Optimal. Leaf size=69 \[ \frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}+2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{1-7 \tan ^2(x)}+1}{\sqrt {3}}\right )+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+2 \log (\cos (x)) \]
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Rubi [A] time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3670, 444, 50, 55, 618, 204, 31} \[ 2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{1-7 \tan ^2(x)}+1}{\sqrt {3}}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+2 \log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 55
Rule 204
Rule 444
Rule 618
Rule 3670
Rubi steps
\begin {align*} \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx &=\operatorname {Subst}\left (\int \frac {x \left (1-7 x^2\right )^{2/3}}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1-7 x)^{2/3}}{1+x} \, dx,x,\tan ^2(x)\right )\\ &=\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}+4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-7 x} (1+x)} \, dx,x,\tan ^2(x)\right )\\ &=2 \log (\cos (x))+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}-3 \operatorname {Subst}\left (\int \frac {1}{2-x} \, dx,x,\sqrt [3]{1-7 \tan ^2(x)}\right )+6 \operatorname {Subst}\left (\int \frac {1}{4+2 x+x^2} \, dx,x,\sqrt [3]{1-7 \tan ^2(x)}\right )\\ &=2 \log (\cos (x))+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}-12 \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,2+2 \sqrt [3]{1-7 \tan ^2(x)}\right )\\ &=2 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{1-7 \tan ^2(x)}}{\sqrt {3}}\right )+2 \log (\cos (x))+3 \log \left (2-\sqrt [3]{1-7 \tan ^2(x)}\right )+\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 42, normalized size = 0.61 \[ -\frac {3}{4} \left (1-7 \tan ^2(x)\right )^{2/3} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{8} (4 \cos (2 x)-3) \sec ^2(x)\right )-1\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \tan (x) \left (1-7 \tan ^2(x)\right )^{2/3} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 3.76, size = 117, normalized size = 1.70 \[ 2 \, \sqrt {3} \arctan \left (\frac {7 \, \sqrt {3} \tan \relax (x)^{2} + 4 \, \sqrt {3} {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} - 16 \, \sqrt {3} {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} - \sqrt {3}}{7 \, \tan \relax (x)^{2} - 65}\right ) + \frac {3}{4} \, {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} + \log \left (\frac {7 \, \tan \relax (x)^{2} + 6 \, {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} - 12 \, {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} + 7}{\tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 79, normalized size = 1.14 \[ 2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left ({\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {3}{4} \, {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} - \log \left ({\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} + 2 \, {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} + 4\right ) + 2 \, \log \left ({\left | {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \tan \relax (x ) \left (1-7 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {2}{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-7 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} \tan \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 101, normalized size = 1.46 \[ 2\,\ln \left (144\,{\left (1-7\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}-288\right )+\frac {3\,{\left (1-7\,{\mathrm {tan}\relax (x)}^2\right )}^{2/3}}{4}+\ln \left (144\,{\left (1-7\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}-72\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )-\ln \left (144\,{\left (1-7\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}-72\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right ) \]
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 \[ \int \left (1 - 7 \tan ^{2}{\relax (x )}\right )^{\frac {2}{3}} \tan {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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