Integrand size = 9, antiderivative size = 5 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=\sec (x)+\tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2748, 3852, 8} \[ \int \sec ^2(x) (1+\sin (x)) \, dx=\tan (x)+\sec (x) \]
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Rule 8
Rule 2748
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \sec (x)+\int \sec ^2(x) \, dx \\ & = \sec (x)-\text {Subst}(\int 1 \, dx,x,-\tan (x)) \\ & = \sec (x)+\tan (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=\sec (x)+\tan (x) \]
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Time = 1.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.60
method | result | size |
default | \(\tan \left (x \right )+\frac {1}{\cos \left (x \right )}\) | \(8\) |
parallelrisch | \(-\frac {2}{\tan \left (\frac {x}{2}\right )-1}\) | \(11\) |
risch | \(\frac {2}{{\mathrm e}^{i x}-i}\) | \(13\) |
norman | \(\frac {-2 \tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )^{3}-2 \tan \left (\frac {x}{2}\right )-2}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (\tan \left (\frac {x}{2}\right )^{2}-1\right )}\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=\frac {\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \]
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Time = 1.14 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.40 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=\tan {\left (x \right )} + \frac {1}{\cos {\left (x \right )}} \]
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none
Time = 0.20 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.40 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=\frac {1}{\cos \left (x\right )} + \tan \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.00 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=-\frac {2}{\tan \left (\frac {1}{2} \, x\right ) - 1} \]
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Time = 26.48 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.00 \[ \int \sec ^2(x) (1+\sin (x)) \, dx=-\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )-1} \]
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