\(\int (a \sin ^3(x))^{3/2} \, dx\) [8]

Optimal result

Integrand size = 10, antiderivative size = 73 \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=-\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)} \]

[Out]

-14/45*a*cos(x)*(a*sin(x)^3)^(1/2)-14/15*a*(sin(1/4*Pi+1/2*x)^2)^(1/2)/sin(1/4*Pi+1/2*x)*EllipticE(cos(1/4*Pi+
1/2*x),2^(1/2))*(a*sin(x)^3)^(1/2)/sin(x)^(3/2)-2/9*a*cos(x)*sin(x)^2*(a*sin(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2719} \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=-\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{9} a \sin ^2(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)} \]

[In]

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

[Out]

(-14*a*Cos[x]*Sqrt[a*Sin[x]^3])/45 - (14*a*EllipticE[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(15*Sin[x]^(3/2)) - (2*a
*Cos[x]*Sin[x]^2*Sqrt[a*Sin[x]^3])/9

Rule 2715

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] && Integ
erQ[2*n]

Rule 2719

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

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 {\left (a \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {9}{2}}(x) \, dx}{\sin ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)}+\frac {\left (7 a \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {5}{2}}(x) \, dx}{9 \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)}+\frac {\left (7 a \sqrt {a \sin ^3(x)}\right ) \int \sqrt {\sin (x)} \, dx}{15 \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\frac {\left (a \sin ^3(x)\right )^{3/2} \left (-168 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+\sqrt {\sin (x)} (-38 \sin (2 x)+5 \sin (4 x))\right )}{180 \sin ^{\frac {9}{2}}(x)} \]

[In]

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

[Out]

((a*Sin[x]^3)^(3/2)*(-168*EllipticE[(Pi - 2*x)/4, 2] + Sqrt[Sin[x]]*(-38*Sin[2*x] + 5*Sin[4*x])))/(180*Sin[x]^
(9/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.45 (sec) , antiderivative size = 302, normalized size of antiderivative = 4.14

method	result	size
default	\(-\frac {\left (\csc ^{2}\left (x \right )\right ) a \left (5 \left (\cos ^{5}\left (x \right )\right ) \sqrt {2}-21 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+42 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )-17 \left (\cos ^{3}\left (x \right )\right ) \sqrt {2}-21 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+33 \cos \left (x \right ) \sqrt {2}-21 \sqrt {2}\right ) \sqrt {a \left (\sin ^{3}\left (x \right )\right )}\, \sqrt {8}}{90}\)	\(302\)

[In]

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

[Out]

-1/90*csc(x)^2*a*(5*cos(x)^5*2^(1/2)-21*(-I*(I-cot(x)+csc(x)))^(1/2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(I*(csc(x)-c
ot(x)))^(1/2)*EllipticF((-I*(I-cot(x)+csc(x)))^(1/2),1/2*2^(1/2))*cos(x)+42*(-I*(I-cot(x)+csc(x)))^(1/2)*(-I*(
I+cot(x)-csc(x)))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*EllipticE((-I*(I-cot(x)+csc(x)))^(1/2),1/2*2^(1/2))*cos(x)-1
7*cos(x)^3*2^(1/2)-21*(-I*(I-cot(x)+csc(x)))^(1/2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*Elli
pticF((-I*(I-cot(x)+csc(x)))^(1/2),1/2*2^(1/2))+42*(-I*(I-cot(x)+csc(x)))^(1/2)*(-I*(I+cot(x)-csc(x)))^(1/2)*(
I*(csc(x)-cot(x)))^(1/2)*EllipticE((-I*(I-cot(x)+csc(x)))^(1/2),1/2*2^(1/2))+33*cos(x)*2^(1/2)-21*2^(1/2))*(a*
sin(x)^3)^(1/2)*8^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\frac {7}{15} i \, \sqrt {2} \sqrt {-i \, a} a {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - \frac {7}{15} i \, \sqrt {2} \sqrt {i \, a} a {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + \frac {2}{45} \, {\left (5 \, a \cos \left (x\right )^{3} - 12 \, a \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \]

[In]

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

[Out]

7/15*I*sqrt(2)*sqrt(-I*a)*a*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) + I*sin(x))) - 7/15*I*sqrt(
2)*sqrt(I*a)*a*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(x) - I*sin(x))) + 2/45*(5*a*cos(x)^3 - 12*a
*cos(x))*sqrt(-(a*cos(x)^2 - a)*sin(x))

Sympy [F]

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

[In]

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

[Out]

Integral((a*sin(x)**3)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*sin(x)^3)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*sin(x)^3)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a*sin(x)^3)^(3/2), x)