Integrand size = 10, antiderivative size = 77 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=-\frac {10 \cos (x)}{21 a \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc (x)}{7 a \sqrt {a \sin ^3(x)}}-\frac {10 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x)}{21 a \sqrt {a \sin ^3(x)}} \]
-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*(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/(a*sin(x)^3)^(1/2)
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=-\frac {2 \sin ^2(x) \left (3 \cot (x)+5 \cos (x) \sin (x)+5 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sin ^{\frac {5}{2}}(x)\right )}{21 \left (a \sin ^3(x)\right )^{3/2}} \]
(-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 |
\(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (\frac {5}{7} \left (-\frac {2}{3} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(x)}\right )-\frac {2 \cos (x)}{7 \sin ^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \sin ^3(x)}}\) |
(((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])
3.1.11.3.1 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[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 = 1.47 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.10
method | result | size |
default | \(\frac {\left (1-\cos \left (x \right )\right )^{2} \left (40 i \left (\csc ^{5}\left (x \right )\right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (1-\cos \left (x \right )\right )^{3}+3 \left (\csc ^{10}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{8}+26 \left (\csc ^{8}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{6}-26 \left (\csc ^{4}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{2}-3 \left (\csc ^{2}\left (x \right )\right )\right ) \sqrt {8}}{336 {\left (\frac {\left (\csc ^{3}\left (x \right )\right ) a \left (1-\cos \left (x \right )\right )^{3}}{{\left (\left (\csc ^{2}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{2}+1\right )}^{3}}\right )}^{\frac {3}{2}} {\left (\left (\csc ^{2}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{2}+1\right )}^{4} \sqrt {\csc \left (x \right ) \left (\left (\csc ^{2}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{2}+1\right ) \left (1-\cos \left (x \right )\right )}\, \sqrt {\left (\csc ^{3}\left (x \right )\right ) \left (1-\cos \left (x \right )\right )^{3}+\csc \left (x \right )-\cot \left (x \right )}}\) | \(239\) |
1/336/(csc(x)^3*a*(1-cos(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 = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.81 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\frac {5 \, {\left (\sqrt {2} \cos \left (x\right )^{4} - 2 \, \sqrt {2} \cos \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {-i \, a} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (\sqrt {2} \cos \left (x\right )^{4} - 2 \, \sqrt {2} \cos \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {i \, a} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{21 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sin \left (x\right )} \]
1/21*(5*(sqrt(2)*cos(x)^4 - 2*sqrt(2)*cos(x)^2 + sqrt(2))*sqrt(-I*a)*sin(x )*weierstrassPInverse(4, 0, cos(x) + I*sin(x)) + 5*(sqrt(2)*cos(x)^4 - 2*s qrt(2)*cos(x)^2 + sqrt(2))*sqrt(I*a)*sin(x)*weierstrassPInverse(4, 0, cos( x) - I*sin(x)) + 2*(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))
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \sin ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\sin \left (x\right )}^3\right )}^{3/2}} \,d x \]