Optimal. Leaf size=125 \[ -\frac {9 \sin ^4(x)}{10 \left (\sin (x) \cos ^5(x)\right )^{2/3}}-\frac {9}{4} \sec ^8(x) \left (\sin (x) \cos ^5(x)\right )^{4/3}+\frac {3}{14} \tan ^4(x) \sqrt [3]{\sin (x) \cos ^5(x)} \sqrt [3]{\tan (x) \sec ^6(x)}+\frac {3}{4} \tan ^2(x) \sqrt [3]{\sin (x) \cos ^5(x)} \sqrt [3]{\tan (x) \sec ^6(x)}+\frac {3}{2} \sqrt [3]{\sin (x) \cos ^5(x)} \sqrt [3]{\tan (x) \sec ^6(x)} \]
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Rubi [A] time = 1.02, antiderivative size = 141, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6719, 6733, 6742, 14} \[ -\frac {9 \sin ^2(x) \cos ^2(x)}{4 \left (\sin (x) \cos ^5(x)\right )^{2/3}}-\frac {9 \sin ^4(x)}{10 \left (\sin (x) \cos ^5(x)\right )^{2/3}}+\frac {3 \sin ^3(x) \cos ^3(x) \sqrt [3]{\tan (x) \sec ^6(x)}}{4 \left (\sin (x) \cos ^5(x)\right )^{2/3}}+\frac {3 \sin ^5(x) \cos (x) \sqrt [3]{\tan (x) \sec ^6(x)}}{14 \left (\sin (x) \cos ^5(x)\right )^{2/3}}+\frac {3 \sin (x) \cos ^5(x) \sqrt [3]{\tan (x) \sec ^6(x)}}{2 \left (\sin (x) \cos ^5(x)\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6719
Rule 6733
Rule 6742
Rubi steps
\begin {align*} \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {-3 x+\sqrt [3]{x \left (1+x^2\right )^3}}{\left (\frac {x}{\left (1+x^2\right )^3}\right )^{2/3} \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=\frac {\left (\cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (-3 x+\sqrt [3]{x \left (1+x^2\right )^3}\right )}{x^{2/3}} \, dx,x,\tan (x)\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=\frac {\left (3 \cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int \left (1+x^6\right ) \left (-3 x^3+\sqrt [3]{x^3 \left (1+x^6\right )^3}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=\frac {\left (3 \cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int \left (-3 x^3+\sqrt [3]{x^3 \left (1+x^6\right )^3}-x^6 \left (3 x^3-\sqrt [3]{\left (x+x^7\right )^3}\right )\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac {9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {\left (3 \cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int \sqrt [3]{x^3 \left (1+x^6\right )^3} \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {\left (3 \cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int x^6 \left (3 x^3-\sqrt [3]{\left (x+x^7\right )^3}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac {9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {\left (3 \cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int \left (3 x^9-x^6 \sqrt [3]{x^3 \left (1+x^6\right )^3}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname {Subst}\left (\int x \left (1+x^6\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac {9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {\left (3 \cos ^4(x) \tan ^{\frac {2}{3}}(x)\right ) \operatorname {Subst}\left (\int x^6 \sqrt [3]{x^3 \left (1+x^6\right )^3} \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \left (x+x^7\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac {9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos ^5(x) \sin (x) \sqrt [3]{\sec ^6(x) \tan (x)}}{2 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos ^3(x) \sin ^3(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{8 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname {Subst}\left (\int x^7 \left (1+x^6\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac {9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos ^5(x) \sin (x) \sqrt [3]{\sec ^6(x) \tan (x)}}{2 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos ^3(x) \sin ^3(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{8 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {\left (3 \cos ^6(x) \sqrt [3]{\tan (x)} \sqrt [3]{\sec ^6(x) \tan (x)}\right ) \operatorname {Subst}\left (\int \left (x^7+x^{13}\right ) \, dx,x,\sqrt [3]{\tan (x)}\right )}{\left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ &=-\frac {9 \cos ^2(x) \sin ^2(x)}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos ^5(x) \sin (x) \sqrt [3]{\sec ^6(x) \tan (x)}}{2 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos ^3(x) \sin ^3(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{4 \left (\cos ^5(x) \sin (x)\right )^{2/3}}+\frac {3 \cos (x) \sin ^5(x) \sqrt [3]{\sec ^6(x) \tan (x)}}{14 \left (\cos ^5(x) \sin (x)\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 58, normalized size = 0.46 \[ -\frac {3 \sin (x) \left (924 \sin (x)+252 \sin (3 x)-5 (158 \cos (x)+57 \cos (3 x)+9 \cos (5 x)) \sqrt [3]{\tan (x) \sec ^6(x)}\right )}{2240 \left (\sin (x) \cos ^5(x)\right )^{2/3}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {-3 \tan (x)+\sqrt [3]{\sec ^6(x) \tan (x)}}{\left (\cos ^5(x) \sin (x)\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.31, size = 56, normalized size = 0.45 \[ -\frac {3 \, \left (\cos \relax (x)^{5} \sin \relax (x)\right )^{\frac {1}{3}} {\left (21 \, {\left (3 \, \cos \relax (x)^{2} + 2\right )} \sin \relax (x) - 5 \, {\left (9 \, \cos \relax (x)^{5} + 3 \, \cos \relax (x)^{3} + 2 \, \cos \relax (x)\right )} \left (\frac {\sin \relax (x)}{\cos \relax (x)^{7}}\right )^{\frac {1}{3}}\right )}}{140 \, \cos \relax (x)^{5}} \]
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 \[ \int \frac {\left (\frac {\sin \relax (x)}{\cos \relax (x)^{7}}\right )^{\frac {1}{3}} - 3 \, \tan \relax (x)}{\left (\cos \relax (x)^{5} \sin \relax (x)\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.32, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {\sin \relax (x )}{\cos \relax (x )^{7}}\right )^{\frac {1}{3}}-3 \tan \relax (x )}{\left (\left (\cos ^{5}\relax (x )\right ) \sin \relax (x )\right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 60, normalized size = 0.48 \[ -\frac {3}{20} \, \tan \relax (x)^{\frac {20}{3}} - \frac {3}{7} \, \tan \relax (x)^{\frac {14}{3}} - \frac {9}{10} \, \tan \relax (x)^{\frac {10}{3}} - \frac {3}{8} \, \tan \relax (x)^{\frac {8}{3}} - \frac {9}{4} \, \tan \relax (x)^{\frac {4}{3}} + \frac {3 \, {\left (14 \, \tan \relax (x)^{7} + 60 \, \tan \relax (x)^{5} + 105 \, \tan \relax (x)^{3} + 140 \, \tan \relax (x)\right )}}{280 \, \tan \relax (x)^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {3\,\mathrm {tan}\relax (x)-{\left (\frac {\sin \relax (x)}{{\cos \relax (x)}^7}\right )}^{1/3}}{{\left ({\cos \relax (x)}^5\,\sin \relax (x)\right )}^{2/3}} \,d x \]
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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