\(\int \frac {1}{(a \csc ^3(x))^{5/2}} \, dx\) [60]

Optimal result

Integrand size = 10, antiderivative size = 123 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}} \]

[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] (verified)

Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2720} \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \sin ^5(x) \cos (x)}{15 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \sin ^3(x) \cos (x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {78 \sin (x) \cos (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{77 a^2 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]

[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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

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 3856

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 4208

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*} \text {integral}& = \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 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},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] (verified)

Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {\sqrt {a \csc ^3(x)} \sin (x) \left (24960 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sqrt {\sin (x)}+19122 \sin (2 x)-4406 \sin (4 x)+826 \sin (6 x)-77 \sin (8 x)\right )}{73920 a^3} \]

[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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\left (77 \cos \left (x \right )^{6} \cot \left (x \right ) \sqrt {2}-322 \cos \left (x \right )^{4} \cot \left (x \right ) \sqrt {2}+195 i \csc \left (x \right ) \cot \left (x \right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+530 \cos \left (x \right )^{2} \cot \left (x \right ) \sqrt {2}+195 i \csc \left (x \right )^{2} \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-480 \cot \left (x \right ) \sqrt {2}\right ) \sqrt {8}}{2310 \sqrt {a \csc \left (x \right )^{3}}\, a^{2}}\)	\(192\)

[In]

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

[Out]

1/2310/(a*csc(x)^3)^(1/2)*(77*cos(x)^6*cot(x)*2^(1/2)-322*cos(x)^4*cot(x)*2^(1/2)+195*I*csc(x)*cot(x)*(I*(-I+c
ot(x)-csc(x)))^(1/2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2)*EllipticF((I*(-I+cot(x)-csc(x)))
^(1/2),1/2*2^(1/2))+530*cos(x)^2*cot(x)*2^(1/2)+195*I*csc(x)^2*(I*(-I+cot(x)-csc(x)))^(1/2)*(-I*(I+cot(x)-csc(
x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2)*EllipticF((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))-480*cot(x)*2^(1/2))
/a^2*8^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {2 \, {\left (77 \, \cos \left (x\right )^{9} - 399 \, \cos \left (x\right )^{7} + 852 \, \cos \left (x\right )^{5} - 1010 \, \cos \left (x\right )^{3} + 480 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} + 195 i \, \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{1155 \, a^{3}} \]

[In]

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

[Out]

-1/1155*(2*(77*cos(x)^9 - 399*cos(x)^7 + 852*cos(x)^5 - 1010*cos(x)^3 + 480*cos(x))*sqrt(-a/((cos(x)^2 - 1)*si
n(x))) + 195*I*sqrt(2*I*a)*weierstrassPInverse(4, 0, cos(x) + I*sin(x)) - 195*I*sqrt(-2*I*a)*weierstrassPInver
se(4, 0, cos(x) - I*sin(x)))/a^3

Sympy [F]

\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral((a*csc(x)**3)**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csc(x)^3)^(-5/2), x)

Giac [F]

\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csc(x)^3)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{5/2}} \,d x \]

[In]

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

[Out]

int(1/(a/sin(x)^3)^(5/2), x)