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

Optimal. Leaf size=778 \[ \frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sin \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \sqrt{b^2-a^2}}-\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \sqrt{b^2-a^2}}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \sin \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \]

[Out]

(e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*n)*Log[1 - (I*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 - (I*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) + (I*b^3*(e*x)^(2*n)*Log[1 - (I*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 - (I*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*Sin[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, (I*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, (I*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, (I*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, (I*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)*Cos[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*S
in[c + d*x^n]))

________________________________________________________________________________________

Rubi [A]  time = 1.3025, antiderivative size = 778, 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 = {4209, 4205, 4191, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sin \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \sqrt{b^2-a^2}}-\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \sqrt{b^2-a^2}}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \sin \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 2*n)/(a + b*Csc[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 - (I*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 - (I*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) + (I*b^3*(e*x)^(2*n)*Log[1 - (I*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 - (I*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*Sin[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, (I*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, (I*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, (I*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, (I*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)*Cos[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*S
in[c + d*x^n]))

Rule 4209

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

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[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 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
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 \csc \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 \csc \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 \csc (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 \sin (c+d x))^2}-\frac{2 b x}{a^2 (b+a \sin (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 \sin (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 \sin (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} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \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}{i a+2 b e^{i (c+d x)}-i 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 \sin (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{\cos (c+d x)}{b+a \sin (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} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \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}{i a+2 b e^{i (c+d x)}-i 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 i 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 i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt{-a^2+b^2} e n}-\frac{\left (4 i 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 i 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 \sin \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{i 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{i 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 \sin \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} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}+\frac{\left (2 i 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 i 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^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 i 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 i 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 i 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{i 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{i 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{i 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{i 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 \sin \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} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \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 i 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 i 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 i 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 i 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{i 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{i 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{i 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{i 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 \sin \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{i 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{i 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} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \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 i 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 i 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{i 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{i 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{i 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{i 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 \sin \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{i 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{i 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{i 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{i 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} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )}\\ \end{align*}

Mathematica [B]  time = 10.3566, size = 2839, normalized size = 3.65 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(b*Cos[c] + a*Sin[d*x^n])*(b + a*Sin[c + d
*x^n]))/(2*a^2*(-a + b)*(a + b)*d*n*(a + b*Csc[c + d*x^n])^2) - (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Cot[c]*Csc[c +
d*x^n]^2*(b + a*Sin[c + d*x^n])^2)/(a^2*(-a^2 + b^2)*d*n*(a + b*Csc[c + d*x^n])^2) + (2*b^3*x^(1 - 2*n)*(e*x)
^(-1 + 2*n)*ArcTanh[(a*Cos[c + d*x^n] + I*(b + a*Sin[c + d*x^n]))/Sqrt[a^2 - b^2]]*Cot[c]*Csc[c + d*x^n]^2*(b
+ a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Csc[c + d*x^n])^2) - (2*b*x^(1 - 2*n)*(e*x)^(-1 + 2
*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-c + Pi/
2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(-c + ArcCos[-(b/a)])*ArcTanh[((a
- b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2
- d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[Sqrt[a^2 -
b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(A
rcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sq
rt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^n
]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - 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 + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + P
i/2 - d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*L
og[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b
^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c +
Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b
^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)
/2]))]))/Sqrt[a^2 - b^2])*(b + a*Sin[c + d*x^n])^2)/((a^2 - b^2)*d^2*n*(a + b*Csc[c + d*x^n])^2) + (b^3*x^(1 -
2*n)*(e*x)^(-1 + 2*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 +
b^2] + (2*(-c + Pi/2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(-c + ArcCos[-
(b/a)])*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a +
b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2
]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos
[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + P
i/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[
b + a*Sin[c + d*x^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - 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 + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2
- b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - 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 + Pi/2 - d*x^n)/2]))/(a*(a
+ b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^
2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] - PolyLog[2, (
(b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[
(-c + Pi/2 - d*x^n)/2]))]))/Sqrt[a^2 - b^2])*(b + a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)*d^2*n*(a + b*Csc[c + d
*x^n])^2) + (x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(-2*b^2*Cos[c] + a^2*d*x^n*Sin[c] -
b^2*d*x^n*Sin[c])*(b + a*Sin[c + d*x^n])^2)/(4*a^2*(a - b)*(a + b)*d*n*(a + b*Csc[c + d*x^n])^2) + (b^2*x^(1
- 2*n)*(e*x)^(-1 + 2*n)*Csc[c]*Csc[c + d*x^n]^2*(-(a*d*x^n*Cos[c]) + a*Log[b + a*Cos[d*x^n]*Sin[c] + a*Cos[c]*
Sin[d*x^n]]*Sin[c] + ((2*I)*a*b*ArcTan[(I*a*Cos[c] - I*(-b + a*Sin[c])*Tan[(d*x^n)/2])/Sqrt[-b^2 + a^2*Cos[c]^
2 + a^2*Sin[c]^2]]*Cos[c])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2])*(b + a*Sin[c + d*x^n])^2)/(a*(a^2 - b^2)*
d^2*n*(a + b*Csc[c + d*x^n])^2*(a^2*Cos[c]^2 + a^2*Sin[c]^2))

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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*csc(c+d*x^n))^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 1.25326, size = 5230, normalized size = 6.72 \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*csc(c+d*x^n))^2,x, algorithm="fricas")

[Out]

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

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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*csc(c+d*x**n))**2,x)

[Out]

Timed out

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

[Out]

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