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 = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 8
Rule 3767
Rule 4594
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*}
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Mathematica [A] time = 0.46, 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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fricas [A] time = 0.82, size = 34, normalized size = 0.97 \[ \frac {a x \sin \left (a x\right ) + \cos \left (a x\right )}{a^{4} x \cos \left (a x\right ) - a^{3} \sin \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 1.51 \[ -\frac {2 \, a x \tan \left (\frac {1}{2} \, a x\right ) - \tan \left (\frac {1}{2} \, a x\right )^{2} + 1}{a^{4} x \tan \left (\frac {1}{2} \, a x\right )^{2} - a^{4} x + 2 \, a^{3} \tan \left (\frac {1}{2} \, a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.23, size = 54, normalized size = 1.54 \[ \frac {\frac {\tan ^{2}\left (\frac {a x}{2}\right )}{a^{3}}-\frac {1}{a^{3}}-\frac {2 x \tan \left (\frac {a x}{2}\right )}{a^{2}}}{a x \left (\tan ^{2}\left (\frac {a x}{2}\right )\right )-a x +2 \tan \left (\frac {a x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 100, normalized size = 2.86 \[ \frac {2 \, {\left (2 \, a x \cos \left (2 \, a x\right ) + {\left (a^{2} x^{2} - 1\right )} \sin \left (2 \, a x\right )\right )}}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} - 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} + 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{{\left (\sin \left (a\,x\right )-a\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.05, size = 112, normalized size = 3.20 \[ - \frac {2 a x \tan {\left (\frac {a x}{2} \right )}}{a^{4} x \tan ^{2}{\left (\frac {a x}{2} \right )} - a^{4} x + 2 a^{3} \tan {\left (\frac {a x}{2} \right )}} + \frac {\tan ^{2}{\left (\frac {a x}{2} \right )}}{a^{4} x \tan ^{2}{\left (\frac {a x}{2} \right )} - a^{4} x + 2 a^{3} \tan {\left (\frac {a x}{2} \right )}} - \frac {1}{a^{4} x \tan ^{2}{\left (\frac {a x}{2} \right )} - a^{4} x + 2 a^{3} \tan {\left (\frac {a x}{2} \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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