3.84 \(\int \frac {(e x)^{-1+3 n}}{(a+b \csc (c+d x^n))^2} \, dx\)

Optimal. Leaf size=1417 \[ -\frac {2 i b^2 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (d x^n+c\right )}}{i b-\sqrt {a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (d x^n+c\right )}}{i b+\sqrt {a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac {4 i b (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}-\frac {2 i b^3 (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}-\frac {4 i b (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}+\frac {2 i b^3 (e x)^{3 n} \text {Li}_3\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {4 b (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}-\frac {2 b^3 (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac {4 b (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}+\frac {2 b^3 (e x)^{3 n} \text {Li}_2\left (\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac {i b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 i b (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}-\frac {i b^3 (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}-\frac {2 i b (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}+\frac {i b^3 (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}-\frac {b^2 (e x)^{3 n} \cos \left (d x^n+c\right ) x^{-n}}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (d x^n+c\right )\right )}+\frac {(e x)^{3 n}}{3 a^2 e n} \]

[Out]

1/3*(e*x)^(3*n)/a^2/e/n+2*I*b*(e*x)^(3*n)*ln(1-I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^
2+b^2)^(1/2)+2*b^2*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))+
2*b^2*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(I*b+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))-4*I*b*(e*x)^(
3*n)*polylog(3,I*a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)-I*b^3*(e*x)^(
3*n)*ln(1-I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)-I*b^2*(e*x)^(3*n)/a^2/(a
^2-b^2)/d/e/n/(x^n)-2*I*b^3*(e*x)^(3*n)*polylog(3,I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3
/2)/d^3/e/n/(x^(3*n))-2*b^3*(e*x)^(3*n)*polylog(2,I*a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3
/2)/d^2/e/n/(x^(2*n))+2*b^3*(e*x)^(3*n)*polylog(2,I*a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3
/2)/d^2/e/n/(x^(2*n))+I*b^3*(e*x)^(3*n)*ln(1-I*a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d
/e/n/(x^n)-2*I*b*(e*x)^(3*n)*ln(1-I*a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-
b^2*(e*x)^(3*n)*cos(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(b+a*sin(c+d*x^n))-2*I*b^2*(e*x)^(3*n)*polylog(2,-a*exp(I
*(c+d*x^n))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))+2*I*b^3*(e*x)^(3*n)*polylog(3,I*a*exp(I*(c+
d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))+4*b*(e*x)^(3*n)*polylog(2,I*a*exp(I*(c+d*
x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)-4*b*(e*x)^(3*n)*polylog(2,I*a*exp(I*(c+d*x^
n))/(b+(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)+4*I*b*(e*x)^(3*n)*polylog(3,I*a*exp(I*(c+d*x^
n))/(b-(-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)-2*I*b^2*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x
^n))/(I*b+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))

________________________________________________________________________________________

Rubi [A]  time = 2.44, antiderivative size = 1417, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {4209, 4205, 4191, 3324, 3323, 2264, 2190, 2531, 2282, 6589, 4521, 2279, 2391} \[ -\frac {2 i b^2 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{i b-\sqrt {a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{i b+\sqrt {a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac {4 i b (e x)^{3 n} \text {PolyLog}\left (3,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}-\frac {2 i b^3 (e x)^{3 n} \text {PolyLog}\left (3,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}-\frac {4 i b (e x)^{3 n} \text {PolyLog}\left (3,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}+\frac {2 i b^3 (e x)^{3 n} \text {PolyLog}\left (3,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {4 b (e x)^{3 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}-\frac {2 b^3 (e x)^{3 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac {4 b (e x)^{3 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}+\frac {2 b^3 (e x)^{3 n} \text {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac {i b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 i b (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}-\frac {i b^3 (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}-\frac {2 i b (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}+\frac {i b^3 (e x)^{3 n} \log \left (1-\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}-\frac {b^2 (e x)^{3 n} \cos \left (d x^n+c\right ) x^{-n}}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (d x^n+c\right )\right )}+\frac {(e x)^{3 n}}{3 a^2 e n} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n])^2,x]

[Out]

(e*x)^(3*n)/(3*a^2*e*n) - (I*b^2*(e*x)^(3*n))/(a^2*(a^2 - b^2)*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(I
*(c + d*x^n)))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(
I*(c + d*x^n)))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) - (I*b^3*(e*x)^(3*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)^(3*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)^(3*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)^(3*n)*Lo
g[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) - ((2*I)*b^2*(e*x)^(3*
n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3*e*n*x^(3*n)) - ((2*I)*b^
2*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3*e*n*x^(3*n))
- (2*b^3*(e*x)^(3*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)) + (4*b*(e*x)^(3*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)) + (2*b^3*(e*x)^(3*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)) - (4*b*(e*x)^(3*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)) - ((2*I)*b^3*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[
-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) + ((4*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n
)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) + ((2*I)*b^3*(e*x)^(3*n)*PolyLog[3, (I*a*E
^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) - ((4*I)*b*(e*x)^(3*n)*Pol
yLog[3, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - (b^2*(e*x)^(
3*n)*Cos[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Sin[c + d*x^n]))

Rule 2190

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 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 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 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 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 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 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 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 4521

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^
2, 2] + b*E^(I*(c + d*x))), x], x] + Dist[I, Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + b*E^
(I*(c + d*x))), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

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}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac {x^{-1+3 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, 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))^2} \, 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^2}+\frac {b^2 x^2}{a^2 (b+a \sin (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {\left (2 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^2 e n}+\frac {\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{(b+a \sin (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {b^2 x^{-n} (e x)^{3 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^{-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^2 e n}-\frac {\left (b^3 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^2 \left (a^2-b^2\right ) e n}+\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x \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)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}-\frac {b^2 x^{-n} (e x)^{3 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^{-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^2 \left (a^2-b^2\right ) e n}+\frac {\left (4 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 )}{a \sqrt {-a^2+b^2} e n}-\frac {\left (4 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 )}{a \sqrt {-a^2+b^2} e n}+\frac {\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b-\sqrt {a^2-b^2}+a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}+\frac {\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b+\sqrt {a^2-b^2}+a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {b^2 x^{-n} (e x)^{3 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^{-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 )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}-\frac {\left (2 i b^3 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 )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {a e^{i (c+d x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {a e^{i (c+d x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {\left (4 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^2 \sqrt {-a^2+b^2} d e n}+\frac {\left (4 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^2 \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 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^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 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^2 \sqrt {-a^2+b^2} d e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{3 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^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{i b-\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 ) d^3 e n}+\frac {\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{i b+\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 ) d^3 e n}-\frac {\left (4 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^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (2 i b^3 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^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n}+\frac {\left (2 i b^3 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^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 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^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{3 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 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^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (4 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^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 b^3 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^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 b^3 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^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 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^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {b^2 x^{-n} (e x)^{3 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^{-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^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 i b^3 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^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^3 e n}\\ &=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 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^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (c+d x^n\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{i \left (c+d x^n\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {2 i b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {b^2 x^{-n} (e x)^{3 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*}

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Mathematica [F]  time = 10.67, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n])^2,x]

[Out]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n])^2, x]

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fricas [C]  time = 0.91, size = 3785, normalized size = 2.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n))^2,x, algorithm="fricas")

[Out]

1/12*(4*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*e^(3*n - 1)*x^(3*n)*sin(d*x^n + c) + 4*(a^4*b - 2*a^2*b^3 + b^5)*d^3*e^(
3*n - 1)*x^(3*n) - 12*(a^3*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*cos(d*x^n + c) + (12*I*(2*a^3*b^2 - a*b^4)*d*e
^(3*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (12*I*a^2*b^3 - 12*I*b^5)*e^(3*n - 1) + (12*I*(2*a^4*b - a^2*b^3)*d*e^(
3*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (12*I*a^3*b^2 - 12*I*a*b^4)*e^(3*n - 1))*sin(d*x^n + c))*dilog(-1/2*(2*(a
*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) + (-2*I*a*sqrt((a^2 - b^2)/a^2) + 2*b)*sin(d*x^n + c) + 2*a)/a +
1) + (12*I*(2*a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (-12*I*a^2*b^3 + 12*I*b^5)*e^(3*n - 1
) + (12*I*(2*a^4*b - a^2*b^3)*d*e^(3*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (-12*I*a^3*b^2 + 12*I*a*b^4)*e^(3*n -
1))*sin(d*x^n + c))*dilog(1/2*(2*(a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) - (-2*I*a*sqrt((a^2 - b^2)/a^2
) + 2*b)*sin(d*x^n + c) - 2*a)/a + 1) + (-12*I*(2*a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (
-12*I*a^2*b^3 + 12*I*b^5)*e^(3*n - 1) + (-12*I*(2*a^4*b - a^2*b^3)*d*e^(3*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (
-12*I*a^3*b^2 + 12*I*a*b^4)*e^(3*n - 1))*sin(d*x^n + c))*dilog(-1/2*(2*(a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x
^n + c) + (2*I*a*sqrt((a^2 - b^2)/a^2) + 2*b)*sin(d*x^n + c) + 2*a)/a + 1) + (-12*I*(2*a^3*b^2 - a*b^4)*d*e^(3
*n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (12*I*a^2*b^3 - 12*I*b^5)*e^(3*n - 1) + (-12*I*(2*a^4*b - a^2*b^3)*d*e^(3*
n - 1)*x^n*sqrt((a^2 - b^2)/a^2) + (12*I*a^3*b^2 - 12*I*a*b^4)*e^(3*n - 1))*sin(d*x^n + c))*dilog(1/2*(2*(a*sq
rt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) - (2*I*a*sqrt((a^2 - b^2)/a^2) + 2*b)*sin(d*x^n + c) - 2*a)/a + 1) +
6*(((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1)*sin(d*x^n + c) + ((2*a
^3*b^2 - a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1))*log(2*a*cos(d*x^n + c) + 2*I*a*s
in(d*x^n + c) + 2*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) + 6*(((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^
3*b^2 - a*b^4)*c)*e^(3*n - 1)*sin(d*x^n + c) + ((2*a^3*b^2 - a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^2*b^3 - b
^5)*c)*e^(3*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) - 6*(((
2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1)*sin(d*x^n + c) + ((2*a^3*b^2
- a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*c)*e^(3*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) - 6*(((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^3*b^2
- a*b^4)*c)*e^(3*n - 1)*sin(d*x^n + c) + ((2*a^3*b^2 - a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*c
)*e^(3*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) + 6*((2*a^3
*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sqrt((a^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*d*e^(3*n - 1)*x^n - ((2*a^3*b^
2 - a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d^2*e^(3*n - 1)
*x^(2*n)*sqrt((a^2 - b^2)/a^2) + 2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n - ((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 -
b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1))*sin(d*x^n + c))*log(1/2*(2*(a*sqrt((a^2 - b^2)/a^2) + I*b)*cos
(d*x^n + c) + (-2*I*a*sqrt((a^2 - b^2)/a^2) + 2*b)*sin(d*x^n + c) + 2*a)/a) - 6*((2*a^3*b^2 - a*b^4)*d^2*e^(3*
n - 1)*x^(2*n)*sqrt((a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*d*e^(3*n - 1)*x^n - ((2*a^3*b^2 - a*b^4)*c^2*sqrt((a^
2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d^2*e^(3*n - 1)*x^(2*n)*sqrt((a^2 - b^
2)/a^2) - 2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n - ((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^3*b^2
- a*b^4)*c)*e^(3*n - 1))*sin(d*x^n + c))*log(-1/2*(2*(a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) - (-2*I*a*
sqrt((a^2 - b^2)/a^2) + 2*b)*sin(d*x^n + c) - 2*a)/a) + 6*((2*a^3*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sqrt((a
^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*d*e^(3*n - 1)*x^n - ((2*a^3*b^2 - a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^2
*b^3 - b^5)*c)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d^2*e^(3*n - 1)*x^(2*n)*sqrt((a^2 - b^2)/a^2) + 2*(a^3*b^2 -
a*b^4)*d*e^(3*n - 1)*x^n - ((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1
))*sin(d*x^n + c))*log(1/2*(2*(a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) + (2*I*a*sqrt((a^2 - b^2)/a^2) +
2*b)*sin(d*x^n + c) + 2*a)/a) - 6*((2*a^3*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sqrt((a^2 - b^2)/a^2) - 2*(a^2*
b^3 - b^5)*d*e^(3*n - 1)*x^n - ((2*a^3*b^2 - a*b^4)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*c)*e^(3*n -
1) + ((2*a^4*b - a^2*b^3)*d^2*e^(3*n - 1)*x^(2*n)*sqrt((a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^
n - ((2*a^4*b - a^2*b^3)*c^2*sqrt((a^2 - b^2)/a^2) + 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1))*sin(d*x^n + c))*log(-
1/2*(2*(a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) - (2*I*a*sqrt((a^2 - b^2)/a^2) + 2*b)*sin(d*x^n + c) - 2
*a)/a) - 12*((2*a^4*b - a^2*b^3)*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2)*sin(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(3*n
- 1)*sqrt((a^2 - b^2)/a^2))*polylog(3, 1/2*(2*(a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) + (2*I*a*sqrt((a
^2 - b^2)/a^2) - 2*b)*sin(d*x^n + c))/a) + 12*((2*a^4*b - a^2*b^3)*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2)*sin(d*x^n
+ c) + (2*a^3*b^2 - a*b^4)*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2))*polylog(3, -1/2*(2*(a*sqrt((a^2 - b^2)/a^2) + I
*b)*cos(d*x^n + c) - (2*I*a*sqrt((a^2 - b^2)/a^2) - 2*b)*sin(d*x^n + c))/a) - 12*((2*a^4*b - a^2*b^3)*e^(3*n -
1)*sqrt((a^2 - b^2)/a^2)*sin(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2))*polylog(3, 1
/2*(2*(a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) + (-2*I*a*sqrt((a^2 - b^2)/a^2) - 2*b)*sin(d*x^n + c))/a)
+ 12*((2*a^4*b - a^2*b^3)*e^(3*n - 1)*sqrt((a^2 - b^2)/a^2)*sin(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(3*n - 1)*
sqrt((a^2 - b^2)/a^2))*polylog(3, -1/2*(2*(a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) - (-2*I*a*sqrt((a^2 -
b^2)/a^2) - 2*b)*sin(d*x^n + c))/a))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^3*n*sin(d*x^n + c) + (a^6*b - 2*a^4*b^3 +
a^2*b^5)*d^3*n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n))^2,x, algorithm="giac")

[Out]

integrate((e*x)^(3*n - 1)/(b*csc(d*x^n + c) + a)^2, x)

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maple [F]  time = 7.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{-1+3 n}}{\left (a +b \csc \left (c +d \,x^{n}\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n))^2,x)

[Out]

int((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n))^2,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csc(c+d*x^n))^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(3*n - 1)/(a + b/sin(c + d*x^n))^2,x)

[Out]

int((e*x)^(3*n - 1)/(a + b/sin(c + d*x^n))^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+3*n)/(a+b*csc(c+d*x**n))**2,x)

[Out]

Integral((e*x)**(3*n - 1)/(a + b*csc(c + d*x**n))**2, x)

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