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

Optimal. Leaf size=19 \[ -\frac{1}{a^2 (a x \sin (a x)+\cos (a x))} \]

[Out]

-(1/(a^2*(Cos[a*x] + a*x*Sin[a*x])))

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Rubi [A]  time = 0.0559611, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {6686} \[ -\frac{1}{a^2 (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a^2*(Cos[a*x] + a*x*Sin[a*x])))

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x \cos (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx &=-\frac{1}{a^2 (\cos (a x)+a x \sin (a x))}\\ \end{align*}

Mathematica [A]  time = 0.0200806, size = 19, normalized size = 1. \[ -\frac{1}{a^2 (a x \sin (a x)+\cos (a x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a^2*(Cos[a*x] + a*x*Sin[a*x])))

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Maple [A]  time = 0.043, size = 20, normalized size = 1.1 \begin{align*} -{\frac{1}{{a}^{2} \left ( \cos \left ( ax \right ) +ax\sin \left ( ax \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.989304, size = 26, normalized size = 1.37 \begin{align*} -\frac{1}{{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((a*x*sin(a*x) + cos(a*x))*a^2)

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Fricas [A]  time = 1.99532, size = 49, normalized size = 2.58 \begin{align*} -\frac{1}{a^{3} x \sin \left (a x\right ) + a^{2} \cos \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a^3*x*sin(a*x) + a^2*cos(a*x))

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Sympy [A]  time = 3.69455, size = 20, normalized size = 1.05 \begin{align*} - \frac{1}{a^{3} x \sin{\left (a x \right )} + a^{2} \cos{\left (a x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a**3*x*sin(a*x) + a**2*cos(a*x))

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Giac [B]  time = 1.16989, size = 54, normalized size = 2.84 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, a x\right )^{2} + 1\right )}}{2 \, a^{3} x \tan \left (\frac{1}{2} \, a x\right ) - a^{2} \tan \left (\frac{1}{2} \, a x\right )^{2} + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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