Optimal. Leaf size=133 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{a^3-b^3}}+\frac {3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}+\frac {\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}} \]
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Rubi [A] time = 0.16, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3670, 444, 55, 617, 204, 31} \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{a^3-b^3}}+\frac {3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}+\frac {\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 444
Rule 617
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan (x)}{\sqrt [3]{a^3+b^3 \tan ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt [3]{a^3+b^3 x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{a^3+b^3 x}} \, dx,x,\tan ^2(x)\right )\\ &=\frac {\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{\left (a^3-b^3\right )^{2/3}+\sqrt [3]{a^3-b^3} x+x^2} \, dx,x,\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a^3-b^3}-x} \, dx,x,\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}\\ &=\frac {\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}}+\frac {3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}\right )}{2 \sqrt [3]{a^3-b^3}}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a^3-b^3}}+\frac {\log (\cos (x))}{2 \sqrt [3]{a^3-b^3}}+\frac {3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )}{4 \sqrt [3]{a^3-b^3}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 105, normalized size = 0.79 \[ \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a^3+b^3 \tan ^2(x)}}{\sqrt [3]{a^3-b^3}}+1}{\sqrt {3}}\right )+3 \log \left (\sqrt [3]{a^3-b^3}-\sqrt [3]{a^3+b^3 \tan ^2(x)}\right )+2 \log (\cos (x))}{4 \sqrt [3]{a^3-b^3}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan (x)}{\sqrt [3]{a^3+b^3 \tan ^2(x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [A] time = 0.76, size = 186, normalized size = 1.40 \[ \frac {3 \, {\left (a^{3} - b^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b^{3} \tan \relax (x)^{2} + a^{3}\right )}^{\frac {1}{3}} + {\left (a^{3} - b^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - b^{3}\right )}^{\frac {1}{3}}}\right )}{2 \, {\left (\sqrt {3} a^{3} - \sqrt {3} b^{3}\right )}} - \frac {\log \left ({\left (b^{3} \tan \relax (x)^{2} + a^{3}\right )}^{\frac {2}{3}} + {\left (b^{3} \tan \relax (x)^{2} + a^{3}\right )}^{\frac {1}{3}} {\left (a^{3} - b^{3}\right )}^{\frac {1}{3}} + {\left (a^{3} - b^{3}\right )}^{\frac {2}{3}}\right )}{4 \, {\left (a^{3} - b^{3}\right )}^{\frac {1}{3}}} + \frac {\log \left ({\left | {\left (b^{3} \tan \relax (x)^{2} + a^{3}\right )}^{\frac {1}{3}} - {\left (a^{3} - b^{3}\right )}^{\frac {1}{3}} \right |}\right )}{2 \, {\left (a^{3} - b^{3}\right )}^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\tan \relax (x )}{\left (a^{3}+b^{3} \left (\tan ^{2}\relax (x )\right )\right )^{\frac {1}{3}}}\, dx\]
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 \[ \int \frac {\tan \relax (x)}{{\left (b^{3} \tan \relax (x)^{2} + a^{3}\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 250, normalized size = 1.88 \[ \frac {\ln \left (\frac {9\,{\left (a^3+b^3\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}}{4}-\frac {9\,a^3-9\,b^3}{4\,{\left (a-b\right )}^{2/3}\,{\left (a^2+a\,b+b^2\right )}^{2/3}}\right )}{2\,{\left (a-b\right )}^{1/3}\,{\left (a^2+a\,b+b^2\right )}^{1/3}}+\frac {\ln \left (\frac {9\,{\left (a^3+b^3\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}}{4}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,a^3-9\,b^3\right )}{16\,{\left (a-b\right )}^{2/3}\,{\left (a^2+a\,b+b^2\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,{\left (a-b\right )}^{1/3}\,{\left (a^2+a\,b+b^2\right )}^{1/3}}-\frac {\ln \left (\frac {9\,{\left (a^3+b^3\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}}{4}-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,a^3-9\,b^3\right )}{16\,{\left (a-b\right )}^{2/3}\,{\left (a^2+a\,b+b^2\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,{\left (a-b\right )}^{1/3}\,{\left (a^2+a\,b+b^2\right )}^{1/3}} \]
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 \[ \int \frac {\tan {\relax (x )}}{\sqrt [3]{a^{3} + b^{3} \tan ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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