Optimal. Leaf size=35 \[ \frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\cot (a x)}{a^3} \]
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Rubi [A] time = 0.0388193, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4594, 3767, 8} \[ \frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\cot (a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 4594
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx &=\frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}+\frac{\int \csc ^2(a x) \, dx}{a^2}\\ &=\frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (a x))}{a^3}\\ &=-\frac{\cot (a x)}{a^3}+\frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}\\ \end{align*}
Mathematica [A] time = 0.464316, size = 32, normalized size = 0.91 \[ \frac{a x \sin (a x)+\cos (a x)}{a^3 (a x \cos (a x)-\sin (a x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.316, size = 54, normalized size = 1.5 \begin{align*}{ \left ({\frac{1}{{a}^{3}} \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}}-{a}^{-3}-2\,{\frac{x\tan \left ( 1/2\,ax \right ) }{{a}^{2}}} \right ) \left ( ax \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}-ax+2\,\tan \left ( 1/2\,ax \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02527, size = 135, normalized size = 3.86 \begin{align*} \frac{2 \,{\left (2 \, a x \cos \left (2 \, a x\right ) +{\left (a^{2} x^{2} - 1\right )} \sin \left (2 \, a x\right )\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^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13257, size = 80, normalized size = 2.29 \begin{align*} \frac{a x \sin \left (a x\right ) + \cos \left (a x\right )}{a^{4} x \cos \left (a x\right ) - a^{3} \sin \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.4182, size = 112, normalized size = 3.2 \begin{align*} - \frac{2 a x \tan{\left (\frac{a x}{2} \right )}}{a^{4} x \tan ^{2}{\left (\frac{a x}{2} \right )} - a^{4} x + 2 a^{3} \tan{\left (\frac{a x}{2} \right )}} + \frac{\tan ^{2}{\left (\frac{a x}{2} \right )}}{a^{4} x \tan ^{2}{\left (\frac{a x}{2} \right )} - a^{4} x + 2 a^{3} \tan{\left (\frac{a x}{2} \right )}} - \frac{1}{a^{4} x \tan ^{2}{\left (\frac{a x}{2} \right )} - a^{4} x + 2 a^{3} \tan{\left (\frac{a x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12798, size = 72, normalized size = 2.06 \begin{align*} -\frac{2 \, a x \tan \left (\frac{1}{2} \, a x\right ) - \tan \left (\frac{1}{2} \, a x\right )^{2} + 1}{a^{4} x \tan \left (\frac{1}{2} \, a x\right )^{2} - a^{4} x + 2 \, a^{3} \tan \left (\frac{1}{2} \, a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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