Optimal. Leaf size=35 \[ \frac {1}{a^2 x}+\frac {\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {4596} \[ \frac {1}{a^2 x}+\frac {\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))} \]
Antiderivative was successfully verified.
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Rule 4596
Rubi steps
\begin {align*} \int \frac {\sin ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac {1}{a^2 x}+\frac {\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 24, normalized size = 0.69 \[ \frac {\cos (a x)}{a^2 x \cos (a x)-a \sin (a x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.95, size = 24, normalized size = 0.69 \[ \frac {\cos \left (a x\right )}{a^{2} x \cos \left (a x\right ) - a \sin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 39, normalized size = 1.11 \[ \frac {\tan \left (\frac {1}{2} \, a x\right )^{2} - 1}{a^{2} x \tan \left (\frac {1}{2} \, a x\right )^{2} - a^{2} x + 2 \, a \tan \left (\frac {1}{2} \, a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.47, size = 77, normalized size = 2.20 \[ \frac {\frac {\tan ^{4}\left (\frac {a x}{2}\right )}{a}+\frac {\tan ^{6}\left (\frac {a x}{2}\right )}{a}-\frac {1}{a}-\frac {\tan ^{2}\left (\frac {a x}{2}\right )}{a}}{\left (1+\tan ^{2}\left (\frac {a x}{2}\right )\right )^{2} \left (a x \left (\tan ^{2}\left (\frac {a x}{2}\right )\right )-a x +2 \tan \left (\frac {a x}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 114, normalized size = 3.26 \[ \frac {a x \cos \left (2 \, a x\right )^{2} + a x \sin \left (2 \, a x\right )^{2} + 2 \, a x \cos \left (2 \, a x\right ) + a x - 2 \, \sin \left (2 \, a x\right )}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} - 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} + 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 24, normalized size = 0.69 \[ -\frac {\cos \left (a\,x\right )}{a\,\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.42, size = 20, normalized size = 0.57 \[ \frac {\cos {\left (a x \right )}}{a^{2} x \cos {\left (a x \right )} - a \sin {\left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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