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

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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.01, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {1+\sin (2 x)} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Sqrt[1 + Sin[2*x]],x]

[Out]

Could not integrate

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fricas [B]  time = 0.93, size = 34, normalized size = 2.12 \[ -\frac {{\left (\cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 1\right )} \sqrt {\sin \left (2 \, x\right ) + 1}}{\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.61, size = 17, normalized size = 1.06 \[ \sqrt {2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + x\right )\right ) \sin \left (-\frac {1}{4} \, \pi + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sgn(cos(-1/4*pi + x))*sin(-1/4*pi + x)

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maple [A]  time = 0.16, size = 22, normalized size = 1.38

method	result	size
default	\(\frac {\left (\sin \left (2 x \right )-1\right ) \sqrt {1+\sin \left (2 x \right )}}{\cos \left (2 x \right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sin \left (2 \, x\right ) + 1}\,{d x} \]

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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mupad [B]  time = 0.23, size = 21, normalized size = 1.31 \[ \frac {\left (\sin \left (2\,x\right )-1\right )\,\sqrt {\sin \left (2\,x\right )+1}}{\cos \left (2\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(2*x) + 1)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sin {\left (2 x \right )} + 1}\, dx \]

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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