Integrand size = 15, antiderivative size = 95 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\frac {\arctan \left (\frac {1-\sqrt [3]{1+2 \cos ^9(x)}}{\sqrt {3} \sqrt [6]{1+2 \cos ^9(x)}}\right )}{3 \sqrt {3}}+\frac {1}{3} \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {1}{9} \text {arctanh}\left (\sqrt {1+2 \cos ^9(x)}\right )-\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6} \]
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Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.71, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3309, 272, 52, 65, 302, 648, 632, 210, 642, 212} \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\frac {\arctan \left (\frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \text {arctanh}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )-\frac {2}{15} \left (2 \cos ^9(x)+1\right )^{5/6}-\frac {1}{18} \log \left (\sqrt [3]{2 \cos ^9(x)+1}-\sqrt [6]{2 \cos ^9(x)+1}+1\right )+\frac {1}{18} \log \left (\sqrt [3]{2 \cos ^9(x)+1}+\sqrt [6]{2 \cos ^9(x)+1}+1\right ) \]
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Rule 52
Rule 65
Rule 210
Rule 212
Rule 272
Rule 302
Rule 632
Rule 642
Rule 648
Rule 3309
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+2 x^9\right )^{5/6}}{x} \, dx,x,\cos (x)\right ) \\ & = -\left (\frac {1}{9} \text {Subst}\left (\int \frac {(1+2 x)^{5/6}}{x} \, dx,x,\cos ^9(x)\right )\right ) \\ & = -\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{x \sqrt [6]{1+2 x}} \, dx,x,\cos ^9(x)\right ) \\ & = -\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}-\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{-\frac {1}{2}+\frac {x^6}{2}} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right ) \\ & = -\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {2}{9} \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {2}{9} \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right ) \\ & = \frac {2}{9} \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}-\frac {1}{18} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {1}{18} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right ) \\ & = \frac {2}{9} \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}-\frac {1}{18} \log \left (1-\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )+\frac {1}{18} \log \left (1+\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [6]{1+2 \cos ^9(x)}\right ) \\ & = \frac {\arctan \left (\frac {1-2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}-\frac {1}{18} \log \left (1-\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )+\frac {1}{18} \log \left (1+\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.62 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\frac {1}{90} \left (10 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )-10 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )+20 \text {arctanh}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-12 \left (1+2 \cos ^9(x)\right )^{5/6}-5 \log \left (1-\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )+5 \log \left (1+\sqrt [6]{1+2 \cos ^9(x)}+\sqrt [3]{1+2 \cos ^9(x)}\right )\right ) \]
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\[\int {\left (1+2 \left (\cos ^{9}\left (x \right )\right )\right )}^{\frac {5}{6}} \tan \left (x \right )d x\]
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Timed out. \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\text {Timed out} \]
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Timed out. \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (72) = 144\).
Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.53 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1\right )}\right ) - \frac {2}{15} \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {5}{6}} + \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} + {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} - {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (72) = 144\).
Time = 0.37 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.54 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1\right )}\right ) - \frac {2}{15} \, {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {5}{6}} + \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} + {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{3}} - {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{9} \, \log \left ({\left | {\left (2 \, \cos \left (x\right )^{9} + 1\right )}^{\frac {1}{6}} - 1 \right |}\right ) \]
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Timed out. \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx=\int \mathrm {tan}\left (x\right )\,{\left (2\,{\cos \left (x\right )}^9+1\right )}^{5/6} \,d x \]
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