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

Optimal. Leaf size=48 \[ \frac{2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{\sqrt{a \sin ^3(x)}}-\frac{2 \sin (x) \cos (x)}{\sqrt{a \sin ^3(x)}} \]

[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]

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Rubi [A]  time = 0.0173165, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2639} \[ \frac{2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right )}{\sqrt{a \sin ^3(x)}}-\frac{2 \sin (x) \cos (x)}{\sqrt{a \sin ^3(x)}} \]

Antiderivative was successfully verified.

[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 3207

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
]])

Rule 2636

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 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \sin ^3(x)}} \, dx &=\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]  time = 0.0265732, size = 37, normalized size = 0.77 \[ \frac{2 \sin ^{\frac{3}{2}}(x) E\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )-\sin (2 x)}{\sqrt{a \sin ^3(x)}} \]

Antiderivative was successfully verified.

[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]

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Maple [C]  time = 0.257, size = 330, normalized size = 6.9 \begin{align*}{\sin \left ( x \right ) \left ( 2\,\sqrt{2}\cos \left ( x \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) -\sqrt{2}\cos \left ( x \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +2\,\sqrt{2}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},1/2\,\sqrt{2} \right ) -\sqrt{2}\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}},{\frac{\sqrt{2}}{2}} \right ) -2 \right ){\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*2^(1/2)*cos(x)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-(I*cos(x)-sin(x)-I)/sin(x
))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-2^(1/2)*cos(x)*(-I*(-1+cos(x))/sin(x))^(1/2
)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-(I*cos(x)-sin(x)-I)/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x)
)^(1/2),1/2*2^(1/2))+2*2^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-(I*cos(x)-si
n(x)-I)/sin(x))^(1/2)*EllipticE(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-2^(1/2)*(-I*(-1+cos(x))/sin(x)
)^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-(I*cos(x)-sin(x)-I)/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/
sin(x))^(1/2),1/2*2^(1/2))-2)*sin(x)/(a*sin(x)^3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*sin(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-(a*cos(x)^2 - a)*sin(x))/((a*cos(x)^2 - a)*sin(x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin ^{3}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*sin(x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sin \left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*sin(x)^3), x)