3.58 \(\int \frac{1}{\sqrt{a \csc ^3(x)}} \, dx\)

Optimal. Leaf size=50 \[ -\frac{2 \text{EllipticF}\left (\frac{\pi }{4}-\frac{x}{2},2\right )}{3 \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}}-\frac{2 \cot (x)}{3 \sqrt{a \csc ^3(x)}} \]

[Out]

(-2*Cot[x])/(3*Sqrt[a*Csc[x]^3]) - (2*EllipticF[Pi/4 - x/2, 2])/(3*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2))

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Rubi [A]  time = 0.0296832, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ -\frac{2 \cot (x)}{3 \sqrt{a \csc ^3(x)}}-\frac{2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{3 \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Csc[x]^3],x]

[Out]

(-2*Cot[x])/(3*Sqrt[a*Csc[x]^3]) - (2*EllipticF[Pi/4 - x/2, 2])/(3*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2))

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

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{1}{\sqrt{a \csc ^3(x)}} \, dx &=\frac{(-\csc (x))^{3/2} \int \frac{1}{(-\csc (x))^{3/2}} \, dx}{\sqrt{a \csc ^3(x)}}\\ &=-\frac{2 \cot (x)}{3 \sqrt{a \csc ^3(x)}}+\frac{(-\csc (x))^{3/2} \int \sqrt{-\csc (x)} \, dx}{3 \sqrt{a \csc ^3(x)}}\\ &=-\frac{2 \cot (x)}{3 \sqrt{a \csc ^3(x)}}+\frac{\int \frac{1}{\sqrt{\sin (x)}} \, dx}{3 \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{2 \cot (x)}{3 \sqrt{a \csc ^3(x)}}-\frac{2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{3 \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0490962, size = 38, normalized size = 0.76 \[ \frac{-\frac{2 \text{EllipticF}\left (\frac{1}{4} (\pi -2 x),2\right )}{\sin ^{\frac{3}{2}}(x)}-2 \cot (x)}{3 \sqrt{a \csc ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Csc[x]^3],x]

[Out]

(-2*Cot[x] - (2*EllipticF[(Pi - 2*x)/4, 2])/Sin[x]^(3/2))/(3*Sqrt[a*Csc[x]^3])

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Maple [C]  time = 0.25, size = 125, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{8}}{ \left ( -6+6\,\cos \left ( x \right ) \right ) \sin \left ( x \right ) } \left ( i\sin \left ( x \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\sqrt{2}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}+2\, \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\cos \left ( x \right ) \right ){\frac{1}{\sqrt{-2\,{\frac{a}{\sin \left ( x \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^3)^(1/2),x)

[Out]

-1/6*8^(1/2)*(I*sin(x)*(-I*(-1+cos(x))/sin(x))^(1/2)*2^(1/2)*(-(I*cos(x)-sin(x)-I)/sin(x))^(1/2)*EllipticF(((I
*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)+2*cos(x)^2-2*cos(x))/(-1+cos(x
))/(-2*a/sin(x)/(cos(x)^2-1))^(1/2)/sin(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \csc \left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(a*csc(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \csc \left (x\right )^{3}}}{a \csc \left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(x)^3)/(a*csc(x)^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \csc ^{3}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**3)**(1/2),x)

[Out]

Integral(1/sqrt(a*csc(x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \csc \left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(a*csc(x)^3), x)