Integrand size = 17, antiderivative size = 11 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=-\log \left (1+\frac {\cos (x)}{x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6844, 31} \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=-\log \left (\frac {\cos (x)}{x}+1\right ) \]
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Rule 31
Rule 6844
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {\cos (x)}{x}\right ) \\ & = -\log \left (1+\frac {\cos (x)}{x}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=\log (x)-\log (x+\cos (x)) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(24\) vs. \(2(11)=22\).
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27
method | result | size |
parallelrisch | \(-\ln \left (\frac {x +\cos \left (x \right )}{1+\cos \left (x \right )}\right )+\ln \left (x \right )+\ln \left (\frac {1}{1+\cos \left (x \right )}\right )\) | \(25\) |
risch | \(i x +\ln \left (x \right )-\ln \left (2 x \,{\mathrm e}^{i x}+{\mathrm e}^{2 i x}+1\right )\) | \(26\) |
norman | \(-\ln \left (x \tan \left (\frac {x}{2}\right )^{2}-\tan \left (\frac {x}{2}\right )^{2}+x +1\right )+\ln \left (x \right )+\ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )\) | \(35\) |
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=-\log \left (x + \cos \left (x\right )\right ) + \log \left (x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=\log {\left (x \right )} - \log {\left (x + \cos {\left (x \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (11) = 22\).
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 5.91 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=-\frac {1}{2} \, \log \left (4 \, x^{2} \cos \left (x\right )^{2} + 4 \, x^{2} \sin \left (x\right )^{2} + 4 \, x \sin \left (2 \, x\right ) \sin \left (x\right ) + 2 \, {\left (2 \, x \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, x \cos \left (x\right ) + \sin \left (2 \, x\right )^{2} + 1\right ) + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (11) = 22\).
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 7.18 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=-\frac {1}{2} \, \log \left (\frac {4 \, {\left (x^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, x \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, x^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )^{4} + x^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, x + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) + \log \left ({\left | x \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (x)+x \sin (x)}{x (x+\cos (x))} \, dx=\ln \left (x\right )-\ln \left (x+\cos \left (x\right )\right ) \]
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