\(\int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx\) [598]

Optimal result

Integrand size = 21, antiderivative size = 34 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{a^2 x}-\frac {\cos (a x)}{a^2 x (\cos (a x)+a x \sin (a x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4693} \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{a^2 x}-\frac {\cos (a x)}{a^2 x (a x \sin (a x)+\cos (a x))} \]

[In]

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

[Out]

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

Rule 4693

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{a^2 x}-\frac {\cos (a x)}{a^2 x (\cos (a x)+a x \sin (a x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin (a x)}{a (\cos (a x)+a x \sin (a x))} \]

[In]

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

[Out]

Sin[a*x]/(a*(Cos[a*x] + a*x*Sin[a*x]))

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91

method	result	size
parallelrisch	\(-\frac {2 \tan \left (\frac {a x}{2}\right )}{\left (-1-2 \tan \left (\frac {a x}{2}\right ) a x +\tan \left (\frac {a x}{2}\right )^{2}\right ) a}\)	\(31\)
risch	\(\frac {1}{a \left (a x +i\right )}-\frac {2 i}{\left (a x +i\right ) \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right ) a}\)	\(55\)
norman	\(\frac {\frac {2 \tan \left (\frac {a x}{2}\right )}{a}+\frac {4 \tan \left (\frac {a x}{2}\right )^{3}}{a}+\frac {2 \tan \left (\frac {a x}{2}\right )^{5}}{a}}{\left (1+\tan \left (\frac {a x}{2}\right )^{2}\right )^{2} \left (1+2 \tan \left (\frac {a x}{2}\right ) a x -\tan \left (\frac {a x}{2}\right )^{2}\right )}\)	\(70\)

[In]

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

[Out]

-2*tan(1/2*a*x)/(-1-2*tan(1/2*a*x)*a*x+tan(1/2*a*x)^2)/a

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin \left (a x\right )}{a^{2} x \sin \left (a x\right ) + a \cos \left (a x\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin {\left (a x \right )}}{a^{2} x \sin {\left (a x \right )} + a \cos {\left (a x \right )}} \]

[In]

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

[Out]

sin(a*x)/(a**2*x*sin(a*x) + a*cos(a*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (34) = 68\).

Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.35 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \cos \left (2 \, a x\right )^{2} + a x \sin \left (2 \, a x\right )^{2} - 2 \, a x \cos \left (2 \, a x\right ) + a x + 2 \, \sin \left (2 \, a x\right )}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} + 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a} \]

[In]

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

[Out]

(a*x*cos(2*a*x)^2 + a*x*sin(2*a*x)^2 - 2*a*x*cos(2*a*x) + a*x + 2*sin(2*a*x))/((a^2*x^2 + (a^2*x^2 + 1)*cos(2*
a*x)^2 + 4*a*x*sin(2*a*x) + (a^2*x^2 + 1)*sin(2*a*x)^2 - 2*(a^2*x^2 - 1)*cos(2*a*x) + 1)*a)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {2 \, \tan \left (\frac {1}{2} \, a x\right )}{2 \, a^{2} x \tan \left (\frac {1}{2} \, a x\right ) - a \tan \left (\frac {1}{2} \, a x\right )^{2} + a} \]

[In]

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

[Out]

2*tan(1/2*a*x)/(2*a^2*x*tan(1/2*a*x) - a*tan(1/2*a*x)^2 + a)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin \left (a\,x\right )}{a\,\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )} \]

[In]

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

[Out]

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