Integrand size = 5, antiderivative size = 2 \[ \int \sec (x) \tan (x) \, dx=\sec (x) \]
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Time = 0.01 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2686, 8} \[ \int \sec (x) \tan (x) \, dx=\sec (x) \]
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Rule 8
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \text {Subst}(\int 1 \, dx,x,\sec (x)) \\ & = \sec (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sec (x) \tan (x) \, dx=\sec (x) \]
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Time = 0.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50
method | result | size |
derivativedivides | \(\sec \left (x \right )\) | \(3\) |
default | \(\sec \left (x \right )\) | \(3\) |
risch | \(\frac {2 \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}+1}\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos \left (x\right )} \]
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Time = 0.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos {\left (x \right )}} \]
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none
Time = 0.19 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos \left (x\right )} \]
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none
Time = 0.31 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \sec (x) \tan (x) \, dx=\frac {1}{\cos \left (x\right )} \]
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Time = 0.33 (sec) , antiderivative size = 12, normalized size of antiderivative = 6.00 \[ \int \sec (x) \tan (x) \, dx=-\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2-1} \]
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