\(\int \frac {(e x)^{-1+2 n}}{(a+b \csc (c+d x^n))^2} \, dx\) [83]

Optimal result

Integrand size = 24, antiderivative size = 778 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sin \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cos \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \sin \left (c+d x^n\right )\right )} \] Output:

1/2*(e*x)^(2*n)/a^2/e/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)+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/(-a^2+b^2)^(1/2)/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)-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/(-a^2+b^2)^(1/2)/d/e/n/(x^n)+b^2*(e*x)^(2*n)*ln(b+a*s 
in(c+d*x^n))/a^2/(a^2-b^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))+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/(-a^2+b^2)^(1/2)/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))-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/(- 
a^2+b^2)^(1/2)/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))
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2839\) vs. \(2(778)=1556\).

Time = 9.43 (sec) , antiderivative size = 2839, normalized size of antiderivative = 3.65 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

-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...
 

Rubi [A] (verified)

Time = 1.47 (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

\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \left (\frac {2 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}+\frac {b^2 \log \left (a \sin \left (c+d x^n\right )+b\right )}{a^2 d^2 \left (a^2-b^2\right )}+\frac {2 i b x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {2 i b x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {b^2 x^n \cos \left (c+d x^n\right )}{a d \left (a^2-b^2\right ) \left (a \sin \left (c+d x^n\right )+b\right )}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^n \log \left (1-\frac {i a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {x^{2 n}}{2 a^2}\right )}{e n}\)

Input:

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

Output:

((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))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

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]
 

rule 4693

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]
 

rule 4697

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]
 

Maple [C] (warning: unable to verify)

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

Time = 1.42 (sec) , antiderivative size = 3887, normalized size of antiderivative = 5.00

method	result	size
risch	\(\text {Expression too large to display}\)	\(3887\)

Input:

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

Output:

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

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. 2435 vs. \(2 (710) = 1420\).

Time = 0.34 (sec) , antiderivative size = 2435, normalized size of antiderivative = 3.13 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-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...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \csc \left (c+d x^n\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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