Optimal. Leaf size=27 \[ \frac{x}{3}+\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin (x) \cos (x)}{2 \sin ^2(x)+1}\right ) \]
[Out]
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Rubi [A] time = 0.0364624, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{x}{3}+\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin (x) \cos (x)}{2 \sin ^2(x)+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[(4 - 3*Cos[x]^2 + 5*Sin[x]^2)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 9.27266, size = 7, normalized size = 0.26 \[ \frac{\operatorname{atan}{\left (3 \tan{\left (x \right )} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4-3*cos(x)**2+5*sin(x)**2),x)
[Out]
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Mathematica [A] time = 0.0159438, size = 9, normalized size = 0.33 \[ \frac{1}{3} \tan ^{-1}(3 \tan (x)) \]
Antiderivative was successfully verified.
[In] Integrate[(4 - 3*Cos[x]^2 + 5*Sin[x]^2)^(-1),x]
[Out]
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Maple [A] time = 0.039, size = 8, normalized size = 0.3 \[{\frac{\arctan \left ( 3\,\tan \left ( x \right ) \right ) }{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(4-3*cos(x)^2+5*sin(x)^2),x)
[Out]
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Maxima [A] time = 1.53778, size = 9, normalized size = 0.33 \[ \frac{1}{3} \, \arctan \left (3 \, \tan \left (x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(3*cos(x)^2 - 5*sin(x)^2 - 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222552, size = 28, normalized size = 1.04 \[ -\frac{1}{6} \, \arctan \left (\frac{10 \, \cos \left (x\right )^{2} - 9}{6 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(3*cos(x)^2 - 5*sin(x)^2 - 4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 30.8507, size = 148, normalized size = 5.48 \[ \frac{\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{- 12 \sqrt{2} + 17}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor }{6 \sqrt{2} \sqrt{- 12 \sqrt{2} + 17} + 9 \sqrt{- 12 \sqrt{2} + 17}} + \frac{\sqrt{- 12 \sqrt{2} + 17} \sqrt{12 \sqrt{2} + 17} \left (\operatorname{atan}{\left (\frac{\tan{\left (\frac{x}{2} \right )}}{\sqrt{12 \sqrt{2} + 17}} \right )} + \pi \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\rfloor \right )}{6 \sqrt{2} \sqrt{- 12 \sqrt{2} + 17} + 9 \sqrt{- 12 \sqrt{2} + 17}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(4-3*cos(x)**2+5*sin(x)**2),x)
[Out]
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GIAC/XCAS [A] time = 0.208632, size = 27, normalized size = 1. \[ \frac{1}{3} \, x - \frac{1}{3} \, \arctan \left (\frac{\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(3*cos(x)^2 - 5*sin(x)^2 - 4),x, algorithm="giac")
[Out]