Integrand size = 10, antiderivative size = 48 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \]
-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)
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x) \]
-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)
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4611, 3042, 4255, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a \csc ^3(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {-a \sec \left (x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \int (-\csc (x))^{3/2}dx}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \int (-\csc (x))^{3/2}dx}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \left (2 \cos (x) \sqrt {-\csc (x)}-\int \frac {1}{\sqrt {-\csc (x)}}dx\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \left (2 \cos (x) \sqrt {-\csc (x)}-\int \frac {1}{\sqrt {-\csc (x)}}dx\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \left (2 \cos (x) \sqrt {-\csc (x)}-\frac {\int \sqrt {\sin (x)}dx}{\sqrt {\sin (x)} \sqrt {-\csc (x)}}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \left (2 \cos (x) \sqrt {-\csc (x)}-\frac {\int \sqrt {\sin (x)}dx}{\sqrt {\sin (x)} \sqrt {-\csc (x)}}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sqrt {a \csc ^3(x)} \left (2 \cos (x) \sqrt {-\csc (x)}+\frac {2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{\sqrt {\sin (x)} \sqrt {-\csc (x)}}\right )}{(-\csc (x))^{3/2}}\) |
(Sqrt[a*Csc[x]^3]*(2*Cos[x]*Sqrt[-Csc[x]] + (2*EllipticE[Pi/4 - x/2, 2])/( Sqrt[-Csc[x]]*Sqrt[Sin[x]])))/(-Csc[x])^(3/2)
3.1.57.3.1 Defintions of rubi rules used
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]
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] + Simp[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[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]
Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 5.62
method | result | size |
default | \(-\frac {\sqrt {a \csc \left (x \right )^{3}}\, \left (-2 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+\sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \cos \left (x \right )-2 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+\sqrt {2}\right ) \sin \left (x \right ) \sqrt {8}}{2}\) | \(270\) |
-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)*Ell ipticF((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)
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 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \, \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - \sqrt {-2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) \]
-2*sqrt(-a/((cos(x)^2 - 1)*sin(x)))*cos(x)*sin(x) - sqrt(2*I*a)*weierstras sZeta(4, 0, weierstrassPInverse(4, 0, cos(x) + I*sin(x))) - sqrt(-2*I*a)*w eierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I*sin(x)))
\[ \int \sqrt {a \csc ^3(x)} \, dx=\int \sqrt {a \csc ^{3}{\left (x \right )}}\, dx \]
\[ \int \sqrt {a \csc ^3(x)} \, dx=\int { \sqrt {a \csc \left (x\right )^{3}} \,d x } \]
\[ \int \sqrt {a \csc ^3(x)} \, dx=\int { \sqrt {a \csc \left (x\right )^{3}} \,d x } \]
Timed out. \[ \int \sqrt {a \csc ^3(x)} \, dx=\int \sqrt {\frac {a}{{\sin \left (x\right )}^3}} \,d x \]