Optimal. Leaf size=20 \[ \frac {1}{a^2 (a x \cos (a x)-\sin (a x))} \]
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Rubi [A] time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 1.00 \[ -\frac {1}{a^2 (\sin (a x)-a x \cos (a x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.75, size = 21, normalized size = 1.05 \[ \frac {1}{a^{3} x \cos \left (a x\right ) - a^{2} \sin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 42, normalized size = 2.10 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, a x\right )^{2} + 1\right )}}{a^{3} x \tan \left (\frac {1}{2} \, a x\right )^{2} - a^{3} x + 2 \, a^{2} \tan \left (\frac {1}{2} \, a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 21, normalized size = 1.05 \[ \frac {1}{a^{2} \left (a x \cos \left (a x \right )-\sin \left (a x \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 1.00 \[ \frac {1}{{\left (a x \cos \left (a x\right ) - \sin \left (a x\right )\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 23, normalized size = 1.15 \[ -\frac {1}{a^2\,\sin \left (a\,x\right )-a^3\,x\,\cos \left (a\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.48, size = 19, normalized size = 0.95 \[ \frac {1}{a^{3} x \cos {\left (a x \right )} - a^{2} \sin {\left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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