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]

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

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Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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
]])

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*}

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Mathematica [A]  time = 0.03, 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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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \relax (x)^{3}}\,{d x} \]

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)

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maple [C]  time = 0.48, size = 124, normalized size = 2.48 \[ -\frac {\left (i \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sin \relax (x ) \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{2}\relax (x )\right ) \sqrt {2}-\cos \relax (x ) \sqrt {2}\right ) \sqrt {a \left (1-\left (\cos ^{2}\relax (x )\right )\right ) \sin \relax (x )}\, \sqrt {8}}{6 \sin \relax (x ) \left (-1+\cos \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \relax (x)^{3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a\,{\sin \relax (x)}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*sin(x)^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin ^{3}{\relax (x )}}\, dx \]

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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