Optimal. Leaf size=20 \[ -\frac {3}{32} \left (\sqrt [3]{1-8 \tan ^2(x)}+1\right )^2 \]
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Rubi [A] time = 0.23, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4342, 6686} \[ -\frac {3}{32} \left (\sqrt [3]{1-8 \tan ^2(x)}+1\right )^2 \]
Antiderivative was successfully verified.
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Rule 4342
Rule 6686
Rubi steps
\begin {align*} \int \frac {\sec ^2(x) \tan (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {x \left (1+\sqrt [3]{1-8 x^2}\right )}{\left (1-8 x^2\right )^{2/3}} \, dx,x,\tan (x)\right )\\ &=-\frac {3}{32} \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )^2\\ \end {align*}
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Mathematica [B] time = 0.21, size = 42, normalized size = 2.10 \[ -\frac {3 (9 \cos (2 x)-7) \left (\sqrt [3]{1-8 \tan ^2(x)}+2\right ) \sec ^2(x)}{64 \left (1-8 \tan ^2(x)\right )^{2/3}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^2(x) \tan (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.56, size = 35, normalized size = 1.75 \[ -\frac {3}{32} \, \left (\frac {9 \, \cos \relax (x)^{2} - 8}{\cos \relax (x)^{2}}\right )^{\frac {2}{3}} - \frac {3}{16} \, \left (\frac {9 \, \cos \relax (x)^{2} - 8}{\cos \relax (x)^{2}}\right )^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 25, normalized size = 1.25 \[ -\frac {3}{32} \, {\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} - \frac {3}{16} \, {\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 26, normalized size = 1.30
method | result | size |
derivativedivides | \(-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {1}{3}}}{16}-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}{32}\) | \(26\) |
default | \(-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {1}{3}}}{16}-\frac {3 \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}{32}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 86, normalized size = 4.30 \[ -\frac {3 \, {\left (\frac {{\left (9 \, \sin \relax (x)^{2} - 1\right )} {\left (3 \, \sin \relax (x) - 1\right )}^{\frac {1}{3}} {\left (\sin \relax (x) + 1\right )}^{\frac {1}{3}} {\left (\sin \relax (x) - 1\right )}^{\frac {1}{3}}}{{\left (3 \, \sin \relax (x) + 1\right )}^{\frac {1}{3}}} + \frac {2 \, {\left (9 \, \sin \relax (x)^{2} - 1\right )} {\left (\sin \relax (x) + 1\right )}^{\frac {2}{3}} {\left (\sin \relax (x) - 1\right )}^{\frac {2}{3}}}{{\left (3 \, \sin \relax (x) + 1\right )}^{\frac {2}{3}}}\right )}}{32 \, {\left (\sin \relax (x)^{2} - 1\right )} {\left (3 \, \sin \relax (x) - 1\right )}^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 43, normalized size = 2.15 \[ -\frac {3\,\left ({\left (18\,{\cos \relax (x)}^2-16\right )}^{1/3}+2\,{\left (2\,{\cos \relax (x)}^2\right )}^{1/3}\right )\,{\left (18\,{\cos \relax (x)}^2-16\right )}^{1/3}}{32\,{\left (2\,{\cos \relax (x)}^2\right )}^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sqrt [3]{1 - 8 \tan ^{2}{\relax (x )}} + 1\right ) \tan {\relax (x )}}{\left (1 - 8 \tan ^{2}{\relax (x )}\right )^{\frac {2}{3}} \cos ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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