3.60 \(\int \frac{1}{(a \csc ^3(x))^{5/2}} \, dx\)

Optimal. Leaf size=123 \[ -\frac{26 \text{EllipticF}\left (\frac{\pi }{4}-\frac{x}{2},2\right )}{77 a^2 \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}}-\frac{26 \cot (x)}{77 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \sin ^5(x) \cos (x)}{15 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \sin ^3(x) \cos (x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{78 \sin (x) \cos (x)}{385 a^2 \sqrt{a \csc ^3(x)}} \]

[Out]

(-26*Cot[x])/(77*a^2*Sqrt[a*Csc[x]^3]) - (26*EllipticF[Pi/4 - x/2, 2])/(77*a^2*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2))
- (78*Cos[x]*Sin[x])/(385*a^2*Sqrt[a*Csc[x]^3]) - (26*Cos[x]*Sin[x]^3)/(165*a^2*Sqrt[a*Csc[x]^3]) - (2*Cos[x]*
Sin[x]^5)/(15*a^2*Sqrt[a*Csc[x]^3])

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Rubi [A]  time = 0.0611232, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ -\frac{26 \cot (x)}{77 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \sin ^5(x) \cos (x)}{15 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \sin ^3(x) \cos (x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{78 \sin (x) \cos (x)}{385 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{77 a^2 \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csc[x]^3)^(-5/2),x]

[Out]

(-26*Cot[x])/(77*a^2*Sqrt[a*Csc[x]^3]) - (26*EllipticF[Pi/4 - x/2, 2])/(77*a^2*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2))
- (78*Cos[x]*Sin[x])/(385*a^2*Sqrt[a*Csc[x]^3]) - (26*Cos[x]*Sin[x]^3)/(165*a^2*Sqrt[a*Csc[x]^3]) - (2*Cos[x]*
Sin[x]^5)/(15*a^2*Sqrt[a*Csc[x]^3])

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}{\left (a \csc ^3(x)\right )^{5/2}} \, dx &=\frac{(-\csc (x))^{3/2} \int \frac{1}{(-\csc (x))^{15/2}} \, dx}{a^2 \sqrt{a \csc ^3(x)}}\\ &=-\frac{2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt{a \csc ^3(x)}}+\frac{\left (13 (-\csc (x))^{3/2}\right ) \int \frac{1}{(-\csc (x))^{11/2}} \, dx}{15 a^2 \sqrt{a \csc ^3(x)}}\\ &=-\frac{26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt{a \csc ^3(x)}}+\frac{\left (39 (-\csc (x))^{3/2}\right ) \int \frac{1}{(-\csc (x))^{7/2}} \, dx}{55 a^2 \sqrt{a \csc ^3(x)}}\\ &=-\frac{78 \cos (x) \sin (x)}{385 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt{a \csc ^3(x)}}+\frac{\left (39 (-\csc (x))^{3/2}\right ) \int \frac{1}{(-\csc (x))^{3/2}} \, dx}{77 a^2 \sqrt{a \csc ^3(x)}}\\ &=-\frac{26 \cot (x)}{77 a^2 \sqrt{a \csc ^3(x)}}-\frac{78 \cos (x) \sin (x)}{385 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt{a \csc ^3(x)}}+\frac{\left (13 (-\csc (x))^{3/2}\right ) \int \sqrt{-\csc (x)} \, dx}{77 a^2 \sqrt{a \csc ^3(x)}}\\ &=-\frac{26 \cot (x)}{77 a^2 \sqrt{a \csc ^3(x)}}-\frac{78 \cos (x) \sin (x)}{385 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt{a \csc ^3(x)}}+\frac{13 \int \frac{1}{\sqrt{\sin (x)}} \, dx}{77 a^2 \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{26 \cot (x)}{77 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{77 a^2 \sqrt{a \csc ^3(x)} \sin ^{\frac{3}{2}}(x)}-\frac{78 \cos (x) \sin (x)}{385 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt{a \csc ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.147149, size = 63, normalized size = 0.51 \[ -\frac{\sin (x) \sqrt{a \csc ^3(x)} \left (24960 \sqrt{\sin (x)} \text{EllipticF}\left (\frac{1}{4} (\pi -2 x),2\right )+19122 \sin (2 x)-4406 \sin (4 x)+826 \sin (6 x)-77 \sin (8 x)\right )}{73920 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csc[x]^3)^(-5/2),x]

[Out]

-(Sqrt[a*Csc[x]^3]*Sin[x]*(24960*EllipticF[(Pi - 2*x)/4, 2]*Sqrt[Sin[x]] + 19122*Sin[2*x] - 4406*Sin[4*x] + 82
6*Sin[6*x] - 77*Sin[8*x]))/(73920*a^3)

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Maple [C]  time = 0.264, size = 158, normalized size = 1.3 \begin{align*} -{\frac{2\,\sqrt{8}}{ \left ( -1155+1155\,\cos \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{7}} \left ( -154\, \left ( \cos \left ( x \right ) \right ) ^{8}+195\,i\sqrt{2}\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 ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}+154\, \left ( \cos \left ( x \right ) \right ) ^{7}+644\, \left ( \cos \left ( x \right ) \right ) ^{6}-644\, \left ( \cos \left ( x \right ) \right ) ^{5}-1060\, \left ( \cos \left ( x \right ) \right ) ^{4}+1060\, \left ( \cos \left ( x \right ) \right ) ^{3}+960\, \left ( \cos \left ( x \right ) \right ) ^{2}-960\,\cos \left ( x \right ) \right ) \left ( -2\,{\frac{a}{\sin \left ( x \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^3)^(5/2),x)

[Out]

-2/1155*8^(1/2)*(-154*cos(x)^8+195*I*2^(1/2)*sin(x)*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)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)+154*cos(x
)^7+644*cos(x)^6-644*cos(x)^5-1060*cos(x)^4+1060*cos(x)^3+960*cos(x)^2-960*cos(x))/(-1+cos(x))/(-2*a/sin(x)/(c
os(x)^2-1))^(5/2)/sin(x)^7

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*csc(x)^3)^(-5/2), 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^{3} \csc \left (x\right )^{9}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csc(x)^3)/(a^3*csc(x)^9), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**3)**(5/2),x)

[Out]

Integral((a*csc(x)**3)**(-5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*csc(x)^3)^(-5/2), x)