Optimal. Leaf size=49 \[ \frac {4}{125 \sqrt [3]{2-5 \sin ^3(x)}}+\frac {2}{125} \left (2-5 \sin ^3(x)\right )^{2/3}-\frac {1}{625} \left (2-5 \sin ^3(x)\right )^{5/3} \]
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Rubi [A]
time = 0.07, 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 = {4419, 272, 45}
\begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4419
Rubi steps
\begin {align*} \int \frac {\cos (x) \sin ^8(x)}{\left (2-5 \sin ^3(x)\right )^{4/3}} \, dx &=\text {Subst}\left (\int \frac {x^8}{\left (2-5 x^3\right )^{4/3}} \, dx,x,\sin (x)\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(2-5 x)^{4/3}} \, dx,x,\sin ^3(x)\right )\\ &=\frac {1}{3} \text {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 = 0.36, size = 30, normalized size = 0.61 \begin {gather*} \frac {36-30 \sin ^3(x)-25 \sin ^6(x)}{625 \sqrt [3]{2-5 \sin ^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.83, size = 0, normalized size = 0.00 \[\int \frac {\cot \left (x \right ) \left (\sin ^{9}\left (x \right )\right )}{\left (2-5 \left (\sin ^{3}\left (x \right )\right )\right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.11, size = 37, normalized size = 0.76 \begin {gather*} -\frac {1}{625} \, {\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac {5}{3}} + \frac {2}{125} \, {\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac {2}{3}} + \frac {4}{125 \, {\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 46, normalized size = 0.94 \begin {gather*} \frac {25 \, \cos \left (x\right )^{6} - 75 \, \cos \left (x\right )^{4} + 75 \, \cos \left (x\right )^{2} + 30 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + 11}{625 \, {\left (5 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + 2\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 37, normalized size = 0.76 \begin {gather*} -\frac {1}{625} \, {\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac {5}{3}} + \frac {2}{125} \, {\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac {2}{3}} + \frac {4}{125 \, {\left (-5 \, \sin \left (x\right )^{3} + 2\right )}^{\frac {1}{3}}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {cot}\left (x\right )\,{\sin \left (x\right )}^9}{{\left (2-5\,{\sin \left (x\right )}^3\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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