Integrand size = 10, antiderivative size = 73 \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=-\frac {14}{45} a \cos (x) \sqrt {a \sin ^3(x)}-\frac {14 a E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{15 \sin ^{\frac {3}{2}}(x)}-\frac {2}{9} a \cos (x) \sin ^2(x) \sqrt {a \sin ^3(x)} \] Output:
-14/45*a*cos(x)*(a*sin(x)^3)^(1/2)-14/15*a*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)
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\frac {\left (a \sin ^3(x)\right )^{3/2} \left (-168 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+\sqrt {\sin (x)} (-38 \sin (2 x)+5 \sin (4 x))\right )}{180 \sin ^{\frac {9}{2}}(x)} \] Input:
Integrate[(a*Sin[x]^3)^(3/2),x]
Output:
((a*Sin[x]^3)^(3/2)*(-168*EllipticE[(Pi - 2*x)/4, 2] + Sqrt[Sin[x]]*(-38*S in[2*x] + 5*Sin[4*x])))/(180*Sin[x]^(9/2))
Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3119}
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.
| \(\displaystyle \int \left (a \sin ^3(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin (x)^3\right )^{3/2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \int \sin ^{\frac {9}{2}}(x)dx}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \int \sin (x)^{9/2}dx}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \left (\frac {7}{9} \int \sin ^{\frac {5}{2}}(x)dx-\frac {2}{9} \sin ^{\frac {7}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \left (\frac {7}{9} \int \sin (x)^{5/2}dx-\frac {2}{9} \sin ^{\frac {7}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\sin (x)}dx-\frac {2}{5} \sin ^{\frac {3}{2}}(x) \cos (x)\right )-\frac {2}{9} \sin ^{\frac {7}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\sin (x)}dx-\frac {2}{5} \sin ^{\frac {3}{2}}(x) \cos (x)\right )-\frac {2}{9} \sin ^{\frac {7}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a \sqrt {a \sin ^3(x)} \left (\frac {7}{9} \left (-\frac {6}{5} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )-\frac {2}{5} \sin ^{\frac {3}{2}}(x) \cos (x)\right )-\frac {2}{9} \sin ^{\frac {7}{2}}(x) \cos (x)\right )}{\sin ^{\frac {3}{2}}(x)}\) |
Input:
Int[(a*Sin[x]^3)^(3/2),x]
Output:
(a*Sqrt[a*Sin[x]^3]*((-2*Cos[x]*Sin[x]^(7/2))/9 + (7*((-6*EllipticE[Pi/4 - x/2, 2])/5 - (2*Cos[x]*Sin[x]^(3/2))/5))/9))/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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 = 1.00 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.41
| method | result | size |
| default | \(-\frac {\csc \left (x \right )^{2} \left (\left (-21 \cos \left (x \right )-21\right ) \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (x \right )+i \csc \left (x \right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}+\left (42 \cos \left (x \right )+42\right ) \sqrt {1-i \cot \left (x \right )+i \csc \left (x \right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (x \right )-i \csc \left (x \right )}+\left (5 \cos \left (x \right )^{5}-17 \cos \left (x \right )^{3}+33 \cos \left (x \right )-21\right ) \sqrt {2}\right ) \sqrt {a \sin \left (x \right )^{3}}\, a \sqrt {8}}{90}\) | \(176\) |
Input:
int((a*sin(x)^3)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/90*csc(x)^2*((-21*cos(x)-21)*EllipticF((1+I*cot(x)-I*csc(x))^(1/2),1/2* 2^(1/2))*(1-I*cot(x)+I*csc(x))^(1/2)*(I*(csc(x)-cot(x)))^(1/2)*(1+I*cot(x) -I*csc(x))^(1/2)+(42*cos(x)+42)*(1-I*cot(x)+I*csc(x))^(1/2)*(I*(csc(x)-cot (x)))^(1/2)*EllipticE((1+I*cot(x)-I*csc(x))^(1/2),1/2*2^(1/2))*(1+I*cot(x) -I*csc(x))^(1/2)+(5*cos(x)^5-17*cos(x)^3+33*cos(x)-21)*2^(1/2))*(a*sin(x)^ 3)^(1/2)*a*8^(1/2)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\frac {14}{15} i \, \sqrt {-\frac {1}{2} i \, a} a {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - \frac {14}{15} i \, \sqrt {\frac {1}{2} i \, a} a {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + \frac {2}{45} \, {\left (5 \, a \cos \left (x\right )^{3} - 12 \, a \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )} \] Input:
integrate((a*sin(x)^3)^(3/2),x, algorithm="fricas")
Output:
14/15*I*sqrt(-1/2*I*a)*a*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, c os(x) + I*sin(x))) - 14/15*I*sqrt(1/2*I*a)*a*weierstrassZeta(4, 0, weierst rassPInverse(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))
\[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\int \left (a \sin ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a*sin(x)**3)**(3/2),x)
Output:
Integral((a*sin(x)**3)**(3/2), x)
\[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\int { \left (a \sin \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a*sin(x)^3)^(3/2),x, algorithm="maxima")
Output:
integrate((a*sin(x)^3)^(3/2), x)
\[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\int { \left (a \sin \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a*sin(x)^3)^(3/2),x, algorithm="giac")
Output:
integrate((a*sin(x)^3)^(3/2), x)
Timed out. \[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\sin \left (x\right )}^3\right )}^{3/2} \,d x \] Input:
int((a*sin(x)^3)^(3/2),x)
Output:
int((a*sin(x)^3)^(3/2), x)
\[ \int \left (a \sin ^3(x)\right )^{3/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (x \right )}\, \sin \left (x \right )^{4}d x \right ) a \] Input:
int((a*sin(x)^3)^(3/2),x)
Output:
sqrt(a)*int(sqrt(sin(x))*sin(x)**4,x)*a