3.12 \(\int \sqrt {\sin (b x)} \, dx\)

Optimal. Leaf size=19 \[ -\frac {2 E\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b} \]

[Out]

-2*(sin(1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/4*Pi+1/2*b*x)*EllipticE(cos(1/4*Pi+1/2*b*x),2^(1/2))/b

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Rubi [A]  time = 0.01, antiderivative size = 19, 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 = {2639} \[ -\frac {2 E\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sin[b*x]],x]

[Out]

(-2*EllipticE[Pi/4 - (b*x)/2, 2])/b

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin {align*} \int \sqrt {\sin (b x)} \, dx &=-\frac {2 E\left (\left .\frac {\pi }{4}-\frac {b x}{2}\right |2\right )}{b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 1.11 \[ -\frac {2 E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-b x\right )\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sin[b*x]],x]

[Out]

(-2*EllipticE[(Pi/2 - b*x)/2, 2])/b

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sin \left (b x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(sin(b*x)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sin \left (b x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(b*x)), x)

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maple [A]  time = 0.06, size = 77, normalized size = 4.05 \[ -\frac {\sqrt {\sin \left (b x \right )+1}\, \sqrt {-2 \sin \left (b x \right )+2}\, \sqrt {-\sin \left (b x \right )}\, \left (2 \EllipticE \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {\sin \left (b x \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (b x \right ) \sqrt {\sin \left (b x \right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(b*x)+1)^(1/2)*(-2*sin(b*x)+2)^(1/2)*(-sin(b*x))^(1/2)*(2*EllipticE((sin(b*x)+1)^(1/2),1/2*2^(1/2))-Ellip
ticF((sin(b*x)+1)^(1/2),1/2*2^(1/2)))/cos(b*x)/sin(b*x)^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(b*x)), x)

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mupad [B]  time = 0.37, size = 15, normalized size = 0.79 \[ -\frac {2\,\mathrm {E}\left (\frac {\pi }{4}-\frac {b\,x}{2}\middle |2\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*ellipticE(pi/4 - (b*x)/2, 2))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sin(b*x)), x)

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