Optimal. Leaf size=47 \[ \frac {3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac {3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]
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Rubi [A] time = 0.14, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6719, 14} \[ \frac {3}{5} \sin (x) \cos ^3(x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}+\frac {3}{11} \sin ^3(x) \cos (x) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6719
Rubi steps
\begin {align*} \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt [3]{x^2 \left (1+x^2\right )^6}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \operatorname {Subst}\left (\int x^{2/3} \left (1+x^2\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac {2}{3}}(x)}\\ &=\frac {\left (\cos ^4(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\right ) \operatorname {Subst}\left (\int \left (x^{2/3}+x^{8/3}\right ) \, dx,x,\tan (x)\right )}{\tan ^{\frac {2}{3}}(x)}\\ &=\frac {3}{5} \cos ^3(x) \sin (x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}+\frac {3}{11} \cos (x) \sin ^3(x) \sqrt [3]{\sec ^{12}(x) \tan ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 63, normalized size = 1.34 \[ \frac {3 \sin (x) \cos (x) \left (8 \left (-\tan ^2(x)\right )^{5/6}+3 \cos (2 x) \left (\left (-\tan ^2(x)\right )^{5/6}-1\right )-3\right ) \sqrt [3]{\tan ^2(x) \sec ^{12}(x)}}{55 \left (-\tan ^2(x)\right )^{5/6}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{\sec ^{12}(x) \tan ^2(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.00, size = 29, normalized size = 0.62 \[ \frac {3}{55} \, {\left (6 \, \cos \relax (x)^{3} + 5 \, \cos \relax (x)\right )} \left (-\frac {\cos \relax (x)^{2} - 1}{\cos \relax (x)^{14}}\right )^{\frac {1}{3}} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {\sin \relax (x)^{2}}{\cos \relax (x)^{14}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[\int \left (\frac {\sin ^{2}\relax (x )}{\cos \relax (x )^{14}}\right )^{\frac {1}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 13, normalized size = 0.28 \[ \frac {3}{11} \, \tan \relax (x)^{\frac {11}{3}} + \frac {3}{5} \, \tan \relax (x)^{\frac {5}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.94, size = 32, normalized size = 0.68 \[ \frac {6\,\sin \left (2\,x\right )\,{\left (1-\cos \left (2\,x\right )\right )}^{1/3}\,\left (3\,\cos \left (2\,x\right )+8\right )}{55\,{\left (\cos \left (2\,x\right )+1\right )}^{7/3}} \]
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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