Integrand size = 10, antiderivative size = 123 \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)} \] Output:
-26/77*a^2*cot(x)*(a*sin(x)^3)^(1/2)+26/77*a^2*InverseJacobiAM(-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*si n(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)
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.53 \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\frac {a \left (-12480 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right )+(-15465 \cos (x)+3657 \cos (3 x)-749 \cos (5 x)+77 \cos (7 x)) \sqrt {\sin (x)}\right ) \left (a \sin ^3(x)\right )^{3/2}}{36960 \sin ^{\frac {9}{2}}(x)} \] Input:
Integrate[(a*Sin[x]^3)^(5/2),x]
Output:
(a*(-12480*EllipticF[(Pi - 2*x)/4, 2] + (-15465*Cos[x] + 3657*Cos[3*x] - 7 49*Cos[5*x] + 77*Cos[7*x])*Sqrt[Sin[x]])*(a*Sin[x]^3)^(3/2))/(36960*Sin[x] ^(9/2))
Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a \sin ^3(x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin (x)^3\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \int \sin ^{\frac {15}{2}}(x)dx}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \int \sin (x)^{15/2}dx}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \int \sin ^{\frac {11}{2}}(x)dx-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \int \sin (x)^{11/2}dx-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \int \sin ^{\frac {7}{2}}(x)dx-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \int \sin (x)^{7/2}dx-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \sin ^{\frac {3}{2}}(x)dx-\frac {2}{7} \sin ^{\frac {5}{2}}(x) \cos (x)\right )-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \sin (x)^{3/2}dx-\frac {2}{7} \sin ^{\frac {5}{2}}(x) \cos (x)\right )-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (x)}}dx-\frac {2}{3} \sqrt {\sin (x)} \cos (x)\right )-\frac {2}{7} \sin ^{\frac {5}{2}}(x) \cos (x)\right )-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (x)}}dx-\frac {2}{3} \sqrt {\sin (x)} \cos (x)\right )-\frac {2}{7} \sin ^{\frac {5}{2}}(x) \cos (x)\right )-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {a^2 \sqrt {a \sin ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (-\frac {2}{3} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )-\frac {2}{3} \sqrt {\sin (x)} \cos (x)\right )-\frac {2}{7} \sin ^{\frac {5}{2}}(x) \cos (x)\right )-\frac {2}{11} \sin ^{\frac {9}{2}}(x) \cos (x)\right )-\frac {2}{15} \sin ^{\frac {13}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
Input:
Int[(a*Sin[x]^3)^(5/2),x]
Output:
(a^2*Sqrt[a*Sin[x]^3]*((-2*Cos[x]*Sin[x]^(13/2))/15 + (13*((-2*Cos[x]*Sin[ x]^(9/2))/11 + (9*((5*((-2*EllipticF[Pi/4 - x/2, 2])/3 - (2*Cos[x]*Sqrt[Si n[x]])/3))/7 - (2*Cos[x]*Sin[x]^(5/2))/7))/11))/15))/Sin[x]^(3/2)
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[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]])
Result contains complex when optimal does not.
Time = 4.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {\sqrt {a \sin \left (x \right )^{3}}\, a^{2} \left (\cot \left (x \right ) \sqrt {2}\, \left (77 \cos \left (x \right )^{6}-322 \cos \left (x \right )^{4}+530 \cos \left (x \right )^{2}-480\right )+i \csc \left (x \right )^{2} \left (195 \cos \left (x \right )+195\right ) \sqrt {1-i \cot \left (x \right )+i \csc \left (x \right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}\right ) \sqrt {8}}{2310}\) | \(117\) |
Input:
int((a*sin(x)^3)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/2310*(a*sin(x)^3)^(1/2)*a^2*(cot(x)*2^(1/2)*(77*cos(x)^6-322*cos(x)^4+53 0*cos(x)^2-480)+I*csc(x)^2*(195*cos(x)+195)*(1-I*cot(x)+I*csc(x))^(1/2)*(I *(csc(x)-cot(x)))^(1/2)*EllipticF((1+I*cot(x)-I*csc(x))^(1/2),1/2*2^(1/2)) *(1+I*cot(x)-I*csc(x))^(1/2))*8^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\frac {2 \, {\left (195 \, \sqrt {-\frac {1}{2} i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 195 \, \sqrt {\frac {1}{2} i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + {\left (77 \, a^{2} \cos \left (x\right )^{7} - 322 \, a^{2} \cos \left (x\right )^{5} + 530 \, a^{2} \cos \left (x\right )^{3} - 480 \, a^{2} \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}\right )}}{1155 \, \sin \left (x\right )} \] Input:
integrate((a*sin(x)^3)^(5/2),x, algorithm="fricas")
Output:
2/1155*(195*sqrt(-1/2*I*a)*a^2*sin(x)*weierstrassPInverse(4, 0, cos(x) + I *sin(x)) + 195*sqrt(1/2*I*a)*a^2*sin(x)*weierstrassPInverse(4, 0, cos(x) - I*sin(x)) + (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)
Timed out. \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a*sin(x)**3)**(5/2),x)
Output:
Timed out
\[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\int { \left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a*sin(x)^3)^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(x)^3)^(5/2), x)
\[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\int { \left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a*sin(x)^3)^(5/2),x, algorithm="giac")
Output:
integrate((a*sin(x)^3)^(5/2), x)
Timed out. \[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\sin \left (x\right )}^3\right )}^{5/2} \,d x \] Input:
int((a*sin(x)^3)^(5/2),x)
Output:
int((a*sin(x)^3)^(5/2), x)
\[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (x \right )}\, \sin \left (x \right )^{7}d x \right ) a^{2} \] Input:
int((a*sin(x)^3)^(5/2),x)
Output:
sqrt(a)*int(sqrt(sin(x))*sin(x)**7,x)*a**2