\(\int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx\) [58]

Optimal result

Integrand size = 10, antiderivative size = 50 \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{3 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)} \]

[Out]

-2/3*cot(x)/(a*csc(x)^3)^(1/2)-2/3*(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 = 0.04 (sec) , antiderivative size = 50, 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, 3854, 3856, 2720} \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{3 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]

[In]

Int[1/Sqrt[a*Csc[x]^3],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\frac {-2 \cot (x)-\frac {2 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right )}{\sin ^{\frac {3}{2}}(x)}}{3 \sqrt {a \csc ^3(x)}} \]

[In]

Integrate[1/Sqrt[a*Csc[x]^3],x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.26

method	result	size
default	\(\frac {\left (-i \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )-i \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (x \right ) \sin \left (x \right )\right ) \sqrt {8}}{6 \sqrt {a \csc \left (x \right )^{3}}\, \left (\cos \left (x \right )-1\right ) \left (\cos \left (x \right )+1\right )}\)	\(163\)

[In]

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

[Out]

1/6*(-I*(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))*cos(x)-I*(I*(-I+cot(x)-csc(x)))^(1/2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-c
sc(x)+cot(x)))^(1/2)*EllipticF((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))+2^(1/2)*cos(x)*sin(x))/(a*csc(x)^3)^(
1/2)/(cos(x)-1)/(cos(x)+1)*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 = 68, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} - i \, \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + i \, \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{3 \, a} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(a*csc(x)**3), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/sqrt(a*csc(x)^3), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/sqrt(a*csc(x)^3), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a/sin(x)^3)^(1/2), x)