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

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \sin \left (b x + a\right )}, x\right ) \]

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

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)

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maple [A]  time = 0.05, size = 98, normalized size = 2.28 \[ -\frac {c \sqrt {-\sin \left (b x +a \right )+1}\, \sqrt {2 \sin \left (b x +a \right )+2}\, \left (\sqrt {\sin }\left (b x +a \right )\right ) \left (2 \EllipticE \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {-\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}\, b} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \sin \left (b x + a\right )}\,{d x} \]

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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mupad [B]  time = 0.40, size = 36, normalized size = 0.84 \[ \frac {2\,\sqrt {c\,\sin \left (a+b\,x\right )}\,\mathrm {E}\left (\frac {a}{2}-\frac {\pi }{4}+\frac {b\,x}{2}\middle |2\right )}{b\,\sqrt {\sin \left (a+b\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x))^(1/2),x)

[Out]

(2*(c*sin(a + b*x))^(1/2)*ellipticE(a/2 - pi/4 + (b*x)/2, 2))/(b*sin(a + b*x)^(1/2))

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

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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