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

Optimal. Leaf size=56 \[ \frac {\sin (a x)}{a^2 x^2}+\frac {\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text {Si}(a x)+\frac {\cos (a x)}{a x} \]

[Out]

cos(a*x)/a/x+Si(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))

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Rubi [A]  time = 0.10, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {4598, 3297, 3299} \[ \frac {\sin (a x)}{a^2 x^2}+\frac {\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text {Si}(a x)+\frac {\cos (a x)}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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])) + SinIntegral[a*x]

Rule 3297

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] - Dist[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 3299

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

Rule 4598

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] - Dist[(b^2*(n - 1))/d^2, In
t[(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]

Rubi steps

\begin {align*} \int \frac {\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx &=\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}\\ &=\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))}-\frac {\int \frac {\cos (a x)}{x^2} \, dx}{a}\\ &=\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))}+\int \frac {\sin (a x)}{x} \, 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)\\ \end {align*}

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Mathematica [C]  time = 7.42, size = 242, normalized size = 4.32 \[ \frac {-i e \text {Ci}(i-a x) (a x \cos (a x)-\sin (a x))+i e \text {Ci}(a x+i) (a x \cos (a x)-\sin (a x))-i e \text {Ei}(-i a x-1) \sin (a x)+i e \text {Ei}(i a x-1) \sin (a x)+i e a x \text {Ei}(-i a x-1) \cos (a x)-i e a x \text {Ei}(i a x-1) \cos (a x)-2 \text {Si}(a x) \sin (a x)-e \text {Si}(i-a x) \sin (a x)+e \text {Si}(a x+i) \sin (a x)+2 a x \text {Si}(a x) \cos (a x)+e a x \text {Si}(i-a x) \cos (a x)-e a x \text {Si}(a x+i) \cos (a x)+\cos (2 a x)+1}{2 a x \cos (a x)-2 \sin (a x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1 + Cos[2*a*x] + I*a*E*x*Cos[a*x]*ExpIntegralEi[-1 - I*a*x] - I*a*E*x*Cos[a*x]*ExpIntegralEi[-1 + I*a*x] - I*
E*CosIntegral[I - a*x]*(a*x*Cos[a*x] - Sin[a*x]) + I*E*CosIntegral[I + a*x]*(a*x*Cos[a*x] - Sin[a*x]) - I*E*Ex
pIntegralEi[-1 - I*a*x]*Sin[a*x] + I*E*ExpIntegralEi[-1 + I*a*x]*Sin[a*x] + 2*a*x*Cos[a*x]*SinIntegral[a*x] -
2*Sin[a*x]*SinIntegral[a*x] + a*E*x*Cos[a*x]*SinIntegral[I - a*x] - E*Sin[a*x]*SinIntegral[I - a*x] - a*E*x*Co
s[a*x]*SinIntegral[I + a*x] + E*Sin[a*x]*SinIntegral[I + a*x])/(2*a*x*Cos[a*x] - 2*Sin[a*x])

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fricas [A]  time = 0.87, size = 45, normalized size = 0.80 \[ \frac {a x \cos \left (a x\right ) \operatorname {Si}\left (a x\right ) + \cos \left (a x\right )^{2} - \sin \left (a x\right ) \operatorname {Si}\left (a x\right )}{a x \cos \left (a x\right ) - \sin \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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))

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giac [C]  time = 0.37, size = 496, normalized size = 8.86 \[ \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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^3*x^3*imag_part(cos_integral(a*x))*tan(1/2*a*x)^4 - a^3*x^3*imag_part(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_integral(a*x)) + a^3*x^3*imag_part(cos_integral(-a*x)) - 2*a^3*x^3*sin_int
egral(a*x) + a*x*imag_part(cos_integral(a*x))*tan(1/2*a*x)^4 - a*x*imag_part(cos_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*i
mag_part(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_integral(a*x)) + a*x*imag_part(co
s_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_in
tegral(-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))

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maple [C]  time = 4.28, size = 108, normalized size = 1.93 \[ \frac {i \Ei \left (1, -i a x \right )}{2}+\frac {i {\mathrm e}^{i a x}}{2 i a x -2}+\frac {i {\mathrm e}^{-i a x}}{2 i a x +2}-\frac {i \Ei \left (1, i a x \right )}{2}+\frac {2 \,{\mathrm e}^{i a x}}{\left (a x +i\right ) \left (a x -i\right ) \left (a x \,{\mathrm e}^{2 i a x}+i {\mathrm e}^{2 i a x}+a x -i\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (a\,x\right )}^3}{x\,{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}{\left (a x \right )}}{x \left (a x \cos {\left (a x \right )} - \sin {\left (a x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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