Integrand size = 12, antiderivative size = 5 \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=\log \left (\tan \left (e^x\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 2700, 29} \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=\log \left (\tan \left (e^x\right )\right ) \]
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Rule 29
Rule 2320
Rule 2700
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \csc (x) \sec (x) \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan \left (e^x\right )\right ) \\ & = \log \left (\tan \left (e^x\right )\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(5)=10\).
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 4.20 \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=2 \left (-\frac {1}{2} \log \left (\cos \left (e^x\right )\right )+\frac {1}{2} \log \left (\sin \left (e^x\right )\right )\right ) \]
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Time = 0.53 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\ln \left (\tan \left ({\mathrm e}^{x}\right )\right )\) | \(5\) |
default | \(\ln \left (\tan \left ({\mathrm e}^{x}\right )\right )\) | \(5\) |
risch | \(-\ln \left ({\mathrm e}^{2 i {\mathrm e}^{x}}+1\right )+\ln \left ({\mathrm e}^{2 i {\mathrm e}^{x}}-1\right )\) | \(22\) |
norman | \(-\ln \left (\tan \left (\frac {{\mathrm e}^{x}}{2}\right )-1\right )-\ln \left (\tan \left (\frac {{\mathrm e}^{x}}{2}\right )+1\right )+\ln \left (\tan \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\) | \(28\) |
parallelrisch | \(-\ln \left (\tan \left (\frac {{\mathrm e}^{x}}{2}\right )-1\right )-\ln \left (\tan \left (\frac {{\mathrm e}^{x}}{2}\right )+1\right )+\ln \left (\tan \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\) | \(28\) |
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (4) = 8\).
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 4.20 \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=-\frac {1}{2} \, \log \left (\cos \left (e^{x}\right )^{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{4} \, \cos \left (e^{x}\right )^{2} + \frac {1}{4}\right ) \]
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\[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=\int e^{x} \csc {\left (e^{x} \right )} \sec {\left (e^{x} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (4) = 8\).
Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 3.80 \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=-\frac {1}{2} \, \log \left (\sin \left (e^{x}\right )^{2} - 1\right ) + \frac {1}{2} \, \log \left (\sin \left (e^{x}\right )^{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (4) = 8\).
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=-\frac {1}{2} \, \log \left ({\left | \sin \left (e^{x}\right )^{2} - 1 \right |}\right ) + \log \left ({\left | \sin \left (e^{x}\right ) \right |}\right ) \]
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Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 8.60 \[ \int e^x \csc \left (e^x\right ) \sec \left (e^x\right ) \, dx=-\ln \left (-16\,{\mathrm {e}}^{2\,x}-16\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}\right )+\ln \left (16\,{\mathrm {e}}^{2\,x}-16\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}\right ) \]
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