3.590 \(\int \frac{x \sin (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx\)

Optimal. Leaf size=20 \[ \frac{1}{a^2 (a x \cos (a x)-\sin (a x))} \]

[Out]

1/(a^2*(a*x*Cos[a*x] - Sin[a*x]))

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Rubi [A]  time = 0.037818, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6686} \[ \frac{1}{a^2 (a x \cos (a x)-\sin (a x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(a^2*(a*x*Cos[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 \sin (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac{1}{a^2 (a x \cos (a x)-\sin (a x))}\\ \end{align*}

Mathematica [A]  time = 0.0309856, size = 20, normalized size = 1. \[ -\frac{1}{a^2 (\sin (a x)-a x \cos (a x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 21, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{2} \left ( ax\cos \left ( ax \right ) -\sin \left ( ax \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01165, size = 27, normalized size = 1.35 \begin{align*} \frac{1}{{\left (a x \cos \left (a x\right ) - \sin \left (a x\right )\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((a*x*cos(a*x) - sin(a*x))*a^2)

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Fricas [A]  time = 1.96554, size = 47, normalized size = 2.35 \begin{align*} \frac{1}{a^{3} x \cos \left (a x\right ) - a^{2} \sin \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^3*x*cos(a*x) - a^2*sin(a*x))

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Sympy [A]  time = 3.4626, size = 19, normalized size = 0.95 \begin{align*} \frac{1}{a^{3} x \cos{\left (a x \right )} - a^{2} \sin{\left (a x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a**3*x*cos(a*x) - a**2*sin(a*x))

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Giac [B]  time = 1.16951, size = 57, normalized size = 2.85 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, a x\right )^{2} + 1\right )}}{a^{3} x \tan \left (\frac{1}{2} \, a x\right )^{2} - a^{3} x + 2 \, a^{2} \tan \left (\frac{1}{2} \, a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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