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

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

[Out]

-2/3*cot(x)/(a*csc(x)^3)^(1/2)-2/3*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/4*Pi+1/2*x)*EllipticF(cos(1/4*Pi+1/2*x),2
^(1/2))/sin(x)^(3/2)/(a*csc(x)^3)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.05, size = 38, normalized size = 0.76 \[ \frac {-2 \cot (x)-\frac {2 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )}{\sin ^{\frac {3}{2}}(x)}}{3 \sqrt {a \csc ^3(x)}} \]

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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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \csc \relax (x)^{3}}}{a \csc \relax (x)^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc \relax (x)^{3}}}\,{d x} \]

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)

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maple [C]  time = 1.09, size = 125, normalized size = 2.50 \[ -\frac {\left (i \sin \relax (x ) \sqrt {2}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-i-\sin \relax (x )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+2 \left (\cos ^{2}\relax (x )\right )-2 \cos \relax (x )\right ) \sqrt {8}}{6 \left (-1+\cos \relax (x )\right ) \sqrt {-\frac {2 a}{\sin \relax (x ) \left (-1+\cos ^{2}\relax (x )\right )}}\, \sin \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc \relax (x)^{3}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\frac {a}{{\sin \relax (x)}^3}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/sin(x)^3)^(1/2),x)

[Out]

int(1/(a/sin(x)^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \csc ^{3}{\relax (x )}}}\, dx \]

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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