Integrand size = 26, antiderivative size = 56 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {\cos (a x)}{a x}+\frac {\sin (a x)}{a^2 x^2}+\frac {\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text {Si}(a x) \]
Result contains complex when optimal does not.
Time = 2.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.86 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {1}{2} \left (\frac {e^{-i a x}}{-i+a x}+\frac {e^{i a x}}{i+a x}-i e \operatorname {CosIntegral}(i-a x)+i e \operatorname {CosIntegral}(i+a x)+i e \operatorname {ExpIntegralEi}(-1-i a x)-i e \operatorname {ExpIntegralEi}(-1+i a x)+\frac {2}{(-i+a x) (i+a x) (a x \cos (a x)-\sin (a x))}+2 \text {Si}(a x)+e \text {Si}(i-a x)-e \text {Si}(i+a x)\right ) \]
(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
Time = 0.43 (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 |
| \(\displaystyle \frac {\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}-\frac {2 \left (\frac {1}{2} a \left (-a \text {Si}(a x)-\frac {\cos (a x)}{x}\right )-\frac {\sin (a x)}{2 x^2}\right )}{a^2}\) |
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
3.6.88.3.1 Defintions of rubi rules used
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]
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]
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]
Result contains complex when optimal does not.
Time = 1.61 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.14
| method | result | size |
| risch | \(\frac {i \left (i \operatorname {Ei}_{1}\left (-i a x \right ) a x -\operatorname {Ei}_{1}\left (-i a x \right )+{\mathrm e}^{i a x}\right )}{2 i a x -2}+\frac {i {\mathrm e}^{-i a x}}{2 i a x +2}-\frac {i \operatorname {Ei}_{1}\left (i a x \right )}{2}+\frac {2 \,{\mathrm e}^{i a x}}{\left (a x +i\right ) \left (a x -i\right ) \left (a x \,{\mathrm e}^{2 i a x}+i {\mathrm e}^{2 i a x}+a x -i\right )}\) | \(120\) |
1/2*I*(I*Ei(1,-I*a*x)*a*x-Ei(1,-I*a*x)+exp(I*a*x))/(-1+I*a*x)+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)/(a*x*exp(2* I*a*x)+I*exp(2*I*a*x)+a*x-I)
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\frac {a x \cos \left (a x\right ) \operatorname {Si}\left (a x\right ) + \cos \left (a x\right )^{2} - \sin \left (a x\right ) \operatorname {Si}\left (a x\right )}{a x \cos \left (a x\right ) - \sin \left (a x\right )} \]
(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))
\[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\int \frac {\sin ^{3}{\left (a x \right )}}{x \left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.36 (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=\frac {a^{3} x^{3} \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} - a^{3} x^{3} \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{3} x^{3} \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 2 \, a^{2} x^{2} \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + 4 \, a^{2} x^{2} \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 2 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{4} - a^{3} x^{3} \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) + a^{3} x^{3} \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) - 2 \, a^{3} x^{3} \operatorname {Si}\left (a x\right ) + a x \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} - a x \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a x \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) - 2 \, a^{2} x^{2} \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) + 4 \, a^{2} x^{2} \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right ) + 4 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{2} - 2 \, a^{2} x^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 2 \, \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + 4 \, \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - 4 \, \tan \left (\frac {1}{2} \, a x\right )^{4} - a x \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) + a x \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) - 2 \, a x \operatorname {Si}\left (a x\right ) + 2 \, \Im \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) + 4 \, \operatorname {Si}\left (a x\right ) \tan \left (\frac {1}{2} \, a x\right ) - 4}{2 \, {\left (a^{3} x^{3} \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{3} - a^{3} x^{3} + a x \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right ) + 2 \, \tan \left (\frac {1}{2} \, a x\right )^{3} - a x + 2 \, \tan \left (\frac {1}{2} \, a x\right )\right )}} \]
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))
Timed out. \[ \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx=\int \frac {{\sin \left (a\,x\right )}^3}{x\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]