Optimal. Leaf size=45 \[ \frac{x}{3}+\frac{1}{3} \tan ^{-1}\left (\frac{\sin (x) \cos (x) \left (\cos ^2(x)+1\right )}{\sqrt{\cos ^4(x)+\cos ^2(x)+1} \cos ^2(x)+1}\right ) \]
[Out]
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Rubi [C] time = 0.752164, antiderivative size = 304, normalized size of antiderivative = 6.76, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\cos ^2(x) \tan ^{-1}\left (\frac{\tan (x)}{\sqrt{\tan ^4(x)+3 \tan ^2(x)+3}}\right ) \sqrt{\tan ^4(x)+3 \tan ^2(x)+3}}{2 \sqrt{\cos ^4(x) \left (\tan ^4(x)+3 \tan ^2(x)+3\right )}}-\frac{\sqrt [4]{3} \cos ^2(x) \left (\tan ^2(x)+\sqrt{3}\right ) \sqrt{\frac{\tan ^4(x)+3 \tan ^2(x)+3}{\left (\tan ^2(x)+\sqrt{3}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\tan (x)}{\sqrt [4]{3}}\right )|\frac{1}{4} \left (2-\sqrt{3}\right )\right )}{2 \left (3-\sqrt{3}\right ) \sqrt{\cos ^4(x) \left (\tan ^4(x)+3 \tan ^2(x)+3\right )}}+\frac{\sqrt [4]{3} \left (1+\sqrt{3}\right ) \cos ^2(x) \left (\tan ^2(x)+\sqrt{3}\right ) \sqrt{\frac{\tan ^4(x)+3 \tan ^2(x)+3}{\left (\tan ^2(x)+\sqrt{3}\right )^2}} \Pi \left (\frac{1}{6} \left (3-2 \sqrt{3}\right );2 \tan ^{-1}\left (\frac{\tan (x)}{\sqrt [4]{3}}\right )|\frac{1}{4} \left (2-\sqrt{3}\right )\right )}{4 \left (3-\sqrt{3}\right ) \sqrt{\cos ^4(x) \left (\tan ^4(x)+3 \tan ^2(x)+3\right )}} \]
Warning: Unable to verify antiderivative.
[In] Int[Cos[x]^2/Sqrt[1 + Cos[x]^2 + Cos[x]^4],x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)**2/(1+cos(x)**2+cos(x)**4)**(1/2),x)
[Out]
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Mathematica [C] time = 3.87583, 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.
[In] Integrate[Cos[x]^2/Sqrt[1 + Cos[x]^2 + Cos[x]^4],x]
[Out]
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Maple [C] time = 0.585, size = 312, normalized size = 6.9 \[ -2\,{\frac{\sqrt{ \left ( \left ( \cos \left ( 2\,x \right ) \right ) ^{2}+4\,\cos \left ( 2\,x \right ) +7 \right ) \left ( \sin \left ( 2\,x \right ) \right ) ^{2}} \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) ^{2}}{ \left ( -1+i\sqrt{3} \right ) \sqrt{ \left ( \cos \left ( 2\,x \right ) -1 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) \left ( \cos \left ( 2\,x \right ) +2+i\sqrt{3} \right ) \left ( i\sqrt{3}-\cos \left ( 2\,x \right ) -2 \right ) }\sin \left ( 2\,x \right ) \sqrt{ \left ( \cos \left ( 2\,x \right ) \right ) ^{2}+4\,\cos \left ( 2\,x \right ) +7}}\sqrt{{\frac{ \left ( -1+i\sqrt{3} \right ) \left ( \cos \left ( 2\,x \right ) -1 \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}}\sqrt{{\frac{\cos \left ( 2\,x \right ) +2+i\sqrt{3}}{ \left ( i\sqrt{3}+3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}}\sqrt{{\frac{i\sqrt{3}-\cos \left ( 2\,x \right ) -2}{ \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( -1+i\sqrt{3} \right ) \left ( \cos \left ( 2\,x \right ) -1 \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+\cos \left ( 2\,x \right ) \right ) }}},{\frac{i\sqrt{3}-3}{-1+i\sqrt{3}}},\sqrt{{\frac{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)^2/(1+cos(x)^2+cos(x)^4)^(1/2),x)
[Out]
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Maxima [A] time = 65.1339, size = 47, normalized size = 1.04 \[ 2 \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + 2 \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281681, size = 45, normalized size = 1. \[ \frac{1}{6} \, \arctan \left (\frac{2 \, \sqrt{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 1} \cos \left (x\right )^{3} \sin \left (x\right )}{2 \, \cos \left (x\right )^{6} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)**2/(1+cos(x)**2+cos(x)**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\cos \left (x\right )^{2}}{\sqrt{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1),x, algorithm="giac")
[Out]