Integrand size = 16, antiderivative size = 10 \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=-\cot (x)+\log (1+\cot (x)) \]
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Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4427, 46} \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=-\cot (x)-\log (\tan (x))+\log (\tan (x)+1) \]
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Rule 46
Rule 4427
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x^2 (1+x)} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,\tan (x)\right ) \\ & = -\cot (x)-\log (\tan (x))+\log (1+\tan (x)) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=-\cot (x)-\log (\sin (x))+\log (\cos (x)+\sin (x)) \]
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Time = 1.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80
method | result | size |
default | \(\ln \left (1+\tan \left (x \right )\right )-\frac {1}{\tan \left (x \right )}-\ln \left (\tan \left (x \right )\right )\) | \(18\) |
risch | \(-\frac {2 i}{{\mathrm e}^{2 i x}-1}-\ln \left ({\mathrm e}^{2 i x}-1\right )+\ln \left (i+{\mathrm e}^{2 i x}\right )\) | \(33\) |
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (10) = 20\).
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 3.60 \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=-\frac {\log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right ) \sin \left (x\right ) - \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 2 \, \cos \left (x\right )}{2 \, \sin \left (x\right )} \]
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\[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=\int \frac {\sec ^{2}{\left (x \right )}}{\left (\tan {\left (x \right )} + 1\right ) \tan ^{2}{\left (x \right )}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.70 \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=-\frac {1}{\tan \left (x\right )} + \log \left (\tan \left (x\right ) + 1\right ) - \log \left (\tan \left (x\right )\right ) \]
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none
Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.90 \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=-\frac {1}{\tan \left (x\right )} + \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) - \log \left ({\left | \tan \left (x\right ) \right |}\right ) \]
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Time = 27.75 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx=2\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (x\right )+1\right )-\frac {1}{\mathrm {tan}\left (x\right )} \]
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