3.9 \(\int \sqrt{a \sin ^3(x)} \, dx\)

Optimal. Leaf size=50 \[ -\frac{2}{3} \cot (x) \sqrt{a \sin ^3(x)}-\frac{2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{3 \sin ^{\frac{3}{2}}(x)} \]

[Out]

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

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Rubi [A]  time = 0.0176481, antiderivative size = 50, 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, 2635, 2641} \[ -\frac{2}{3} \cot (x) \sqrt{a \sin ^3(x)}-\frac{2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{3 \sin ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2641

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

Rubi steps

\begin{align*} \int \sqrt{a \sin ^3(x)} \, dx &=\frac{\sqrt{a \sin ^3(x)} \int \sin ^{\frac{3}{2}}(x) \, dx}{\sin ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} \cot (x) \sqrt{a \sin ^3(x)}+\frac{\sqrt{a \sin ^3(x)} \int \frac{1}{\sqrt{\sin (x)}} \, dx}{3 \sin ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} \cot (x) \sqrt{a \sin ^3(x)}-\frac{2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{3 \sin ^{\frac{3}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0305087, size = 41, normalized size = 0.82 \[ -\frac{2 \sqrt{a \sin ^3(x)} \left (F\left (\left .\frac{1}{4} (\pi -2 x)\right |2\right )+\sqrt{\sin (x)} \cos (x)\right )}{3 \sin ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.271, size = 118, normalized size = 2.4 \begin{align*} -{\frac{\sqrt{8}}{6\,\sin \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) } \left ( i\sqrt{{\frac{-i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}}\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 ) \sqrt{{\frac{i\cos \left ( x \right ) +\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}+ \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{2}-\cos \left ( x \right ) \sqrt{2} \right ) \sqrt{a \left ( \sin \left ( x \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*8^(1/2)*(I*(-I*(-1+cos(x))/sin(x))^(1/2)*sin(x)*(-(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))*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)+cos(x)^2*2^(1/2)-cos(x)*2^(1/2))*(a*si
n(x)^3)^(1/2)/sin(x)/(-1+cos(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*sin(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-{\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((a*sin(x)^3)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*sin(x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*sin(x)^3), x)