Optimal. Leaf size=27 \[ \frac {3}{2} \log \left (1-\sqrt [3]{1-8 \tan ^2(x)}\right )-\log (\tan (x)) \]
________________________________________________________________________________________
Rubi [A] time = 0.97, 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 = {4366, 6725, 514, 444, 57, 618, 204, 31, 55} \[ \frac {3}{2} \log \left (1-\sqrt [3]{9-8 \sec ^2(x)}\right )-\frac {1}{2} \log \left (1-\sec ^2(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 55
Rule 57
Rule 204
Rule 444
Rule 514
Rule 618
Rule 4366
Rule 6725
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 &=-\operatorname {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 )\\ &=-\operatorname {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 )\\ &=\operatorname {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 )+\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{9-\frac {8}{x^2}} x \left (-1+x^2\right )} \, dx,x,\cos (x)\right )\\ &=\operatorname {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 )+\operatorname {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} \operatorname {Subst}\left (\int \frac {1}{(9-8 x)^{2/3} (1-x)} \, dx,x,\sec ^2(x)\right )\right )-\frac {1}{2} \operatorname {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} \operatorname {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*}
________________________________________________________________________________________
Mathematica [B] time = 4.30, size = 58, normalized size = 2.15 \[ \frac {1}{4} \left (5 \log \left (1-\sqrt [3]{1-8 \tan ^2(x)}\right )-\log \left (\left (1-8 \tan ^2(x)\right )^{2/3}+\sqrt [3]{1-8 \tan ^2(x)}+1\right )-2 \log (\tan (x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc (x) \sec (x) \left (1+\sqrt [3]{1-8 \tan ^2(x)}\right )}{\left (1-8 \tan ^2(x)\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 7.06, size = 93, normalized size = 3.44 \[ -\frac {1}{2} \, \log \left (\frac {16 \, {\left (145 \, \cos \relax (x)^{4} - 200 \, \cos \relax (x)^{2} + 3 \, {\left (11 \, \cos \relax (x)^{4} - 8 \, \cos \relax (x)^{2}\right )} \left (\frac {9 \, \cos \relax (x)^{2} - 8}{\cos \relax (x)^{2}}\right )^{\frac {2}{3}} + 3 \, {\left (19 \, \cos \relax (x)^{4} - 16 \, \cos \relax (x)^{2}\right )} \left (\frac {9 \, \cos \relax (x)^{2} - 8}{\cos \relax (x)^{2}}\right )^{\frac {1}{3}} + 64\right )}}{\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.84, size = 40, normalized size = 1.48 \[ -\frac {1}{2} \, \log \left ({\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} + {\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \log \left ({\left | {\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.78, size = 0, normalized size = 0.00 \[\int \frac {\cot \relax (x ) \left (1+\left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {1}{3}}\right )}{\cos \relax (x )^{2} \left (1-8 \left (\tan ^{2}\relax (x )\right )\right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left ({\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {1}{3}} + 1\right )} \cot \relax (x)}{{\left (-8 \, \tan \relax (x)^{2} + 1\right )}^{\frac {2}{3}} \cos \relax (x)^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {cot}\relax (x)\,\left ({\left (1-8\,{\mathrm {tan}\relax (x)}^2\right )}^{1/3}+1\right )}{{\cos \relax (x)}^2\,{\left (1-8\,{\mathrm {tan}\relax (x)}^2\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sqrt [3]{1 - 8 \tan ^{2}{\relax (x )}} + 1\right ) \cot {\relax (x )}}{\left (1 - 8 \tan ^{2}{\relax (x )}\right )^{\frac {2}{3}} \cos ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________