3.3.37 \(\int \sqrt {1+\cos (x)} \, dx\) [237]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2725} \begin {gather*} \frac {2 \sin (x)}{\sqrt {\cos (x)+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cos[x]],x]

[Out]

(2*Sin[x])/Sqrt[1 + Cos[x]]

Rule 2725

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x
]])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \sqrt {1+\cos (x)} \, dx &=\frac {2 \sin (x)}{\sqrt {1+\cos (x)}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.33 \begin {gather*} 2 \sqrt {1+\cos (x)} \tan \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cos[x]],x]

[Out]

2*Sqrt[1 + Cos[x]]*Tan[x/2]

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Mathics [A]
time = 2.37, size = 12, normalized size = 1.00 \begin {gather*} 2 \sqrt {1+\text {Cos}\left [x\right ]} \text {Tan}\left [\frac {x}{2}\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(1+Cos[x])^(1/2),x]')

[Out]

2 Sqrt[1 + Cos[x]] Tan[x / 2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).
time = 0.04, size = 22, normalized size = 1.83

method	result	size
default	\(\frac {4 \cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sqrt {2}}{\sqrt {2 \cos \left (x \right )+2}}\)	\(22\)
risch	\(-\frac {i \sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{i x}\right )^{2} {\mathrm e}^{-i x}}\, \left ({\mathrm e}^{i x}-1\right )}{1+{\mathrm e}^{i x}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cos(x))^(1/2),x,method=_RETURNVERBOSE)

[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 = 0.39, size = 9, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(2)*sin(1/2*x)

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Fricas [A]
time = 0.31, size = 10, normalized size = 0.83 \begin {gather*} \frac {2 \, \sin \left (x\right )}{\sqrt {\cos \left (x\right ) + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))^(1/2),x, algorithm="fricas")

[Out]

2*sin(x)/sqrt(cos(x) + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (12) = 24\)
time = 0.37, size = 36, normalized size = 3.00 \begin {gather*} 2 \sqrt {1 - \frac {\tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {1}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1}} \tan {\left (\frac {x}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))**(1/2),x)

[Out]

2*sqrt(1 - tan(x/2)**2/(tan(x/2)**2 + 1) + 1/(tan(x/2)**2 + 1))*tan(x/2)

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Giac [A]
time = 0.00, size = 18, normalized size = 1.50 \begin {gather*} 2 \sqrt {2} \mathrm {sign}\left (\cos \left (\frac {x}{2}\right )\right ) \sin \left (\frac {x}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cos(x))^(1/2),x)

[Out]

2*sqrt(2)*sgn(cos(1/2*x))*sin(1/2*x)

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Mupad [B]
time = 0.15, size = 10, normalized size = 0.83 \begin {gather*} \frac {2\,\sin \left (x\right )}{\sqrt {\cos \left (x\right )+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x) + 1)^(1/2),x)

[Out]

(2*sin(x))/(cos(x) + 1)^(1/2)

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