3.20 \(\int \sqrt{c \csc (a+b x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0181133, 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 = {3771, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \csc (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*Csc[a + b*x]],x]

[Out]

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

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

Rubi steps

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

Mathematica [A]  time = 0.0355421, size = 42, normalized size = 0.98 \[ -\frac{2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right ) \sqrt{c \csc (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*Csc[a + b*x]],x]

[Out]

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

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Maple [C]  time = 0.196, size = 165, normalized size = 3.8 \begin{align*}{\frac{-i\sqrt{2} \left ( -1+\cos \left ( bx+a \right ) \right ) \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2}}{b \left ( \sin \left ( bx+a \right ) \right ) ^{2}}\sqrt{{\frac{c}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*csc(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*csc(b*x + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c*csc(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*csc(b*x + a)), x)