Optimal. Leaf size=33 \[ \frac{\tan (a x)}{a^3}-\frac{x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]
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Rubi [A] time = 0.0379801, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4595, 3767, 8} \[ \frac{\tan (a x)}{a^3}-\frac{x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]
Antiderivative was successfully verified.
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Rule 4595
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx &=-\frac{x \sec (a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac{\int \sec ^2(a x) \, dx}{a^2}\\ &=-\frac{x \sec (a x)}{a^2 (\cos (a x)+a x \sin (a x))}-\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (a x))}{a^3}\\ &=-\frac{x \sec (a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac{\tan (a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.417018, size = 31, normalized size = 0.94 \[ \frac{\sin (a x)-a x \cos (a x)}{a^3 (a x \sin (a x)+\cos (a x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.372, size = 53, normalized size = 1.6 \begin{align*}{ \left ({\frac{x}{{a}^{2}} \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}}-{\frac{x}{{a}^{2}}}+2\,{\frac{\tan \left ( 1/2\,ax \right ) }{{a}^{3}}} \right ) \left ( 2\,\tan \left ( 1/2\,ax \right ) xa- \left ( \tan \left ({\frac{ax}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02277, size = 135, normalized size = 4.09 \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 = 1.97615, size = 81, normalized size = 2.45 \begin{align*} -\frac{a x \cos \left (a x\right ) - \sin \left (a x\right )}{a^{4} x \sin \left (a x\right ) + a^{3} \cos \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.1095, size = 109, normalized size = 3.3 \begin{align*} \frac{a x \tan ^{2}{\left (\frac{a x}{2} \right )}}{2 a^{4} x \tan{\left (\frac{a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac{a x}{2} \right )} + a^{3}} - \frac{a x}{2 a^{4} x \tan{\left (\frac{a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac{a x}{2} \right )} + a^{3}} + \frac{2 \tan{\left (\frac{a x}{2} \right )}}{2 a^{4} x \tan{\left (\frac{a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac{a x}{2} \right )} + a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15183, size = 70, normalized size = 2.12 \begin{align*} \frac{a x \tan \left (\frac{1}{2} \, a x\right )^{2} - a x + 2 \, \tan \left (\frac{1}{2} \, a x\right )}{2 \, a^{4} x \tan \left (\frac{1}{2} \, a x\right ) - a^{3} \tan \left (\frac{1}{2} \, a x\right )^{2} + a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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