Optimal. Leaf size=19 \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]
[Out]
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Rubi [B] time = 1.01033, antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 25, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \log \left (\tan ^2\left (\frac{x}{2}\right )+\tan \left (\frac{x}{2}\right )+2\right )-\log \left (1-2 \tan \left (\frac{x}{2}\right )\right )-\log \left (\tan \left (\frac{x}{2}\right )+1\right ) \]
Antiderivative was successfully verified.
[In] Int[(2 + Cos[x] + 5*Sin[x])/(4*Cos[x] - 2*Sin[x] + Cos[x]*Sin[x] - 2*Sin[x]^2),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((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)**2),x)
[Out]
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Mathematica [A] time = 0.059474, size = 19, normalized size = 1. \[ \log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1) \]
Antiderivative was successfully verified.
[In] Integrate[(2 + Cos[x] + 5*Sin[x])/(4*Cos[x] - 2*Sin[x] + Cos[x]*Sin[x] - 2*Sin[x]^2),x]
[Out]
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Maple [A] time = 0.105, size = 35, normalized size = 1.8 \[ -\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) -\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) +\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+\tan \left ({\frac{x}{2}} \right ) +2 \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)^2),x)
[Out]
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Maxima [A] time = 1.59504, size = 72, normalized size = 3.79 \[ -\log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 2\right ) - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((cos(x) + 5*sin(x) + 2)/(cos(x)*sin(x) - 2*sin(x)^2 + 4*cos(x) - 2*sin(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247355, size = 58, normalized size = 3.05 \[ -\frac{1}{2} \, \log \left (2 \, \cos \left (x\right )^{2} - \frac{1}{2} \,{\left (3 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) - \frac{3}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (\frac{1}{2} \,{\left (\cos \left (x\right ) + 3\right )} \sin \left (x\right ) + \frac{3}{2} \, \cos \left (x\right ) + \frac{5}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((cos(x) + 5*sin(x) + 2)/(cos(x)*sin(x) - 2*sin(x)^2 + 4*cos(x) - 2*sin(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.92217, size = 32, normalized size = 1.68 \[ - \log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{2} \right )} - \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} + \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + \tan{\left (\frac{x}{2} \right )} + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)**2),x)
[Out]
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GIAC/XCAS [A] time = 0.222127, size = 49, normalized size = 2.58 \[{\rm ln}\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \tan \left (\frac{1}{2} \, x\right ) + 2\right ) -{\rm ln}\left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) -{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((cos(x) + 5*sin(x) + 2)/(cos(x)*sin(x) - 2*sin(x)^2 + 4*cos(x) - 2*sin(x)),x, algorithm="giac")
[Out]