Integrand size = 10, antiderivative size = 123 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}} \]
-26/77*cot(x)/a^2/(a*csc(x)^3)^(1/2)-26/77*(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))/a^2/sin(x)^(3/2)/(a*cs c(x)^3)^(1/2)-78/385*cos(x)*sin(x)/a^2/(a*csc(x)^3)^(1/2)-26/165*cos(x)*si n(x)^3/a^2/(a*csc(x)^3)^(1/2)-2/15*cos(x)*sin(x)^5/a^2/(a*csc(x)^3)^(1/2)
Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {\sqrt {a \csc ^3(x)} \sin (x) \left (24960 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sqrt {\sin (x)}+19122 \sin (2 x)-4406 \sin (4 x)+826 \sin (6 x)-77 \sin (8 x)\right )}{73920 a^3} \]
-1/73920*(Sqrt[a*Csc[x]^3]*Sin[x]*(24960*EllipticF[(Pi - 2*x)/4, 2]*Sqrt[S in[x]] + 19122*Sin[2*x] - 4406*Sin[4*x] + 826*Sin[6*x] - 77*Sin[8*x]))/a^3
Time = 0.62 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 4611, 3042, 4256, 3042, 4256, 3042, 4256, 3042, 4256, 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.
| \(\displaystyle \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-a \sec \left (x+\frac {\pi }{2}\right )^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{15/2}}dx}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{15/2}}dx}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \int \frac {1}{(-\csc (x))^{11/2}}dx+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \int \frac {1}{(-\csc (x))^{11/2}}dx+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \int \frac {1}{(-\csc (x))^{7/2}}dx+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \int \frac {1}{(-\csc (x))^{7/2}}dx+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \frac {1}{(-\csc (x))^{3/2}}dx+\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}\right )+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \frac {1}{(-\csc (x))^{3/2}}dx+\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}\right )+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {-\csc (x)}dx+\frac {2 \cos (x)}{3 \sqrt {-\csc (x)}}\right )+\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}\right )+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {-\csc (x)}dx+\frac {2 \cos (x)}{3 \sqrt {-\csc (x)}}\right )+\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}\right )+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (x)} \sqrt {-\csc (x)} \int \frac {1}{\sqrt {\sin (x)}}dx+\frac {2 \cos (x)}{3 \sqrt {-\csc (x)}}\right )+\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}\right )+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\sin (x)} \sqrt {-\csc (x)} \int \frac {1}{\sqrt {\sin (x)}}dx+\frac {2 \cos (x)}{3 \sqrt {-\csc (x)}}\right )+\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}\right )+\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}\right )+\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {(-\csc (x))^{3/2} \left (\frac {2 \cos (x)}{15 (-\csc (x))^{13/2}}+\frac {13}{15} \left (\frac {2 \cos (x)}{11 (-\csc (x))^{9/2}}+\frac {9}{11} \left (\frac {2 \cos (x)}{7 (-\csc (x))^{5/2}}+\frac {5}{7} \left (\frac {2 \cos (x)}{3 \sqrt {-\csc (x)}}-\frac {2}{3} \sqrt {\sin (x)} \sqrt {-\csc (x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )\right )\right )\right )\right )}{a^2 \sqrt {a \csc ^3(x)}}\) |
((-Csc[x])^(3/2)*((2*Cos[x])/(15*(-Csc[x])^(13/2)) + (13*((2*Cos[x])/(11*( -Csc[x])^(9/2)) + (9*((2*Cos[x])/(7*(-Csc[x])^(5/2)) + (5*((2*Cos[x])/(3*S qrt[-Csc[x]]) - (2*Sqrt[-Csc[x]]*EllipticF[Pi/4 - x/2, 2]*Sqrt[Sin[x]])/3) )/7))/11))/15))/(a^2*Sqrt[a*Csc[x]^3])
3.1.60.3.1 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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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 = 2.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(\frac {\left (77 \cos \left (x \right )^{6} \cot \left (x \right ) \sqrt {2}-322 \cos \left (x \right )^{4} \cot \left (x \right ) \sqrt {2}+195 i \csc \left (x \right ) \cot \left (x \right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+530 \cos \left (x \right )^{2} \cot \left (x \right ) \sqrt {2}+195 i \csc \left (x \right )^{2} \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-480 \cot \left (x \right ) \sqrt {2}\right ) \sqrt {8}}{2310 \sqrt {a \csc \left (x \right )^{3}}\, a^{2}}\) | \(192\) |
1/2310/(a*csc(x)^3)^(1/2)*(77*cos(x)^6*cot(x)*2^(1/2)-322*cos(x)^4*cot(x)* 2^(1/2)+195*I*csc(x)*cot(x)*(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))+530*cos(x)^2*cot(x)*2^(1/2)+195*I*csc(x)^2*(I*(-I+cot(x) -csc(x)))^(1/2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(-I*(-csc(x)+cot(x)))^(1/2)*E llipticF((I*(-I+cot(x)-csc(x)))^(1/2),1/2*2^(1/2))-480*cot(x)*2^(1/2))/a^2 *8^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=-\frac {2 \, {\left (77 \, \cos \left (x\right )^{9} - 399 \, \cos \left (x\right )^{7} + 852 \, \cos \left (x\right )^{5} - 1010 \, \cos \left (x\right )^{3} + 480 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} + 195 i \, \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{1155 \, a^{3}} \]
-1/1155*(2*(77*cos(x)^9 - 399*cos(x)^7 + 852*cos(x)^5 - 1010*cos(x)^3 + 48 0*cos(x))*sqrt(-a/((cos(x)^2 - 1)*sin(x))) + 195*I*sqrt(2*I*a)*weierstrass PInverse(4, 0, cos(x) + I*sin(x)) - 195*I*sqrt(-2*I*a)*weierstrassPInverse (4, 0, cos(x) - I*sin(x)))/a^3
\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{5/2}} \,d x \]