\(\int (a \csc ^3(x))^{5/2} \, dx\) [55]

Optimal result

Integrand size = 10, antiderivative size = 123 \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+\frac {154}{195} a^2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \]

[Out]

-154/585*a^2*cot(x)*(a*csc(x)^3)^(1/2)-22/117*a^2*cot(x)*csc(x)^2*(a*csc(x)^3)^(1/2)-2/13*a^2*cot(x)*csc(x)^4*
(a*csc(x)^3)^(1/2)-154/195*a^2*cos(x)*sin(x)*(a*csc(x)^3)^(1/2)+154/195*a^2*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/
4*Pi+1/2*x)*EllipticE(cos(1/4*Pi+1/2*x),2^(1/2))*sin(x)^(3/2)*(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, 3853, 3856, 2719} \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)}+\frac {154}{195} a^2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)} \]

[In]

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

[Out]

(-154*a^2*Cot[x]*Sqrt[a*Csc[x]^3])/585 - (22*a^2*Cot[x]*Csc[x]^2*Sqrt[a*Csc[x]^3])/117 - (2*a^2*Cot[x]*Csc[x]^
4*Sqrt[a*Csc[x]^3])/13 - (154*a^2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x])/195 + (154*a^2*Sqrt[a*Csc[x]^3]*EllipticE[Pi
/4 - x/2, 2]*Sin[x]^(3/2))/195

Rule 2719

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

Rule 3853

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 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 {\left (a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{15/2} \, dx}{(-\csc (x))^{3/2}} \\ & = -\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (11 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{11/2} \, dx}{13 (-\csc (x))^{3/2}} \\ & = -\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{7/2} \, dx}{117 (-\csc (x))^{3/2}} \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{3/2} \, dx}{195 (-\csc (x))^{3/2}} \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{195 (-\csc (x))^{3/2}} \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {1}{195} \left (77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+\frac {154}{195} a^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] (verified)

Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.47 \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=\frac {\left (a \csc ^3(x)\right )^{5/2} \left (29568 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {15}{2}}(x)-9414 \sin (2 x)+5346 \sin (4 x)-1694 \sin (6 x)+231 \sin (8 x)\right )}{37440} \]

[In]

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

[Out]

((a*Csc[x]^3)^(5/2)*(29568*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(15/2) - 9414*Sin[2*x] + 5346*Sin[4*x] - 1694*Sin
[6*x] + 231*Sin[8*x]))/37440

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.02 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.50

method	result	size
default	\(\frac {a^{2} \sqrt {a \csc \left (x \right )^{3}}\, \left (\sin \left (x \right ) \left (-231 \cos \left (x \right )-231\right ) \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+\sin \left (x \right ) \left (462 \cos \left (x \right )+462\right ) \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-231 \sin \left (x \right )-77 \cot \left (x \right )-55 \csc \left (x \right )^{2} \cot \left (x \right )-45 \cot \left (x \right ) \csc \left (x \right )^{4}\right )\right ) \sqrt {8}}{1170}\)	\(185\)

[In]

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

[Out]

1/1170*a^2*(a*csc(x)^3)^(1/2)*(sin(x)*(-231*cos(x)-231)*(-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))*(I*(-I+cot(x)-csc(x)))^(1/2)+sin(x)*(462*cos(x)+462)*(-
I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2)*(I*(-I+cot(x)-csc(x)))^(1/2)*EllipticE((I*(-I+cot(x)-cs
c(x)))^(1/2),1/2*2^(1/2))+2^(1/2)*(-231*sin(x)-77*cot(x)-55*csc(x)^2*cot(x)-45*cot(x)*csc(x)^4))*8^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.31 \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=-\frac {231 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sqrt {2 i \, a} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sqrt {-2 i \, a} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - 2 \, {\left (231 \, a^{2} \cos \left (x\right )^{7} - 770 \, a^{2} \cos \left (x\right )^{5} + 902 \, a^{2} \cos \left (x\right )^{3} - 408 \, a^{2} \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{585 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]

[In]

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

[Out]

-1/585*(231*(a^2*cos(x)^4 - 2*a^2*cos(x)^2 + a^2)*sqrt(2*I*a)*sin(x)*weierstrassZeta(4, 0, weierstrassPInverse
(4, 0, cos(x) + I*sin(x))) + 231*(a^2*cos(x)^4 - 2*a^2*cos(x)^2 + a^2)*sqrt(-2*I*a)*sin(x)*weierstrassZeta(4,
0, weierstrassPInverse(4, 0, cos(x) - I*sin(x))) - 2*(231*a^2*cos(x)^7 - 770*a^2*cos(x)^5 + 902*a^2*cos(x)^3 -
408*a^2*cos(x))*sqrt(-a/((cos(x)^2 - 1)*sin(x))))/((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x))

Sympy [F]

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

[In]

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

[Out]

Integral((a*csc(x)**3)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*csc(x)^3)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*csc(x)^3)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a/sin(x)^3)^(5/2), x)