Optimal. Leaf size=39 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (\sin ^2(x)+1\right ) \cos (x)}{2 \sqrt {1-\sin ^6(x)}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.28, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3216, 1996, 1904, 206} \[ \frac {\tanh ^{-1}\left (\frac {\cos (x) \left (6-3 \cos ^2(x)\right )}{2 \sqrt {3} \sqrt {\cos ^6(x)-3 \cos ^4(x)+3 \cos ^2(x)}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1904
Rule 1996
Rule 3216
Rubi steps
\begin {align*} \int \frac {\sin (x)}{\sqrt {1-\sin ^6(x)}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\left (1-x^2\right )^3}} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {3 x^2-3 x^4+x^6}} \, dx,x,\cos (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {\cos (x) \left (6-3 \cos ^2(x)\right )}{\sqrt {3 \cos ^2(x)-3 \cos ^4(x)+\cos ^6(x)}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\cos (x) \left (6-3 \cos ^2(x)\right )}{2 \sqrt {3} \sqrt {3 \cos ^2(x)-3 \cos ^4(x)+\cos ^6(x)}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 65, normalized size = 1.67 \[ -\frac {\cos (x) \sqrt {-8 \cos (2 x)+\cos (4 x)+15} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} (\cos (2 x)-3)}{\sqrt {-8 \cos (2 x)+\cos (4 x)+15}}\right )}{4 \sqrt {6-6 \sin ^6(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 63, normalized size = 1.62 \[ \frac {1}{12} \, \sqrt {3} \log \left (\frac {7 \, \cos \relax (x)^{5} - 24 \, \cos \relax (x)^{3} - 4 \, \sqrt {\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2}} {\left (\sqrt {3} \cos \relax (x)^{2} - 2 \, \sqrt {3}\right )} + 24 \, \cos \relax (x)}{\cos \relax (x)^{5}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.35, size = 67, normalized size = 1.72 \[ -\frac {\sqrt {3} \log \left (\cos \relax (x)^{2} + \sqrt {3} - \sqrt {\cos \relax (x)^{4} - 3 \, \cos \relax (x)^{2} + 3}\right ) - \sqrt {3} \log \left (-\cos \relax (x)^{2} + \sqrt {3} + \sqrt {\cos \relax (x)^{4} - 3 \, \cos \relax (x)^{2} + 3}\right )}{6 \, \mathrm {sgn}\left (\cos \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 67, normalized size = 1.72 \[ -\frac {\sqrt {\cos ^{4}\relax (x )-3 \left (\cos ^{2}\relax (x )\right )+3}\, \sqrt {3}\, \arctanh \left (\frac {\left (\cos ^{2}\relax (x )-2\right ) \sqrt {3}}{2 \sqrt {\cos ^{4}\relax (x )-3 \left (\cos ^{2}\relax (x )\right )+3}}\right ) \cos \relax (x )}{6 \sqrt {\cos ^{6}\relax (x )-3 \left (\cos ^{4}\relax (x )\right )+3 \left (\cos ^{2}\relax (x )\right )}} \]
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 \[ \int \frac {\sin \relax (x)}{\sqrt {-\sin \relax (x)^{6} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sin \relax (x)}{\sqrt {1-{\sin \relax (x)}^6}} \,d x \]
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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