3.237 \(\int \sqrt{1+\cos (x)} \, dx\)

Optimal. Leaf size=12 \[ \frac{2 \sin (x)}{\sqrt{\cos (x)+1}} \]

[Out]

(2*Sin[x])/Sqrt[1 + Cos[x]]

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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]

(2*Sin[x])/Sqrt[1 + Cos[x]]

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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]

2*sin(x)/sqrt(cos(x) + 1)

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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]

2*Sqrt[1 + Cos[x]]*Tan[x/2]

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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]

2*cos(1/2*x)*sin(1/2*x)*2^(1/2)/(cos(1/2*x)^2)^(1/2)

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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]

2*sqrt(2)*sin(x)/(sqrt(sin(x)^2/(cos(x) + 1)^2 + 1)*(cos(x) + 1))

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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]

2*sin(x)/sqrt(cos(x) + 1)

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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]

Integral(sqrt(cos(x) + 1), x)

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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]

integrate(sqrt(cos(x) + 1), x)