\(\int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx\) [531]

Optimal result

Integrand size = 26, antiderivative size = 56 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {\cos (a x)}{a x}+\frac {\sin (a x)}{a^2 x^2}+\frac {\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text {Si}(a x) \] Output:

cos(a*x)/a/x+sin(a*x)/a^2/x^2+sin(a*x)^2/a^2/x^2/(a*x*cos(a*x)-sin(a*x))+S 
i(a*x)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.78 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.86 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {1}{2} \left (\frac {e^{-i a x}}{-i+a x}+\frac {e^{i a x}}{i+a x}-i e \operatorname {CosIntegral}(i-a x)+i e \operatorname {CosIntegral}(i+a x)+i e \operatorname {ExpIntegralEi}(-1-i a x)-i e \operatorname {ExpIntegralEi}(-1+i a x)+\frac {2}{(-i+a x) (i+a x) (a x \cos (a x)-\sin (a x))}+2 \text {Si}(a x)+e \text {Si}(i-a x)-e \text {Si}(i+a x)\right ) \] Input:

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

Output:

(1/(E^(I*a*x)*(-I + a*x)) + E^(I*a*x)/(I + a*x) - I*E*CosIntegral[I - a*x] 
 + I*E*CosIntegral[I + a*x] + I*E*ExpIntegralEi[-1 - I*a*x] - I*E*ExpInteg 
ralEi[-1 + I*a*x] + 2/((-I + a*x)*(I + a*x)*(a*x*Cos[a*x] - Sin[a*x])) + 2 
*SinIntegral[a*x] + E*SinIntegral[I - a*x] - E*SinIntegral[I + a*x])/2
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5109, 3042, 3778, 3042, 3778, 25, 3042, 3780}

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.

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

\(\Big \downarrow \) 5109

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}-\frac {2 \left (\frac {1}{2} a \left (-a \text {Si}(a x)-\frac {\cos (a x)}{x}\right )-\frac {\sin (a x)}{2 x^2}\right )}{a^2}\)

Input:

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

Output:

Sin[a*x]^2/(a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])) - (2*(-1/2*Sin[a*x]/x^2 + ( 
a*(-(Cos[a*x]/x) - a*SinIntegral[a*x]))/2))/a^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 5109

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.14

method	result	size
risch	\(\frac {i \left (i \operatorname {expIntegral}_{1}\left (-i a x \right ) a x +{\mathrm e}^{i a x}-\operatorname {expIntegral}_{1}\left (-i a x \right )\right )}{2 i a x -2}+\frac {i {\mathrm e}^{-i a x}}{2 i a x +2}-\frac {i \operatorname {expIntegral}_{1}\left (i a x \right )}{2}+\frac {2 \,{\mathrm e}^{i a x}}{\left (a x +i\right ) \left (a x -i\right ) \left ({\mathrm e}^{2 i a x} a x +i {\mathrm e}^{2 i a x}+a x -i\right )}\)	\(120\)

Input:

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

Output:

1/2*I*(I*Ei(1,-I*a*x)*a*x+exp(I*a*x)-Ei(1,-I*a*x))/(I*a*x-1)+1/2*I*exp(-I* 
a*x)/(I*a*x+1)-1/2*I*Ei(1,I*a*x)+2*exp(I*a*x)/(a*x+I)/(a*x-I)/(exp(2*I*a*x 
)*a*x+I*exp(2*I*a*x)+a*x-I)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {a x \cos \left (a x\right ) \operatorname {Si}\left (a x\right ) + \cos \left (a x\right )^{2} - \sin \left (a x\right ) \operatorname {Si}\left (a x\right )}{a x \cos \left (a x\right ) - \sin \left (a x\right )} \] Input:

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

Output:

(a*x*cos(a*x)*sin_integral(a*x) + cos(a*x)^2 - sin(a*x)*sin_integral(a*x)) 
/(a*x*cos(a*x) - sin(a*x))
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(sin(a*x)^3/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 [C] (verification not implemented)

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

Time = 0.35 (sec) , antiderivative size = 496, normalized size of antiderivative = 8.86 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(a^3*x^3*imag_part(cos_integral(a*x))*tan(1/2*a*x)^4 - a^3*x^3*imag_pa 
rt(cos_integral(-a*x))*tan(1/2*a*x)^4 + 2*a^3*x^3*sin_integral(a*x)*tan(1/ 
2*a*x)^4 + 2*a^2*x^2*imag_part(cos_integral(a*x))*tan(1/2*a*x)^3 - 2*a^2*x 
^2*imag_part(cos_integral(-a*x))*tan(1/2*a*x)^3 + 4*a^2*x^2*sin_integral(a 
*x)*tan(1/2*a*x)^3 - 2*a^2*x^2*tan(1/2*a*x)^4 - a^3*x^3*imag_part(cos_inte 
gral(a*x)) + a^3*x^3*imag_part(cos_integral(-a*x)) - 2*a^3*x^3*sin_integra 
l(a*x) + a*x*imag_part(cos_integral(a*x))*tan(1/2*a*x)^4 - a*x*imag_part(c 
os_integral(-a*x))*tan(1/2*a*x)^4 + 2*a*x*sin_integral(a*x)*tan(1/2*a*x)^4 
 + 2*a^2*x^2*imag_part(cos_integral(a*x))*tan(1/2*a*x) - 2*a^2*x^2*imag_pa 
rt(cos_integral(-a*x))*tan(1/2*a*x) + 4*a^2*x^2*sin_integral(a*x)*tan(1/2* 
a*x) + 4*a^2*x^2*tan(1/2*a*x)^2 - 2*a^2*x^2 + 2*imag_part(cos_integral(a*x 
))*tan(1/2*a*x)^3 - 2*imag_part(cos_integral(-a*x))*tan(1/2*a*x)^3 + 4*sin 
_integral(a*x)*tan(1/2*a*x)^3 - 4*tan(1/2*a*x)^4 - a*x*imag_part(cos_integ 
ral(a*x)) + a*x*imag_part(cos_integral(-a*x)) - 2*a*x*sin_integral(a*x) + 
2*imag_part(cos_integral(a*x))*tan(1/2*a*x) - 2*imag_part(cos_integral(-a* 
x))*tan(1/2*a*x) + 4*sin_integral(a*x)*tan(1/2*a*x) - 4)/(a^3*x^3*tan(1/2* 
a*x)^4 + 2*a^2*x^2*tan(1/2*a*x)^3 - a^3*x^3 + a*x*tan(1/2*a*x)^4 + 2*a^2*x 
^2*tan(1/2*a*x) + 2*tan(1/2*a*x)^3 - a*x + 2*tan(1/2*a*x))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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