Integrand size = 19, antiderivative size = 9 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {1}{3} \arcsin \left (\cos ^3(x)\right ) \]
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Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4420, 281, 222} \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {1}{3} \arcsin \left (\cos ^3(x)\right ) \]
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Rule 222
Rule 281
Rule 4420
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^6}} \, dx,x,\cos (x)\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\cos ^3(x)\right )\right ) \\ & = -\frac {1}{3} \arcsin \left (\cos ^3(x)\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.64 (sec) , antiderivative size = 162, normalized size of antiderivative = 18.00 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {i \cos ^2(x) \operatorname {EllipticPi}\left (\frac {3}{2}+\frac {i \sqrt {3}}{2},i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {3}}} \tan (x)\right ),\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}\right ) \sin (x) \sqrt {1-\frac {2 i \tan ^2(x)}{-3 i+\sqrt {3}}} \sqrt {1+\frac {2 i \tan ^2(x)}{3 i+\sqrt {3}}}}{\sqrt {2} \sqrt {-\frac {i}{-3 i+\sqrt {3}}} \sqrt {1-\cos ^6(x)}} \]
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\[\int \frac {\cos \left (x \right )^{2} \sin \left (x \right )}{\sqrt {1-\cos \left (x \right )^{6}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (7) = 14\).
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 3.22 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {-\cos \left (x\right )^{6} + 1} \cos \left (x\right )^{3}}{2 \, \cos \left (x\right )^{6} - 1}\right ) \]
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Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (7) = 14\).
Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.00 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\frac {1}{3} \, \arctan \left (\frac {\sqrt {-\cos \left (x\right )^{6} + 1}}{\cos \left (x\right )^{3}}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=-\frac {1}{3} \, \arcsin \left (\cos \left (x\right )^{3}\right ) \]
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Timed out. \[ \int \frac {\cos ^2(x) \sin (x)}{\sqrt {1-\cos ^6(x)}} \, dx=\int \frac {{\cos \left (x\right )}^2\,\sin \left (x\right )}{\sqrt {1-{\cos \left (x\right )}^6}} \,d x \]
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