\(\int (e x)^{-1+3 n} (a+b \csc (c+d x^n)) \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 221 \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \] Output:

1/3*a*(e*x)^(3*n)/e/n-2*b*(e*x)^(3*n)*arctanh(exp(I*(c+d*x^n)))/d/e/n/(x^n 
)+2*I*b*(e*x)^(3*n)*polylog(2,-exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))-2*I*b*( 
e*x)^(3*n)*polylog(2,exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))-2*b*(e*x)^(3*n)*p 
olylog(3,-exp(I*(c+d*x^n)))/d^3/e/n/(x^(3*n))+2*b*(e*x)^(3*n)*polylog(3,ex 
p(I*(c+d*x^n)))/d^3/e/n/(x^(3*n))
 

Mathematica [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2010, 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.

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

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (a (e x)^{3 n-1}+b (e x)^{3 n-1} \csc \left (c+d x^n\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (d x^n+c\right )}\right )}{d^2 e n}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

Maple [F]

\[\int \left (e x \right )^{-1+3 n} \left (a +b \csc \left (c +d \,x^{n}\right )\right )d x\]

Input:

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

Output:

int((e*x)^(-1+3*n)*(a+b*csc(c+d*x^n)),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. 557 vs. \(2 (211) = 422\).

Time = 0.12 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.52 \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\frac {2 \, a d^{3} e^{3 \, n - 1} x^{3 \, n} - 3 \, b d^{2} e^{3 \, n - 1} x^{2 \, n} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, b d^{2} e^{3 \, n - 1} x^{2 \, n} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) + 3 \, b c^{2} e^{3 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) + 3 \, b c^{2} e^{3 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right )}{6 \, d^{3} n} \] Input:

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

Output:

1/6*(2*a*d^3*e^(3*n - 1)*x^(3*n) - 3*b*d^2*e^(3*n - 1)*x^(2*n)*log(cos(d*x 
^n + c) + I*sin(d*x^n + c) + 1) - 3*b*d^2*e^(3*n - 1)*x^(2*n)*log(cos(d*x^ 
n + c) - I*sin(d*x^n + c) + 1) - 6*I*b*d*e^(3*n - 1)*x^n*dilog(cos(d*x^n + 
 c) + I*sin(d*x^n + c)) + 6*I*b*d*e^(3*n - 1)*x^n*dilog(cos(d*x^n + c) - I 
*sin(d*x^n + c)) - 6*I*b*d*e^(3*n - 1)*x^n*dilog(-cos(d*x^n + c) + I*sin(d 
*x^n + c)) + 6*I*b*d*e^(3*n - 1)*x^n*dilog(-cos(d*x^n + c) - I*sin(d*x^n + 
 c)) + 3*b*c^2*e^(3*n - 1)*log(-1/2*cos(d*x^n + c) + 1/2*I*sin(d*x^n + c) 
+ 1/2) + 3*b*c^2*e^(3*n - 1)*log(-1/2*cos(d*x^n + c) - 1/2*I*sin(d*x^n + c 
) + 1/2) + 6*b*e^(3*n - 1)*polylog(3, cos(d*x^n + c) + I*sin(d*x^n + c)) + 
 6*b*e^(3*n - 1)*polylog(3, cos(d*x^n + c) - I*sin(d*x^n + c)) - 6*b*e^(3* 
n - 1)*polylog(3, -cos(d*x^n + c) + I*sin(d*x^n + c)) - 6*b*e^(3*n - 1)*po 
lylog(3, -cos(d*x^n + c) - I*sin(d*x^n + c)) + 3*(b*d^2*e^(3*n - 1)*x^(2*n 
) - b*c^2*e^(3*n - 1))*log(-cos(d*x^n + c) + I*sin(d*x^n + c) + 1) + 3*(b* 
d^2*e^(3*n - 1)*x^(2*n) - b*c^2*e^(3*n - 1))*log(-cos(d*x^n + c) - I*sin(d 
*x^n + c) + 1))/(d^3*n)
 

Sympy [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{3 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\frac {e^{3 n} \left (x^{3 n} a +3 \left (\int \frac {x^{3 n} \csc \left (x^{n} d +c \right )}{x}d x \right ) b n \right )}{3 e n} \] Input:

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

Output:

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