Optimal. Leaf size=12 \[ \frac{2 \sin (x)}{\sqrt{\cos (x)+1}} \]
[Out]
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Rubi [A] time = 0.0124195, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 \sin (x)}{\sqrt{\cos (x)+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + Cos[x]],x]
[Out]
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Rubi in Sympy [A] time = 0.496345, size = 12, normalized size = 1. \[ \frac{2 \sin{\left (x \right )}}{\sqrt{\cos{\left (x \right )} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+cos(x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0061168, size = 16, normalized size = 1.33 \[ 2 \sqrt{\cos (x)+1} \tan \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + Cos[x]],x]
[Out]
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Maple [B] time = 0.021, size = 22, normalized size = 1.8 \[ 2\,{\frac{\cos \left ( x/2 \right ) \sin \left ( x/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( x/2 \right ) \right ) ^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+cos(x))^(1/2),x)
[Out]
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Maxima [A] time = 1.59605, size = 38, normalized size = 3.17 \[ \frac{2 \, \sqrt{2} \sin \left (x\right )}{\sqrt{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}{\left (\cos \left (x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(cos(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210835, size = 14, normalized size = 1.17 \[ \frac{2 \, \sin \left (x\right )}{\sqrt{\cos \left (x\right ) + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(cos(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\cos{\left (x \right )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+cos(x))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\cos \left (x\right ) + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(cos(x) + 1),x, algorithm="giac")
[Out]