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

Optimal. Leaf size=34 \[ \frac{1}{a^2 x}-\frac{\cos (a x)}{a^2 x (a x \sin (a x)+\cos (a x))} \]

[Out]

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

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Rubi [A]  time = 0.0224119, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4597} \[ \frac{1}{a^2 x}-\frac{\cos (a x)}{a^2 x (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[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 4597

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*} \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))}\\ \end{align*}

Mathematica [A]  time = 0.220609, size = 22, normalized size = 0.65 \[ \frac{\sin (a x)}{a (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[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]))

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Maple [B]  time = 0.663, size = 70, normalized size = 2.1 \begin{align*}{ \left ( 2\,{\frac{\tan \left ( 1/2\,ax \right ) }{a}}+4\,{\frac{ \left ( \tan \left ( 1/2\,ax \right ) \right ) ^{3}}{a}}+2\,{\frac{ \left ( \tan \left ( 1/2\,ax \right ) \right ) ^{5}}{a}} \right ) \left ( 1+ \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2} \right ) ^{-2} \left ( 2\,\tan \left ( 1/2\,ax \right ) xa- \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2/a*tan(1/2*a*x)+4/a*tan(1/2*a*x)^3+2/a*tan(1/2*a*x)^5)/(1+tan(1/2*a*x)^2)^2/(2*tan(1/2*a*x)*x*a-tan(1/2*a*x)
^2+1)

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Maxima [B]  time = 1.06031, size = 154, normalized size = 4.53 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.02329, size = 54, normalized size = 1.59 \begin{align*} \frac{\sin \left (a x\right )}{a^{2} x \sin \left (a x\right ) + a \cos \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [A]  time = 3.09642, size = 20, normalized size = 0.59 \begin{align*} \frac{\sin{\left (a x \right )}}{a^{2} x \sin{\left (a x \right )} + a \cos{\left (a x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.14777, size = 43, normalized size = 1.26 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)