Integrand size = 21, antiderivative size = 34 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{a^2 x}-\frac {\cos (a x)}{a^2 x (\cos (a x)+a x \sin (a x))} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4693} \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {1}{a^2 x}-\frac {\cos (a x)}{a^2 x (a x \sin (a x)+\cos (a x))} \]
[In]
[Out]
Rule 4693
Rubi steps \begin{align*} \text {integral}& = \frac {1}{a^2 x}-\frac {\cos (a x)}{a^2 x (\cos (a x)+a x \sin (a x))} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin (a x)}{a (\cos (a x)+a x \sin (a x))} \]
[In]
[Out]
Time = 2.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91
method | result | size |
parallelrisch | \(-\frac {2 \tan \left (\frac {a x}{2}\right )}{\left (-1-2 \tan \left (\frac {a x}{2}\right ) a x +\tan \left (\frac {a x}{2}\right )^{2}\right ) a}\) | \(31\) |
risch | \(\frac {1}{a \left (a x +i\right )}-\frac {2 i}{\left (a x +i\right ) \left (a x \,{\mathrm e}^{2 i a x}-a x +i {\mathrm e}^{2 i a x}+i\right ) a}\) | \(55\) |
norman | \(\frac {\frac {2 \tan \left (\frac {a x}{2}\right )}{a}+\frac {4 \tan \left (\frac {a x}{2}\right )^{3}}{a}+\frac {2 \tan \left (\frac {a x}{2}\right )^{5}}{a}}{\left (1+\tan \left (\frac {a x}{2}\right )^{2}\right )^{2} \left (1+2 \tan \left (\frac {a x}{2}\right ) a x -\tan \left (\frac {a x}{2}\right )^{2}\right )}\) | \(70\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin \left (a x\right )}{a^{2} x \sin \left (a x\right ) + a \cos \left (a x\right )} \]
[In]
[Out]
Time = 2.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin {\left (a x \right )}}{a^{2} x \sin {\left (a x \right )} + a \cos {\left (a x \right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (34) = 68\).
Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.35 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {a x \cos \left (2 \, a x\right )^{2} + a x \sin \left (2 \, a x\right )^{2} - 2 \, a x \cos \left (2 \, a x\right ) + a x + 2 \, \sin \left (2 \, a x\right )}{{\left (a^{2} x^{2} + {\left (a^{2} x^{2} + 1\right )} \cos \left (2 \, a x\right )^{2} + 4 \, a x \sin \left (2 \, a x\right ) + {\left (a^{2} x^{2} + 1\right )} \sin \left (2 \, a x\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \cos \left (2 \, a x\right ) + 1\right )} a} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {2 \, \tan \left (\frac {1}{2} \, a x\right )}{2 \, a^{2} x \tan \left (\frac {1}{2} \, a x\right ) - a \tan \left (\frac {1}{2} \, a x\right )^{2} + a} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx=\frac {\sin \left (a\,x\right )}{a\,\left (\cos \left (a\,x\right )+a\,x\,\sin \left (a\,x\right )\right )} \]
[In]
[Out]