Integrand size = 23, antiderivative size = 32 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=-\frac {1}{4} \cot ^2\left (\frac {\pi }{4}+2 x\right )+\frac {1}{2} \log \left (\tan \left (\frac {\pi }{4}+2 x\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2700, 14} \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=\frac {1}{2} \log \left (\tan \left (2 x+\frac {\pi }{4}\right )\right )-\frac {1}{4} \cot ^2\left (2 x+\frac {\pi }{4}\right ) \]
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Rule 14
Rule 2700
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x^3} \, dx,x,\tan \left (\frac {\pi }{4}+2 x\right )\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {1}{x}\right ) \, dx,x,\tan \left (\frac {\pi }{4}+2 x\right )\right ) \\ & = -\frac {1}{4} \cot ^2\left (\frac {\pi }{4}+2 x\right )+\frac {1}{2} \log \left (\tan \left (\frac {\pi }{4}+2 x\right )\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=-\frac {1}{4} \csc ^2\left (\frac {\pi }{4}+2 x\right )-\frac {1}{2} \log \left (\cos \left (\frac {1}{4} (\pi +8 x)\right )\right )+\frac {1}{2} \log \left (\sin \left (\frac {\pi }{4}+2 x\right )\right ) \]
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Time = 0.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-\frac {1}{4 \sin \left (\frac {\pi }{4}+2 x \right )^{2}}+\frac {\ln \left (\tan \left (\frac {\pi }{4}+2 x \right )\right )}{2}\) | \(25\) |
default | \(-\frac {1}{4 \sin \left (\frac {\pi }{4}+2 x \right )^{2}}+\frac {\ln \left (\tan \left (\frac {\pi }{4}+2 x \right )\right )}{2}\) | \(25\) |
risch | \(\frac {i {\mathrm e}^{4 i x}}{\left (i {\mathrm e}^{4 i x}-1\right )^{2}}+\frac {\ln \left (i {\mathrm e}^{4 i x}-1\right )}{2}-\frac {\ln \left (i {\mathrm e}^{4 i x}+1\right )}{2}\) | \(48\) |
parallelrisch | \(\ln \left (\sqrt {\tan }\left (\frac {\pi }{8}+x \right )\right )+\ln \left (\frac {1}{\sqrt {\tan \left (\frac {\pi }{8}+x \right )-1}}\right )+\ln \left (\frac {1}{\sqrt {\tan \left (\frac {\pi }{8}+x \right )+1}}\right )-\frac {\left (\tan ^{2}\left (\frac {\pi }{8}+x \right )\right )}{16}-\frac {\left (\cot ^{2}\left (\frac {\pi }{8}+x \right )\right )}{16}\) | \(53\) |
norman | \(\frac {-\frac {1}{16}-\frac {\left (\tan ^{4}\left (\frac {\pi }{8}+x \right )\right )}{16}}{\tan \left (\frac {\pi }{8}+x \right )^{2}}+\frac {\ln \left (\tan \left (\frac {\pi }{8}+x \right )\right )}{2}-\frac {\ln \left (\tan \left (\frac {\pi }{8}+x \right )-1\right )}{2}-\frac {\ln \left (\tan \left (\frac {\pi }{8}+x \right )+1\right )}{2}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=-\frac {{\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2} - 1\right )} \log \left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2}\right ) - {\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2} - 1\right )} \log \left (-\frac {1}{4} \, \cos \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2} + \frac {1}{4}\right ) - 1}{4 \, {\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.57 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=- \frac {\log {\left (\tan {\left (x + \frac {\pi }{8} \right )} - 1 \right )}}{2} - \frac {\log {\left (\tan {\left (x + \frac {\pi }{8} \right )} + 1 \right )}}{2} + \frac {\log {\left (\tan {\left (x + \frac {\pi }{8} \right )} \right )}}{2} - \frac {\tan ^{2}{\left (x + \frac {\pi }{8} \right )}}{16} - \frac {1}{16 \tan ^{2}{\left (x + \frac {\pi }{8} \right )}} \]
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none
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=-\frac {1}{4 \, \sin \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2}} - \frac {1}{4} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2} - 1\right ) + \frac {1}{4} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 2 \, x\right )^{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.12 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=-\frac {{\left (\frac {4 \, {\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) - 1\right )}}{\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1} - 1\right )} {\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right )}}{16 \, {\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) - 1\right )}} + \frac {\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) - 1}{16 \, {\left (\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right )}} + \frac {1}{4} \, \log \left (-\frac {\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) - 1}{\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1}\right ) - \frac {1}{2} \, \log \left ({\left | -\frac {\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) - 1}{\cos \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1} - 1 \right |}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \csc ^3\left (\frac {\pi }{4}+2 x\right ) \sec \left (\frac {\pi }{4}+2 x\right ) \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {\Pi }{4}+2\,x\right )\right )}{2}-\frac {1}{4\,{\sin \left (\frac {\Pi }{4}+2\,x\right )}^2} \]
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