x/e*(x^n)^2*(e^n)^2*a*exp(-1/2*I*Pi*csgn(I*e*x)*(2*n*csgn(I*e)*csgn(I*x)-2*n*csg
n(I*x)*csgn(I*e*x)-2*n*csgn(I*e)*csgn(I*e*x)+2*n*csgn(I*e*x)^2-csgn(I*e*x)^2)))-2*b^2/a^2/(a^2-b^2)/d^2*(e^n)^
2/e/n*ln(exp(I*x^n*d))*exp(-1/2*I*csgn(I*e*x)*Pi*(-1+2*n)*(-csgn(I*e*x)+csgn(I*x))*(-csgn(I*e*x)+csgn(I*e)))+b
^2/a^2/(a^2-b^2)/d^2*(e^n)^2/e/n*ln(exp(2*I*(c+d*x^n))*a+2*b*exp(I*(c+d*x^n))+a)*exp(-1/2*I*csgn(I*e*x)*Pi*(-1
+2*n)*(-csgn(I*e*x)+csgn(I*x))*(-csgn(I*e*x)+csgn(I*e)))-I*b^3/a^2/(a^2-b^2)/d*(e^n)^2/e/n*x^n/(exp(2*I*c)*b^2
-a^2*exp(2*I*c))^(1/2)*ln((-a*exp(I*(d*x^n+2*c))-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))/(-exp(I*c)*
b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*x)*
csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I
*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c))-2*b/(a^2-b^2)/d^2*(e^n)^2/e/n/(exp(2*I*c)*
b^2-a^2*exp(2*I*c))^(1/2)*dilog((a*exp(I*(d*x^n+2*c))+exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))/(exp(I
*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I
*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*cs
gn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c))+2*I*b/(a^2-b^2)/d*(e^n)^2/e/n*x^n/(exp
(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)*ln((-a*exp(I*(d*x^n+2*c))-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))/
(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n
*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x
)+Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c))+b^3/a^2/(a^2-b^2)/d^2*(e^n)^2/e
/n/(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)*dilog((a*exp(I*(d*x^n+2*c))+exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c)
)^(1/2))/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x
)-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*cs
gn(I*e*x)+Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c))+2*b/(a^2-b^2)/d^2*(e^n)
^2/e/n/(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)*dilog((-a*exp(I*(d*x^n+2*c))-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2
*I*c))^(1/2))/(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn
(I*e*x)-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I
*x)*csgn(I*e*x)+Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c))-b^3/a^2/(a^2-b^2)
/d^2*(e^n)^2/e/n/(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)*dilog((-a*exp(I*(d*x^n+2*c))-exp(I*c)*b+(exp(2*I*c)*b^2
-a^2*exp(2*I*c))^(1/2))/(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn
(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I
*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c))+I*b^3/a
^2/(a^2-b^2)/d*(e^n)^2/e/n*x^n/(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)*ln((a*exp(I*(d*x^n+2*c))+exp(I*c)*b+(exp(
2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*csgn(
I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-
Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3-2*c)
)-2*I*b/(a^2-b^2)/d*(e^n)^2/e/n*x^n/(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)*ln((a*exp(I*(d*x^n+2*c))+exp(I*c)*b+
(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))*exp(-1/2*I*(2*Pi*n*
csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*
x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3
-2*c))

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [B]  time = 3.33648, size = 5435, normalized size = 7.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d^2*e^(2*n - 1)*x^(2*n)*cos(d*x^n + c) + 2*(a^4*b - 2*a^2*b^3 + b^5)*d^2*e^(2
*n - 1)*x^(2*n) + 4*(a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sin(d*x^n + c) - 2*((2*a^4*b - a^2*b^3)*e^(2*n - 1)*sq
rt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*(a
*sqrt(-(a^2 - b^2)/a^2) + b)*cos(d*x^n + c) + (2*I*a*sqrt(-(a^2 - b^2)/a^2) + 2*I*b)*sin(d*x^n + c) + 2*a)/a +
1) - 2*((2*a^4*b - a^2*b^3)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n -
1)*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2) + b)*cos(d*x^n + c) + (-2*I*a*sqrt(-(a^2 -
b^2)/a^2) - 2*I*b)*sin(d*x^n + c) + 2*a)/a + 1) + 2*((2*a^4*b - a^2*b^3)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2)*co
s(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*dilog(1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2)
- b)*cos(d*x^n + c) - (2*I*a*sqrt(-(a^2 - b^2)/a^2) - 2*I*b)*sin(d*x^n + c) - 2*a)/a + 1) + 2*((2*a^4*b - a^2*
b^3)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2
))*dilog(1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2) - b)*cos(d*x^n + c) - (-2*I*a*sqrt(-(a^2 - b^2)/a^2) + 2*I*b)*sin(d*
x^n + c) - 2*a)/a + 1) + ((2*a^3*b^2 - 2*a*b^4 - 2*I*(2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)
*cos(d*x^n + c) + (2*a^2*b^3 - 2*b^5 - 2*I*(2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1))*log(2*a*
cos(d*x^n + c) + 2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) + ((2*a^3*b^2 - 2*a*b^4 + 2*I*(2*a^4
*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)*cos(d*x^n + c) + (2*a^2*b^3 - 2*b^5 + 2*I*(2*a^3*b^2 - a*b
^4)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1))*log(2*a*cos(d*x^n + c) - 2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^
2)/a^2) + 2*b) + ((2*a^3*b^2 - 2*a*b^4 - 2*I*(2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)*cos(d*x
^n + c) + (2*a^2*b^3 - 2*b^5 - 2*I*(2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1))*log(-2*a*cos(d*x
^n + c) + 2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) + ((2*a^3*b^2 - 2*a*b^4 + 2*I*(2*a^4*b - a^
2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)*cos(d*x^n + c) + (2*a^2*b^3 - 2*b^5 + 2*I*(2*a^3*b^2 - a*b^4)*c*s
qrt(-(a^2 - b^2)/a^2))*e^(2*n - 1))*log(-2*a*cos(d*x^n + c) - 2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2
) - 2*b) + (-2*I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - 2*I*(2*a^3*b^2 - a*b^4)*c*e^(2
*n - 1)*sqrt(-(a^2 - b^2)/a^2) + (-2*I*(2*a^4*b - a^2*b^3)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - 2*I*(2*a
^4*b - a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*cos(d*x^n + c))*log(1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2) + b
)*cos(d*x^n + c) + (2*I*a*sqrt(-(a^2 - b^2)/a^2) + 2*I*b)*sin(d*x^n + c) + 2*a)/a) + (2*I*(2*a^3*b^2 - a*b^4)*
d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + 2*I*(2*a^3*b^2 - a*b^4)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2) + (2*I
*(2*a^4*b - a^2*b^3)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + 2*I*(2*a^4*b - a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a
^2 - b^2)/a^2))*cos(d*x^n + c))*log(1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2) + b)*cos(d*x^n + c) + (-2*I*a*sqrt(-(a^2
- b^2)/a^2) - 2*I*b)*sin(d*x^n + c) + 2*a)/a) + (-2*I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/
a^2) - 2*I*(2*a^3*b^2 - a*b^4)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2) + (-2*I*(2*a^4*b - a^2*b^3)*d*e^(2*n - 1)*
x^n*sqrt(-(a^2 - b^2)/a^2) - 2*I*(2*a^4*b - a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*cos(d*x^n + c))*log
(-1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2) - b)*cos(d*x^n + c) - (2*I*a*sqrt(-(a^2 - b^2)/a^2) - 2*I*b)*sin(d*x^n + c)
- 2*a)/a) + (2*I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + 2*I*(2*a^3*b^2 - a*b^4)*c*e^(
2*n - 1)*sqrt(-(a^2 - b^2)/a^2) + (2*I*(2*a^4*b - a^2*b^3)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + 2*I*(2*a
^4*b - a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*cos(d*x^n + c))*log(-1/2*(2*(a*sqrt(-(a^2 - b^2)/a^2) -
b)*cos(d*x^n + c) - (-2*I*a*sqrt(-(a^2 - b^2)/a^2) + 2*I*b)*sin(d*x^n + c) - 2*a)/a))/((a^7 - 2*a^5*b^2 + a^3*
b^4)*d^2*n*cos(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2*n)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{2 \, n - 1}}{{\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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