Integrand size = 10, antiderivative size = 77 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=-\frac {10 \cos (x)}{21 a \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}}-\frac {10 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x)}{21 a \sqrt {a \sin ^3(x)}} \] Output:
-10/21*cos(x)/a/(a*sin(x)^3)^(1/2)-2/7*cot(x)*csc(x)/a/(a*sin(x)^3)^(1/2)+ 10/21*InverseJacobiAM(-1/4*Pi+1/2*x,2^(1/2))*sin(x)^(3/2)/a/(a*sin(x)^3)^( 1/2)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=-\frac {2 \sin ^2(x) \left (3 \cot (x)+5 \cos (x) \sin (x)+5 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sin ^{\frac {5}{2}}(x)\right )}{21 \left (a \sin ^3(x)\right )^{3/2}} \] Input:
Integrate[(a*Sin[x]^3)^(-3/2),x]
Output:
(-2*Sin[x]^2*(3*Cot[x] + 5*Cos[x]*Sin[x] + 5*EllipticF[(Pi - 2*x)/4, 2]*Si n[x]^(5/2)))/(21*(a*Sin[x]^3)^(3/2))
Time = 0.34 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3686, 3042, 3116, 3042, 3116, 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.
| \(\displaystyle \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sin (x)^3\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {9}{2}}(x)}dx}{a \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin (x)^{9/2}}dx}{a \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {5}{7} \int \frac {1}{\sin ^{\frac {5}{2}}(x)}dx-\frac {2 \cos (x)}{7 \sin ^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {5}{7} \int \frac {1}{\sin (x)^{5/2}}dx-\frac {2 \cos (x)}{7 \sin ^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (x)}}dx-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(x)}\right )-\frac {2 \cos (x)}{7 \sin ^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (x)}}dx-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(x)}\right )-\frac {2 \cos (x)}{7 \sin ^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \sin ^3(x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {5}{7} \left (-\frac {2}{3} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(x)}\right )-\frac {2 \cos (x)}{7 \sin ^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \sin ^3(x)}}\) |
Input:
Int[(a*Sin[x]^3)^(-3/2),x]
Output:
(((5*((-2*EllipticF[Pi/4 - x/2, 2])/3 - (2*Cos[x])/(3*Sin[x]^(3/2))))/7 - (2*Cos[x])/(7*Sin[x]^(7/2)))*Sin[x]^(3/2))/(a*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[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[(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 = 0.92 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(\frac {\left (i \sin \left (x \right ) \left (5 \cos \left (x \right )+5\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}\, \sqrt {1-i \cot \left (x \right )+i \csc \left (x \right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}+\cot \left (x \right ) \csc \left (x \right ) \sqrt {2}\, \left (5 \cos \left (x \right )^{2}-8\right )\right ) \sqrt {8}}{42 \sqrt {a \sin \left (x \right )^{3}}\, a}\) | \(105\) |
Input:
int(1/(a*sin(x)^3)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/42/(a*sin(x)^3)^(1/2)/a*(I*sin(x)*(5*cos(x)+5)*EllipticF((1+I*cot(x)-I*c sc(x))^(1/2),1/2*2^(1/2))*(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)+cot(x)*csc(x)*2^(1/2)*(5*cos(x)^2-8))*8^ (1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\frac {2 \, {\left (5 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sqrt {-\frac {1}{2} i \, a} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sqrt {\frac {1}{2} i \, a} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + {\left (5 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}\right )}}{21 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sin \left (x\right )} \] Input:
integrate(1/(a*sin(x)^3)^(3/2),x, algorithm="fricas")
Output:
2/21*(5*(cos(x)^4 - 2*cos(x)^2 + 1)*sqrt(-1/2*I*a)*sin(x)*weierstrassPInve rse(4, 0, cos(x) + I*sin(x)) + 5*(cos(x)^4 - 2*cos(x)^2 + 1)*sqrt(1/2*I*a) *sin(x)*weierstrassPInverse(4, 0, cos(x) - I*sin(x)) + (5*cos(x)^3 - 8*cos (x))*sqrt(-(a*cos(x)^2 - a)*sin(x)))/((a^2*cos(x)^4 - 2*a^2*cos(x)^2 + a^2 )*sin(x))
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \sin ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a*sin(x)**3)**(3/2),x)
Output:
Integral((a*sin(x)**3)**(-3/2), x)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a*sin(x)^3)^(3/2),x, algorithm="maxima")
Output:
integrate((a*sin(x)^3)^(-3/2), x)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a*sin(x)^3)^(3/2),x, algorithm="giac")
Output:
integrate((a*sin(x)^3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\sin \left (x\right )}^3\right )}^{3/2}} \,d x \] Input:
int(1/(a*sin(x)^3)^(3/2),x)
Output:
int(1/(a*sin(x)^3)^(3/2), x)
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (x \right )}}{\sin \left (x \right )^{5}}d x \right )}{a^{2}} \] Input:
int(1/(a*sin(x)^3)^(3/2),x)
Output:
(sqrt(a)*int(sqrt(sin(x))/sin(x)**5,x))/a**2