Optimal. Leaf size=19 \[ -\frac{1}{a^2 (a x \sin (a x)+\cos (a x))} \]
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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.
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Rule 6686
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.
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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.
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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.
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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.
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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.
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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.
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