Integrand size = 10, antiderivative size = 48 \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=-\frac {2 \cos (x) \sin (x)}{\sqrt {a \sin ^3(x)}}+\frac {2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{\sqrt {a \sin ^3(x)}} \] Output:
-2*cos(x)*sin(x)/(a*sin(x)^3)^(1/2)+2*EllipticE(cos(1/4*Pi+1/2*x),2^(1/2)) *sin(x)^(3/2)/(a*sin(x)^3)^(1/2)
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\frac {2 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x)-\sin (2 x)}{\sqrt {a \sin ^3(x)}} \] Input:
Integrate[1/Sqrt[a*Sin[x]^3],x]
Output:
(2*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2) - Sin[2*x])/Sqrt[a*Sin[x]^3]
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3686, 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.
\(\displaystyle \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a \sin (x)^3}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {3}{2}}(x)}dx}{\sqrt {a \sin ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin (x)^{3/2}}dx}{\sqrt {a \sin ^3(x)}}\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (-\int \sqrt {\sin (x)}dx-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )}{\sqrt {a \sin ^3(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (-\int \sqrt {\sin (x)}dx-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )}{\sqrt {a \sin ^3(x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sin ^{\frac {3}{2}}(x) \left (2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )-\frac {2 \cos (x)}{\sqrt {\sin (x)}}\right )}{\sqrt {a \sin ^3(x)}}\) |
Input:
Int[1/Sqrt[a*Sin[x]^3],x]
Output:
((2*EllipticE[Pi/4 - x/2, 2] - (2*Cos[x])/Sqrt[Sin[x]])*Sin[x]^(3/2))/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 = 0.68 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.15
method | result | size |
default | \(-\frac {\sin \left (x \right ) \left (-2 \sqrt {1+i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {1-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \left (\cos \left (x \right )+1\right ) \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {1-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \left (\cos \left (x \right )+1\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\right ) \sqrt {8}}{2 \sqrt {a \sin \left (x \right )^{3}}}\) | \(151\) |
Input:
int(1/(a*sin(x)^3)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*sin(x)*(-2*(1+I*(-csc(x)+cot(x)))^(1/2)*(1-I*(-csc(x)+cot(x)))^(1/2)* (-I*(-csc(x)+cot(x)))^(1/2)*(cos(x)+1)*EllipticE((1+I*cot(x)-I*csc(x))^(1/ 2),1/2*2^(1/2))+(1+I*(-csc(x)+cot(x)))^(1/2)*(1-I*(-csc(x)+cot(x)))^(1/2)* (-I*(-csc(x)+cot(x)))^(1/2)*(cos(x)+1)*EllipticF((1+I*cot(x)-I*csc(x))^(1/ 2),1/2*2^(1/2))+2^(1/2))/(a*sin(x)^3)^(1/2)*8^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=-\frac {2 \, {\left ({\left (i \, \cos \left (x\right )^{2} - i\right )} \sqrt {-\frac {1}{2} i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + {\left (-i \, \cos \left (x\right )^{2} + i\right )} \sqrt {\frac {1}{2} i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \cos \left (x\right )\right )}}{a \cos \left (x\right )^{2} - a} \] Input:
integrate(1/(a*sin(x)^3)^(1/2),x, algorithm="fricas")
Output:
-2*((I*cos(x)^2 - I)*sqrt(-1/2*I*a)*weierstrassZeta(4, 0, weierstrassPInve rse(4, 0, cos(x) + I*sin(x))) + (-I*cos(x)^2 + I)*sqrt(1/2*I*a)*weierstras sZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I*sin(x))) - sqrt(-(a*cos(x )^2 - a)*sin(x))*cos(x))/(a*cos(x)^2 - a)
\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \sin ^{3}{\left (x \right )}}}\, dx \] Input:
integrate(1/(a*sin(x)**3)**(1/2),x)
Output:
Integral(1/sqrt(a*sin(x)**3), x)
\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right )^{3}}} \,d x } \] Input:
integrate(1/(a*sin(x)^3)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(a*sin(x)^3), x)
\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right )^{3}}} \,d x } \] Input:
integrate(1/(a*sin(x)^3)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(a*sin(x)^3), x)
Timed out. \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int \frac {1}{\sqrt {a\,{\sin \left (x\right )}^3}} \,d x \] Input:
int(1/(a*sin(x)^3)^(1/2),x)
Output:
int(1/(a*sin(x)^3)^(1/2), x)
\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (x \right )}}{\sin \left (x \right )^{2}}d x \right )}{a} \] Input:
int(1/(a*sin(x)^3)^(1/2),x)
Output:
(sqrt(a)*int(sqrt(sin(x))/sin(x)**2,x))/a