Integrand size = 18, antiderivative size = 33 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {x \sec (a x)}{a^2 (\cos (a x)+a x \sin (a x))}+\frac {\tan (a x)}{a^3} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4691, 3852, 8} \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\tan (a x)}{a^3}-\frac {x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))} \]
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Rule 8
Rule 3852
Rule 4691
Rubi steps \begin{align*} \text {integral}& = -\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 {\text {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*}
Time = 1.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {-a x \cos (a x)+\sin (a x)}{a^3 (\cos (a x)+a x \sin (a x))} \]
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Result contains complex when optimal does not.
Time = 1.89 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {2 i \left (a x -i\right )}{a^{3} \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right )}\) | \(40\) |
parallelrisch | \(\frac {-a x \tan \left (\frac {a x}{2}\right )^{2}+a x -2 \tan \left (\frac {a x}{2}\right )}{a^{3} \left (-1-2 \tan \left (\frac {a x}{2}\right ) a x +\tan \left (\frac {a x}{2}\right )^{2}\right )}\) | \(47\) |
norman | \(\frac {\frac {x \tan \left (\frac {a x}{2}\right )^{2}}{a^{2}}+\frac {2 \tan \left (\frac {a x}{2}\right )}{a^{3}}-\frac {x}{a^{2}}}{1+2 \tan \left (\frac {a x}{2}\right ) a x -\tan \left (\frac {a x}{2}\right )^{2}}\) | \(53\) |
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none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {a x \cos \left (a x\right ) - \sin \left (a x\right )}{a^{4} x \sin \left (a x\right ) + a^{3} \cos \left (a x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (31) = 62\).
Time = 2.41 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.30 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \tan ^{2}{\left (\frac {a x}{2} \right )}}{2 a^{4} x \tan {\left (\frac {a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac {a x}{2} \right )} + a^{3}} - \frac {a x}{2 a^{4} x \tan {\left (\frac {a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac {a x}{2} \right )} + a^{3}} + \frac {2 \tan {\left (\frac {a x}{2} \right )}}{2 a^{4} x \tan {\left (\frac {a x}{2} \right )} - a^{3} \tan ^{2}{\left (\frac {a x}{2} \right )} + a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (33) = 66\).
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.03 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=-\frac {2 \, {\left (2 \, a x \cos \left (2 \, a x\right ) + {\left (a^{2} x^{2} - 1\right )} \sin \left (2 \, a x\right )\right )}}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} + 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a^{3}} \]
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none
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \tan \left (\frac {1}{2} \, a x\right )^{2} - a x + 2 \, \tan \left (\frac {1}{2} \, a x\right )}{2 \, a^{4} x \tan \left (\frac {1}{2} \, a x\right ) - a^{3} \tan \left (\frac {1}{2} \, a x\right )^{2} + a^{3}} \]
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Timed out. \[ \int \frac {x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx=\int \frac {x^2}{{\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )}^2} \,d x \]
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