Integrand size = 7, antiderivative size = 16 \[ \int \sec (x) \tan ^2(x) \, dx=-\frac {1}{2} \text {arctanh}(\sin (x))+\frac {1}{2} \sec (x) \tan (x) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2691, 3855} \[ \int \sec (x) \tan ^2(x) \, dx=\frac {1}{2} \tan (x) \sec (x)-\frac {1}{2} \text {arctanh}(\sin (x)) \]
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Rule 2691
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \sec (x) \tan (x)-\frac {1}{2} \int \sec (x) \, dx \\ & = -\frac {1}{2} \text {arctanh}(\sin (x))+\frac {1}{2} \sec (x) \tan (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sec (x) \tan ^2(x) \, dx=-\frac {1}{2} \text {arctanh}(\sin (x))+\frac {1}{2} \sec (x) \tan (x) \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {\sin ^{3}\left (x \right )}{2 \cos \left (x \right )^{2}}+\frac {\sin \left (x \right )}{2}-\frac {\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}\) | \(24\) |
risch | \(-\frac {i \left ({\mathrm e}^{3 i x}-{\mathrm e}^{i x}\right )}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {\ln \left (i+{\mathrm e}^{i x}\right )}{2}+\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{2}\) | \(49\) |
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \sec (x) \tan ^2(x) \, dx=-\frac {\cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, \sin \left (x\right )}{4 \, \cos \left (x\right )^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \sec (x) \tan ^2(x) \, dx=\frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{4} - \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{4} - \frac {\sin {\left (x \right )}}{2 \sin ^{2}{\left (x \right )} - 2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \sec (x) \tan ^2(x) \, dx=-\frac {\sin \left (x\right )}{2 \, {\left (\sin \left (x\right )^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\sin \left (x\right ) - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \sec (x) \tan ^2(x) \, dx=-\frac {\sin \left (x\right )}{2 \, {\left (\sin \left (x\right )^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \sec (x) \tan ^2(x) \, dx=\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\mathrm {tan}\left (\frac {x}{2}\right )}{{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right )}^2}-\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]
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