3.21 \(\int \frac{1}{\sqrt{c \csc (a+b x)}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0180977, 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, 2639} \[ \frac{2 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*Csc[a + b*x]],x]

[Out]

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

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

Mathematica [A]  time = 0.031684, size = 42, normalized size = 0.98 \[ -\frac{2 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c*Csc[a + b*x]],x]

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.255, size = 521, normalized size = 12.1 \begin{align*} -{\frac{\sqrt{2}}{b\sin \left ( bx+a \right ) } \left ( 2\,\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\cos \left ( bx+a \right ){\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \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 ) }}}-{\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 ) \sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\cos \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 ) }}}+2\,\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \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 ) }}}-{\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 ) \sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\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{2}\cos \left ( bx+a \right ) -\sqrt{2} \right ){\frac{1}{\sqrt{{\frac{c}{\sin \left ( bx+a \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(c*csc(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*csc(b*x + a))/(c*csc(b*x + a)), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \csc{\left (a + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(c*csc(a + b*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(c*csc(b*x + a)), x)