Optimal. Leaf size=27 \[ -\log (\tan (x))+\frac {3}{2} \log \left (1-\sqrt [3]{1-8 \tan ^2(x)}\right ) \]
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Rubi [A]
time = 0.63, antiderivative size = 35, normalized size of antiderivative = 1.30, number of steps
used = 15, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {4451, 6857,
528, 455, 59, 632, 210, 31, 57} \begin {gather*} \frac {3}{2} \log \left (1-\sqrt [3]{9-8 \sec ^2(x)}\right )-\frac {1}{2} \log \left (1-\sec ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 59
Rule 210
Rule 455
Rule 528
Rule 632
Rule 4451
Rule 6857
Rubi steps
\begin {align*} \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx &=-\text {Subst}\left (\int \frac {1+\sqrt [3]{9-\frac {8}{x^2}}}{\left (9-\frac {8}{x^2}\right )^{2/3} x \left (1-x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (-\frac {1}{\left (9-\frac {8}{x^2}\right )^{2/3} x \left (-1+x^2\right )}-\frac {1}{\sqrt [3]{9-\frac {8}{x^2}} x \left (-1+x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{\left (9-\frac {8}{x^2}\right )^{2/3} x \left (-1+x^2\right )} \, dx,x,\cos (x)\right )+\text {Subst}\left (\int \frac {1}{\sqrt [3]{9-\frac {8}{x^2}} x \left (-1+x^2\right )} \, dx,x,\cos (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{\left (9-\frac {8}{x^2}\right )^{2/3} \left (1-\frac {1}{x^2}\right ) x^3} \, dx,x,\cos (x)\right )+\text {Subst}\left (\int \frac {1}{\sqrt [3]{9-\frac {8}{x^2}} \left (1-\frac {1}{x^2}\right ) x^3} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(9-8 x)^{2/3} (1-x)} \, dx,x,\sec ^2(x)\right )\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{9-8 x} (1-x)} \, dx,x,\sec ^2(x)\right )\\ &=-\log (\tan (x))-2 \left (\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{9-8 \sec ^2(x)}\right )\right )\\ &=\frac {3}{2} \log \left (1-\sqrt [3]{9-8 \sec ^2(x)}\right )-\log (\tan (x))\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(27)=54\).
time = 2.85, size = 58, normalized size = 2.15 \begin {gather*} \frac {1}{4} \left (-2 \log (\tan (x))+5 \log \left (1-\sqrt [3]{1-8 \tan ^2(x)}\right )-\log \left (1+\sqrt [3]{1-8 \tan ^2(x)}+\left (1-8 \tan ^2(x)\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {\cot \left (x \right ) \left (1+\left (1-8 \left (\tan ^{2}\left (x \right )\right )\right )^{\frac {1}{3}}\right )}{\cos \left (x \right )^{2} \left (1-8 \left (\tan ^{2}\left (x \right )\right )\right )^{\frac {2}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (23) = 46\).
time = 1.71, size = 93, normalized size = 3.44 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {16 \, {\left (145 \, \cos \left (x\right )^{4} - 200 \, \cos \left (x\right )^{2} + 3 \, {\left (11 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2}\right )} \left (\frac {9 \, \cos \left (x\right )^{2} - 8}{\cos \left (x\right )^{2}}\right )^{\frac {2}{3}} + 3 \, {\left (19 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2}\right )} \left (\frac {9 \, \cos \left (x\right )^{2} - 8}{\cos \left (x\right )^{2}}\right )^{\frac {1}{3}} + 64\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}\right ) \end {gather*}
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 \begin {gather*} \int \frac {\left (\sqrt [3]{1 - 8 \tan ^{2}{\left (x \right )}} + 1\right ) \cot {\left (x \right )}}{\left (1 - 8 \tan ^{2}{\left (x \right )}\right )^{\frac {2}{3}} \cos ^{2}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 40, normalized size = 1.48 \begin {gather*} -\frac {1}{2} \, \log \left ({\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {2}{3}} + {\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \log \left ({\left | {\left (-8 \, \tan \left (x\right )^{2} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \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 = -0.04 \begin {gather*} \int \frac {\mathrm {cot}\left (x\right )\,\left ({\left (1-8\,{\mathrm {tan}\left (x\right )}^2\right )}^{1/3}+1\right )}{{\cos \left (x\right )}^2\,{\left (1-8\,{\mathrm {tan}\left (x\right )}^2\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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