Optimal. Leaf size=71 \[ -\frac {\cot (x)}{3}-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}} \]
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Rubi [A] time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3211, 3181, 203, 3175, 3767, 8} \[ -\frac {\cot (x)}{3}-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 203
Rule 3175
Rule 3181
Rule 3211
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{1-\cos ^6(x)} \, dx &=\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} \operatorname {Subst}\left (\int \frac {1}{1+\left (1+\sqrt [3]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+\left (1-(-1)^{2/3}\right ) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {1}{3} \operatorname {Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [3]{-1}} \cot (x)\right )}{3 \sqrt {1+\sqrt [3]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{2/3}} \cot (x)\right )}{3 \sqrt {1-(-1)^{2/3}}}-\frac {\cot (x)}{3}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 117, normalized size = 1.65 \[ \frac {\sin (x) (8 \cos (2 x)+\cos (4 x)+15) \left (6 \cos (x)+i \sqrt [4]{-3} \left (\sqrt {3}+3 i\right ) \sin (x) \tan ^{-1}\left (\frac {1}{2} \sqrt [4]{-\frac {1}{3}} \left (\sqrt {3}-i\right ) \tan (x)\right )+\sqrt [4]{-3} \left (\sqrt {3}-3 i\right ) \sin (x) \tan ^{-1}\left (\frac {(-1)^{3/4} \left (\sqrt {3}+i\right ) \tan (x)}{2 \sqrt [4]{3}}\right )\right )}{144 \left (\cos ^6(x)-1\right )} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 199, normalized size = 2.80 \[ \frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (-\frac {3^{\frac {3}{4}} {\left (3^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} + 4 \, \tan \relax (x)\right )}}{3 \, {\left (\sqrt {6} + \sqrt {2}\right )}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} + \frac {1}{18} \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {3^{\frac {3}{4}} {\left (3^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )} - 4 \, \tan \relax (x)\right )}}{3 \, {\left (\sqrt {6} + \sqrt {2}\right )}}\right )\right )} \sqrt {6 \, \sqrt {3} + 9} - \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (\frac {1}{2} \, {\left (\sqrt {6} 3^{\frac {1}{4}} - 3^{\frac {1}{4}} \sqrt {2}\right )} \tan \relax (x) + \tan \relax (x)^{2} + \sqrt {3}\right ) + \frac {1}{36} \, \sqrt {6 \, \sqrt {3} - 9} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} 3^{\frac {1}{4}} - 3^{\frac {1}{4}} \sqrt {2}\right )} \tan \relax (x) + \tan \relax (x)^{2} + \sqrt {3}\right ) - \frac {1}{3 \, \tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 233, normalized size = 3.28 \[ -\frac {\sqrt {3}\, \sqrt {2 \sqrt {3}-3}\, \ln \left (\tan ^{2}\relax (x )+\tan \relax (x ) \sqrt {2 \sqrt {3}-3}+\sqrt {3}\right )}{36}+\frac {\arctan \left (\frac {2 \tan \relax (x )+\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right )}{3 \sqrt {2 \sqrt {3}+3}}+\frac {\arctan \left (\frac {2 \tan \relax (x )+\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right ) \sqrt {3}}{6 \sqrt {2 \sqrt {3}+3}}+\frac {\sqrt {3}\, \sqrt {2 \sqrt {3}-3}\, \ln \left (\tan ^{2}\relax (x )-\tan \relax (x ) \sqrt {2 \sqrt {3}-3}+\sqrt {3}\right )}{36}+\frac {\arctan \left (\frac {2 \tan \relax (x )-\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right )}{3 \sqrt {2 \sqrt {3}+3}}+\frac {\arctan \left (\frac {2 \tan \relax (x )-\sqrt {2 \sqrt {3}-3}}{\sqrt {2 \sqrt {3}+3}}\right ) \sqrt {3}}{6 \sqrt {2 \sqrt {3}+3}}-\frac {1}{3 \tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.29, size = 95, normalized size = 1.34 \[ -\frac {1}{3\,\mathrm {tan}\relax (x)}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {3^{1/4}\,\sqrt {6}\,\mathrm {tan}\relax (x)\,\left (\frac {1}{27}-\frac {1}{27}{}\mathrm {i}\right )}{-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}}\right )\,\left (3^{1/4}\,\left (1+1{}\mathrm {i}\right )+3^{3/4}\,\left (-1+1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {3^{1/4}\,\sqrt {6}\,\mathrm {tan}\relax (x)\,\left (\frac {1}{27}+\frac {1}{27}{}\mathrm {i}\right )}{\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}}\right )\,\left (3^{1/4}\,\left (1-\mathrm {i}\right )+3^{3/4}\,\left (-1-\mathrm {i}\right )\right )\,1{}\mathrm {i}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 23.61, size = 728, normalized size = 10.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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