Optimal. Leaf size=95 \[ -\frac {2}{15} \left (2 \cos ^9(x)+1\right )^{5/6}+\frac {\tan ^{-1}\left (\frac {1-\sqrt [3]{2 \cos ^9(x)+1}}{\sqrt {3} \sqrt [6]{2 \cos ^9(x)+1}}\right )}{3 \sqrt {3}}+\frac {1}{3} \tanh ^{-1}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )-\frac {1}{9} \tanh ^{-1}\left (\sqrt {2 \cos ^9(x)+1}\right ) \]
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Rubi [A] time = 0.26, antiderivative size = 162, normalized size of antiderivative = 1.71, number of steps used = 14, number of rules used = 10, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3230, 266, 50, 63, 296, 634, 618, 204, 628, 206} \[ -\frac {2}{15} \left (2 \cos ^9(x)+1\right )^{5/6}-\frac {1}{18} \log \left (\sqrt [3]{2 \cos ^9(x)+1}-\sqrt [6]{2 \cos ^9(x)+1}+1\right )+\frac {1}{18} \log \left (\sqrt [3]{2 \cos ^9(x)+1}+\sqrt [6]{2 \cos ^9(x)+1}+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \tanh ^{-1}\left (\sqrt [6]{2 \cos ^9(x)+1}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 204
Rule 206
Rule 266
Rule 296
Rule 618
Rule 628
Rule 634
Rule 3230
Rubi steps
\begin {align*} \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1+2 x^9\right )^{5/6}}{x} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{9} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {2}{9} \operatorname {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} \operatorname {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} \tanh ^{-1}\left (\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {2}{15} \left (1+2 \cos ^9(x)\right )^{5/6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [6]{1+2 \cos ^9(x)}\right )\\ &=\frac {2}{9} \tanh ^{-1}\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} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [6]{1+2 \cos ^9(x)}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [6]{1+2 \cos ^9(x)}\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [6]{1+2 \cos ^9(x)}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \tanh ^{-1}\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*}
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Mathematica [A] time = 0.14, size = 154, normalized size = 1.62 \[ \frac {1}{90} \left (-12 \left (2 \cos ^9(x)+1\right )^{5/6}-5 \log \left (\sqrt [3]{2 \cos ^9(x)+1}-\sqrt [6]{2 \cos ^9(x)+1}+1\right )+5 \log \left (\sqrt [3]{2 \cos ^9(x)+1}+\sqrt [6]{2 \cos ^9(x)+1}+1\right )+10 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [6]{2 \cos ^9(x)+1}}{\sqrt {3}}\right )-10 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{2 \cos ^9(x)+1}+1}{\sqrt {3}}\right )+20 \tanh ^{-1}\left (\sqrt [6]{2 \cos ^9(x)+1}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+2 \cos ^9(x)\right )^{5/6} \tan (x) \, dx \]
Verification is Not applicable to the result.
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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 = 1.24, size = 146, normalized size = 1.54 \[ -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} - 1\right )}\right ) - \frac {2}{15} \, {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {5}{6}} + \frac {1}{18} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{3}} + {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{3}} - {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{9} \, \log \left ({\left | {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[\int \left (1+2 \left (\cos ^{9}\relax (x )\right )\right )^{\frac {5}{6}} \tan \relax (x )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 145, normalized size = 1.53 \[ -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} - 1\right )}\right ) - \frac {2}{15} \, {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {5}{6}} + \frac {1}{18} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{3}} + {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{18} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{3}} - {\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} + 1\right ) - \frac {1}{9} \, \log \left ({\left (2 \, \cos \relax (x)^{9} + 1\right )}^{\frac {1}{6}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {tan}\relax (x)\,{\left (2\,{\cos \relax (x)}^9+1\right )}^{5/6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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