3.28 \(\int \sqrt{c \sin (a+b x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{b \sqrt{\sin (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0183664, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2640, 2639} \[ \frac{2 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \sin (a+b x)}}{b \sqrt{\sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*Sin[a + b*x]],x]

[Out]

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

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [A]  time = 0.0211787, size = 42, normalized size = 0.98 \[ -\frac{2 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right ) \sqrt{c \sin (a+b x)}}{b \sqrt{\sin (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*Sin[a + b*x]],x]

[Out]

(-2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[c*Sin[a + b*x]])/(b*Sqrt[Sin[a + b*x]])

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Maple [A]  time = 0.036, size = 98, normalized size = 2.3 \begin{align*} -{\frac{c}{b\cos \left ( bx+a \right ) }\sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2}\sqrt{\sin \left ( bx+a \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*(-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)/(c*sin(b*x+a))^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c \sin \left (b x + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*sin(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c*sin(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a)), x)