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

Optimal result

Integrand size = 10, antiderivative size = 34 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)}-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, 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 = {3282, 3286, 2718} \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=-\frac {1}{3} \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {2}{3} a \cot (x) \sqrt {a \sin ^2(x)} \]

[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 2718

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

Rule 3282

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 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 {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] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{12} a (-9 \cos (x)+\cos (3 x)) \csc (x) \sqrt {a \sin ^2(x)} \]

[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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71

method	result	size
default	\(-\frac {a^{2} \sin \left (x \right ) \cos \left (x \right ) \left (\sin ^{2}\left (x \right )+2\right )}{3 \sqrt {a \left (\sin ^{2}\left (x \right )\right )}}\)	\(24\)
risch	\(\frac {i a \,{\mathrm e}^{4 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}-\frac {3 i a \,{\mathrm e}^{2 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {3 i \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, a}{8 \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i a \,{\mathrm e}^{-2 i x} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}\)	\(145\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\frac {{\left (a \cos \left (x\right )^{3} - 3 \, a \cos \left (x\right )\right )} \sqrt {-a \cos \left (x\right )^{2} + a}}{3 \, \sin \left (x\right )} \]

[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)

Sympy [F]

\[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\int \left (a \sin ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((a*sin(x)**2)**(3/2), x)

Maxima [F]

\[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\int { \left (a \sin \left (x\right )^{2}\right )^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((a*sin(x)^2)^(3/2), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{3} \, {\left ({\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) + 2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )\right )} a^{\frac {3}{2}} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (a \sin ^2(x)\right )^{3/2} \, dx=\int {\left (a\,{\sin \left (x\right )}^2\right )}^{3/2} \,d x \]

[In]

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

[Out]

int((a*sin(x)^2)^(3/2), x)