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

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

[Out]

-26/77*cot(x)/a^2/(a*csc(x)^3)^(1/2)-26/77*(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))/a^2/sin(x)^(3/2)/(a*csc(x)^3)^(1/2)-78/385*cos(x)*sin(x)/a^2/(a*csc(x)^3)^(1/2)-26/165*cos(x)*
sin(x)^3/a^2/(a*csc(x)^3)^(1/2)-2/15*cos(x)*sin(x)^5/a^2/(a*csc(x)^3)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3769, 3771, 2641} \[ -\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \sin ^5(x) \cos (x)}{15 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \sin ^3(x) \cos (x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {78 \sin (x) \cos (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{77 a^2 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]

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 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}{\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 = 0.14, size = 63, normalized size = 0.51 \[ -\frac {\sin (x) \sqrt {a \csc ^3(x)} \left (19122 \sin (2 x)-4406 \sin (4 x)+826 \sin (6 x)-77 \sin (8 x)+24960 \sqrt {\sin (x)} F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right )}{73920 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/73920*(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] + 826*Sin[6*x] - 77*Sin[8*x]))/a^3

________________________________________________________________________________________

fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \csc \relax (x)^{3}}}{a^{3} \csc \relax (x)^{9}}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]

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)

________________________________________________________________________________________

maple [C]  time = 0.79, size = 158, normalized size = 1.28 \[ -\frac {2 \left (-154 \left (\cos ^{8}\relax (x )\right )+195 i \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sin \relax (x )+154 \left (\cos ^{7}\relax (x )\right )+644 \left (\cos ^{6}\relax (x )\right )-644 \left (\cos ^{5}\relax (x )\right )-1060 \left (\cos ^{4}\relax (x )\right )+1060 \left (\cos ^{3}\relax (x )\right )+960 \left (\cos ^{2}\relax (x )\right )-960 \cos \relax (x )\right ) \sqrt {8}}{1155 \left (-1+\cos \relax (x )\right ) \left (-\frac {2 a}{\sin \relax (x ) \left (-1+\cos ^{2}\relax (x )\right )}\right )^{\frac {5}{2}} \sin \relax (x )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155*(-154*cos(x)^8+195*I*2^(1/2)*sin(x)*EllipticF(((I*cos(x)-I+sin(x))/sin(x))^(1/2),1/2*2^(1/2))*((-I*cos
(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)-I+sin(x))/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/sin(x)/(-1+cos(x)^2
)*a)^(5/2)/sin(x)^7*8^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\sin \relax (x)}^3}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a/sin(x)^3)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \csc ^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]

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)

________________________________________________________________________________________