3.57 \(\int \sqrt{a \csc ^3(x)} \, dx\)

Optimal. Leaf size=48 \[ 2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)} \]

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2)

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Rubi [A]  time = 0.0270866, antiderivative size = 48, 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, 3768, 3771, 2639} \[ 2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[Pi/4 - x/2, 2]*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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
 1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 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 \sqrt{a \csc ^3(x)} \, dx &=\frac{\sqrt{a \csc ^3(x)} \int (-\csc (x))^{3/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)-\frac{\sqrt{a \csc ^3(x)} \int \frac{1}{\sqrt{-\csc (x)}} \, dx}{(-\csc (x))^{3/2}}\\ &=-2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)-\left (\sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)\right ) \int \sqrt{\sin (x)} \, dx\\ &=-2 \cos (x) \sqrt{a \csc ^3(x)} \sin (x)+2 \sqrt{a \csc ^3(x)} E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sin ^{\frac{3}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0410909, size = 46, normalized size = 0.96 \[ 2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right ) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)

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Maple [C]  time = 0.278, size = 343, normalized size = 7.2 \begin{align*}{\frac{\sqrt{8}\sin \left ( x \right ) }{4} \left ( 2\,\cos \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 ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) -\cos \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 ) }}}\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 ) +2\,\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 ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \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 ) }}}\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 ) -2 \right ) \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((a*csc(x)^3)^(1/2),x)

[Out]

1/4*8^(1/2)*(2*cos(x)*(-I*(-1+cos(x))/sin(x))^(1/2)*2^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-(I*cos(x)-sin
(x)-I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-cos(x)*(-I*(-1+cos(x))/sin(x))^
(1/2)*2^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(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))+2*(-I*(-1+cos(x))/sin(x))^(1/2)*2^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-
(I*cos(x)-sin(x)-I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-(-I*(-1+cos(x))/si
n(x))^(1/2)*2^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(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))-2)*sin(x)*(-2*a/sin(x)/(cos(x)^2-1))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*csc(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*csc(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csc(x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*csc(x)^3), x)