Optimal. Leaf size=125 \[ -\frac {9 \sin ^4(x)}{10 \left (\cos ^5(x) \sin (x)\right )^{2/3}}-\frac {9}{4} \sec ^8(x) \left (\cos ^5(x) \sin (x)\right )^{4/3}+\frac {3}{2} \sqrt [3]{\cos ^5(x) \sin (x)} \sqrt [3]{\sec ^6(x) \tan (x)}+\frac {3}{4} \sqrt [3]{\cos ^5(x) \sin (x)} \tan ^2(x) \sqrt [3]{\sec ^6(x) \tan (x)}+\frac {3}{14} \sqrt [3]{\cos ^5(x) \sin (x)} \tan ^4(x) \sqrt [3]{\sec ^6(x) \tan (x)} \]
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Rubi [A]
time = 0.65, 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 = {6851, 6865,
6874, 14} \begin {gather*} -\frac {9 \sin ^4(x)}{10 \left (\sin (x) \cos ^5(x)\right )^{2/3}}-\frac {9 \sin ^2(x) \cos ^2(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}}+\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 6851
Rule 6865
Rule 6874
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 &=\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.24, size = 58, normalized size = 0.46 \begin {gather*} -\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]{\sec ^6(x) \tan (x)}\right )}{2240 \left (\cos ^5(x) \sin (x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.06, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {\sin \left (x \right )}{\cos \left (x \right )^{7}}\right )^{\frac {1}{3}}-3 \tan \left (x \right )}{\left (\left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )\right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.80, size = 60, normalized size = 0.48 \begin {gather*} -\frac {3}{20} \, \tan \left (x\right )^{\frac {20}{3}} - \frac {3}{7} \, \tan \left (x\right )^{\frac {14}{3}} - \frac {9}{10} \, \tan \left (x\right )^{\frac {10}{3}} - \frac {3}{8} \, \tan \left (x\right )^{\frac {8}{3}} - \frac {9}{4} \, \tan \left (x\right )^{\frac {4}{3}} + \frac {3 \, {\left (14 \, \tan \left (x\right )^{7} + 60 \, \tan \left (x\right )^{5} + 105 \, \tan \left (x\right )^{3} + 140 \, \tan \left (x\right )\right )}}{280 \, \tan \left (x\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 56, normalized size = 0.45 \begin {gather*} -\frac {3 \, \left (\cos \left (x\right )^{5} \sin \left (x\right )\right )^{\frac {1}{3}} {\left (21 \, {\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) - 5 \, {\left (9 \, \cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{3} + 2 \, \cos \left (x\right )\right )} \left (\frac {\sin \left (x\right )}{\cos \left (x\right )^{7}}\right )^{\frac {1}{3}}\right )}}{140 \, \cos \left (x\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int -\frac {3\,\mathrm {tan}\left (x\right )-{\left (\frac {\sin \left (x\right )}{{\cos \left (x\right )}^7}\right )}^{1/3}}{{\left ({\cos \left (x\right )}^5\,\sin \left (x\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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