Integrand size = 9, antiderivative size = 10 \[ \int (-\cot (x)+\tan (x))^2 \, dx=-4 x-\cot (x)+\tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {472, 209} \[ \int (-\cot (x)+\tan (x))^2 \, dx=-4 x+\tan (x)-\cot (x) \]
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Rule 209
Rule 472
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2 \left (1+x^2\right )} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {4}{1+x^2}\right ) \, dx,x,\tan (x)\right ) \\ & = -\cot (x)+\tan (x)-4 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = -4 x-\cot (x)+\tan (x) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.90 \[ \int (-\cot (x)+\tan (x))^2 \, dx=-2 x-\arctan (\tan (x))-\cot (x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(x)\right )+\tan (x) \]
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Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
default | \(-4 x -\cot \left (x \right )+\tan \left (x \right )\) | \(11\) |
parallelrisch | \(-4 x -\cot \left (x \right )+\tan \left (x \right )\) | \(11\) |
norman | \(\frac {-1+\tan ^{2}\left (x \right )-4 x \tan \left (x \right )}{\tan \left (x \right )}\) | \(17\) |
parts | \(-\cot \left (x \right )+\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (x \right )\right )+\tan \left (x \right )-\arctan \left (\tan \left (x \right )\right )-2 x\) | \(24\) |
risch | \(-4 x -\frac {4 i}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}\) | \(26\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.90 \[ \int (-\cot (x)+\tan (x))^2 \, dx=-\frac {4 \, x \tan \left (x\right ) - \tan \left (x\right )^{2} + 1}{\tan \left (x\right )} \]
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Time = 0.14 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int (-\cot (x)+\tan (x))^2 \, dx=- 4 x + \tan {\left (x \right )} - \frac {1}{\tan {\left (x \right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int (-\cot (x)+\tan (x))^2 \, dx=-4 \, x - \frac {1}{\tan \left (x\right )} + \tan \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int (-\cot (x)+\tan (x))^2 \, dx=-4 \, x - \frac {1}{\tan \left (x\right )} + \tan \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int (-\cot (x)+\tan (x))^2 \, dx=\mathrm {tan}\left (x\right )-4\,x-\frac {1}{\mathrm {tan}\left (x\right )} \]
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