3.2 \(\int (a \sin ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac{2}{3} a \cot (x) \sqrt{a \sin ^2(x)} \]

[Out]

(-2*a*Cot[x]*Sqrt[a*Sin[x]^2])/3 - (Cot[x]*(a*Sin[x]^2)^(3/2))/3

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Rubi [A]  time = 0.0185186, antiderivative size = 34, 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 = {3203, 3207, 2638} \[ -\frac{1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac{2}{3} a \cot (x) \sqrt{a \sin ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sin[x]^2)^(3/2),x]

[Out]

(-2*a*Cot[x]*Sqrt[a*Sin[x]^2])/3 - (Cot[x]*(a*Sin[x]^2)^(3/2))/3

Rule 3203

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x]
+ Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] &&  !IntegerQ[p] &&
GtQ[p, 1]

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 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0378638, size = 26, normalized size = 0.76 \[ \frac{1}{12} a (\cos (3 x)-9 \cos (x)) \csc (x) \sqrt{a \sin ^2(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[x]^2)^(3/2),x]

[Out]

(a*(-9*Cos[x] + Cos[3*x])*Csc[x]*Sqrt[a*Sin[x]^2])/12

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Maple [A]  time = 0.628, size = 24, normalized size = 0.7 \begin{align*} -{\frac{{a}^{2}\cos \left ( x \right ) \sin \left ( x \right ) \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) }{3}{\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (x\right )^{2}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(x)^2)^(3/2), x)

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Fricas [A]  time = 1.55797, size = 81, normalized size = 2.38 \begin{align*} \frac{{\left (a \cos \left (x\right )^{3} - 3 \, a \cos \left (x\right )\right )} \sqrt{-a \cos \left (x\right )^{2} + a}}{3 \, \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sin(x)**2)**(3/2), x)

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Giac [A]  time = 1.22319, size = 32, normalized size = 0.94 \begin{align*} \frac{1}{3} \,{\left ({\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) + 2 \, \mathrm{sgn}\left (\sin \left (x\right )\right )\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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