Integrand size = 10, antiderivative size = 71 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x) \]
-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)
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.65 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sin ^{\frac {9}{2}}(x)+22 \sin (2 x)-5 \sin (4 x)\right ) \]
-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]))
Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4611, 3042, 4255, 3042, 4255, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \left (a \csc ^3(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (-a \sec \left (x+\frac {\pi }{2}\right )^3\right )^{3/2}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \int (-\csc (x))^{9/2}dx}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \int (-\csc (x))^{9/2}dx}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {5}{7} \int (-\csc (x))^{5/2}dx+\frac {2}{7} \cos (x) (-\csc (x))^{7/2}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {5}{7} \int (-\csc (x))^{5/2}dx+\frac {2}{7} \cos (x) (-\csc (x))^{7/2}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {-\csc (x)}dx+\frac {2}{3} \cos (x) (-\csc (x))^{3/2}\right )+\frac {2}{7} \cos (x) (-\csc (x))^{7/2}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {-\csc (x)}dx+\frac {2}{3} \cos (x) (-\csc (x))^{3/2}\right )+\frac {2}{7} \cos (x) (-\csc (x))^{7/2}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (x)} \sqrt {-\csc (x)} \int \frac {1}{\sqrt {\sin (x)}}dx+\frac {2}{3} \cos (x) (-\csc (x))^{3/2}\right )+\frac {2}{7} \cos (x) (-\csc (x))^{7/2}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (x)} \sqrt {-\csc (x)} \int \frac {1}{\sqrt {\sin (x)}}dx+\frac {2}{3} \cos (x) (-\csc (x))^{3/2}\right )+\frac {2}{7} \cos (x) (-\csc (x))^{7/2}\right )}{(-\csc (x))^{3/2}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle -\frac {a \sqrt {a \csc ^3(x)} \left (\frac {2}{7} \cos (x) (-\csc (x))^{7/2}+\frac {5}{7} \left (\frac {2}{3} \cos (x) (-\csc (x))^{3/2}-\frac {2}{3} \sqrt {\sin (x)} \sqrt {-\csc (x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )\right )\right )}{(-\csc (x))^{3/2}}\) |
-((a*Sqrt[a*Csc[x]^3]*((2*Cos[x]*(-Csc[x])^(7/2))/7 + (5*((2*Cos[x]*(-Csc[ x])^(3/2))/3 - (2*Sqrt[-Csc[x]]*EllipticF[Pi/4 - x/2, 2]*Sqrt[Sin[x]])/3)) /7))/(-Csc[x])^(3/2))
3.1.56.3.1 Defintions of rubi rules used
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]
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 = 1.37 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.37
method | result | size |
default | \(\frac {\left (\frac {a \sin \left (x \right )^{3} \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right )^{3}}{\left (1-\cos \left (x \right )\right )^{3}}\right )^{\frac {3}{2}} \left (1-\cos \left (x \right )\right )^{2} \left (40 i \csc \left (x \right )^{5} \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (1-\cos \left (x \right )\right )^{3}+3 \csc \left (x \right )^{10} \left (1-\cos \left (x \right )\right )^{8}+26 \csc \left (x \right )^{8} \left (1-\cos \left (x \right )\right )^{6}-26 \csc \left (x \right )^{4} \left (1-\cos \left (x \right )\right )^{2}-3 \csc \left (x \right )^{2}\right ) \sqrt {8}}{336 \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right )^{4} \sqrt {\csc \left (x \right ) \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right ) \left (1-\cos \left (x \right )\right )}\, \sqrt {\csc \left (x \right )^{3} \left (1-\cos \left (x \right )\right )^{3}+\csc \left (x \right )-\cot \left (x \right )}}\) | \(239\) |
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/(csc(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)-c ot(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)
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 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {5 \, {\left (i \, a \cos \left (x\right )^{2} - i \, a\right )} \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (-i \, a \cos \left (x\right )^{2} + i \, a\right )} \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{3} - 8 \, a \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{21 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]
-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)
\[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int \left (a \csc ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int { \left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
\[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int { \left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{3/2} \,d x \]