Optimal. Leaf size=49 \[ -\frac {1}{625} \left (2-5 \sin ^3(x)\right )^{5/3}+\frac {2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}+\frac {4}{125 \sqrt [3]{2-5 \sin ^3(x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4334, 266, 43} \[ -\frac {1}{625} \left (2-5 \sin ^3(x)\right )^{5/3}+\frac {2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}+\frac {4}{125 \sqrt [3]{2-5 \sin ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4334
Rubi steps
\begin {align*} \int \frac {\cos (x) \sin ^8(x)}{\left (2-5 \sin ^3(x)\right )^{4/3}} \, dx &=\operatorname {Subst}\left (\int \frac {x^8}{\left (2-5 x^3\right )^{4/3}} \, dx,x,\sin (x)\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{(2-5 x)^{4/3}} \, dx,x,\sin ^3(x)\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {4}{25 (2-5 x)^{4/3}}-\frac {4}{25 \sqrt [3]{2-5 x}}+\frac {1}{25} (2-5 x)^{2/3}\right ) \, dx,x,\sin ^3(x)\right )\\ &=\frac {4}{125 \sqrt [3]{2-5 \sin ^3(x)}}+\frac {2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}-\frac {1}{625} \left (2-5 \sin ^3(x)\right )^{5/3}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 30, normalized size = 0.61 \[ \frac {-25 \sin ^6(x)-30 \sin ^3(x)+36}{625 \sqrt [3]{2-5 \sin ^3(x)}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos (x) \sin ^8(x)}{\left (2-5 \sin ^3(x)\right )^{4/3}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.27, size = 46, normalized size = 0.94 \[ \frac {25 \, \cos \relax (x)^{6} - 75 \, \cos \relax (x)^{4} + 75 \, \cos \relax (x)^{2} + 30 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x) + 11}{625 \, {\left (5 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x) + 2\right )}^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 37, normalized size = 0.76 \[ -\frac {1}{625} \, {\left (-5 \, \sin \relax (x)^{3} + 2\right )}^{\frac {5}{3}} + \frac {2}{125} \, {\left (-5 \, \sin \relax (x)^{3} + 2\right )}^{\frac {2}{3}} + \frac {4}{125 \, {\left (-5 \, \sin \relax (x)^{3} + 2\right )}^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.96, size = 0, normalized size = 0.00 \[\int \frac {\cot \relax (x ) \left (\sin ^{9}\relax (x )\right )}{\left (2-5 \left (\sin ^{3}\relax (x )\right )\right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 37, normalized size = 0.76 \[ -\frac {1}{625} \, {\left (-5 \, \sin \relax (x)^{3} + 2\right )}^{\frac {5}{3}} + \frac {2}{125} \, {\left (-5 \, \sin \relax (x)^{3} + 2\right )}^{\frac {2}{3}} + \frac {4}{125 \, {\left (-5 \, \sin \relax (x)^{3} + 2\right )}^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cot}\relax (x)\,{\sin \relax (x)}^9}{{\left (2-5\,{\sin \relax (x)}^3\right )}^{4/3}} \,d x \]
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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