\(\int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx\) [10]

Optimal result

Integrand size = 10, antiderivative size = 48 \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=-\frac {2 \cos (x) \sin (x)}{\sqrt {a \sin ^3(x)}}+\frac {2 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{\sqrt {a \sin ^3(x)}} \]

[Out]

-2*cos(x)*sin(x)/(a*sin(x)^3)^(1/2)+2*(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*sin(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2716, 2719} \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\frac {2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{\sqrt {a \sin ^3(x)}}-\frac {2 \sin (x) \cos (x)}{\sqrt {a \sin ^3(x)}} \]

[In]

Int[1/Sqrt[a*Sin[x]^3],x]

[Out]

(-2*Cos[x]*Sin[x])/Sqrt[a*Sin[x]^3] + (2*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2))/Sqrt[a*Sin[x]^3]

Rule 2716

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] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2719

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]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[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
]])

Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {3}{2}}(x)} \, dx}{\sqrt {a \sin ^3(x)}} \\ & = -\frac {2 \cos (x) \sin (x)}{\sqrt {a \sin ^3(x)}}-\frac {\sin ^{\frac {3}{2}}(x) \int \sqrt {\sin (x)} \, dx}{\sqrt {a \sin ^3(x)}} \\ & = -\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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\frac {2 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x)-\sin (2 x)}{\sqrt {a \sin ^3(x)}} \]

[In]

Integrate[1/Sqrt[a*Sin[x]^3],x]

[Out]

(2*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2) - Sin[2*x])/Sqrt[a*Sin[x]^3]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.17 (sec) , antiderivative size = 274, normalized size of antiderivative = 5.71

method	result	size
default	\(\frac {\left (2 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )-\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+2 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right ) \sin \left (x \right ) \sqrt {8}}{2 \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}\)	\(274\)

[In]

int(1/(a*sin(x)^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(2*(-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-co
t(x)+csc(x)))^(1/2),1/2*2^(1/2))*cos(x)-(-I*(I-cot(x)+csc(x)))^(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))*cos(x)+2*(-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))-(-I*(I-co
t(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))-2^(1/2))*sin(x)/(a*sin(x)^3)^(1/2)*8^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\frac {{\left (-i \, \sqrt {2} \cos \left (x\right )^{2} + i \, \sqrt {2}\right )} \sqrt {-i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + {\left (i \, \sqrt {2} \cos \left (x\right )^{2} - i \, \sqrt {2}\right )} \sqrt {i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + 2 \, \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \cos \left (x\right )}{a \cos \left (x\right )^{2} - a} \]

[In]

integrate(1/(a*sin(x)^3)^(1/2),x, algorithm="fricas")

[Out]

((-I*sqrt(2)*cos(x)^2 + I*sqrt(2))*sqrt(-I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) + I*sin(x
))) + (I*sqrt(2)*cos(x)^2 - I*sqrt(2))*sqrt(I*a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I*si
n(x))) + 2*sqrt(-(a*cos(x)^2 - a)*sin(x))*cos(x))/(a*cos(x)^2 - a)

Sympy [F]

\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \sin ^{3}{\left (x \right )}}}\, dx \]

[In]

integrate(1/(a*sin(x)**3)**(1/2),x)

[Out]

Integral(1/sqrt(a*sin(x)**3), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right )^{3}}} \,d x } \]

[In]

integrate(1/(a*sin(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(a*sin(x)^3), x)

Giac [F]

\[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (x\right )^{3}}} \,d x } \]

[In]

integrate(1/(a*sin(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(a*sin(x)^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a \sin ^3(x)}} \, dx=\int \frac {1}{\sqrt {a\,{\sin \left (x\right )}^3}} \,d x \]

[In]

int(1/(a*sin(x)^3)^(1/2),x)

[Out]

int(1/(a*sin(x)^3)^(1/2), x)