\(\int \sqrt {a \csc ^3(x)} \, dx\) [57]

Optimal result

Integrand size = 10, antiderivative size = 48 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \]

[Out]

-2*cos(x)*sin(x)*(a*csc(x)^3)^(1/2)+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 = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \[ \int \sqrt {a \csc ^3(x)} \, dx=2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)} \]

[In]

Int[Sqrt[a*Csc[x]^3],x]

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2)

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 {\sqrt {a \csc ^3(x)} \int (-\csc (x))^{3/2} \, dx}{(-\csc (x))^{3/2}} \\ & = -2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {\sqrt {a \csc ^3(x)} \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{(-\csc (x))^{3/2}} \\ & = -2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\left (\sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx \\ & = -2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+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 = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x) \]

[In]

Integrate[Sqrt[a*Csc[x]^3],x]

[Out]

-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 5.62

method	result	size
default	\(-\frac {\sqrt {a \csc \left (x \right )^{3}}\, \left (-2 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+\sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \cos \left (x \right )-2 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+\sqrt {2}\right ) \sin \left (x \right ) \sqrt {8}}{2}\)	\(270\)

[In]

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

[Out]

-1/2*(a*csc(x)^3)^(1/2)*(-2*(-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)-csc(x)))^(1/2),1/2*2^(1/2))*cos(x)+(-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)*cos(x)-2*(-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)-csc(x)))^(
1/2),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))*(I*(-I+cot(x)-csc(x)))^(1/2)+2^(1/2))*sin(x)*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 = 64, normalized size of antiderivative = 1.33 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \, \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - \sqrt {-2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) \]

[In]

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

[Out]

-2*sqrt(-a/((cos(x)^2 - 1)*sin(x)))*cos(x)*sin(x) - sqrt(2*I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0
, cos(x) + I*sin(x))) - sqrt(-2*I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I*sin(x)))

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a*csc(x)**3), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(a*csc(x)^3), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(a*csc(x)^3), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a/sin(x)^3)^(1/2), x)