Optimal. Leaf size=45 \[ \frac{x}{3}+\frac{1}{3} \tan ^{-1}\left (\frac{\sin (x) \cos (x) \left (\cos ^2(x)+1\right )}{\sqrt{\cos ^4(x)+\cos ^2(x)+1} \cos ^2(x)+1}\right ) \]
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Rubi [C] time = 0.491171, antiderivative size = 289, normalized size of antiderivative = 6.42, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6719, 1216, 1103, 1706} \[ \frac{\cos ^2(x) \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{\tan ^4(x)+3 \tan ^2(x)+3}}\right ) \sqrt{\tan ^4(x)+3 \tan ^2(x)+3}}{2 \sqrt{\cos ^4(x) \left (\tan ^4(x)+3 \tan ^2(x)+3\right )}}-\frac{\left (1+\sqrt{3}\right ) \cos ^2(x) \left (\tan ^2(x)+\sqrt{3}\right ) \sqrt{\frac{\tan ^4(x)+3 \tan ^2(x)+3}{\left (\tan ^2(x)+\sqrt{3}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\tan (x)}{\sqrt [4]{3}}\right )|\frac{1}{4} \left (2-\sqrt{3}\right )\right )}{4 \sqrt [4]{3} \sqrt{\cos ^4(x) \left (\tan ^4(x)+3 \tan ^2(x)+3\right )}}+\frac{\left (2+\sqrt{3}\right ) \cos ^2(x) \left (\tan ^2(x)+\sqrt{3}\right ) \sqrt{\frac{\tan ^4(x)+3 \tan ^2(x)+3}{\left (\tan ^2(x)+\sqrt{3}\right )^2}} \Pi \left (\frac{1}{6} \left (3-2 \sqrt{3}\right );2 \tan ^{-1}\left (\frac{\tan (x)}{\sqrt [4]{3}}\right )|\frac{1}{4} \left (2-\sqrt{3}\right )\right )}{4 \sqrt [4]{3} \sqrt{\cos ^4(x) \left (\tan ^4(x)+3 \tan ^2(x)+3\right )}} \]
Warning: Unable to verify antiderivative.
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Rule 6719
Rule 1216
Rule 1103
Rule 1706
Rubi steps
\begin{align*} \int \frac{\cos ^2(x)}{\sqrt{1+\cos ^2(x)+\cos ^4(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2 \sqrt{\frac{3+3 x^2+x^4}{\left (1+x^2\right )^2}}} \, dx,x,\tan (x)\right )\\ &=\frac{\left (\cos ^2(x) \sqrt{3+3 \tan ^2(x)+\tan ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{3+3 x^2+x^4}} \, dx,x,\tan (x)\right )}{\sqrt{\cos ^4(x) \left (3+3 \tan ^2(x)+\tan ^4(x)\right )}}\\ &=\frac{\left (\left (-1-\sqrt{3}\right ) \cos ^2(x) \sqrt{3+3 \tan ^2(x)+\tan ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+3 x^2+x^4}} \, dx,x,\tan (x)\right )}{2 \sqrt{\cos ^4(x) \left (3+3 \tan ^2(x)+\tan ^4(x)\right )}}+\frac{\left (\left (3+\sqrt{3}\right ) \cos ^2(x) \sqrt{3+3 \tan ^2(x)+\tan ^4(x)}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{3}}}{\left (1+x^2\right ) \sqrt{3+3 x^2+x^4}} \, dx,x,\tan (x)\right )}{2 \sqrt{\cos ^4(x) \left (3+3 \tan ^2(x)+\tan ^4(x)\right )}}\\ &=\frac{\tan ^{-1}\left (\frac{\tan (x)}{\sqrt{3+3 \tan ^2(x)+\tan ^4(x)}}\right ) \cos ^2(x) \sqrt{3+3 \tan ^2(x)+\tan ^4(x)}}{2 \sqrt{\cos ^4(x) \left (3+3 \tan ^2(x)+\tan ^4(x)\right )}}-\frac{\left (1+\sqrt{3}\right ) \cos ^2(x) F\left (2 \tan ^{-1}\left (\frac{\tan (x)}{\sqrt [4]{3}}\right )|\frac{1}{4} \left (2-\sqrt{3}\right )\right ) \left (\sqrt{3}+\tan ^2(x)\right ) \sqrt{\frac{3+3 \tan ^2(x)+\tan ^4(x)}{\left (\sqrt{3}+\tan ^2(x)\right )^2}}}{4 \sqrt [4]{3} \sqrt{\cos ^4(x) \left (3+3 \tan ^2(x)+\tan ^4(x)\right )}}+\frac{\left (2+\sqrt{3}\right ) \cos ^2(x) \Pi \left (\frac{1}{6} \left (3-2 \sqrt{3}\right );2 \tan ^{-1}\left (\frac{\tan (x)}{\sqrt [4]{3}}\right )|\frac{1}{4} \left (2-\sqrt{3}\right )\right ) \left (\sqrt{3}+\tan ^2(x)\right ) \sqrt{\frac{3+3 \tan ^2(x)+\tan ^4(x)}{\left (\sqrt{3}+\tan ^2(x)\right )^2}}}{4 \sqrt [4]{3} \sqrt{\cos ^4(x) \left (3+3 \tan ^2(x)+\tan ^4(x)\right )}}\\ \end{align*}
Mathematica [C] time = 2.02207, size = 159, normalized size = 3.53 \[ -\frac{2 i \cos ^2(x) \sqrt{1-\frac{2 i \tan ^2(x)}{\sqrt{3}-3 i}} \sqrt{1+\frac{2 i \tan ^2(x)}{\sqrt{3}+3 i}} \Pi \left (\frac{3}{2}+\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{3}}} \tan (x)\right )|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right )}{\sqrt{-\frac{i}{\sqrt{3}-3 i}} \sqrt{8 \cos (2 x)+\cos (4 x)+15}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.421, size = 312, normalized size = 6.9 \begin{align*} -2\,{\frac{\sqrt{ \left ( \left ( \cos \left ( 2\,x \right ) \right ) ^{2}+4\,\cos \left ( 2\,x \right ) +7 \right ) \left ( \sin \left ( 2\,x \right ) \right ) ^{2}} \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) ^{2}}{ \left ( -1+i\sqrt{3} \right ) \sqrt{ \left ( \cos \left ( 2\,x \right ) -1 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) \left ( \cos \left ( 2\,x \right ) +2+i\sqrt{3} \right ) \left ( i\sqrt{3}-\cos \left ( 2\,x \right ) -2 \right ) }\sin \left ( 2\,x \right ) \sqrt{ \left ( \cos \left ( 2\,x \right ) \right ) ^{2}+4\,\cos \left ( 2\,x \right ) +7}}\sqrt{{\frac{ \left ( -1+i\sqrt{3} \right ) \left ( \cos \left ( 2\,x \right ) -1 \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}}\sqrt{{\frac{\cos \left ( 2\,x \right ) +2+i\sqrt{3}}{ \left ( i\sqrt{3}+3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}}\sqrt{{\frac{i\sqrt{3}-\cos \left ( 2\,x \right ) -2}{ \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( -1+i\sqrt{3} \right ) \left ( \cos \left ( 2\,x \right ) -1 \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}},{\frac{i\sqrt{3}-3}{-1+i\sqrt{3}}},\sqrt{{\frac{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (x\right )^{2}}{\sqrt{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.84118, size = 105, normalized size = 2.33 \begin{align*} \frac{1}{6} \, \arctan \left (\frac{2 \, \sqrt{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 1} \cos \left (x\right )^{3} \sin \left (x\right )}{2 \, \cos \left (x\right )^{6} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (x\right )^{2}}{\sqrt{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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