\(\int \frac {\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx\) [597]

Optimal result

Integrand size = 24, antiderivative size = 56 \[ \int \frac {\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\cos (a x)}{a^2 x^2}+\operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{a x}-\frac {\cos ^2(a x)}{a^2 x^2 (\cos (a x)+a x \sin (a x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4695, 3378, 3383} \[ \int \frac {\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\cos (a x)}{a^2 x^2}-\frac {\cos ^2(a x)}{a^2 x^2 (a x \sin (a x)+\cos (a x))}+\operatorname {CosIntegral}(a x)-\frac {\sin (a x)}{a x} \]

[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 3378

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 3383

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]

Rule 4695

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),
Int[(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
]

Rubi steps \begin{align*} \text {integral}& = -\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}+\operatorname {CosIntegral}(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] (verified)

Result contains complex when optimal does not.

Time = 1.86 (sec) , antiderivative size = 181, normalized size of antiderivative = 3.23 \[ \int \frac {\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{2} \left (2 \operatorname {CosIntegral}(a x)-e \operatorname {CosIntegral}(i-a x)-e \operatorname {CosIntegral}(i+a x)+\frac {i \cos (a x)+e (i+a x) \operatorname {ExpIntegralEi}(-1+i a x)-\sin (a x)}{i+a x}-\frac {i \cos (a x)+e (i-a x) \operatorname {ExpIntegralEi}(-1-i a x)+\sin (a x)}{-i+a x}-\frac {2}{(-i+a x) (i+a x) (\cos (a x)+a x \sin (a x))}-i e \text {Si}(i-a x)-i e \text {Si}(i+a x)\right ) \]

[In]

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

[Out]

(2*CosIntegral[a*x] - E*CosIntegral[I - a*x] - E*CosIntegral[I + a*x] + (I*Cos[a*x] + E*(I + a*x)*ExpIntegralE
i[-1 + I*a*x] - Sin[a*x])/(I + a*x) - (I*Cos[a*x] + E*(I - a*x)*ExpIntegralEi[-1 - I*a*x] + Sin[a*x])/(-I + a*
x) - 2/((-I + a*x)*(I + a*x)*(Cos[a*x] + a*x*Sin[a*x])) - I*E*SinIntegral[I - a*x] - I*E*SinIntegral[I + a*x])
/2

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.38 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.12

method	result	size
risch	\(-\frac {i \operatorname {Ei}_{1}\left (-i a x \right ) a x -\operatorname {Ei}_{1}\left (-i a x \right )+{\mathrm e}^{i a x}}{2 \left (i a x -1\right )}+\frac {{\mathrm e}^{-i a x}}{2 i a x +2}-\frac {\operatorname {Ei}_{1}\left (i a x \right )}{2}-\frac {2 i {\mathrm e}^{i a x}}{\left (a x +i\right ) \left (a x -i\right ) \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right )}\)	\(119\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \operatorname {Ci}\left (a x\right ) \sin \left (a x\right ) + \cos \left (a x\right )^{2} + \cos \left (a x\right ) \operatorname {Ci}\left (a x\right ) - 1}{a x \sin \left (a x\right ) + \cos \left (a x\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [C] (verification not implemented)

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

Time = 0.34 (sec) , antiderivative size = 366, normalized size of antiderivative = 6.54 \[ \int \frac {\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx=\frac {2 \, a^{3} x^{3} \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + 2 \, a^{3} x^{3} \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} - a^{2} x^{2} \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} - a^{2} x^{2} \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{3} x^{3} \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) + 2 \, a^{3} x^{3} \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) - 8 \, a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{2} + 2 \, a x \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + 2 \, a x \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{3} + a^{2} x^{2} \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) + a^{2} x^{2} \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) - \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} - \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a x \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) + 2 \, a x \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) \tan \left (\frac {1}{2} \, a x\right ) - 2 \, \tan \left (\frac {1}{2} \, a x\right )^{4} - 12 \, \tan \left (\frac {1}{2} \, a x\right )^{2} + \Re \left ( \operatorname {Ci}\left (a x\right ) \right ) + \Re \left ( \operatorname {Ci}\left (-a x\right ) \right ) - 2}{2 \, {\left (2 \, a^{3} x^{3} \tan \left (\frac {1}{2} \, a x\right )^{3} - a^{2} x^{2} \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a^{3} x^{3} \tan \left (\frac {1}{2} \, a x\right ) + 2 \, a x \tan \left (\frac {1}{2} \, a x\right )^{3} + a^{2} x^{2} - \tan \left (\frac {1}{2} \, a x\right )^{4} + 2 \, a x \tan \left (\frac {1}{2} \, a x\right ) + 1\right )}} \]

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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