\(\int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx\) [535]

Optimal result

Integrand size = 24, antiderivative size = 104 \[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=-\frac {2 x \text {arctanh}\left (e^{i a x}\right )}{a^3}-\frac {\csc (a x)}{a^4}-\frac {x \cot (a x) \csc (a x)}{a^3}+\frac {i \operatorname {PolyLog}\left (2,-e^{i a x}\right )}{a^4}-\frac {i \operatorname {PolyLog}\left (2,e^{i a x}\right )}{a^4}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))} \] Output:

-2*x*arctanh(exp(I*a*x))/a^3-csc(a*x)/a^4-x*cot(a*x)*csc(a*x)/a^3+I*polylo 
g(2,-exp(I*a*x))/a^4-I*polylog(2,exp(I*a*x))/a^4+x^2*csc(a*x)^2/a^2/(a*x*c 
os(a*x)-sin(a*x))
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.51 \[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\frac {\csc (a x)+a^2 x^2 \csc (a x)-a x \log \left (1-e^{i a x}\right )+a^2 x^2 \cot (a x) \log \left (1-e^{i a x}\right )+a x \log \left (1+e^{i a x}\right )-a^2 x^2 \cot (a x) \log \left (1+e^{i a x}\right )+i (-1+a x \cot (a x)) \operatorname {PolyLog}\left (2,-e^{i a x}\right )-i (-1+a x \cot (a x)) \operatorname {PolyLog}\left (2,e^{i a x}\right )}{a^4 (-1+a x \cot (a x))} \] Input:

Integrate[(x^3*Csc[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]
 

Output:

(Csc[a*x] + a^2*x^2*Csc[a*x] - a*x*Log[1 - E^(I*a*x)] + a^2*x^2*Cot[a*x]*L 
og[1 - E^(I*a*x)] + a*x*Log[1 + E^(I*a*x)] - a^2*x^2*Cot[a*x]*Log[1 + E^(I 
*a*x)] + I*(-1 + a*x*Cot[a*x])*PolyLog[2, -E^(I*a*x)] - I*(-1 + a*x*Cot[a* 
x])*PolyLog[2, E^(I*a*x)])/(a^4*(-1 + a*x*Cot[a*x]))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5111, 3042, 4673, 3042, 4671, 2715, 2838}

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.

\(\displaystyle \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx\)

\(\Big \downarrow \) 5111

\(\displaystyle \frac {2 \int x \csc ^3(a x)dx}{a^2}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \int x \csc (a x)^3dx}{a^2}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\)

\(\Big \downarrow \) 4673

\(\displaystyle \frac {2 \left (\frac {1}{2} \int x \csc (a x)dx-\frac {\csc (a x)}{2 a^2}-\frac {x \cot (a x) \csc (a x)}{2 a}\right )}{a^2}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {1}{2} \int x \csc (a x)dx-\frac {\csc (a x)}{2 a^2}-\frac {x \cot (a x) \csc (a x)}{2 a}\right )}{a^2}+\frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {2 \left (\frac {1}{2} \left (-\frac {\int \log \left (1-e^{i a x}\right )dx}{a}+\frac {\int \log \left (1+e^{i a x}\right )dx}{a}-\frac {2 x \text {arctanh}\left (e^{i a x}\right )}{a}\right )-\frac {\csc (a x)}{2 a^2}-\frac {x \cot (a x) \csc (a x)}{2 a}\right )}{a^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {2 \left (\frac {1}{2} \left (\frac {i \int e^{-i a x} \log \left (1-e^{i a x}\right )de^{i a x}}{a^2}-\frac {i \int e^{-i a x} \log \left (1+e^{i a x}\right )de^{i a x}}{a^2}-\frac {2 x \text {arctanh}\left (e^{i a x}\right )}{a}\right )-\frac {\csc (a x)}{2 a^2}-\frac {x \cot (a x) \csc (a x)}{2 a}\right )}{a^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac {2 \left (\frac {1}{2} \left (\frac {i \operatorname {PolyLog}\left (2,-e^{i a x}\right )}{a^2}-\frac {i \operatorname {PolyLog}\left (2,e^{i a x}\right )}{a^2}-\frac {2 x \text {arctanh}\left (e^{i a x}\right )}{a}\right )-\frac {\csc (a x)}{2 a^2}-\frac {x \cot (a x) \csc (a x)}{2 a}\right )}{a^2}\)

Input:

Int[(x^3*Csc[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]
 

Output:

(2*(-1/2*Csc[a*x]/a^2 - (x*Cot[a*x]*Csc[a*x])/(2*a) + ((-2*x*ArcTanh[E^(I* 
a*x)])/a + (I*PolyLog[2, -E^(I*a*x)])/a^2 - (I*PolyLog[2, E^(I*a*x)])/a^2) 
/2))/a^2 + (x^2*Csc[a*x]^2)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))
 

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 5111

Int[(Csc[(a_.)*(x_)]^(n_.)*((b_.)*(x_))^(m_.))/(Cos[(a_.)*(x_)]*(d_.)*(x_) 
+ (c_.)*Sin[(a_.)*(x_)])^2, x_Symbol] :> Simp[b*(b*x)^(m - 1)*(Csc[a*x]^(n 
+ 1)/(a*d*(c*Sin[a*x] + d*x*Cos[a*x]))), x] + Simp[b^2*((n + 1)/d^2)   Int[ 
(b*x)^(m - 2)*Csc[a*x]^(n + 2), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && E 
qQ[a*c + d, 0] && EqQ[m, n + 2]
 

Maple [F]

\[\int \frac {x^{3} \csc \left (a x \right )}{\left (a x \cos \left (a x \right )-\sin \left (a x \right )\right )^{2}}d x\]

Input:

int(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x)
 

Output:

int(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,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. 295 vs. \(2 (89) = 178\).

Time = 0.09 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.84 \[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\frac {2 \, a^{2} x^{2} - {\left (i \, a x \cos \left (a x\right ) - i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) - {\left (-i \, a x \cos \left (a x\right ) + i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) - {\left (i \, a x \cos \left (a x\right ) - i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (-\cos \left (a x\right ) + i \, \sin \left (a x\right )\right ) - {\left (-i \, a x \cos \left (a x\right ) + i \, \sin \left (a x\right )\right )} {\rm Li}_2\left (-\cos \left (a x\right ) - i \, \sin \left (a x\right )\right ) - {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) - {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (-\cos \left (a x\right ) + i \, \sin \left (a x\right ) + 1\right ) + {\left (a^{2} x^{2} \cos \left (a x\right ) - a x \sin \left (a x\right )\right )} \log \left (-\cos \left (a x\right ) - i \, \sin \left (a x\right ) + 1\right ) + 2}{2 \, {\left (a^{5} x \cos \left (a x\right ) - a^{4} \sin \left (a x\right )\right )}} \] Input:

integrate(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="fricas")
 

Output:

1/2*(2*a^2*x^2 - (I*a*x*cos(a*x) - I*sin(a*x))*dilog(cos(a*x) + I*sin(a*x) 
) - (-I*a*x*cos(a*x) + I*sin(a*x))*dilog(cos(a*x) - I*sin(a*x)) - (I*a*x*c 
os(a*x) - I*sin(a*x))*dilog(-cos(a*x) + I*sin(a*x)) - (-I*a*x*cos(a*x) + I 
*sin(a*x))*dilog(-cos(a*x) - I*sin(a*x)) - (a^2*x^2*cos(a*x) - a*x*sin(a*x 
))*log(cos(a*x) + I*sin(a*x) + 1) - (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log( 
cos(a*x) - I*sin(a*x) + 1) + (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log(-cos(a* 
x) + I*sin(a*x) + 1) + (a^2*x^2*cos(a*x) - a*x*sin(a*x))*log(-cos(a*x) - I 
*sin(a*x) + 1) + 2)/(a^5*x*cos(a*x) - a^4*sin(a*x))
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\int \frac {x^{3} \csc {\left (a x \right )}}{\left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \] Input:

integrate(x**3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))**2,x)
 

Output:

Integral(x**3*csc(a*x)/(a*x*cos(a*x) - sin(a*x))**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\int { \frac {x^{3} \csc \left (a x\right )}{{\left (a x \cos \left (a x\right ) - \sin \left (a x\right )\right )}^{2}} \,d x } \] Input:

integrate(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x, algorithm="giac")
 

Output:

integrate(x^3*csc(a*x)/(a*x*cos(a*x) - sin(a*x))^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\int \frac {x^3}{\sin \left (a\,x\right )\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \] Input:

int(x^3/(sin(a*x)*(sin(a*x) - a*x*cos(a*x))^2),x)
 

Output:

int(x^3/(sin(a*x)*(sin(a*x) - a*x*cos(a*x))^2), x)
 

Reduce [F]

\[ \int \frac {x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx=\int \frac {\csc \left (a x \right ) x^{3}}{\cos \left (a x \right )^{2} a^{2} x^{2}-2 \cos \left (a x \right ) \sin \left (a x \right ) a x +\sin \left (a x \right )^{2}}d x \] Input:

int(x^3*csc(a*x)/(a*x*cos(a*x)-sin(a*x))^2,x)
 

Output:

int((csc(a*x)*x**3)/(cos(a*x)**2*a**2*x**2 - 2*cos(a*x)*sin(a*x)*a*x + sin 
(a*x)**2),x)