3.235 \(\int \sqrt{1+\sin (x)} \, dx\)

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

[Out]

(-2*Cos[x])/Sqrt[1 + Sin[x]]

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Rubi [A]  time = 0.0063663, 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, Rules used = {2646} \[ -\frac{2 \cos (x)}{\sqrt{\sin (x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sin[x]],x]

[Out]

(-2*Cos[x])/Sqrt[1 + Sin[x]]

Rule 2646

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

Mathematica [B]  time = 0.011964, size = 40, normalized size = 3.33 \[ \frac{2 \sqrt{\sin (x)+1} \left (\sin \left (\frac{x}{2}\right )-\cos \left (\frac{x}{2}\right )\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sin[x]],x]

[Out]

(2*(-Cos[x/2] + Sin[x/2])*Sqrt[1 + Sin[x]])/(Cos[x/2] + Sin[x/2])

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Maple [A]  time = 0.027, size = 17, normalized size = 1.4 \begin{align*} 2\,{\frac{ \left ( -1+\sin \left ( x \right ) \right ) \sqrt{1+\sin \left ( x \right ) }}{\cos \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+sin(x))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin \left (x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(x) + 1), x)

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Fricas [B]  time = 1.65851, size = 88, normalized size = 7.33 \begin{align*} -\frac{2 \,{\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )} \sqrt{\sin \left (x\right ) + 1}}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin{\left (x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sin(x) + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin \left (x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+sin(x))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sin(x) + 1), x)