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

Optimal. Leaf size=338 \[ \frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 e n \sqrt{b^2-a^2}}-\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{2 n}}{2 a e n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.620692, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4209, 4205, 4191, 3323, 2264, 2190, 2279, 2391} \[ \frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 e n \sqrt{b^2-a^2}}-\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{2 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{2 n}}{2 a e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac{x^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx}{e}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{a+b \csc (c+d x)} \, 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}-\frac{b x}{a (b+a \sin (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{\left (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 e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}-\frac{\left (2 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 e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}+\frac{\left (2 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 )}{\sqrt{-a^2+b^2} e n}-\frac{\left (2 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 )}{\sqrt{-a^2+b^2} e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}+\frac{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 \sqrt{-a^2+b^2} d e n}-\frac{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 \sqrt{-a^2+b^2} d e n}-\frac{\left (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 \sqrt{-a^2+b^2} d e n}+\frac{\left (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 \sqrt{-a^2+b^2} d e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}+\frac{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 \sqrt{-a^2+b^2} d e n}-\frac{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 \sqrt{-a^2+b^2} d e n}-\frac{\left (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 \sqrt{-a^2+b^2} d^2 e n}+\frac{\left (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 \sqrt{-a^2+b^2} d^2 e n}\\ &=\frac{(e x)^{2 n}}{2 a e n}+\frac{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 \sqrt{-a^2+b^2} d e n}-\frac{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 \sqrt{-a^2+b^2} d e n}+\frac{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 \sqrt{-a^2+b^2} d^2 e n}-\frac{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 \sqrt{-a^2+b^2} d^2 e n}\\ \end{align*}

Mathematica [B]  time = 5.10782, size = 1003, normalized size = 2.97 \[ \frac{(e x)^{2 n} \csc \left (d x^n+c\right ) \left (1-\frac{2 b x^{-2 n} \left (\frac{\pi \tan ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} \left (d x^n+c\right )\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+\frac{2 \left (c-\cos ^{-1}\left (-\frac{b}{a}\right )\right ) \tanh ^{-1}\left (\frac{(a-b) \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )+\left (-2 d x^n-2 c+\pi \right ) \tanh ^{-1}\left (\frac{(a+b) \tan \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )-2 i \tanh ^{-1}\left (\frac{(a-b) \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{(a+b) \left (a-b-i \sqrt{a^2-b^2}\right ) \left (i \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )+1\right )}{a \left (a+b+\sqrt{a^2-b^2} \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )\right )}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 i \left (\tanh ^{-1}\left (\frac{(a+b) \tan \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )-\tanh ^{-1}\left (\frac{(a-b) \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )\right )\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{a^2-b^2} e^{-\frac{1}{2} i \left (d x^n+c\right )}}{\sqrt{2} \sqrt{a} \sqrt{b+a \sin \left (d x^n+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 i \tanh ^{-1}\left (\frac{(a-b) \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac{(a+b) \tan \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (-\frac{(-1)^{3/4} \sqrt{a^2-b^2} e^{\frac{1}{2} i \left (d x^n+c\right )}}{\sqrt{2} \sqrt{a} \sqrt{b+a \sin \left (d x^n+c\right )}}\right )-\left (\cos ^{-1}\left (-\frac{b}{a}\right )+2 i \tanh ^{-1}\left (\frac{(a-b) \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )}{\sqrt{a^2-b^2}}\right )\right ) \log \left (\frac{i \left (i b+\sqrt{a^2-b^2}\right ) \left (a+b+\sqrt{a^2-b^2} \tan \left (\frac{1}{4} \left (2 d x^n+2 c-\pi \right )\right )\right )}{a \left (a+b+\sqrt{a^2-b^2} \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )\right )}+1\right )+i \left (\text{PolyLog}\left (2,\frac{\left (b-i \sqrt{a^2-b^2}\right ) \left (a+b+\sqrt{a^2-b^2} \tan \left (\frac{1}{4} \left (2 d x^n+2 c-\pi \right )\right )\right )}{a \left (a+b+\sqrt{a^2-b^2} \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (b+i \sqrt{a^2-b^2}\right ) \left (a+b+\sqrt{a^2-b^2} \tan \left (\frac{1}{4} \left (2 d x^n+2 c-\pi \right )\right )\right )}{a \left (a+b+\sqrt{a^2-b^2} \cot \left (\frac{1}{4} \left (2 d x^n+2 c+\pi \right )\right )\right )}\right )\right )}{\sqrt{a^2-b^2}}\right )}{d^2}\right ) \left (b+a \sin \left (d x^n+c\right )\right )}{2 a e n \left (a+b \csc \left (d x^n+c\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((e*x)^(2*n)*Csc[c + d*x^n]*(1 - (2*b*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^
2] + (2*(c - ArcCos[-(b/a)])*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] + (-2*c + Pi - 2*d
*x^n)*ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] - (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a - b
)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 - b^2])*(1 + I*Cot[(2*c + Pi
+ 2*d*x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))] + (ArcCos[-(b/a)] + (2*I)*(-ArcTan
h[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] + ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^n)/4])/Sqrt[
a^2 - b^2]]))*Log[((-1)^(1/4)*Sqrt[a^2 - b^2])/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Sin[c + d*x^n
]])] + (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] - (2*I)*ArcTanh[
((a + b)*Tan[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)
))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^n]]))] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2
*d*x^n)/4])/Sqrt[a^2 - b^2]])*Log[1 + (I*(I*b + Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*
x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*
(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]
))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^n)/4]))/(a*(a + b + S
qrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))]))/Sqrt[a^2 - b^2]))/(d^2*x^(2*n)))*(b + a*Sin[c + d*x^n]))/(2*a*
e*n*(a + b*Csc[c + d*x^n]))

________________________________________________________________________________________

Maple [C]  time = 0.658, size = 1332, normalized size = 3.9 \begin{align*} \text{result 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*csc(c+d*x^n)),x)

[Out]

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

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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)),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 0.952718, size = 2865, normalized size = 8.48 \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)),x, algorithm="fricas")

[Out]

-1/4*(2*a*b*c*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*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*b*c*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*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*b*c*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*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*b*c*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*log(-2*a*c
os(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^2 - b^2)*d^2*e^(2*n - 1)*x^(2
*n) - 2*I*a*b*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*b*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*d
ilog(-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*b*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)*co
s(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*b*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*(a*b*d*e^(2*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + a*b*c*e^(2*n - 1)*sqrt((a^2
- b^2)/a^2))*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*b*d*e^(2*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + a*b*c*e^(2*n - 1)*sqrt((a^2 - b^2
)/a^2))*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*b*d*e^(2*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + a*b*c*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2))
*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*b*d*e^(2*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + a*b*c*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2))*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^3 - a*b^2)*d^2*n)

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

[Out]

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

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

[Out]

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