Integrand size = 24, antiderivative size = 778 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )} \]
1/2*(e*x)^(2*n)/a^2/e/n+b^2*(e*x)^(2*n)*ln(b+a*sin(c+d*x^n))/a^2/(a^2-b^2) /d^2/e/n/(x^(2*n))-I*b^3*(e*x)^(2*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^3*(e*x)^(2*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)-b^3*( e*x)^(2*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))+b^3*(e*x)^(2*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))-b^2*(e*x)^( 2*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*(e*x)^( 2*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*I*b*(e*x)^(2*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*(e*x)^(2*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)-2*b *(e*x)^(2*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)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2839\) vs. \(2(778)=1556\).
Time = 11.17 (sec) , antiderivative size = 2839, normalized size of antiderivative = 3.65 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\text {Result too large to show} \]
-1/2*(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(b *Cos[c] + a*Sin[d*x^n])*(b + a*Sin[c + d*x^n]))/(a^2*(-a + b)*(a + b)*d*n* (a + b*Csc[c + d*x^n])^2) - (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Cot[c]*Csc[c + d*x^n]^2*(b + a*Sin[c + d*x^n])^2)/(a^2*(-a^2 + b^2)*d*n*(a + b*Csc[c + d *x^n])^2) + (2*b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*ArcTanh[(a*Cos[c + d*x^n] + I*(b + a*Sin[c + d*x^n]))/Sqrt[a^2 - b^2]]*Cot[c]*Csc[c + d*x^n]^2*(b + a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Csc[c + d*x^n])^2 ) - (2*b*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b* Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-c + Pi/2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*( -c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^ n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqr t[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[( (a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Ta n[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2) *(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^n]])] - (Ar cCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan...
Time = 1.43 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4697, 4693, 3042, 4679, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(e x)^{2 n-1}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 4697 |
\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \int \frac {x^{2 n-1}}{\left (a+b \csc \left (d x^n+c\right )\right )^2}dx}{e}\) |
\(\Big \downarrow \) 4693 |
\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \int \frac {x^n}{\left (a+b \csc \left (d x^n+c\right )\right )^2}dx^n}{e n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \int \frac {x^n}{\left (a+b \csc \left (d x^n+c\right )\right )^2}dx^n}{e n}\) |
\(\Big \downarrow \) 4679 |
\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \int \left (-\frac {2 b x^n}{a^2 \left (b+a \sin \left (d x^n+c\right )\right )}+\frac {x^n}{a^2}+\frac {b^2 x^n}{a^2 \left (b+a \sin \left (d x^n+c\right )\right )^2}\right )dx^n}{e n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \left (\frac {2 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}+\frac {b^2 \log \left (a \sin \left (c+d x^n\right )+b\right )}{a^2 d^2 \left (a^2-b^2\right )}+\frac {2 i b x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {2 i b x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {b^2 x^n \cos \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a \sin \left (c+d x^n\right )+b\right )}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {x^{2 n}}{2 a^2}\right )}{e n}\) |
((e*x)^(2*n)*(x^(2*n)/(2*a^2) - (I*b^3*x^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) + ((2*I)*b*x^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 ) + (I*b^3*x^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) - ((2*I)*b*x^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) + (b^2*Log[b + a*Sin[c + d *x^n]])/(a^2*(a^2 - b^2)*d^2) - (b^3*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) + (2*b*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) + (b^3*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) - (2*b*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) - (b^2*x^n*Cos[c + d*x^n])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*x^n]))))/(e*n*x^(2*n))
3.1.83.3.1 Defintions of rubi rules used
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*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 0]
Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(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]
Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x _Symbol] :> Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.14 (sec) , antiderivative size = 3867, normalized size of antiderivative = 4.97
1/2/a^2/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)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*l n(e)+2*ln(x)))-2*I*b^2/a^2/(-a^2+b^2)/d/n*x^n/(2*b*exp(I*(c+d*x^n))-I*a*ex p(2*I*(c+d*x^n))+I*a)*(e^n)^2*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*( b*exp(1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e)*csgn(I*x)*csg n(I*e*x)-2*Pi*n*csgn(I*e*x)^3+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e)* csgn(I*e*x)^2+Pi*csgn(I*e*x)^3-Pi*csgn(I*x)*csgn(I*e*x)^2+2*d*x^n+2*c))+I* exp(1/2*I*Pi*csgn(I*e*x)*(-2*csgn(I*e*x)^2*n+2*csgn(I*e)*csgn(I*e*x)*n+2*c sgn(I*x)*csgn(I*e*x)*n-2*csgn(I*e)*csgn(I*x)*n+csgn(I*e*x)^2-csgn(I*e)*csg n(I*e*x)-csgn(I*x)*csgn(I*e*x)))*a)/e-2/d/(a^2-b^2)^2*b*(a^2*exp(2*I*c)-ex p(2*I*c)*b^2)^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csg n(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*P i*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))*x^n*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)))+1/a^2/d/(a^2-b^2)^2*b^3*(a^2* exp(2*I*c)-exp(2*I*c)*b^2)^(1/2)/n/e*(e^n)^2*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))*x^...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2435 vs. \(2 (710) = 1420\).
Time = 0.52 (sec) , antiderivative size = 2435, normalized size of antiderivative = 3.13 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]
1/2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*e^(2*n - 1)*x^(2*n)*sin(d*x^n + c) + (a ^4*b - 2*a^2*b^3 + b^5)*d^2*e^(2*n - 1)*x^(2*n) - 2*(a^3*b^2 - a*b^4)*d*e^ (2*n - 1)*x^n*cos(d*x^n + c) + ((2*I*a^4*b - I*a^2*b^3)*e^(2*n - 1)*sqrt(( a^2 - b^2)/a^2)*sin(d*x^n + c) + (2*I*a^3*b^2 - I*a*b^4)*e^(2*n - 1)*sqrt( (a^2 - b^2)/a^2))*dilog(((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) + ((2*I*a^4*b - I*a^2*b^3)*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*sin(d*x^n + c) + (2*I*a^3*b^ 2 - I*a*b^4)*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2))*dilog(-((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) + ((-2*I*a^4*b + I*a^2*b^3)*e^(2*n - 1)*sqrt((a^2 - b^2)/a ^2)*sin(d*x^n + c) + (-2*I*a^3*b^2 + I*a*b^4)*e^(2*n - 1)*sqrt((a^2 - b^2) /a^2))*dilog(((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) + ((-2*I*a^4*b + I*a^2*b^ 3)*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2)*sin(d*x^n + c) + (-2*I*a^3*b^2 + I*a* b^4)*e^(2*n - 1)*sqrt((a^2 - b^2)/a^2))*dilog(-((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) + ((a^3*b^2 - a*b^4 - (2*a^4*b - a^2*b^3)*c*sqrt((a^2 - b^2)/a^2) )*e^(2*n - 1)*sin(d*x^n + c) + (a^2*b^3 - b^5 - (2*a^3*b^2 - a*b^4)*c*sqrt ((a^2 - b^2)/a^2))*e^(2*n - 1))*log(2*a*cos(d*x^n + c) + 2*I*a*sin(d*x^n + c) + 2*a*sqrt((a^2 - b^2)/a^2) + 2*I*b) + ((a^3*b^2 - a*b^4 - (2*a^4*b...
\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{2 n - 1}}{\left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{2 \, n - 1}}{{\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
-1/2*(4*a*b^3*e^(2*n)*x^n*cos(d*x^n + c) - (a^4 - a^2*b^2)*d*e^(2*n)*x^(2* n)*cos(2*d*x^n + 2*c)^2 - 4*(a^2*b^2 - b^4)*d*e^(2*n)*x^(2*n)*cos(d*x^n + c)^2 - (a^4 - a^2*b^2)*d*e^(2*n)*x^(2*n)*sin(2*d*x^n + 2*c)^2 - 4*(a^2*b^2 - b^4)*d*e^(2*n)*x^(2*n)*sin(d*x^n + c)^2 - 4*(a^3*b - a*b^3)*d*e^(2*n)*x ^(2*n)*sin(d*x^n + c) - (a^4 - a^2*b^2)*d*e^(2*n)*x^(2*n) + 2*(2*a*b^3*e^( 2*n)*x^n*cos(d*x^n + c) + 2*(a^3*b - a*b^3)*d*e^(2*n)*x^(2*n)*sin(d*x^n + c) + (a^4 - a^2*b^2)*d*e^(2*n)*x^(2*n))*cos(2*d*x^n + 2*c) - 2*((a^6 - a^4 *b^2)*d*e*n*cos(2*d*x^n + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*e*n*cos(d*x^n + c)^2 + 4*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c)*sin(2*d*x^n + 2*c) + (a^6 - a^4*b^2)*d*e*n*sin(2*d*x^n + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*e*n*sin(d *x^n + c)^2 + 4*(a^5*b - a^3*b^3)*d*e*n*sin(d*x^n + c) + (a^6 - a^4*b^2)*d *e*n - 2*(2*(a^5*b - a^3*b^3)*d*e*n*sin(d*x^n + c) + (a^6 - a^4*b^2)*d*e*n )*cos(2*d*x^n + 2*c))*integrate(-2*(a^2*b^4*e^(2*n)*x^n*cos(2*c)*sin(2*d*x ^n) + a^2*b^4*e^(2*n)*x^n*cos(2*d*x^n)*sin(2*c) - 2*(a^3*b^3 - a*b^5)*e^(2 *n)*x^n*cos(d*x^n)*cos(c) + 2*(a^3*b^3 - a*b^5)*e^(2*n)*x^n*sin(d*x^n)*sin (c) - (a^3*b^3*e^(2*n)*x^n*cos(d*x^n + c) + (2*a^5*b - a^3*b^3)*d*e^(2*n)* x^(2*n)*sin(d*x^n + c))*cos(2*d*x^n + 2*c) + ((a^3*b^3 - a*b^5)*e^(2*n)*x^ n + (a*b^5*e^(2*n)*x^n*cos(2*c) - (2*a^3*b^3 - a*b^5)*d*e^(2*n)*x^(2*n)*si n(2*c))*cos(2*d*x^n) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(2*n)*x^(2*n)* cos(c) + (a^2*b^4 - b^6)*e^(2*n)*x^n*sin(c))*cos(d*x^n) - (a*b^5*e^(2*n...
\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{2 \, n - 1}}{{\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{2\,n-1}}{{\left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )}^2} \,d x \]