Optimal. Leaf size=19 \[ -\frac {1}{a^2 (a x \sin (a x)+\cos (a x))} \]
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Rubi [A] time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.00 \[ -\frac {1}{a^2 (a x \sin (a x)+\cos (a x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 22, normalized size = 1.16 \[ -\frac {1}{a^{3} x \sin \left (a x\right ) + a^{2} \cos \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 40, normalized size = 2.11 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, a x\right )^{2} + 1\right )}}{2 \, a^{3} x \tan \left (\frac {1}{2} \, a x\right ) - a^{2} \tan \left (\frac {1}{2} \, a x\right )^{2} + a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 20, normalized size = 1.05 \[ -\frac {1}{a^{2} \left (\cos \left (a x \right )+a x \sin \left (a x \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 1.00 \[ -\frac {1}{{\left (a x \sin \left (a x\right ) + \cos \left (a x\right )\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 22, normalized size = 1.16 \[ -\frac {1}{a^2\,\cos \left (a\,x\right )+a^3\,x\,\sin \left (a\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.01, size = 20, normalized size = 1.05 \[ - \frac {1}{a^{3} x \sin {\left (a x \right )} + a^{2} \cos {\left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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