Integrand size = 10, antiderivative size = 123 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{195 a^2 \sqrt {a \sin ^3(x)}} \] Output:
-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*EllipticE(cos(1/4*Pi+1/2*x),2^(1/2))*sin (x)^(3/2)/a^2/(a*sin(x)^3)^(1/2)
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {2 \left (\cot (x) \left (77+55 \csc ^2(x)+45 \csc ^4(x)\right )+231 \cos (x) \sin (x)-231 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x)\right )}{585 a^2 \sqrt {a \sin ^3(x)}} \] Input:
Integrate[(a*Sin[x]^3)^(-5/2),x]
Output:
(-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 = 0.47 (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 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )-\frac {2 \cos (x)}{5 \sin ^{\frac {5}{2}}(x)}\right )-\frac {2 \cos (x)}{9 \sin ^{\frac {9}{2}}(x)}\right )-\frac {2 \cos (x)}{13 \sin ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \sin ^3(x)}}\) |
Input:
Int[(a*Sin[x]^3)^(-5/2),x]
Output:
(((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])
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 = 1.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(\frac {\left (\sin \left (x \right ) \left (462 \cos \left (x \right )+462\right ) \sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}\, \sqrt {1-i \cot \left (x \right )+i \csc \left (x \right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right )+\sin \left (x \right ) \left (-231 \cos \left (x \right )-231\right ) \sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}\, \sqrt {1-i \cot \left (x \right )+i \csc \left (x \right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-231 \sin \left (x \right )-77 \cot \left (x \right )-55 \cot \left (x \right ) \csc \left (x \right )^{2}-45 \cot \left (x \right ) \csc \left (x \right )^{4}\right )\right ) \sqrt {8}}{1170 \sqrt {a \sin \left (x \right )^{3}}\, a^{2}}\) | \(185\) |
Input:
int(1/(a*sin(x)^3)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/1170/(a*sin(x)^3)^(1/2)/a^2*(sin(x)*(462*cos(x)+462)*(1+I*cot(x)-I*csc(x ))^(1/2)*(1-I*cot(x)+I*csc(x))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*EllipticE(( 1+I*cot(x)-I*csc(x))^(1/2),1/2*2^(1/2))+sin(x)*(-231*cos(x)-231)*(1+I*cot( x)-I*csc(x))^(1/2)*(1-I*cot(x)+I*csc(x))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*E llipticF((1+I*cot(x)-I*csc(x))^(1/2),1/2*2^(1/2))+2^(1/2)*(-231*sin(x)-77* cot(x)-55*cot(x)*csc(x)^2-45*cot(x)*csc(x)^4))*8^(1/2)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {2 \, {\left (231 \, {\left (i \, \cos \left (x\right )^{8} - 4 i \, \cos \left (x\right )^{6} + 6 i \, \cos \left (x\right )^{4} - 4 i \, \cos \left (x\right )^{2} + i\right )} \sqrt {-\frac {1}{2} i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (-i \, \cos \left (x\right )^{8} + 4 i \, \cos \left (x\right )^{6} - 6 i \, \cos \left (x\right )^{4} + 4 i \, \cos \left (x\right )^{2} - i\right )} \sqrt {\frac {1}{2} i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - {\left (231 \, \cos \left (x\right )^{7} - 770 \, \cos \left (x\right )^{5} + 902 \, \cos \left (x\right )^{3} - 408 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}\right )}}{585 \, {\left (a^{3} \cos \left (x\right )^{8} - 4 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} - 4 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )}} \] Input:
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="fricas")
Output:
-2/585*(231*(I*cos(x)^8 - 4*I*cos(x)^6 + 6*I*cos(x)^4 - 4*I*cos(x)^2 + I)* sqrt(-1/2*I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) + I* sin(x))) + 231*(-I*cos(x)^8 + 4*I*cos(x)^6 - 6*I*cos(x)^4 + 4*I*cos(x)^2 - I)*sqrt(1/2*I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I*sin(x))) - (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)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \sin ^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a*sin(x)**3)**(5/2),x)
Output:
Integral((a*sin(x)**3)**(-5/2), x)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(x)^3)^(-5/2), x)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="giac")
Output:
integrate((a*sin(x)^3)^(-5/2), x)
Timed out. \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\sin \left (x\right )}^3\right )}^{5/2}} \,d x \] Input:
int(1/(a*sin(x)^3)^(5/2),x)
Output:
int(1/(a*sin(x)^3)^(5/2), x)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (x \right )}}{\sin \left (x \right )^{8}}d x \right )}{a^{3}} \] Input:
int(1/(a*sin(x)^3)^(5/2),x)
Output:
(sqrt(a)*int(sqrt(sin(x))/sin(x)**8,x))/a**3