3.527 \(\int \frac{\csc (e+f x)}{\sqrt{d \csc (e+f x)}} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{d f} \]

[Out]

(2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(d*f)

________________________________________________________________________________________

Rubi [A]  time = 0.0214527, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {16, 3771, 2641} \[ \frac{2 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{d f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]/Sqrt[d*Csc[e + f*x]],x]

[Out]

(2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[Sin[e + f*x]])/(d*f)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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

Mathematica [A]  time = 0.0166597, size = 45, normalized size = 0.98 \[ -\frac{2 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right ) \sqrt{d \csc (e+f x)}}{d f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]/Sqrt[d*Csc[e + f*x]],x]

[Out]

(-2*Sqrt[d*Csc[e + f*x]]*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]])/(d*f)

________________________________________________________________________________________

Maple [C]  time = 0.109, size = 165, normalized size = 3.6 \begin{align*}{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}\sqrt{2}}{f \left ( \sin \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{\frac{d}{\sin \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)/(d*csc(f*x+e))^(1/2),x)

[Out]

-I/f*(-1+cos(f*x+e))*((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)*(-(I*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^
(1/2)*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(1/
2))*(cos(f*x+e)+1)^2*2^(1/2)/(d/sin(f*x+e))^(1/2)/sin(f*x+e)^3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)/(d*csc(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)/sqrt(d*csc(f*x + e)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)/(d*csc(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*csc(f*x + e))/d, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)/(d*csc(f*x+e))**(1/2),x)

[Out]

Integral(csc(e + f*x)/sqrt(d*csc(e + f*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)/(d*csc(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(csc(f*x + e)/sqrt(d*csc(f*x + e)), x)