\(\int (a \sin ^3(x))^{5/2} \, dx\) [7]

Optimal result

Integrand size = 10, antiderivative size = 123 \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)} \]

[Out]

-26/77*a^2*cot(x)*(a*sin(x)^3)^(1/2)-26/77*a^2*(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)-78/385*a^2*cos(x)*sin(x)*(a*sin(x)^3)^(1/2)-26/165*a^2*cos
(x)*sin(x)^3*(a*sin(x)^3)^(1/2)-2/15*a^2*cos(x)*sin(x)^5*(a*sin(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2720} \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=-\frac {26}{165} a^2 \sin ^3(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \sin ^5(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)} \]

[In]

Int[(a*Sin[x]^3)^(5/2),x]

[Out]

(-26*a^2*Cot[x]*Sqrt[a*Sin[x]^3])/77 - (26*a^2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(77*Sin[x]^(3/2)) -
(78*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^3])/385 - (26*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^3])/165 - (2*a^2*Cos[x]*Si
n[x]^5*Sqrt[a*Sin[x]^3])/15

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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {15}{2}}(x) \, dx}{\sin ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (13 a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {11}{2}}(x) \, dx}{15 \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (39 a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {7}{2}}(x) \, dx}{55 \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (39 a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {3}{2}}(x) \, dx}{77 \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (13 a^2 \sqrt {a \sin ^3(x)}\right ) \int \frac {1}{\sqrt {\sin (x)}} \, dx}{77 \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.53 \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\frac {a \left (-12480 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right )+(-15465 \cos (x)+3657 \cos (3 x)-749 \cos (5 x)+77 \cos (7 x)) \sqrt {\sin (x)}\right ) \left (a \sin ^3(x)\right )^{3/2}}{36960 \sin ^{\frac {9}{2}}(x)} \]

[In]

Integrate[(a*Sin[x]^3)^(5/2),x]

[Out]

(a*(-12480*EllipticF[(Pi - 2*x)/4, 2] + (-15465*Cos[x] + 3657*Cos[3*x] - 749*Cos[5*x] + 77*Cos[7*x])*Sqrt[Sin[
x]])*(a*Sin[x]^3)^(3/2))/(36960*Sin[x]^(9/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\sqrt {a \left (\sin ^{3}\left (x \right )\right )}\, \left (77 \left (\cos ^{6}\left (x \right )\right ) \cot \left (x \right ) \sqrt {2}-322 \left (\cos ^{4}\left (x \right )\right ) \cot \left (x \right ) \sqrt {2}+195 i \cot \left (x \right ) \csc \left (x \right ) \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}+195 i \left (\csc ^{2}\left (x \right )\right ) \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}+530 \left (\cos ^{2}\left (x \right )\right ) \cot \left (x \right ) \sqrt {2}-480 \cot \left (x \right ) \sqrt {2}\right ) a^{2} \sqrt {8}}{2310}\)	\(192\)

[In]

int((a*sin(x)^3)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/2310*(a*sin(x)^3)^(1/2)*(77*cos(x)^6*cot(x)*2^(1/2)-322*cos(x)^4*cot(x)*2^(1/2)+195*I*cot(x)*csc(x)*(-I*(I+c
ot(x)-csc(x)))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*EllipticF((-I*(I-cot(x)+csc(x)))^(1/2),1/2*2^(1/2))*(-I*(I-cot(
x)+csc(x)))^(1/2)+195*I*csc(x)^2*(-I*(I+cot(x)-csc(x)))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*EllipticF((-I*(I-cot(x
)+csc(x)))^(1/2),1/2*2^(1/2))*(-I*(I-cot(x)+csc(x)))^(1/2)+530*cos(x)^2*cot(x)*2^(1/2)-480*cot(x)*2^(1/2))*a^2
*8^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\frac {195 \, \sqrt {2} \sqrt {-i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 195 \, \sqrt {2} \sqrt {i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{7} - 322 \, a^{2} \cos \left (x\right )^{5} + 530 \, a^{2} \cos \left (x\right )^{3} - 480 \, a^{2} \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{1155 \, \sin \left (x\right )} \]

[In]

integrate((a*sin(x)^3)^(5/2),x, algorithm="fricas")

[Out]

1/1155*(195*sqrt(2)*sqrt(-I*a)*a^2*sin(x)*weierstrassPInverse(4, 0, cos(x) + I*sin(x)) + 195*sqrt(2)*sqrt(I*a)
*a^2*sin(x)*weierstrassPInverse(4, 0, cos(x) - I*sin(x)) + 2*(77*a^2*cos(x)^7 - 322*a^2*cos(x)^5 + 530*a^2*cos
(x)^3 - 480*a^2*cos(x))*sqrt(-(a*cos(x)^2 - a)*sin(x)))/sin(x)

Sympy [F(-1)]

Timed out. \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a*sin(x)**3)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\int { \left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((a*sin(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sin(x)^3)^(5/2), x)

Giac [F]

\[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\int { \left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((a*sin(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*sin(x)^3)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\sin \left (x\right )}^3\right )}^{5/2} \,d x \]

[In]

int((a*sin(x)^3)^(5/2),x)

[Out]

int((a*sin(x)^3)^(5/2), x)