3.81 \(\int \frac{(e x)^{-1+3 n}}{a+b \csc (c+d x^n)} \, dx\)

Optimal. Leaf size=499 \[ \frac{2 i b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 e n \sqrt{b^2-a^2}}-\frac{2 i b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 e n \sqrt{b^2-a^2}}+\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 e n \sqrt{b^2-a^2}}-\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{3 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{3 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{3 n}}{3 a e n} \]

[Out]

(e*x)^(3*n)/(3*a*e*n) + (I*b*(e*x)^(3*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)^(3*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) + (2*b*(e*x)^(3*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)) - (2*b*(e*x)^(3*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)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])
/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2
+ b^2])])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n))

________________________________________________________________________________________

Rubi [A]  time = 0.941027, antiderivative size = 499, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4209, 4205, 4191, 3323, 2264, 2190, 2531, 2282, 6589} \[ \frac{2 i b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 e n \sqrt{b^2-a^2}}-\frac{2 i b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 e n \sqrt{b^2-a^2}}+\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 e n \sqrt{b^2-a^2}}-\frac{2 b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{3 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{3 n} \log \left (1-\frac{i a e^{i \left (c+d x^n\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{3 n}}{3 a e n} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n]),x]

[Out]

(e*x)^(3*n)/(3*a*e*n) + (I*b*(e*x)^(3*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)^(3*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) + (2*b*(e*x)^(3*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)) - (2*b*(e*x)^(3*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)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])
/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2
+ b^2])])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{(e x)^{-1+3 n}}{a+b \csc \left (c+d x^n\right )} \, dx &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac{x^{-1+3 n}}{a+b \csc \left (c+d x^n\right )} \, dx}{e}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \csc (c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \left (\frac{x^2}{a}-\frac{b x^2}{a (b+a \sin (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{\left (b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a \sin (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}-\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{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)^{3 n}}{3 a e n}+\frac{\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{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^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{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)^{3 n}}{3 a e n}+\frac{i b x^{-n} (e x)^{3 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)^{3 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 (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \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 (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \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)^{3 n}}{3 a e n}+\frac{i b x^{-n} (e x)^{3 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)^{3 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{2 b x^{-2 n} (e x)^{3 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{2 b x^{-2 n} (e x)^{3 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{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\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^2 e n}+\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\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^2 e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}+\frac{i b x^{-n} (e x)^{3 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)^{3 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{2 b x^{-2 n} (e x)^{3 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{2 b x^{-2 n} (e x)^{3 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{\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b-\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^3 e n}-\frac{\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{b+\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^3 e n}\\ &=\frac{(e x)^{3 n}}{3 a e n}+\frac{i b x^{-n} (e x)^{3 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)^{3 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{2 b x^{-2 n} (e x)^{3 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{2 b x^{-2 n} (e x)^{3 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{2 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\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^3 e n}-\frac{2 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\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^3 e n}\\ \end{align*}

Mathematica [F]  time = 1.97034, size = 0, normalized size = 0. \[ \int \frac{(e x)^{-1+3 n}}{a+b \csc \left (c+d x^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n]),x]

[Out]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n]), x]

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Maple [F]  time = 0.733, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{-1+3\,n}}{a+b\csc \left ( c+d{x}^{n} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n)),x)

[Out]

int((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n)),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+3*n)/(a+b*csc(c+d*x^n)),x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 0.880113, size = 3856, normalized size = 7.73 \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+3*n)/(a+b*csc(c+d*x^n)),x, algorithm="fricas")

[Out]

-1/12*(-12*I*a*b*d*e^(3*n - 1)*x^n*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) - 12*I*a*b*d*e^(3*n - 1)*x^n*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) + 12*I*a*b*d*e^(3*n - 1)*x^n*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) + 12*I*a
*b*d*e^(3*n - 1)*x^n*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) - 6*a*b*c^2*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2)*lo
g(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) - 6*a*b*c^2*e^(3*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) + 6*a*b*c^
2*e^(3*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) + 6*a*b*c^2*e^(3*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*s
qrt((a^2 - b^2)/a^2) - 2*I*b) - 4*(a^2 - b^2)*d^3*e^(3*n - 1)*x^(3*n) - 12*a*b*e^(3*n - 1)*sqrt((a^2 - b^2)/a^
2)*polylog(3, -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)*si
n(d*x^n + c))/a) + 12*a*b*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2)*polylog(3, 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))/a) - 12*a*b*e^(3*n - 1)*sqrt((a^2 - b^2)
/a^2)*polylog(3, -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))/a) + 12*a*b*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2)*polylog(3, 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))/a) + 6*(a*b*d^2*e^(3*n - 1)*x^(2*n)*sqr
t((a^2 - b^2)/a^2) - a*b*c^2*e^(3*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) - 6*(a*b*d^2*e^(3*n - 1)*x^(2*n)*s
qrt((a^2 - b^2)/a^2) - a*b*c^2*e^(3*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) + 6*(a*b*d^2*e^(3*n - 1)*x^(2*n)*
sqrt((a^2 - b^2)/a^2) - a*b*c^2*e^(3*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) - 6*(a*b*d^2*e^(3*n - 1)*x^(2*
n)*sqrt((a^2 - b^2)/a^2) - a*b*c^2*e^(3*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^3*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{3 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+3*n)/(a+b*csc(c+d*x**n)),x)

[Out]

Integral((e*x)**(3*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 )^{3 \, 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+3*n)/(a+b*csc(c+d*x^n)),x, algorithm="giac")

[Out]

integrate((e*x)^(3*n - 1)/(b*csc(d*x^n + c) + a), x)