3.1.20 \(\int \sqrt {\sin (a+b x)} \, dx\) [20]

Optimal. Leaf size=21 \[ \frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, 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 = {2719} \begin {gather*} \frac {2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sin[a + b*x]],x]

[Out]

(2*EllipticE[(a - Pi/2 + b*x)/2, 2])/b

Rule 2719

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

Rubi steps

\begin {align*} \int \sqrt {\sin (a+b x)} \, dx &=\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.14 \begin {gather*} -\frac {2 E\left (\left .\frac {1}{2} \left (-a+\frac {\pi }{2}-b x\right )\right |2\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sin[a + b*x]],x]

[Out]

(-2*EllipticE[(-a + Pi/2 - b*x)/2, 2])/b

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Maple [A]
time = 0.09, size = 91, normalized size = 4.33

method	result	size
default	\(-\frac {\sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, \left (2 \EllipticE \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\)	\(91\)
risch	\(-\frac {i \sqrt {2}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{-i \left (b x +a \right )}}}{b}+\frac {i \left (\frac {2 i \left (i-i {\mathrm e}^{2 i \left (b x +a \right )}\right )}{\sqrt {{\mathrm e}^{i \left (b x +a \right )} \left (i-i {\mathrm e}^{2 i \left (b x +a \right )}\right )}}-\frac {\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}\, \sqrt {-2 \,{\mathrm e}^{i \left (b x +a \right )}+2}\, \sqrt {-{\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{i \left (b x +a \right )}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-i {\mathrm e}^{3 i \left (b x +a \right )}+i {\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{-i \left (b x +a \right )}}\, \sqrt {-i \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{i \left (b x +a \right )}}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(b*x+a)+1)^(1/2)*(-2*sin(b*x+a)+2)^(1/2)*(-sin(b*x+a))^(1/2)*(2*EllipticE((sin(b*x+a)+1)^(1/2),1/2*2^(1/2
))-EllipticF((sin(b*x+a)+1)^(1/2),1/2*2^(1/2)))/cos(b*x+a)/sin(b*x+a)^(1/2)/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(b*x + a)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 63, normalized size = 3.00 \begin {gather*} \frac {i \, \sqrt {2} \sqrt {-i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} \sqrt {i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*sqrt(2)*sqrt(-I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) + I*sin(b*x + a))) - I*sqrt(2
)*sqrt(I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*x + a) - I*sin(b*x + a))))/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sin {\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sin(a + b*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sin(b*x + a)), x)

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Mupad [B]
time = 0.37, size = 18, normalized size = 0.86 \begin {gather*} \frac {2\,\mathrm {E}\left (\frac {a}{2}-\frac {\pi }{4}+\frac {b\,x}{2}\middle |2\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^(1/2),x)

[Out]

(2*ellipticE(a/2 - pi/4 + (b*x)/2, 2))/b

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