Integrand size = 10, antiderivative size = 123 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{195 a^2 \sqrt {a \sin ^3(x)}} \]
-154/585*cot(x)/a^2/(a*sin(x)^3)^(1/2)-22/117*cot(x)*csc(x)^2/a^2/(a*sin(x )^3)^(1/2)-2/13*cot(x)*csc(x)^4/a^2/(a*sin(x)^3)^(1/2)-154/195*cos(x)*sin( x)/a^2/(a*sin(x)^3)^(1/2)+154/195*(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^2/(a*sin(x)^3)^( 1/2)
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {2 \left (\cot (x) \left (77+55 \csc ^2(x)+45 \csc ^4(x)\right )+231 \cos (x) \sin (x)-231 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x)\right )}{585 a^2 \sqrt {a \sin ^3(x)}} \]
(-2*(Cot[x]*(77 + 55*Csc[x]^2 + 45*Csc[x]^4) + 231*Cos[x]*Sin[x] - 231*Ell ipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)))/(585*a^2*Sqrt[a*Sin[x]^3])
Time = 0.46 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3686, 3042, 3116, 3042, 3116, 3042, 3116, 3042, 3116, 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.
| \(\displaystyle \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sin (x)^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {15}{2}}(x)}dx}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin (x)^{15/2}}dx}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \int \frac {1}{\sin ^{\frac {11}{2}}(x)}dx-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \int \frac {1}{\sin (x)^{11/2}}dx-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \int \frac {1}{\sin ^{\frac {7}{2}}(x)}dx-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \int \frac {1}{\sin (x)^{7/2}}dx-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\sin ^{\frac {3}{2}}(x)}dx-\frac {2 \cos (x)}{5 \sin ^{\frac {5}{2}}(x)}\right )-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\sin (x)^{3/2}}dx-\frac {2 \cos (x)}{5 \sin ^{\frac {5}{2}}(x)}\right )-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-\int \sqrt {\sin (x)}dx-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )-\frac {2 \cos (x)}{5 \sin ^{\frac {5}{2}}(x)}\right )-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (-\int \sqrt {\sin (x)}dx-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )-\frac {2 \cos (x)}{5 \sin ^{\frac {5}{2}}(x)}\right )-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )-\frac {2 \cos (x)}{5 \sin ^{\frac {5}{2}}(x)}\right )-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
(((11*((7*((3*(2*EllipticE[Pi/4 - x/2, 2] - (2*Cos[x])/Sqrt[Sin[x]]))/5 - (2*Cos[x])/(5*Sin[x]^(5/2))))/9 - (2*Cos[x])/(9*Sin[x]^(9/2))))/13 - (2*Co s[x])/(13*Sin[x]^(13/2)))*Sin[x]^(3/2))/(a^2*Sqrt[a*Sin[x]^3])
3.1.12.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.55
| method | result | size |
| default | \(-\frac {\left (231 \sin \left (x \right ) \cos \left (x \right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-462 \sin \left (x \right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+231 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )-462 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )+231 \sin \left (x \right ) \sqrt {2}+77 \cot \left (x \right ) \sqrt {2}+55 \left (\csc ^{2}\left (x \right )\right ) \cot \left (x \right ) \sqrt {2}+45 \cot \left (x \right ) \left (\csc ^{4}\left (x \right )\right ) \sqrt {2}\right ) \sqrt {8}}{1170 \sqrt {a \left (\sin ^{3}\left (x \right )\right )}\, a^{2}}\) | \(314\) |
-1/1170/(a*sin(x)^3)^(1/2)/a^2*(231*sin(x)*cos(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))-462*sin(x)*(-I*(I-cot(x)+csc(x)))^(1 /2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*EllipticE((-I*( I-cot(x)+csc(x)))^(1/2),1/2*2^(1/2))*cos(x)+231*(-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))*sin(x)-462*(-I*(I-cot(x)+csc(x)))^(1/2 )*(-I*(I+cot(x)-csc(x)))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*EllipticE((-I*(I- cot(x)+csc(x)))^(1/2),1/2*2^(1/2))*sin(x)+231*sin(x)*2^(1/2)+77*cot(x)*2^( 1/2)+55*csc(x)^2*cot(x)*2^(1/2)+45*cot(x)*csc(x)^4*2^(1/2))*8^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {231 \, {\left (i \, \sqrt {2} \cos \left (x\right )^{8} - 4 i \, \sqrt {2} \cos \left (x\right )^{6} + 6 i \, \sqrt {2} \cos \left (x\right )^{4} - 4 i \, \sqrt {2} \cos \left (x\right )^{2} + i \, \sqrt {2}\right )} \sqrt {-i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (-i \, \sqrt {2} \cos \left (x\right )^{8} + 4 i \, \sqrt {2} \cos \left (x\right )^{6} - 6 i \, \sqrt {2} \cos \left (x\right )^{4} + 4 i \, \sqrt {2} \cos \left (x\right )^{2} - i \, \sqrt {2}\right )} \sqrt {i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - 2 \, {\left (231 \, \cos \left (x\right )^{7} - 770 \, \cos \left (x\right )^{5} + 902 \, \cos \left (x\right )^{3} - 408 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{585 \, {\left (a^{3} \cos \left (x\right )^{8} - 4 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} - 4 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )}} \]
-1/585*(231*(I*sqrt(2)*cos(x)^8 - 4*I*sqrt(2)*cos(x)^6 + 6*I*sqrt(2)*cos(x )^4 - 4*I*sqrt(2)*cos(x)^2 + I*sqrt(2))*sqrt(-I*a)*weierstrassZeta(4, 0, w eierstrassPInverse(4, 0, cos(x) + I*sin(x))) + 231*(-I*sqrt(2)*cos(x)^8 + 4*I*sqrt(2)*cos(x)^6 - 6*I*sqrt(2)*cos(x)^4 + 4*I*sqrt(2)*cos(x)^2 - I*sqr t(2))*sqrt(I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I *sin(x))) - 2*(231*cos(x)^7 - 770*cos(x)^5 + 902*cos(x)^3 - 408*cos(x))*sq rt(-(a*cos(x)^2 - a)*sin(x)))/(a^3*cos(x)^8 - 4*a^3*cos(x)^6 + 6*a^3*cos(x )^4 - 4*a^3*cos(x)^2 + a^3)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \sin ^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\sin \left (x\right )}^3\right )}^{5/2}} \,d x \]