\(\int (a \csc ^3(x))^{3/2} \, dx\) [56]

Optimal result

Integrand size = 10, antiderivative size = 71 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x) \]

[Out]

-10/21*a*cos(x)*(a*csc(x)^3)^(1/2)-2/7*a*cot(x)*csc(x)*(a*csc(x)^3)^(1/2)-10/21*a*(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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2720} \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sin ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sqrt {a \csc ^3(x)} \]

[In]

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

[Out]

(-10*a*Cos[x]*Sqrt[a*Csc[x]^3])/21 - (2*a*Cot[x]*Csc[x]*Sqrt[a*Csc[x]^3])/7 - (10*a*Sqrt[a*Csc[x]^3]*EllipticF
[Pi/4 - x/2, 2]*Sin[x]^(3/2))/21

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 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 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{9/2} \, dx}{(-\csc (x))^{3/2}} \\ & = -\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {\left (5 a \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{5/2} \, dx}{7 (-\csc (x))^{3/2}} \\ & = -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {\left (5 a \sqrt {a \csc ^3(x)}\right ) \int \sqrt {-\csc (x)} \, dx}{21 (-\csc (x))^{3/2}} \\ & = -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}+\frac {1}{21} \left (5 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sin (x)}} \, dx \\ & = -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.65 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sin ^{\frac {9}{2}}(x)+22 \sin (2 x)-5 \sin (4 x)\right ) \]

[In]

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

[Out]

-1/84*((a*Csc[x]^3)^(3/2)*(40*EllipticF[(Pi - 2*x)/4, 2]*Sin[x]^(9/2) + 22*Sin[2*x] - 5*Sin[4*x]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.37 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.37

method	result	size
default	\(\frac {\left (\frac {a \sin \left (x \right )^{3} \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right )^{3}}{\left (1-\cos \left (x \right )\right )^{3}}\right )^{\frac {3}{2}} \left (1-\cos \left (x \right )\right )^{2} \left (40 i \csc \left (x \right )^{5} \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (1-\cos \left (x \right )\right )^{3}+3 \csc \left (x \right )^{10} \left (1-\cos \left (x \right )\right )^{8}+26 \csc \left (x \right )^{8} \left (1-\cos \left (x \right )\right )^{6}-26 \csc \left (x \right )^{4} \left (1-\cos \left (x \right )\right )^{2}-3 \csc \left (x \right )^{2}\right ) \sqrt {8}}{336 \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right )^{4} \sqrt {\csc \left (x \right ) \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right ) \left (1-\cos \left (x \right )\right )}\, \sqrt {\csc \left (x \right )^{3} \left (1-\cos \left (x \right )\right )^{3}+\csc \left (x \right )-\cot \left (x \right )}}\)	\(239\)

[In]

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

[Out]

1/336*(a/(1-cos(x))^3*sin(x)^3*(csc(x)^2*(1-cos(x))^2+1)^3)^(3/2)*(1-cos(x))^2/(csc(x)^2*(1-cos(x))^2+1)^4/(cs
c(x)*(csc(x)^2*(1-cos(x))^2+1)*(1-cos(x)))^(1/2)/(csc(x)^3*(1-cos(x))^3+csc(x)-cot(x))^(1/2)*(40*I*csc(x)^5*(-
I*(I-cot(x)+csc(x)))^(1/2)*2^(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))*(1-cos(x))^3+3*csc(x)^10*(1-cos(x))^8+26*csc(x)^8*(1-cos(x))^6-26*csc(x)^4*(1-
cos(x))^2-3*csc(x)^2)*8^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.39 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {5 \, {\left (i \, a \cos \left (x\right )^{2} - i \, a\right )} \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (-i \, a \cos \left (x\right )^{2} + i \, a\right )} \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{3} - 8 \, a \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{21 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]

[In]

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

[Out]

-1/21*(5*(I*a*cos(x)^2 - I*a)*sqrt(2*I*a)*weierstrassPInverse(4, 0, cos(x) + I*sin(x)) + 5*(-I*a*cos(x)^2 + I*
a)*sqrt(-2*I*a)*weierstrassPInverse(4, 0, cos(x) - I*sin(x)) + 2*(5*a*cos(x)^3 - 8*a*cos(x))*sqrt(-a/((cos(x)^
2 - 1)*sin(x))))/(cos(x)^2 - 1)

Sympy [F]

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

[In]

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

[Out]

Integral((a*csc(x)**3)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*csc(x)^3)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*csc(x)^3)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a/sin(x)^3)^(3/2), x)