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 = 0.0163047, 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 \[ -\frac{\cos (2 x)}{\sqrt{\sin (2 x)+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + Sin[2*x]],x]

[Out]

-(Cos[2*x]/Sqrt[1 + Sin[2*x]])

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Rubi in Sympy [A]  time = 0.509548, size = 15, normalized size = 0.94 \[ - \frac{\cos{\left (2 x \right )}}{\sqrt{\sin{\left (2 x \right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+sin(2*x))**(1/2),x)

[Out]

-cos(2*x)/sqrt(sin(2*x) + 1)

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Mathematica [A]  time = 0.024504, 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 = 0.045, size = 22, normalized size = 1.4 \[{\frac{-1+\sin \left ( 2\,x \right ) }{\cos \left ( 2\,x \right ) }\sqrt{1+\sin \left ( 2\,x \right ) }} \]

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 [A]  time = 1.58069, size = 73, normalized size = 4.56 \[ -\frac{1}{\sqrt{\frac{\sin \left (2 \, x\right )^{2}}{{\left (\cos \left (2 \, x\right ) + 1\right )}^{2}} + 1}} + \frac{\sin \left (2 \, x\right )}{\sqrt{\frac{\sin \left (2 \, x\right )^{2}}{{\left (\cos \left (2 \, x\right ) + 1\right )}^{2}} + 1}{\left (\cos \left (2 \, x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sin(2*x) + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211955, size = 46, normalized size = 2.88 \[ -\frac{{\left (\cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 1\right )} \sqrt{\sin \left (2 \, x\right ) + 1}}{\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sin(2*x) + 1),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. \[ \int \sqrt{\sin{\left (2 x \right )} + 1}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sin \left (2 \, x\right ) + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sin(2*x) + 1),x, algorithm="giac")

[Out]

integrate(sqrt(sin(2*x) + 1), x)