\(\int \frac {(e x)^{-1+2 n}}{a+b \csc (c+d x^n)} \, dx\) [80]

Optimal result

Integrand size = 24, antiderivative size = 338 \[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\frac {(e x)^{2 n}}{2 a e n}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4294, 4290, 4276, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d e n \sqrt {b^2-a^2}}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{2 n}}{2 a e n} \]

[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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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 3404

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 4276

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 4290

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 4294

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1003\) vs. \(2(338)=676\).

Time = 6.35 (sec) , antiderivative size = 1003, normalized size of antiderivative = 2.97 \[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\frac {(e x)^{2 n} \csc \left (c+d x^n\right ) \left (1-\frac {2 b x^{-2 n} \left (\frac {\pi \arctan \left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {2 \left (c-\arccos \left (-\frac {b}{a}\right )\right ) \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-2 c+\pi -2 d x^n\right ) \text {arctanh}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )-\left (\arccos \left (-\frac {b}{a}\right )-2 i \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\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 (1+i \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )+\left (\arccos \left (-\frac {b}{a}\right )+2 i \left (-\text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+\text {arctanh}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\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 (c+d x^n\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \sin \left (c+d x^n\right )}}\right )+\left (\arccos \left (-\frac {b}{a}\right )+2 i \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )-2 i \text {arctanh}\left (\frac {(a+b) \tan \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\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 (c+d x^n\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \sin \left (c+d x^n\right )}}\right )-\left (\arccos \left (-\frac {b}{a}\right )+2 i \text {arctanh}\left (\frac {(a-b) \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (1+\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 c-\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )+i \left (\operatorname {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 c-\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )-\operatorname {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 c-\pi +2 d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )}\right )\right )}{\sqrt {a^2-b^2}}\right )}{d^2}\right ) \left (b+a \sin \left (c+d x^n\right )\right )}{2 a e n \left (a+b \csc \left (c+d x^n\right )\right )} \]

[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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.34 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.28

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{2 a n}+\frac {i \left (i x^{n} d \ln \left (\frac {-i {\mathrm e}^{i c} b -a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}+\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}{-i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right )-i x^{n} d \ln \left (\frac {i {\mathrm e}^{i c} b +a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}+\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}{i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right )+\operatorname {dilog}\left (-\frac {i {\mathrm e}^{i c} b}{-i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}-\frac {a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}}{-i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}+\frac {\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}{-i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right )-\operatorname {dilog}\left (\frac {i {\mathrm e}^{i c} b}{i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}+\frac {a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}}{i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}+\frac {\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}{i {\mathrm e}^{i c} b +\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right )\right ) \sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}\, e^{2 n} b \,{\mathrm e}^{-\frac {i \left (2 \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )-2 \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}-2 \pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}+2 \pi n \operatorname {csgn}\left (i e x \right )^{3}-\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi \operatorname {csgn}\left (i e x \right )^{3}+2 c \right )}{2}}}{\left (a^{2}-b^{2}\right ) d^{2} n e a}\)	\(769\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1235 vs. \(2 (302) = 604\).

Time = 0.44 (sec) , antiderivative size = 1235, normalized size of antiderivative = 3.65 \[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(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) + 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) - 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*si
n(d*x^n + c) + 2*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) - 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) - (a^2 - b^2)*d^2*e^(2*n - 1)*x^(2*n) - I*a*
b*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*dilog(((a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) + (I*a*sqrt((a^2 - b
^2)/a^2) - b)*sin(d*x^n + c) - a)/a + 1) - I*a*b*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*dilog(-((a*sqrt((a^2 - b^2)
/a^2) + I*b)*cos(d*x^n + c) - (I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) + a)/a + 1) + I*a*b*e^(2*n - 1)*s
qrt((a^2 - b^2)/a^2)*dilog(((a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) + (-I*a*sqrt((a^2 - b^2)/a^2) - b)*
sin(d*x^n + c) - a)/a + 1) + I*a*b*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*dilog(-((a*sqrt((a^2 - b^2)/a^2) - I*b)*c
os(d*x^n + c) - (-I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) + a)/a + 1) + (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(-((a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) +
(I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) - a)/a) - (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(((a*sqrt((a^2 - b^2)/a^2) + I*b)*cos(d*x^n + c) - (I*a*sqrt((a^2 - b^2
)/a^2) - b)*sin(d*x^n + c) + a)/a) + (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(-((a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) + (-I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x
^n + c) - a)/a) - (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(
((a*sqrt((a^2 - b^2)/a^2) - I*b)*cos(d*x^n + c) - (-I*a*sqrt((a^2 - b^2)/a^2) - b)*sin(d*x^n + c) + a)/a))/((a
^3 - a*b^2)*d^2*n)

Sympy [F]

\[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int \frac {\left (e x\right )^{2 n - 1}}{a + b \csc {\left (c + d x^{n} \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int { \frac {\left (e x\right )^{2 \, n - 1}}{b \csc \left (d x^{n} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int { \frac {\left (e x\right )^{2 \, n - 1}}{b \csc \left (d x^{n} + c\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{-1+2 n}}{a+b \csc \left (c+d x^n\right )} \, dx=\int \frac {{\left (e\,x\right )}^{2\,n-1}}{a+\frac {b}{\sin \left (c+d\,x^n\right )}} \,d x \]

[In]

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

[Out]

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