3.390 \(\int \sqrt{1+\sin (2 x)} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\cos (2 x)}{\sqrt{\sin (2 x)+1}} \]

[Out]

-(Cos[2*x]/Sqrt[1 + Sin[2*x]])

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Rubi [A]  time = 0.0086676, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2646} \[ -\frac{\cos (2 x)}{\sqrt{\sin (2 x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sin[2*x]],x]

[Out]

-(Cos[2*x]/Sqrt[1 + Sin[2*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 (2 x)} \, dx &=-\frac{\cos (2 x)}{\sqrt{1+\sin (2 x)}}\\ \end{align*}

Mathematica [A]  time = 0.0138993, size = 25, normalized size = 1.56 \[ \frac{\sqrt{\sin (2 x)+1} (\sin (x)-\cos (x))}{\sin (x)+\cos (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sin[2*x]],x]

[Out]

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

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Maple [A]  time = 0.033, size = 22, normalized size = 1.4 \begin{align*}{\frac{-1+\sin \left ( 2\,x \right ) }{\cos \left ( 2\,x \right ) }\sqrt{1+\sin \left ( 2\,x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin \left (2 \, x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(2*x) + 1), x)

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Fricas [B]  time = 1.80544, size = 99, normalized size = 6.19 \begin{align*} -\frac{{\left (\cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 1\right )} \sqrt{\sin \left (2 \, x\right ) + 1}}{\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin{\left (2 x \right )} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sin(2*x) + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin \left (2 \, x\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(2*x) + 1), x)