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

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

[Out]

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

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Rubi [A]  time = 0.09317, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4599, 3297, 3302} \[ \frac{\cos (a x)}{a^2 x^2}-\frac{\cos ^2(a x)}{a^2 x^2 (a x \sin (a x)+\cos (a x))}+\text{CosIntegral}(a x)-\frac{\sin (a x)}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4599

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

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 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [C]  time = 7.47208, size = 237, normalized size = 4.23 \[ \frac{-e a x \text{CosIntegral}(a x+i) \sin (a x)-e \text{CosIntegral}(a x+i) \cos (a x)+2 \text{CosIntegral}(a x) (a x \sin (a x)+\cos (a x))-e \text{CosIntegral}(-a x+i) (a x \sin (a x)+\cos (a x))+e a x \text{ExpIntegralEi}(-1-i a x) \sin (a x)+e a x \text{ExpIntegralEi}(-1+i a x) \sin (a x)+e \text{ExpIntegralEi}(-1-i a x) \cos (a x)+e \text{ExpIntegralEi}(-1+i a x) \cos (a x)-i e a x \text{Si}(i-a x) \sin (a x)-i e a x \text{Si}(a x+i) \sin (a x)-i e \text{Si}(i-a x) \cos (a x)-i e \text{Si}(a x+i) \cos (a x)+\cos (2 a x)-1}{2 (a x \sin (a x)+\cos (a x))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 1.171, size = 106, normalized size = 1.9 \begin{align*} -{\frac{{{\rm e}^{iax}}}{-2+2\,iax}}-{\frac{{\it Ei} \left ( 1,-iax \right ) }{2}}+{\frac{{{\rm e}^{-iax}}}{2+2\,iax}}-{\frac{{\it Ei} \left ( 1,iax \right ) }{2}}-{\frac{2\,i{{\rm e}^{iax}}}{ \left ( ax+i \right ) \left ( ax-i \right ) \left ( ax{{\rm e}^{2\,iax}}-ax+i{{\rm e}^{2\,iax}}+i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.21616, size = 219, normalized size = 3.91 \begin{align*} \frac{{\left (\operatorname{Ci}\left (a x\right ) + \operatorname{Ci}\left (-a x\right )\right )} \cos \left (a x\right ) + 2 \, \cos \left (a x\right )^{2} +{\left (a x \operatorname{Ci}\left (a x\right ) + a x \operatorname{Ci}\left (-a x\right )\right )} \sin \left (a x\right ) - 2}{2 \,{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((cos_integral(a*x) + cos_integral(-a*x))*cos(a*x) + 2*cos(a*x)^2 + (a*x*cos_integral(a*x) + a*x*cos_integ
ral(-a*x))*sin(a*x) - 2)/(a*x*sin(a*x) + cos(a*x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.28893, size = 494, normalized size = 8.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^3*x^3*real_part(cos_integral(a*x))*tan(1/2*a*x)^3 + 2*a^3*x^3*real_part(cos_integral(-a*x))*tan(1/2*a
*x)^3 - a^2*x^2*real_part(cos_integral(a*x))*tan(1/2*a*x)^4 - a^2*x^2*real_part(cos_integral(-a*x))*tan(1/2*a*
x)^4 + 2*a^3*x^3*real_part(cos_integral(a*x))*tan(1/2*a*x) + 2*a^3*x^3*real_part(cos_integral(-a*x))*tan(1/2*a
*x) - 8*a^2*x^2*tan(1/2*a*x)^2 + 2*a*x*real_part(cos_integral(a*x))*tan(1/2*a*x)^3 + 2*a*x*real_part(cos_integ
ral(-a*x))*tan(1/2*a*x)^3 + a^2*x^2*real_part(cos_integral(a*x)) + a^2*x^2*real_part(cos_integral(-a*x)) - rea
l_part(cos_integral(a*x))*tan(1/2*a*x)^4 - real_part(cos_integral(-a*x))*tan(1/2*a*x)^4 + 2*a*x*real_part(cos_
integral(a*x))*tan(1/2*a*x) + 2*a*x*real_part(cos_integral(-a*x))*tan(1/2*a*x) - 2*tan(1/2*a*x)^4 - 12*tan(1/2
*a*x)^2 + real_part(cos_integral(a*x)) + real_part(cos_integral(-a*x)) - 2)/(2*a^3*x^3*tan(1/2*a*x)^3 - a^2*x^
2*tan(1/2*a*x)^4 + 2*a^3*x^3*tan(1/2*a*x) + 2*a*x*tan(1/2*a*x)^3 + a^2*x^2 - tan(1/2*a*x)^4 + 2*a*x*tan(1/2*a*
x) + 1)