Optimal. Leaf size=20 \[ \frac{1}{a^2 (a x \cos (a x)-\sin (a x))} \]
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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.
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Rule 6686
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.
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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.
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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.
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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.
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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.
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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.
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