Integrand size = 10, antiderivative size = 71 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3} \]
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Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3290, 3260, 209, 3254, 3852, 8} \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3} \]
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Rule 8
Rule 209
Rule 3254
Rule 3260
Rule 3290
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {1}{1-\cos ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1+\sqrt [3]{-1} \cos ^2(x)} \, dx+\frac {1}{3} \int \frac {1}{1-(-1)^{2/3} \cos ^2(x)} \, dx \\ & = \frac {1}{3} \int \csc ^2(x) \, dx-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\left (1-(-1)^{2/3}\right ) x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {1}{3} \text {Subst}(\int 1 \, dx,x,\cot (x)) \\ & = -\frac {\arctan \left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\arctan \left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.65 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=\frac {(15+8 \cos (2 x)+\cos (4 x)) \sin (x) \left (6 \cos (x)+i \sqrt [4]{-3} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (-i+\sqrt {3}\right ) \tan (x)\right ) \sin (x)+\sqrt [4]{-3} \left (-3 i+\sqrt {3}\right ) \arctan \left (\frac {(-1)^{3/4} \left (i+\sqrt {3}\right ) \tan (x)}{2 \sqrt [4]{3}}\right ) \sin (x)\right )}{144 \left (-1+\cos ^6(x)\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.41 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {2 i}{3 \left ({\mathrm e}^{2 i x}-1\right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3888 \textit {\_Z}^{4}+108 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+1296 i \textit {\_R}^{3}-216 \textit {\_R}^{2}-1\right )\right )\) | \(51\) |
default | \(-\frac {\sqrt {3}\, \left (\frac {\sqrt {2 \sqrt {3}-3}\, \ln \left (\tan ^{2}\left (x \right )+\tan \left (x \right ) \sqrt {2 \sqrt {3}-3}+\sqrt {3}\right )}{2}+\frac {2 \left (-\sqrt {3}-\frac {3}{2}\right ) \arctan \left (\frac {2 \tan \left (x \right )+\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right )}{\sqrt {2 \sqrt {3}+3}}\right )}{18}-\frac {\sqrt {3}\, \left (-\frac {\sqrt {2 \sqrt {3}-3}\, \ln \left (\tan ^{2}\left (x \right )-\tan \left (x \right ) \sqrt {2 \sqrt {3}-3}+\sqrt {3}\right )}{2}+\frac {2 \left (-\sqrt {3}-\frac {3}{2}\right ) \arctan \left (\frac {2 \tan \left (x \right )-\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right )}{\sqrt {2 \sqrt {3}+3}}\right )}{18}-\frac {1}{3 \tan \left (x \right )}\) | \(173\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.37 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {\sqrt {6} \sqrt {i \, \sqrt {3} - 3} \log \left (\sqrt {6} {\left (i \, \sqrt {3} - 3\right )}^{\frac {3}{2}} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} + 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} + 3\right ) \sin \left (x\right ) - \sqrt {6} \sqrt {i \, \sqrt {3} - 3} \log \left (\sqrt {6} \sqrt {i \, \sqrt {3} - 3} {\left (-i \, \sqrt {3} + 3\right )} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} + 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} + 3\right ) \sin \left (x\right ) + \sqrt {6} \sqrt {-i \, \sqrt {3} - 3} \log \left (\sqrt {6} {\left (i \, \sqrt {3} + 3\right )} \sqrt {-i \, \sqrt {3} - 3} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} - 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} - 3\right ) \sin \left (x\right ) - \sqrt {6} \sqrt {-i \, \sqrt {3} - 3} \log \left (\sqrt {6} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {3}{2}} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left (-i \, \sqrt {3} - 2\right )} \cos \left (x\right )^{2} - 3 i \, \sqrt {3} - 3\right ) \sin \left (x\right ) + 24 \, \cos \left (x\right )}{72 \, \sin \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (66) = 132\).
Time = 8.91 (sec) , antiderivative size = 728, normalized size of antiderivative = 10.25 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{1-\cos ^6(x)} \, dx=\int { -\frac {1}{\cos \left (x\right )^{6} - 1} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (49) = 98\).
Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.80 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=\frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (-\frac {3^{\frac {3}{4}} {\left (3^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} + 4 \, \tan \left (x\right )\right )}}{3 \, {\left (\sqrt {6} + \sqrt {2}\right )}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} + \frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {3^{\frac {3}{4}} {\left (3^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} - 4 \, \tan \left (x\right )\right )}}{3 \, {\left (\sqrt {6} + \sqrt {2}\right )}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} - \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (\frac {1}{2} \, {\left (\sqrt {6} 3^{\frac {1}{4}} - 3^{\frac {1}{4}} \sqrt {2}\right )} \tan \left (x\right ) + \tan \left (x\right )^{2} + \sqrt {3}\right ) + \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} 3^{\frac {1}{4}} - 3^{\frac {1}{4}} \sqrt {2}\right )} \tan \left (x\right ) + \tan \left (x\right )^{2} + \sqrt {3}\right ) - \frac {1}{3 \, \tan \left (x\right )} \]
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Time = 3.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.34 \[ \int \frac {1}{1-\cos ^6(x)} \, dx=-\frac {1}{3\,\mathrm {tan}\left (x\right )}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {3^{1/4}\,\sqrt {6}\,\mathrm {tan}\left (x\right )\,\left (\frac {1}{27}-\frac {1}{27}{}\mathrm {i}\right )}{-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}}\right )\,\left (3^{1/4}\,\left (1+1{}\mathrm {i}\right )+3^{3/4}\,\left (-1+1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {3^{1/4}\,\sqrt {6}\,\mathrm {tan}\left (x\right )\,\left (\frac {1}{27}+\frac {1}{27}{}\mathrm {i}\right )}{\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}}\right )\,\left (3^{1/4}\,\left (1-\mathrm {i}\right )+3^{3/4}\,\left (-1-\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36} \]
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